Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 4 (2008), 018, 29 pages Tw o-Field In tegrable Ev olutionary Syste ms of the Thi rd Orde r and Their D if feren tial Substit utions Ana toly G. MESHKO V and Maxim Ju. BALAKHN E V Or el State T e chnic al University, Or el, Russia E-mail: a meshkov@or el.ru, maxib al@yan dex.ru Received Octob er 04, 2007, in f inal form January 17, 2 008; Published online F ebruar y 09, 2008 Original article is av ailable at http: //www .emis .de/journals/SIGMA/2008/018/ Abstract. A list o f fo rty third-o rder ex actly integrable tw o-f ield evolutionary systems is presented. Dif feren tial substitutions c onnecting v arious systems from the list are found. It is prov ed tha t all the systems ca n b e obta ine d from only tw o of them. Examples of zer o curv ature representations with 4 × 4 ma trices are presented. Key wor ds: int egrability; symmetry; conserv ation law; dif ferential substitutions; zero cur- v atur e re pr esentation 2000 Mathematics Subje ct Classific atio n: 37 K10; 35Q 53; 37K2 0 1 In tro duction W e use th e term “in tegrabilit y” in the meaning that a system or equation under consideration p ossesses a Lax representa tion or a zero curv ature representa tion. Suc h systems can b e solv ed b y the inv erse sp ectral transform metho d (IS T ) [1, 2]. E x actly in tegrable evolutio n systems are of in terest b oth for mathematics and app lications. In particular, systems of the follo wing form u t = u xxx + F ( u, v , u x , v x , u xx , v xx ) , v t = a v xxx + G ( u, v , u x , v x , u xx , v xx ) , (1.1) where a is a constant, excite great interest since ab out 1980. The p ap er [3] is d ev oted to construc- tion of systems of the f orm (1.1) among others . Nine inte grable systems of the f orm (1.1 ) and their Lax r epresent ations h a v e b een obtained in the pap er. In particular, it conta ins a complete list of three in tegrable systems (1.1) satisfying the conditions a ( a − 1) 6 = 0 and ord( F , G ) < 2. Here ord = order, ord f < n means that f do e s not dep e nd on u n , v n , u n +1 , v n +1 , . . . . Here and in what f ollo w s, the notations u n = ∂ n u/∂ x n , v n = ∂ n v /∂ x n are used. Tw o of th e three m en tioned systems can b e written in th e follo wing form u t = u 3 + v u 1 , v t = − 1 2 v 3 + u u 1 − v v 1 , (1.2) u t = u 3 + v u 1 , v t = − 1 2 v 3 − v v 1 + u 1 (1.3) and the thir d system is presented b elo w (see (3.46a)). System (1.2) w as found in dep end en tly in [4] and the soliton solutions were constructed there. This system is called as th e Drin feld– Sok olo v–Hirota–Satsuma system. This pap er con tains t w o results: (i) a list of integrable systems of th e form (1.1) with smo oth functions F , G and a = − 1 / 2; (ii) dif ferential substitutions that allo w to connect an y equation from the list with (1.2) or (1.3). There are many articles dealing with in tegrable systems, but some of them (see, e.g., [5, 6, 7]) consider multi- comp onent systems. Other pap ers (see, e.g., [8, 9, 10, 11, 12]) conta in t w o- comp onent systems redu cible to a triangular form. The triangular form is brief ly considered b elo w. There was p o ssibly only one serious attempt [13] to classify in tegrable systems of the form (1.1) usin g the Painlev ´ e test. Unfortunately , f ifteen s y s tems presente d in [13] conta in a large 2 A.G. Meshko v and M.Ju . Balakhnev n um b er of constants some of whic h can b e remo v ed b y scaling and linear transformations. Note that there are tw o- f ield inte grable ev olutionary systems u t = A u 3 + H ( u , u 1 , u 2 ) with a n on - diagonal main matrix A . F or example, an integrable ev olutio nary system with the Jord an main matrix is found in [14]. Moreo v er, ab o ut 50 tw o- f ield integ rable systems of the form (1.1) with a = 1 can b e extracted from pap ers [15, 16, 17, 18, 19] that deal with vec tor evo lutionary equ ations. P artial solutions of the classif ication problem for a = 0 and ord G 6 1 hav e b een obtained in [20], and in [21] for div ergen t systems with a 6 = 1. A complete list of integ rable systems of the form (1.1) do es not exist to da y b ecause the problem is to o cumbers ome and the set of in tegrable systems is v ery large. Our to ol is th e symm etry metho d presented in man y pap ers. W e shall p oi n t out pioneer or review p ap ers only . In [22] the notions of formal symmetry and canonical conserv ed densit y f or a scalar evo lution equation are introd u ced. T hese to ols w ere applied to classif icatio n of the Kd V- t yp e equations in [23]. A complete theory of formal symmetries and f ormal conserv ation la ws for scalar equations has b een present ed in [24]. A generaliz ed th eory was develo p ed for ev olutionary systems in [25]. Review p ap er [26] cont ains b oth general theorems of the symmetry metho d and classif ication results on integrable equations: the th ir d and f ifth ord er scalar equations, Sc hr¨ odinger-t yp e systems, Burgers-t yp e equ ations and sys tems. Review pap er [27] is dev oted to higher symm etries, exact integ rabilit y and related pr oblems. Pec uliarities of sys tems (1.1) h av e b een d iscussed in [21]. F or the sak e of completeness, the main p o in ts of th e symm etry metho d and some results necessary for un d erstanding of this pap er are considered in the S ections 2–4. Brief ly sp ea king, the symmetry metho d deals with the so-called canonical conserv atio n la ws D t ρ n = D x θ n , D t ˜ ρ n = D x ˜ θ n , n = 0 , 1 , 2 , . . . , (1.4) where D t is the ev olutionary deriv ative and D x is the total d eriv ativ e with resp ect to x . In particular, ρ 0 = − F u 2 / 3, ˜ ρ 0 = − G v 2 / (3 a ). The recur sion r elations for the canonical conserved densities ρ n and ˜ ρ n are presen ted in S ection 2. All canonical conserv ed densities are expr essed in terms of functions F and G . Th at is wh y equ ations (1.4) imp ose great restrictions on th e forms of F and G . Equations (1.4) are solv able in the jet space if f E α D t ρ n = 0 , E α D t ˜ ρ n = 0 , α = 1 , 2 , n = 0 , 1 , 2 , . . . (1.5) (see [28 ], f or example). Here E α ≡ δ δ u α = ∞ X n =0 ( − D x ) n ∂ ∂ u α n , ( u 1 = u, u 2 = v ) , is the Euler op erator. Conserv ation la w with ρ = D x χ, θ = D t χ is called trivial and the conserve d d ensit y of the form ρ = D x χ is called trivial to o. Th is can b e wr itten in the f orm ρ ∈ Im D x , where Im = Image. If ρ 1 − ρ 2 ∈ Im D x , then the densities ρ 1 and ρ 2 are said to b e equiv alent. There are a lot of systems in the follo wing form u t = u xxx + F ( u, u x , u xx ) , v t = a v xxx + G ( u, v , u x , v x , u xx , v xx ) , satisfying the integrabilit y conditions (1.4). Su c h systems con taining one indep e ndent equation are said to b e triangular. It follo ws from the integrabilit y conditions that the equation for u m ust b e on e of the kno wn integ rable equations (Kd V, m KdV etc). The second equ ation is usually linear with resp ect to v , v x and v xx . T riangular systems do not p ossess any Lax repr esen tations and are not in tegrable in this sense. Therefore triangular systems and those reducible to the triangular form ha v e b een omitted as trivial. In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 3 The system of t w o indep e ndent equations u t = u xxx + F ( u, u x , u xx ) , v t = a v xxx + G ( v , v x , v xx ) , will b e called disinte grated. It is obvio us that the disintegrat ed form is a partial case of the triangular form. Therefore the disin tegrated systems and those reducible to them ha v e b een omitted. System (1.1 ) will b e called redu cible if it is triangular or can b e redu ced to triangular or disin tegrated form. Otherw ise, the sy s tem will b e called irreducible. Our compu tations sho w that for irreducible integrable systems (1.1 ) parameter a m ust b elong to th e follo wing set: A = n 0 , − 2 , − 1 2 , − 7 2 + 3 2 √ 5 , − 7 2 − 3 2 √ 5 o . These v alues were foun d f irst in [3] and we re rep e ated in [29 ]. The v alue of a is alw a ys def in ed at the end of computations w hen functions F and G h a v e b een found and only some co ef f icien ts are to b e sp ecif ied from the fifth or sev en th in tegrabilit y conditions (see example in S ection 3.1). This means that it is en ough to v erify conditions (1.5) for n = 0 , . . . , 7 and α = 1 , 2 to obtain F , G and a . But for absolute certaint y we ha v e v erif ied conditions (1.5 ) f or n = 8 , 9 and α = 1 , 2 for eac h sy s tem. The p resen ted set A consists of zero and tw o pairs ( a, a − 1 ). The transformation t ′ = at , u ′ = v , v ′ = u c hanges the p arameter a 6 = 0 in (1.1) in to a − 1 . That is why one ought to consider the v alues  0 , − 1 2 , − 7 2 + 3 2 √ 5  of th e p arameter a . I n tegrable systems with a = 0 w ere menti oned ab o ve, see also [30 ]. This pap er is devo ted to in v estigat ion of the case a = − 1 / 2 only . Th e case a = − 7 2 + 3 2 √ 5 will b e presente d in another pap e r. Section 2 con tains recursion formulas for the canonical d en sities. T he origin of the n otion, some examples and a preliminary classif ication are considered. A list of fort y in tegrable systems and an example of compu tations are pr esen ted in Section 3. Section 4 con tains dif f eren tial sub stitutions that connect all systems from th e list. The metho d of computations an d an example are considered. I t is s ho wn that all sys tems from the list present ed in Section 3 can b e obtained f r om (1.2) and (1.3) by dif feren tial s ubstitutions. Section 5 is devot ed to zero curv ature represen tations. The zero curv atur e representat ions for systems (1.2) and (1.3) are obtained from the Drin feld–Sok olo v L -op erators. A metho d of obtaining zero curv a ture representa tions for other systems is demons trated. 2 Canonical densities One of the m ain ob jects of the symmetry app roac h to classif icatio n of integ rable equations is the inf inite set of the canonical conserv ed densities. Let us demonstrate how canonical conserved densities can b e obtained the from the asymp totic expan s ions for eigenfunctions of the Lax op erators. Th e simp lest Lax equations concerned with the KdV equation u t = 6 uu x − u xxx tak e the follo wing form ψ xx − uψ − µ 2 ψ = 0 , (2.1) ψ t = − 4 ψ xxx + 6 uψ x + 3 u x ψ + 4 µ 3 ψ . (2.2) Here u is a solution of the KdV equ ation and µ is a parameter. The standard substitution ψ = exp  Z ρ dx  4 A.G. Meshko v and M.Ju . Balakhnev reduces equations (2.1) and (2.2) to the Riccati form ρ x + ρ 2 − u − µ 2 = 0 , (2.3) ∂ t Z ρ dx = − 4( ∂ x + ρ ) 2 ρ + 6 uρ + 3 u x + 4 µ 3 . (2.4) Dif ferent iating temp oral equation (2.4) with resp e ct to x one can rewrite it, usin g (2.3 ), as the con tin uit y equation: ρ t = ∂ x [(2 u − 4 µ 2 ) ρ − u x ] . (2.5) T o construct an asymptotic exp an s ion one ough t to set ρ = µ + ∞ X n =0 ρ n ( − 2 µ ) − n . (2.6) Then equ ation (2.3) results in the follo wing w ell known recursion formula [2] ρ n +1 = D x ρ n + n − 1 X i =1 ρ i ρ n − i , n = 1 , 2 , . . . , ρ 0 = 0 , ρ 1 = − u, (2.7) and (2.5) results in inf inite sequ ence of conserv ation la ws: D t ρ n = D x (2 uρ n − ρ n +2 ) , n > 0 . (2.8) W e change here ∂ t → D t and ∂ x → D x b ecause u is a solution of the KdV equation. The obtained conserv ation laws are canonical. I t is easy to obtain sev eral f irs t canonical d ensities: ρ 2 = − u 1 , ρ 3 = u 2 − u 2 , ρ 4 = D x (2 u 2 − u 2 ) , . . . . It is sho wn in [2] that all ev en canonical densities are trivial. Note that if one c hooses an- other asymptotic expansion, for example, in p o wers of µ − 1 instead of (2.6), then another set of canonical densities is obtained, which is equiv ale n t to the previous set. The canonical den s ities that follo w from (2.7 ) can also b e obtained by using the temp o ral equation (2.4) only . Indeed, setting ∂ t R ρ dx = θ one obtains f rom (2.4) − 4( ∂ x + ρ ) 2 ρ + 6 uρ + 3 u x + 4 µ 3 = θ . (2.9 ) Using the same expansions as ab ov e ρ = µ + ∞ X n =0 ρ n ( − 2 µ ) − n , θ = ∞ X n =0 θ n ( − 2 µ ) − n , one can obtain from (2.9) the follo wing recursion relation: ρ n +2 = 2 uρ n + 2 n +1 X i =0 ρ i ρ n − i +1 − 4 3 n X i,j =0 ρ i ρ j ρ n − i − j − 1 3 θ n + 2 D x ρ n +1 − n X i =0 ρ i ρ n − i ! − 4 3 D 2 x ρ n − uδ n, − 1 + u 1 δ n 0 , n = − 2 , − 1 , 0 , . . . , where δ i,k is the Kronec k er delta. The obtained relation p ro vides ρ 0 = 0, ρ 1 = − u , ρ 2 = − u 1 − θ 0 / 3, etc. As D t ρ 0 = D x θ 0 and ρ 0 = 0, then θ 0 = 0. The higher canonical d ensities ρ n , In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 5 n > 2 dep end on θ n − 2 . The f luxes θ n m ust b e def ined no w from equ ations (1.4). F or example, θ 1 = u 2 − 3 u 2 . The traditional metho d to obtain the canonical d en sities for an evol ution s ystem [25] u t = K ( u , u x , . . . , u n ) , u ( t, x ) ∈ R m , m > 1 , u α k = ∂ k x u α . (2.10) consists, brief ly , in the follo wing. The main idea is to use the linearized equation ( D t − K ∗ ) ψ = 0 (2.11) or its adjoin t ( D t + K + ∗ ) ϕ = 0 (2.12) as the temp oral Lax equation. Here ( K ∗ ψ ) α = X n,β ∂ K α ∂ u β n D n x ψ β , ( K + ∗ ϕ ) α = X n,β ( − D x ) n ∂ K β ∂ u α n ϕ β , D t = ∂ ∂ t + X n,α D n x ( K α ) ∂ ∂ u α n , D x = ∂ ∂ x + X n,α u α n +1 ∂ ∂ u α n . The sp atial Lax op erator (formal symmetry) was introdu ced in [25] as the inf inite op erat or series R = N X k = −∞ R k D k x , N > 0 , (2.13) comm uting with D t − K ∗ . R k are matrix coef f icien ts dep end ing on u , u x , . . . . It was sh o wn that T r res R (res R = R − 1 ) is the conserve d density for system (2.10). Canonical densities h av e b een def in ed by the formulas ρ n = T r res R n , n = 1 , 2 , . . . , see [26] for details. Op erations with op erator series (2.13) are not so simp le, th erefore w e u se an alternativ e metho d for obtaining the canonical densities. It was prop ose d in [32] heuristically and we present the follo wing explanation (see also [33]). Observ ation. O ne can obtain equation (2.9 ) from (2.2 ) by the follo wing substitution ψ = e ω , ω = Z ρ dx + θ dt, (2.14) where ρ dx + θ dt is the smo oth closed 1-form, that is, D t ρ = D x θ . This imp lies e − ω D t e ω = D t + θ , e − ω D x e ω = D x + ρ and so (2.9) follo ws. Another w a y to obtain the same equ ation is to prolong the op erators D t → ∂ t + θ , D x → ∂ x + ρ in (2.2 ) formally and to set ψ = 1. F or systems, one m ust s et ψ α = 1 f or a f ixed α only . W e shall apply this metho d to system (1.1) no w. The linearized system (1.1) with prolonged op erators D x → D x + ρ , D t → D t + θ tak es the follo wing form:  ( D x + ρ ) 3 + F u + F u 1 ( D x + ρ ) + F u 2 ( D x + ρ ) 2 − D t − θ  Ψ 1 +  F v + F v 1 ( D x + ρ ) + F v 2 ( D x + ρ ) 2  Ψ 2 = 0 , 6 A.G. Meshko v and M.Ju . Balakhnev  G u + G u 1 ( D x + ρ ) + G u 2 ( D x + ρ ) 2  Ψ 1 + a ( D x + ρ ) 3 Ψ 2 +  G v + G v 1 ( D x + ρ ) + G v 2 ( D x + ρ ) 2 − D t − θ  Ψ 2 = 0 . (2.15) If one sets h ere Ψ 1 = 1, then th e f irst equation tak es the follo wing form ( D x + ρ ) 2 ρ + F u + F u 1 ρ + F u 2 ( D x + ρ ) ρ − θ +  F v + F v 1 ( D x + ρ ) + F v 2 ( D x + ρ ) 2  Ψ 2 = 0 . It is obvious fr om this equation that the follo wing forms of the asymp totic expansions are acceptable: ρ = µ − 1 + ∞ X n =0 ρ n µ n , θ = µ − 3 + ∞ X n =0 θ n µ n , Ψ 2 = ∞ X n =0 ρ n µ n . Here µ is a complex p arameter. Then, after some simple calculations, the follo wing recursion relations are obtained ( n > − 1): ρ n +2 = 1 3 θ n − n +1 X i =0 ρ i ρ n − i +1 − 1 3 n X i + j =0 ρ i ρ j ρ n − i − j − 1 3 F u 1 ( δ n, − 1 + ρ n ) − 1 3 F u δ n, 0 − 1 3 ( F v + F v 1 D x + F v 2 D 2 x ) ϕ n − 1 3 F u 2 D x ρ n + 2 ρ n +1 + n X i =0 ρ i ρ n − i ! − 1 3 F v 2   ϕ n +2 + 2 n X i =0 ρ i ϕ n − i +1 + n X i + j =0 ρ i ρ j ϕ n − i − j   − 1 3 F v 2 2 D x ϕ n +1 + n X i =0 ρ i D x ϕ n − i + D x n X i =0 ρ i ϕ n − i ! − D x " ρ n +1 + 1 3 D x ρ n + 1 2 n X i =0 ρ i ρ n − i # − 1 3 F v 1 ϕ n +1 + n X i =0 ρ i ϕ n − i ! , (1 − a ) ϕ n +3 = G u δ n, 0 + G u 2 ( D x ρ n + 2 ρ n +1 + n X i =0 ρ i ρ n − i ) + G u 1 ( δ n, − 1 + ρ n ) − n X i =0 θ i ϕ n − i + G v ϕ n + G v 1 D x ϕ n + ϕ n +1 + n X i =0 ρ i ϕ n − i ! − D t ϕ n + G v 2 2 D x ϕ n +1 + n X i =0 ρ i D x ϕ n − i + D x n X i =0 ρ i ϕ n − i ! + G v 2   ϕ n +2 + D 2 x ϕ n + 2 n +1 X i =0 ρ i ϕ n − i +1 + n X i + j =0 ρ i ρ j ϕ n − i − j   + a D 3 x ϕ n + 3 a D 2 x ϕ n +1 + 6 a n +1 X i =0 ρ i D x ϕ n − i +1 + 3 a n +2 X i =0 ρ i ϕ n − i +2 + 3 a D x ϕ n +2 + 3 a n X i + j =0 ρ i ρ j D x ϕ n − i − j + 3 a n X i =0 ϕ n − i +1 D x ρ i + 3 2 a n X i + j =0 ϕ n − i − j D x ( ρ i ρ j ) + 3 a n +1 X i + j =0 ρ i ρ j ϕ n − i − j +1 In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 7 + 3 a D x n X i =0 ρ i ϕ n − i + a n X i =0 ϕ n − i D 2 x ρ i + a n X i + j + k =0 ρ i ρ j ρ k ϕ n − i − j − k . Here δ i,k is the K ronec k er delta, F u 1 = ∂ F /∂ u 1 and so on. F rom th e recursion relations it is ob vious why the v alue a = 1 is singular. S ome of in itial elemen ts of the sequence { ρ n , ϕ n } read ρ 0 = − 1 3 F u 2 , ϕ 0 = 0 , ϕ 1 = 1 1 − a G u 2 , others are in tro duced via the δ -symb ols. If one sets in (2.15) Ψ 2 = 1 and a 6 = 0, then one more pair of recursion relations for { ˜ ρ n , ˜ ϕ n } is obtained. Th ese recursion relations giv e u s any desired num b er of canonical densities. As an example, w e present here some more canonical densities: ρ 0 = − 1 3 F u 2 , ρ 1 = 1 9 F 2 u 2 − 1 3 F u 1 + 1 3 b F v 2 G u 2 + 1 3 D x F u 2 , ˜ ρ 0 = − 1 3 a G v 2 , ˜ ρ 1 = 1 9 a 2 G 2 v 2 − 1 3 a G v 1 − 1 3 a b F v 2 G u 2 + 1 3 a D x G v 2 , (2.16) where b = a − 1. The tilde denotes another sequen ce of canonical d en sities. F urther canonical densities are to o cumb ersome, therefore we d o not present them here. T o simplify in v estiga tion of the inte grabilit y cond itions, an additional requ iremen t is alw a ys imp osed. Th is is the existence of a formal conserv ation law [25, 26]. A formal conserv ation la w is an op erato r series N in p ow ers of D − 1 x . An equation f or the f orm al conserv ation la w can b e written in the follo wing op erator form ( D t − K ∗ ) N = N ( D t + K ∗ + ) . (2.17) The f orm of this equation coincides with the form of the equation for the No ether op erator [34]. That is a formal conserv ation la w ma y b e called a formal No e ther op e rator. If ( D t − K ∗ , L ) is the Lax p air for an equation, then ( D t + K ∗ + , L + ) is obviously the Lax pair for th e same equation. Hence, canonical d ensities obtained from (2.11) must b e equiv alen t to canonical densities obtained from (2.12). It was shown in [21] that the f irst sequence of th e canonical d ensities ρ n for system (1.1) obtained from (2.11) is equiv alent to the f irst sequence of the canonical d ensities τ n obtained from (2.12) and the second sequence of the canonical densities ˜ ρ n is equ iv alen t to the second sequence of the canonical densities ˜ τ n . Hence, ρ n − τ n ∈ Im D x and ˜ ρ n − ˜ τ n ∈ Im D x , or E α ( ρ n − τ n ) = 0 , E α ( ˜ ρ n − ˜ τ n ) = 0 , α = 1 , 2 , n = 0 , 1 , 2 , . . . . (2.18) Equations (1.4) (or (1.5)) and (2.18) are said to b e the necessary conditions of in tegrabilit y . W e shall refer to it simply as the int egrabilit y conditions f or b revit y . Our computations hav e sh o wn that τ 0 = − ρ 0 , ˜ τ 0 = − ˜ ρ 0 , τ 1 = ρ 1 , ˜ τ 1 = ˜ ρ 1 . (2.19) Other “adjoin t” canonical densities τ i and ˜ τ k essen tially dif fer f rom the “main” canonical d en- sities ρ i and ˜ ρ k . All canonical dens ities can b e obtained usin g the Maple routines cd and acd from the p ac k age JET (see [36]). These r outines generate the “main” and th e “adjoint” canon- ical densities, corr esp ondingly , for almost any ev olutionary system (an exclusion is the case of m ultiple ro ots of the main matrix of the system un d er consideration). Th us, according to (2.16 ) an d (2.19) we ha v e F u 2 ∈ Im D x and G v 2 ∈ Im D x ( a 6 = 0). This implies the follo wing lemma. 8 A.G. Meshko v and M.Ju . Balakhnev Lemma 1. System (1.1) with a ( a − 1) 6 = 0 satisfying the zer oth inte gr ability c onditions (2.18) r e ad s u t = u 3 − 3 2 f u 2 D x f + 3 4 f f u 1 u 2 2 + F 1 ( u, v , u 1 , v 1 , v 2 ) , v t = av 3 − 3 a 2 g v 2 D x g + 3 a 4 g g v 1 v 2 2 + G 1 ( u, v , u 1 , v 1 , u 2 ) . (2.20) wher e ord( f , g ) 6 1 . Indeed, one ma y set F u 2 = − 3 / 2 D x ln f and G v 2 = − 3 / 2 aD x ln g , where ord( f , g ) 6 1 b ecause ord( F , G ) 6 2. Then equations (2.20) follo w. F rom higher inte grabilit y conditions one more lemma follo w s. Lemma 2. Supp ose system (2.20 ) is irr e ducible and satisfies the fol l owing ei g ht inte gr ability c onditions ρ 2 − τ 2 ∈ Im D x , ˜ ρ 2 − ˜ τ 2 ∈ Im D x and D t ρ n ∈ Im D x , D t ˜ ρ n ∈ Im D x , wher e n = 1 , 3 , 5 . Then the system must have the fol lo wing form u t = u 3 − 3 2 f u 2 D x f + 3 4 f f u 1 u 2 2 + f 1 v 2 2 + f 2 v 2 + f 3 , v t = av 3 − 3 a 2 g v 2 D x g + 3 a 4 g g v 1 v 2 2 + g 1 u 2 2 + g 2 u 2 + g 3 , a 6 = 0 , (2.21) wher e ord( f , g , f i , g j ) 6 1 . A sc heme of th e pro of has b e en pr esen ted in [21]. 3 List of in tegrable systems As it is sho wn in S ection 2 the problem of the classif ication of inte grable systems (1.1) is reduced to in v estiga tion of system (2.21). Th at is why it is necessary to start by inv estig ating its symmetry prop ertie s. Lemma 3. System (2.21) ar e invariant under any p oint tr ansformation of the form ( a ) t ′ = α 3 t + β , x ′ = αx + γ t + δ, α 6 = 0 , u ′ = u, v ′ = v , ( b ) u ′ = h 1 ( u ) , v ′ = h 2 ( v ) , and under the fol lo wing p ermutation tr ansfor mation ( c ) t ′ = at, u ′ = v , v ′ = u, wher e α , β , γ and δ ar e c onstants, h i ar e arbitr ary smo oth functions. The classif ication of systems of t yp e (2.21) has b een p erformed by mo d ulo of the presen ted transformations. Moreo v er, some systems (2.21) admit inv ertible con tact transformations. An ef fectiv e to ol for s earc hing s uc h contac t transformations is inv estig ation of the canonical conserved d ensities. F or example, system (3.24) from the next section has the f irst canonical conserved density of the follo wing form: ρ 1 =  v 1 − 2 3 ue v  2 + 2 c 2 1 e − 2 v . It is ob vious that the b est v ariables for th at s y s tem are U = e − v and V = v 1 − 2 3 ue v . In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 9 This is an inv ertible con tac t transformation. In terms of U and V the system tak es the f ollo wing simple f orm: U t = D x  U 2 + 3 2 U V 1 − 3 4 U V 2 + 1 2 c 2 1 U 3  , V t = 1 4 D x ( V 3 − 2 V 2 ) − 3 2 c 2 1 D x (2 U U 1 + U 2 V ) . If c 1 6 = 0 this system can b e redu ced to (3.10) by scaling, otherwise the system is triangular: the equation for V will b e indep en den t single mKdV. Moreo v er, the equation f or U b e comes linear. That is why c 1 6 = 0 in (3.24). Canonical densities for th e tr iangular s ystems con tain only one highest order term in the second p o w er as in the considered example ρ = V 2 or ρ = V 2 x + · · · , or ρ = V 2 xx + · · · etc. T rian- gular systems and those redu cible to the triangular form h a v e b een omitted in th e classif icatio n pro cess as trivial. T o classify in tegrable systems (1.1) with a ( a − 1) 6 = 0 one m ust solve a h uge num b er of large ov erd etermin ed partial dif feren tial systems for eigh t unknown fun ctions of fou r v a riables. This work h as r equired p o werful computers and has tak en ab out six y ears. All the calculatio ns ha v e b een p e rformed in the inte ractiv e mo de of op eration b ecause automatic solving of large systems of p artial dif ferenti al equations is still imp ossible. Th e pac k age p dsol v e fr om th e excellen t system Maple mak es errors solving some s in gle partial dif ferenti al equations. Th e pac k age diff alg cannot op erate with large systems b eca use its algorithms are to o cum b ersome. Th us, one has to solv e complicated p roblems in the interactiv e m o de. Hence, to obtain a true solution on e must en ter true d ata! Un der s uc h circums tances errors are p robable. The longer the computations the more probable are errors. Th is is th e reason why we cannot state with conf idence that all compu tations h a v e b een p recise all these six y ears. That is why the statemen t on completeness of th e obtained set of in tegrable systems is formulated as a hyp othesis. In this and in the follo wing sections c , c i , k , k i are arbitrary constan ts. Hyp othesis. Su pp ose system (2.21) with a = − 1 / 2 is irr e ducible. If the system has infinitely many c anonic al c onservation laws, then it c an b e r e duc e d by an appr opriate p oint tr an sformation to one of the fol lowing systems: u t = u 3 + v u 1 , v t = − 1 2 v 3 + u u 1 − v v 1 ; (3.1) u t = u 3 + v 1 u 1 , v t = − 1 2 v 3 + 1 2 ( u 2 − v 2 1 ); (3.2) u t = u 3 + v u 1 , v t = − 1 2 v 3 − v v 1 + u 1 ; (3.3) u t = u 3 + v u 1 + v 1 u, v t = − 1 2 v 3 − v v 1 + u ; (3.4) u t = u 3 + v 1 u 1 , v t = − 1 2 v 3 − 1 2 v 2 1 + u ; (3.5) u t = u 3 + u u 1 + v 1 , v t = − 1 2 v 3 + 3 2 u 1 u 2 − u v 1 ; (3.6) u t = u 3 + v 2 + k u 1 , v t = − 1 2 v 3 + 3 2 uu 2 + 3 4 u 2 1 + 1 3 u 3 + k  u 2 − v 1  ; (3.7) u t = u 3 + 3 2 v v 2 + 3 4 v 2 1 + 1 3 v 3 − k  v 2 + u 1  , v t = − 1 2 v 3 + u 2 + k v 1 ; (3.8) u t = u 3 − 3 2 u 1 v 2 − 3 4 u 1 v 2 1 + 1 4 u 3 1 , v t = − 1 2 v 3 + 3 2 u 1 u 2 − 3 4 u 2 1 v 1 + 1 4 v 3 1 ; (3.9) u t =  u 2 − 3 2 uv 1 − 3 4 uv 2 + 1 4 u 3  x , v t =  − 1 2 v 2 + 3 2 uu 1 − 3 4 u 2 v + 1 4 v 3  x ; (3.10) u t = u 3 − 3 2 v 2 − 3 2 u 1 v 1 − 1 2 u 3 1 , v t = − 1 2 v 3 + 3 2  v 1 − u 2 + 1 2 u 2 1  2 − 3 4 v 2 1 ; (3.11) u t =  u 2 − 3 2 v 1 − 3 2 uv − 1 2 u 3  x , v t =  − 1 2 v 2 + 3 2  v − u 1 + 1 2 u 2  2 − 3 4 v 2  x ; (3.1 2) u t = u 3 − 3 g v 2 − 3 u 1 ( u 1 + v 1 ) − 3 2 v 2 1 − 6 v 1 g 2 − c 1 g 3 − 3 g 4 , v t = − 1 2 v 3 − 3 4 c 1 u 2 + 3 u 2 1 − 3 2 v 2 1 − 6 u 1 g 2 + c 1 g 3 + 3 g 4 , g = u + v ; (3.13) u t = u 3 − 3 u 1 v 1 + ( u − 3 v 2 ) u 1 , v t = − 1 2 v 3 + 1 2 u 2 − u 1 v − ( u − 3 v 2 ) v 1 ; (3.14) u t = u 3 − 3 u 1 v 2 + uu 1 − 3 u 1 v 2 1 , v t = − 1 2 v 3 + 1 2 u 1 − u v 1 + v 3 1 ; (3.15) 10 A.G. Meshko v and M.Ju . Balakhnev u t = u 3 +  k + p u 2 + v 1  u 1 , v t = − 1 2 v 3 + 3 8 (2 u u 1 + v 2 ) 2 u 2 + v 1 − 3 u u 2 − k (2 u 2 + v 1 ) − 2 3 ( u 2 + v 1 ) 3 / 2 ; (3.16) u t = u 3 − 3 4 (2 v v 1 + u 2 ) 2 v 2 + u 1 + 3 v v 2 + 3 2 v 2 1 + 2 3 v 3 − k (2 v 2 + u 1 ) , v t = − 1 2 v 3 + 1 2 u 2 + k v 1 ; (3.17) u t = u 3 + u 1 ( u 1 + v 2 ) √ u + v 1 − 4 3 u 1 v 1 + c 1 u 1 √ u + v 1 , v t = − 1 2 v 3 − 3 2 u 2 + 3 8 ( u 1 + v 2 ) 2 u + v 1 + 2 3 v 2 1 − 4 3 u 2 − 2 u 1 √ u + v 1 − 2 3 c 1 ( u + v 1 ) 3 / 2 ; (3.18) u t = u 3 + u 1 p u + v 1 − k u 1 , v t = − 1 2 v 3 − 3 2 u 2 + 3 8 ( u 1 + v 2 ) 2 u + v 1 − 2 3 ( u + v 1 ) 3 / 2 + 2 k u + k v 1 ; (3.19) u t = u 3 + uv 1 + ( u 2 + v ) u 1 , v t = − 1 2 v 3 + 3 u 1 u 2 − ( u 2 + v ) v 1 ; (3.20) u t = u 3 + 3( u + k ) v 2 + 3 u 1 ( v 1 + u 2 ) , v t = − 1 2 v 3 − 3 2 uu 2 − 3 2 ( v 1 + u 2 ) 2 − 3 4 u 2 1 + k u 3 + 3 4 u 4 ; (3.21) u t = u 3 − 3 2 v 2 − 3 2 u 1 v 1 − 1 2 u 3 1 − 3 u 1 ( c 1 e u + 2 c 2 e 2 u ) , v t v = − 1 2 v 3 + 3 2  1 2 u 2 1 − u 2 + v 1 + c 1 e u + 2 c 2 e 2 u  2 − 3 4 v 2 1 − 3 2 c 1 u 2 e u + 3 4 c 2 1 e 2 u + 2 c 1 c 2 e 3 u , c 1 6 = 0 or c 2 6 = 0; (3.22) u t = u 3 − 3 2 u 1 v 2 − 3 4 u 1 v 2 1 + u u 1 − c 2 u 1 e − 2 v , v t = − 1 2 v 3 + 1 4 v 3 1 + u 1 − u v 1 + c 2 v 1 e − 2 v ; (3.23) u t = u 3 − 3 2 u 1 v 2 − 3 4 u 1 v 2 1 + u 1 e v ( u 1 + 2 u v 1 ) − 1 3 u 2 u 1 e 2 v − 3 2 c 2 1 u 1 e − 2 v , v t = − 1 2 v 3 + v 3 1 4 + u 2 e v + 1 3 u e 2 v (2 u 1 + u v 1 ) + 3 2 c 2 1 v 1 e − 2 v , c 1 6 = 0; (3.24) u t = u 3 + 3 u 2 v 1 + 3 2 u 1 v 2 + 9 4 u 1 v 2 1 − uu 1 e 2 v − e − 3 v , v t = − 1 2 v 3 + 1 4 v 3 1 + ( u 1 + uv 1 ) e 2 v ; (3.25) u t = u 3 + 3 u 2 v 1 + 3 2 u 1 v 2 + 9 4 u 1 v 2 1 − uu 1 e 2 v − 1 4 u 1 e − 2 v , v t = − 1 2 v 3 + 1 4 v 3 1 + ( u 1 + uv 1 ) e 2 v + 1 4 v 1 e − 2 v ; (3.26) u t = u 3 + 3 2 u 1 v 2 + 3 u 2 v 1 + 9 4 u 1 v 2 1 − c 2 1 u 1 e − 2 v − 1 2 u 1 e 2 v ( u 2 + c 2 ) , v t = − 1 2 v 3 + 1 4 v 3 1 + c 2 1 v 1 e − 2 v + 1 2 e 2 v (2 uu 1 + u 2 v 1 + c 2 v 1 ); (3.27) u t = u 3 + 3 2 u 1 v 2 + 3 u 2 v 1 + 9 4 u 1 v 2 1 − 1 3 e 2 v  u 1 (6 u 2 + c 1 ) + 4 uv 1 (2 u 2 + c 1 )  + e v  v 2 (2 u 2 + c 1 ) + ( u 1 + 2 u v 1 ) 2 + 2 c 1 v 2 1  , v t = − 1 2 v 3 + 1 4 v 3 1 + 1 3 e 2 v  4 u u 1 + (6 u 2 + c 1 ) v 1  + e v ( u 2 + 2 u 1 v 1 ) ; (3.28) u t = u 3 + 3 2 u 1 v 2 + 3 u 2 v 1 + 9 4 u 1 v 2 1 + 3 uv 2 ( c 1 ue v + c 2 ) + c 1 ( c 2 1 − 1) u 4 e 3 v − 3 4 u 2 e 2 v  u 1 (1 + 5 c 2 1 ) + 8 c 2 1 uv 1 + 2 c 2 u (1 − 3 c 2 1 )  − 3 c 2 2 ( u 1 + 2 uv 1 ) + 3 2 c 1 e v ( u 1 + 2 uv 1 − 2 c 2 u ) 2 + 3 2 c 2 v 1 (2 u 1 + 3 uv 1 ) , v t = − 1 2 v 3 + 1 4 v 3 1 + 3 2 c 1 e v ( u 2 + 2 u 1 v 1 ) + c 1 (1 − c 2 1 ) u 3 e 3 v + 6 c 1 c 2 ue v ( v 1 − c 2 ) + 3 4 ue 2 v  2 u 1 (1 + c 2 1 ) + uv 1 (1 + 5 c 2 1 ) + 2 c 2 u (1 − 3 c 2 1 )  + 3 2 c 2 v 1 (2 c 2 − v 1 ); (3.29) u t = u 3 + 3 2 u 1 v 2 + 3 u 2 v 1 + 9 4 u 1 v 2 1 + 3 e v ( u 2 + c )( v 2 + 2 v 2 1 ) + 3 2 e v u 1 ( u 1 + 4 uv 1 ) − 3 2 e 2 v  (3 u 2 + c ) u 1 + 4( u 2 + c ) uv 1  , v t = − 1 2 v 3 + 1 4 v 3 1 + 3 2 e v ( u 2 + 2 u 1 v 1 ) + 3 2 e 2 v  2 uu 1 + (3 u 2 + c ) v 1  ; (3.30) In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 11 u t = u 3 − 3 2 u 1 v 2 − 3 4 u 1 v 2 1 + 1 4 u 3 1 − c 1 e − 2 v u 1 − c 2 ( u 1 + 2 v 1 ) e 2( u + v ) + c 3 ( u 1 − 2 v 1 ) e 2( v − u ) , v t = − 1 2 v 3 + 3 2 u 1 u 2 − 3 4 u 2 1 v 1 + 1 4 v 3 1 + c 1 e − 2 v v 1 +  c 2 e 2( u + v ) − c 3 e 2( v − u )  v 1 ; (3.31) u t = u 3 − 3 2 u 1 v 2 − 3 4 u 1 v 2 1 + 1 4 u 3 1 +  c 2 e u + c 3 e − u − 3 c 2 1 e − 2 v  u 1 , v t = − 1 2 v 3 + 3 2 u 1 u 2 − 3 4 u 2 1 v 1 + 1 4 v 3 1 +  c 2 e u − c 3 e − u  u 1 +  3 c 2 1 e − 2 v − c 2 e u − c 3 e − u  v 1 ; (3.32) u t = u 3 − 3 2 u 1 v 2 − 3 4 u 1 v 2 1 + 1 4 u 3 1 + 3 k ( u 2 1 − 2 v 2 ) e − v − 3( c 2 − 3 k 2 ) u 1 e − 2 v + 8 k ( k 2 − c 2 ) e − 3 v , v t = − 1 2 v 3 + 3 2 u 1 u 2 − 3 4 u 2 1 v 1 + 1 4 v 3 1 + 3 k ( u 2 − 2 u 1 v 1 ) e − v + 3( c 2 − 3 k 2 ) v 1 e − 2 v ; (3.33) u t = u 3 − 3 2 u 1 v 2 − 3 4 u 1 v 2 1 + 1 4 u 3 1 − 3 c 2 1 u 1 e 2( u + v ) + 3 c 1 u 1 ( u 1 + 2 v 1 ) e u + v + ( c 2 e − u − 3 c 2 3 e − 2 v ) u 1 , v t = − 1 2 v 3 + 3 2 u 1 u 2 − 3 4 u 2 1 v 1 + 1 4 v 3 1 + 3 c 2 1 (2 u 1 + v 1 ) e 2( u + v ) + 3 c 1 ( u 2 + u 2 1 ) e u + v − c 2 ( u 1 + v 1 ) e − u + 3 c 2 3 v 1 e − 2 v ; (3.34) u t = u 3 − 3 2 u 1 v 2 − 3 4 u 1 v 2 1 + 1 4 u 3 1 + 3 c 2 u 1 ( u 1 + 2 v 1 ) e u + v − 4 c 1 c 2 e 3( u + v ) + 3[( c 1 − c 2 2 ) u 1 + 2 c 1 v 1 ] e 2( u + v ) , v t = − 1 2 v 3 + 3 2 u 1 u 2 − 3 4 u 2 1 v 1 + 1 4 v 3 1 + 3 c 2 ( u 2 + u 2 1 ) e u + v + 4 c 1 c 2 e 3( u + v ) + 3[2 c 2 2 u 1 − ( c 1 − c 2 2 ) v 1 ] e 2( u + v ) ; (3.35) u t = u 3 − 3 2 u 1 v 2 − 3 4 u 1 v 2 1 + 1 4 u 3 1 + 2 3 c 2 1 u 1 e − 2 v + c 1 (2 v 2 − u 2 1 ) e − v − 2 c 1 c 2 ( u 1 + 2 v 1 ) e u + 3 c 2 u 1 ( u 1 + 2 v 1 ) e u + v − 3 c 2 2 u 1 e 2( u + v ) , v t = − 1 2 v 3 + 3 2 u 1 u 2 − 3 4 u 2 1 v 1 + 1 4 v 3 1 − 2 3 c 2 1 v 1 e − 2 v + c 1 (2 u 1 v 1 − u 2 ) e − v − 2 c 1 c 2 ( u 1 − v 1 ) e u + 3 c 2 ( u 2 + u 2 1 ) e u + v + 3 c 2 2 (2 u 1 + v 1 ) e 2( u + v ) ; (3.36) u t = u 3 − 3 2 u 1 v 2 − 3 4 u 1 v 2 1 + 1 4 u 3 1 − 3 u 1  c 2 1 e 2( u + v ) + c 2 2 e 2( v − u ) + 2 c 1 c 2 e 2 v  − 3 c 2 3 u 1 e − 2 v + 3 c 1 u 1 ( u 1 + 2 v 1 ) e u + v − 3 c 2 u 1 ( u 1 − 2 v 1 ) e v − u , v t = − 1 2 v 3 + 3 2 u 1 u 2 − 3 4 u 2 1 v 1 + 1 4 v 3 1 + 3 c 2 1 (2 u 1 + v 1 ) e 2( u + v ) + 6 c 1 c 2 v 1 e 2 v + 3 c 2 2 ( v 1 − 2 u 1 ) e 2( v − u ) + 3 c 1 ( u 2 + u 2 1 ) e u + v + 3 c 2 ( u 2 1 − u 2 ) e v − u + 3 c 2 3 v 1 e − 2 v ; (3.37) u t = u 3 − 3 2 u 1 v 2 − 3 4 u 1 v 2 1 + 1 4 u 3 1 − 6 c 3 1 e 3( u + v ) − 3 4 c 2 1 (5 u 1 − 8 v 1 ) e 2( u + v ) + 9 2 c 1 u 1 ( u 1 + 2 v 1 ) e u + v + 2 c 2 1 c 2 e 2 u + v + 2 3 c 1 c 2 2 e u − v − 1 2 c 1 c 2 (7 u 1 + 12 v 1 ) e u + c 2 (2 v 2 − u 2 1 ) e − v + 11 12 c 2 2 u 1 e − 2 v − 2 9 c 3 2 e − 3 v , v t = − 1 2 v 3 + 3 2 u 1 u 2 − 3 4 u 2 1 v 1 + 1 4 v 3 1 + 6 c 3 1 e 3( u + v ) + 3 4 c 2 1 (18 u 1 + 5 v 1 ) e 2( u + v ) + 9 2 c 1 ( u 2 + u 2 1 ) e u + v − 4 c 2 1 c 2 e 2 u + v + 2 3 c 1 c 2 2 e u − v − 7 2 c 1 c 2 ( u 1 − v 1 ) e u + c 2 (2 u 1 v 1 − u 2 ) e − v − 11 12 c 2 2 v 1 e − 2 v ; (3.38) u t = u 3 − 3 2 u 1 v 2 − 3 4 u 1 v 2 1 + 1 4 u 3 1 + c 3 e − v ( u 2 1 − 2 v 2 ) + 2 3 c 2 3 u 1 e − 2 v + c 1  3 u 1 e u + v + 2 c 3 e u  ( u 1 + 2 v 1 ) − c 2 (3 u 1 e v − u + 2 c 3 e − u )( u 1 − 2 v 1 ) − 3 u 1 ( c 1 e u + c 2 e − u ) 2 e 2 v , v t = − 1 2 v 3 + 3 2 u 1 u 2 − 3 4 u 2 1 v 1 + 1 4 v 3 1 + c 3 e − v ( u 2 − 2 u 1 v 1 ) − 2 3 c 2 3 v 1 e − 2 v + 3 c 2 1 e 2( u + v ) (2 u 1 + v 1 ) − 3 c 2 2 e 2( v − u ) (2 u 1 − v 1 ) + 3 c 1 e u + v ( u 2 + u 2 1 ) − 3 c 2 e v − u ( u 2 − u 2 1 ) + 2 c 2 c 3 e − u ( u 1 + v 1 ) + 2 c 1 c 3 e u ( u 1 − v 1 ) + 6 c 1 c 2 v 1 e 2 v ; (3.39) 12 A.G. Meshko v and M.Ju . Balakhnev u t = u 3 − 3 4 (2 g 3 − u 2 + 2 g v 1 ) 2 u 1 − g 2 + 3 g ( u 2 − v 2 ) − 6 u 2 1 − 9 u 1 v 1 − 3 2 v 2 1 − 3(5 g 2 + 4 cg + c 2 ) u 1 − 6 g 2 v 1 + 2 cg 2 (8 g + 3 c ) + 9 g 4 , v t = − 1 2 v 3 + 3 4 (2 g 3 − u 2 + 2 g v 1 ) 2 u 1 − g 2 − 3(3 g + c ) u 2 − 3 2 v 2 1 + 3(9 g 2 + 8 cg + 2 c 2 ) u 1 + 3(6 g 2 + 4 cg + c 2 ) v 1 − 2 cg 2 (8 g + 3 c ) − 9 g 4 , g = u + v ; (3.40) Remark 1. S ystems (3.1), (3.3), (3.6) and (3.20) were p rop osed in [3], where system (3.20) is giv en w ith a m isprint. Sys tem (3.10) wa s presented in [29]. Remark 2. T en pairs of in tegrabilit y conditions (for ρ 0 – ρ 9 and ˜ ρ 0 – ˜ ρ 9 ) h a v e b een v erif ied for eac h system (3.1)–(3.40), and n on trivial higher conserved densities w ith orders 2, 3, 4 and 5 ha v e b een found. Remark 3. It is sho wn in [21] that systems (3.10 ) and (3.12) are un ique diverge n t systems of the form (2.21) that satisfy the integ rabilit y conditions. Remark 4. S ystem (3.22) is a mo d if ication of (3.11), sys tems (3.31)–(3.39) are mo d if ications of (3.9). Remark 5. Canonical densities for s y s tem (3.25) d ep end on the n onlo cal v ariable w = D − 1 x e − v . Remark 6. Many of the systems p ossess d iscrete symmetries. Th ey are: u → − u for (3.1), (3.2 ), (3.9), (3.10), (3.16), (3.20) and (3.27 ); u → − u , v → v + π i for (3.24); u → iu , v → v − i 2 π , c 1 → − c 1 for (3.28 ); { u → − u, v → v + π i } ∪ { v → v + π i, c 1 → − c 1 } ∪ { u → − u, c 1 → − c 1 } for (3.29); { u → − u, v → v + π i } ∪ { u → iu, v → v − i 2 π } for (3.30); u → − u , c 2 → c 3 , c 3 → c 1 for (3.32 ); u → − u , k → − k f or (3.33); u → − u , c 1 → c 2 , c 2 → c 1 for (3.37 ); u → − u , c 3 → − c 3 , c 2 → c 1 , c 1 → c 2 for (3.39 ). Also, systems (3.1), (3.2), (3.9), (3.10) and (3.27) p reserv e the r eal sh ap e u nder the transf or- mation u → iu . System (3.33) k eeps the real shap e under the transf ormation u → iu , k → ik . 3.1 Example of computations Let us consider the simplest case of system (1.1): u t = u 3 + f 1 ( u, v ) u 1 + f 2 ( u, v ) v 1 , v t = av 3 + g 1 ( u, v ) u 1 + g 2 ( u, v ) v 1 , (3.41) where a ( a − 1) 6 = 0. F orm ulas (2.16) are red uced now to the follo wing ρ 0 = 0 , ˜ ρ 0 = 0 , ρ 1 = − 1 3 f 1 , ˜ ρ 1 = − 1 3 a g 2 . (3.42) The further canonical densities read ρ 2 = − 1 3 ( f 1 ,u u 1 + f 2 ,u v 1 ) + 1 3 D x f 1 , ˜ ρ 2 = − 1 3 a ( g 1 ,v u 1 + g 2 ,v v 1 ) + 1 3 a D x g 2 , τ 2 = 1 3 ( f 1 ,u u 1 + f 2 ,u v 1 ) , ˜ τ 2 = 1 3 a ( g 1 ,v u 1 + g 2 ,v v 1 ) , (3.43) where indices after commas denote deriv at iv es. In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 13 The f irst int egrabilit y condition (1.5) for ρ 1 can b e split with r esp ect to u 3 , v 3 , u 2 and v 2 . This provides the follo wing equations f 1 ,uv = 0 , f 1 ,uuu = 0 , f 1 ,vv v = 0 , or f 1 ( u, v ) = c 1 u 2 + c 2 u + c 3 v 2 + c 4 v + c 5 . Analogously , condition (1.5) for ˜ ρ 1 implies g 2 ( u, v ) = b 1 u 2 + b 2 u + b 3 v 2 + b 4 v + b 5 . It is obvio us from (3.43) that the second inte grabilit y conditions (2.18) are τ 2 ∈ Im D x and ˜ τ 2 ∈ Im D x . These conditions provide f 2 ,uu = g 1 ,vv = 0 or f 2 ( u, v ) = uf 3 ( v ) + f 4 ( v ), g 1 ( u, v ) = v g 3 ( u ) + g 4 ( u ). Th us, system (3.41) tak es the follo wing form: u t = u 3 + ( c 1 u 2 + c 2 u + c 3 v 2 + c 4 v + c 5 ) u 1 + ( uf 3 ( v ) + f 4 ( v )) v 1 , v t = av 3 + ( v g 3 ( u ) + g 4 ( u )) u 1 + ( b 1 u 2 + b 2 u + b 3 v 2 + b 4 v + b 5 ) v 1 . (3.44) No w one can obtain θ 2 = D − 1 x D t ρ 2 and ˜ θ 2 = D − 1 x D t ˜ ρ 2 in an explicit form. The expr es- sions D t ρ 1 and D t ˜ ρ 1 are not the total deriv ativ es yet: D t ρ 1 = D x h 1 ( u i , v j ) + R 1 ( u, v , u 1 , v 1 ) = D x θ 1 , D t ˜ ρ 1 = D x ˜ h 1 ( u i , v j ) + ˜ R 1 ( u, v , u 1 , v 1 ) = D x ˜ θ 1 . Therefore, we ha v e s et θ 1 = h 1 ( u i , v j ) + q 1 ( u, v ), ˜ θ 1 = ˜ h 1 ( u i , v j ) + ˜ q 1 ( u, v ), w here q 1 and ˜ q 1 are unknown fun ctions and D x q 1 = R 1 , D x ˜ q 1 = ˜ R 1 . This tric k allo ws us to ev aluate ρ 4 , τ 4 , ˜ ρ 4 , ˜ τ 4 and verify the fourth integrabilit y conditions (2.18). Th ese conditions imply f ′′ 3 = g ′′ 3 = 0, hence f 3 = a 1 v + a 2 , g 3 = a 3 u + a 4 . T o simp lify the fu rther analysis one must list all ir reducible cases of f 1 (or g 2 ). Let u s tak e f 1 = c 1 u 2 + c 2 u + c 3 v 2 + c 4 v + c 5 for d ef initeness. Lemma 4. Using c omplex dilatations of u and v , tr anslations u → u + λ 1 , v → v + λ 2 and the Galilei tr ansfo rmation u t → u t + αu x , v t → v t + αv x one c an r e duc e f 1 to one of the fol lowing forms: 1) u 2 + v 2 ; 2) u 2 + αv ; 3) v 2 + αu ; 4) u + v ; 5) u ; 6) v ; 7) f 1 = 0 , wher e α is any c on stant. Mor e over, in the c ases 4–7 the function g 2 must b e line ar ( b 1 = b 3 = 0) b e c ause otherwise the p ermutation u ↔ v gives one of the c ases 1–3 . In cases 1 and 3 contradicti ons follo w fr om the integrabilit y conditions (1.5) with n = 1 , 3 and (2.18) w ith n = 2 , 4. In case 2 the in tegrabilit y conditions (1.5) with n = 1 , 3 , 5 and (2.18) with n = 2 , 4 are satisf ied if f system (3.44) is r educed to a pair of ind ep endent equ ations. Thus, a non trivial in tegrable system (3.41) must b el ong to the follo w ing class: u t = u 3 + ( c 2 u + c 4 v ) u 1 + ( u ( a 1 v + a 2 ) + f 4 ( v )) v 1 , v t = av 3 + ( v ( a 3 u + a 4 ) + g 4 ( u )) u 1 + ( b 2 u + b 4 v + b 5 ) v 1 , (3.45) and only th e follo wing cases are p o ssible: 4) c 2 = c 4 = 1; 5 ) c 2 = 1 , c 4 = 0; 6 ) c 2 = 0 , c 4 = 1; 7) c 2 = c 4 = 0 . In case 4 the in tegrabilit y conditions (1.5) with n = 1 , . . . , 5 provide the functions g 4 = k 1 u + k 2 , f 4 = k 3 v + k 4 , the co ef f icien ts a 3 = 0, a 4 = − 1 + a 2 + b 2 , b 4 = b 2 = ( a + 1) a 2 − 2 a − 1 and the follo wing equations: ( a + 1)(2 a 2 + aa 2 − 2 a − 1) = 0 , a 2 ( a 2 − 2) − a (4 a 2 2 − 5 − 7 a 2 ) + 16 a 2 − 14 a 2 2 − 3 = 0 , 14 A.G. Meshko v and M.Ju . Balakhnev a 5 ( a 2 − 2) − a 4 (4 a 2 2 − 7 a 2 − 6) − 2 a 3 (16 a 2 2 − 5 a 2 − 37) − a 2 (177 a 2 2 − 224 a 2 − 41) − a (236 a 2 2 − 353 a 2 + 102) − a 2 (37 a 2 − 53) − 17 = 0 . Using th e pac k age Gro eb n er in Maple, one can obtain a 2 = (1 − a ) / 3, a 2 + 7 a + 1 = 0 or a = (3 c − 7) / 2, c 2 = 5. Then, the remaining co ef f icien ts are also determined and we obtain u t = u 3 + ( u + v ) u 1 + 1 2  (3 − c ) u + (5 c − 11) v  v 1 , c 2 = 5 , v t = 1 2 v 3 (3 c − 7) +  ( c + 2) u − v  u 1 + 1 2 ( c − 3)( u + v ) v 1 . (3.46) The follo wing substitution V = 1 6 ( c + 1)( v − u ) , U = 1 12 ( c + 3) u + 1 6 v reduces system (3.46) to the third Drinfeld–Soko lo v system U t = − 8 U 3 + 3 V 3 + 6( V − 8 U ) U 1 + 12 U V 1 , V t = 12 U 3 − 2 V 3 + 48 V U 1 + 12(2 U − V ) V 1 (3.46a ) that has b een presented f irst in [3]. Scaling t → − 1 2 t, U = → 1 6 U, V = → − 1 3 V giv es more s ymmetric form of system (3.46a) U t = 4 U 3 + 3 V 3 + (4 U + V ) U 1 + 2 U V 1 , V t = 3 U 3 + V 3 − 4 V U 1 − 2( V + U ) V 1 (3.46b) that w as found in [14]. In case 5 the equations a 1 = a 2 = 0, g ′ 4 f ′ 4 = 0, g ′′′ 4 = 0 follo w from the in tegrabilit y condi- tions (1.5) with n = 1 , . . . , 5. This implies f 4 6 = 0 b ecause otherwise the f ir st equ ation of (3.45) will b e indep enden t. Hence, there are t w o br anc hes (1) f 4 = 1 and (2) f ′ 4 6 = 0, g ′ 4 = 0. Along the f irst br anc h, if one use additionally the integ rabilit y conditions (1.5) with n = 7 and s olv es a large p olynomial system for constant s, one can obtain th e follo wing sy s tem: u t = u 3 + uu 1 + v 1 , v t = − 2 v 3 − uv 1 , that can b e transformed to (1.3) by a scaling. Along the second branc h the in tegrabilit y conditions (1.5) with n = 1 , . . . , 5 p r o vide the follo wing system u t = u 3 + uu 1 − v v 1 , v t = − 2 v 3 − uv 1 , that is equ iv alen t to (1.2). Case 6 is sy m metric to case 5: one can obtain f ′′′ 4 = 0, g ′′ 4 = 0, a 3 = a 4 = b 2 = 0 from the in tegrabilit y conditions (1.5) with n = 1 , . . . , 5. Hence g 4 6 = 0 and we ha v e t w o branches g 4 = 1 or g 4 = u . Using the add itional in tegrabilit y conditions (1.5) with n = 6 , 7 one can obtain equations (1.2) and (1.3). There are many br anc hes in case 7 b u t all of them pr o vide linear or triangular sys tems only . As one can see, classif ication of int egrable systems of the form (3.41) is a suf f icien tly lab orious task. System (2.21) conta ins eight unkno wn functions dep end ing on four v ariables, therefore classif ication of these systems is muc h m ore dif f icult. In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 15 4 Dif feren tial substitutions A dif fer ential sub s titution is a pair of equations u = f ( U, V , U x , V x , . . . , U n , V n ) , v = g ( U, V , U x , V x , . . . , U n , V n ) , (4.1) where f and g are some smo o th functions. Definition 1. If for an y solution ( U, V ) of a system (Σ) form ulas (4.1) pro vide a solution ( u, v ) of system (1.1), then one sa ys that system (1.1) admits sub stitution (4.1). In all cases th at w e kno w, the new systems (Σ) b elong to the same class (1.1) U t = U xxx + P ( U, V , U x , V x , U xx , V xx ) , V t = a V xxx + Q ( U, V , U x , V x , U xx , V xx ) , ( S ) with some smo o th fu nctions P and Q . There exist some group-theoretical explanation of this fact for K dV t yp e equations [35]. O ur attempts to in tro duce another parameter a ′ 6 = a in ( S ) had no success. Substituting (4.1) into (1.1) one obtains the follo wing equations  D 3 x f + F ( f , g , D x f , D x g , D 2 x f , D 2 x g ) − ∂ t f  S = 0 ,  aD 3 x g + G ( f , g , D x f , D x g , D 2 x f , D 2 x g ) − ∂ t g  S = 0 . (4 .2) It is obvio us that transition to the manifold ( S ) in (4.2) is equiv alent to a replacemen t of ∂ t b y the ev olutionary dif feren tiation D t p erformed in accordance with ( S ): D t f = D 3 x f + F ( f , g , D x f , D x g , D 2 x f , D 2 x g ) , D t g = aD 3 x g + G ( f , g , D x f , D x g , D 2 x f , D 2 x g ) , (4.3) where D t f = n X i =1 ∂ f ∂ U i D i x ( U 3 + P ) + n X i =1 ∂ f ∂ V i D i x ( aV 3 + Q ) . Another wa y to obtain (4.3) is to d if ferenti ate equations (4.1 ) w ith resp e ct to t in accordance with (1.1 ) and ( S ) and exclude u and v by us ing (4.1). This algorithm and man y others are co ded in Maple, see for example [36]. T o f ind th e admissib le fun ctions f , g , P , Q from (4.3) one can u se the follo wing easily pr o v able form ula: ∂ ∂ U k D m x f = m X s =0  m s  D m − s x ∂ f ∂ U k − s , ∂ f ∂ U − i ≡ 0 for i > 0 , and the analogous form ula f or ∂ /∂ V k . Dif feren tiating (4.3) with resp ect to U n +3 and V n +3 , one obtains ∂ f ∂ V n = 0 , ∂ g ∂ U n = 0 . (4.4) Other corollaries of (4.3) are to o cumb ersome to consider them in th e general form. Let us consider, as an example, the f irs t order d if ferentia l su b stitutions for sy s tem (1.2 ) u t = u 3 + v u 1 , v t = − 1 2 v 3 + u u 1 − v v 1 . (4.5) 16 A.G. Meshko v and M.Ju . Balakhnev According to (4.4) one has f = f ( U, V , U 1 ) , g = g ( U, V , V 1 ), hence equ ations (4.3) no w read D 3 x f − f U ( U 3 + P ) − f V ( Q − V 3 / 2) − f U 1 ( U 4 + D x P ) + g D x f = 0 , g U ( U 3 + P ) + g V ( Q − V 3 / 2) + g V 1 ( D x Q − V 4 / 2) + 1 2 D 3 x g − f D x f + g D x g = 0 . (4.6) Dif ferent iating (4.6) with r esp ect to U 3 and V 3 one can obtain four equations: ∂ f ∂ U 1 ∂ P ∂ U 2 = 3 D x ∂ f ∂ U 1 , ∂ f ∂ U 1 ∂ P ∂ V 2 = 3 2 ∂ f ∂ V , ∂ g ∂ V 1 ∂ Q ∂ U 2 = − 3 2 ∂ g ∂ U , ∂ g ∂ V 1 ∂ Q ∂ V 2 = − 3 2 D x ∂ g ∂ V 1 . (4.7) Let us consider some corollaries of these equations. 1. If ∂ f /∂ U 1 = 0 and ∂ g/∂ V 1 = 0, then u = f ( U ), v = g ( V ) is a trivial p oin t tran s formation. 2. If ∂ f /∂ U 1 = 0, then u = f ( U ) and one can s et f ( U ) = U b y mo dulo of the p oi n t transformation. In this case P = g U 1 from the f irst of equ ations (4.6). 3. If ∂ g/∂ V 1 = 0, then v = g ( V ) and one can set g ( V ) = V by mo d ulo of the p oin t transfor- mation. In this case Q = f D x f − V V 1 from the second of equations (4.6). 4. If ( ∂ f /∂ U 1 )( ∂ g/∂ V 1 ) 6 = 0, then one can f ind P and Q as p o lynomials of U 2 and V 2 from equations (4.7). In v estigati on of cases 2–4 provides seven n ontrivial solutions of equations (4.6) (see b elo w (3.1) → (3.2 ), . . . , (3.1) → (3.17)). Note that inte gr able system (3.6 ) admits str ange dif f eren tial substitutions that generate non- inte gr able systems. F or example, sy s tem (3.6 ) admits th e follo wing d if ferentia l su bstitution: u = 3 2 V 2 − 3 4 V 2 1 − 3 2 U 1 e V , v = 9 4  − V 4 + V 1 V 3 + V 2 1 V 2 − 1 4 V 4 1 − U 2 1 e 2 V + e V ( U 3 + 2 U 2 V 1 + 3 U 1 V 2 )  , (4.8) so that the functions U and V s atisfy the follo wing s y s tem: U t = U 3 + 3 2 U 2 V 1 + 3 4 U 1 V 2 1 − U 1 e − V f ′′ ( U ) + f ( U ) , V t = − 1 2 V 3 + 1 4 V 3 1 + 3 2 e V ( U 2 + U 1 V 1 ) − f ′ ( U ) with arbitrary function f . This system do es not satisfy the in tegrabilit y conditions (1.4). T o comprehend this unusual phenomenon w e ev aluate V 2 , V 3 and V 4 from the f irst equation (4.8) V 2 = 2 3 u + U 1 e V + 1 2 V 2 1 , V 3 = 2 3 u 1 + D x ( U 1 e V ) + V 1 V 2 , V 4 = 2 3 u 2 + D 2 x ( U 1 e V ) + D x ( V 1 V 2 ) , and substitute them in to the second one. T he r esult is v = − u 2 − 3 2 u 2 . It is easily v erif ied th at the obtained constraint is a redu ction of system (3.6) in to the single KdV equ ation u t = − 1 / 2 u 3 − uu 1 . T his m eans that u sing a subs titution lik e (4.8) we are trying to construct an int egrable sys tem from th e single Kd V equation. Ther e are other su c h examples for system (3.6). Note that the redu ction obtained ab o v e f ollo w s f rom the reduction u = const for s ystem (3.3) (see (3.3) and (3.3) → (3.6)). In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 17 T o organize the p resen ted list of systems we ha v e compu ted admiss ib le dif feren tial s ubstitu- tions for eac h system and present the results in this section. Th e f ormula u ′ = f ( u, v , u x , v x , . . . ) , v ′ = g ( u, v , u x , v x , . . . ) (A) → (B) will denote th at if u ′ and v ′ are su bstituted in to system (A), then system (B) follo w s for u and v . W e sa y in this case th at system (B) is obtained from system (A) by the dif ferentia l su b stitution. Substitution (A) → (B) establishes an in terrelation b etw een the sets of solutions of sys- tems (A) and (B): ( u, v ) 7→ ( u ′ , v ′ ) is a single v alued map. And con v ersely , if for some solu- tion ( u ′ , v ′ ) of sy s tem (A) one solv es the system of tw o ordin ary dif ferenti al equ ations (A) → (B) for u and v , then one or more solutions of system (B) are obtained. Of course, explicit solutions can b e obtained v ery rarely wh en the su bstitution is linear or inv ertible (see b elo w). Let us consider the follo wing simple example: u ′ = u 1 , v ′ = v 1 . (3.10) → (3.9) This substitution is p ossib le for any diverge n t sys tem u t =  u 2 + F ( u, v , u 1 , v 1 )  x , v t =  − 1 2 v 2 + G ( u, v , u 1 , v 1 )  x . It pr o duces the system u t = u 3 + F ( u 1 , v 1 , u 2 , v 2 ), v t = − v 2 / 2 + G ( u 1 , v 1 , u 2 , v 2 ) without u 0 and v 0 . The inv erse transf ormation is quasi-lo cal u = D − 1 x u ′ , v = D − 1 x v ′ . This is a well kno wn fact, that is wh y the substitutions ( u, v ) → ( u 1 , v 1 ) are not written for the d iv ergen t sy s tems b elo w. In some cases analogous subs titutions are not so ob vious and w e present them lik ewise (3.1) → (3.2 ), for example. Theorem 1. Differ ential substitutions pr e sente d b elow c onne ct al l systems f r om the list of Se c- tion 3 with systems (1.2) and (1.3) . Systems (1.2) and (1.3) ar e also implicitly c onne cte d with e ach other. The pro of can b e obtained by a direct verif icati on. List of the substitutions: u ′ = u, v ′ = v 1 ; (3.1) → (3.2) u ′ = 3 √ 2 ( u 2 − u 1 v 1 ) , v ′ = 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ); (3.1) → (3.9) u ′ = 3 √ 2 ( u 1 − u v ) , v ′ = 3 2 v 1 − 3 4 ( u 2 + v 2 ); (3.1) → (3.10) u ′ = 3 4 √ 2  u 2 1 − 2 u 2 + 2 v 1  , v ′ = 3 2 v 1 ; (3.1) → (3.11) u ′ = 3 4 √ 2  u 2 − 2 u 1 + 2 v  , v ′ = 3 2 v ; (3.1) → (3.12) u ′ = 3 √ 2  ( u + v ) 2 − u 1  + 3 16 √ 2 c 2 1 , v ′ = 3 v 1 + 3 2 c 1 ( u + v ); (3.1) → (3.13) u ′ = √ 2 u, v ′ = − 3 v 1 + u − 3 v 2 ; (3.1) → (3.14) u ′ = u, v ′ = k + p u 2 + v 1 ; (3.1) → (3.16) u ′ = p v 2 + u 1 , v ′ = v − k ; (3.1) → (3.17) u ′ = √ 2  4 3 u + 3 16 c 2 1  , v ′ = u 1 + v 2 √ u + v 1 − 4 3 v 1 + c 1 √ u + v 1 ; (3.1) → (3.18) u ′ = 3 2 √ 2  1 2 u 2 1 − u 2 + v 1 + c 1 e u + 2 c 2 e 2 u  , v ′ = 3 2 ( v 1 + c 1 e u ); (3.1) → (3.22) u ′ = √ 2 u, v ′ = − 3 2 v 2 − 3 4 v 2 1 + u − c 2 e − 2 v ; (3.1) → (3.23) u ′ = 2 c 1 u, v ′ = − 3 2 v 2 − 3 4 v 2 1 − 1 3 u 2 e 2 v − 3 2 c 2 1 e − 2 v + e v ( u 1 + 2 uv 1 ); (3.1) → (3.24) u ′ = √ 3 u 1 e v + 2 c 1 u, 18 A.G. Meshko v and M.Ju . Balakhnev v ′ = 3 2 v 2 − 3 4 v 2 1 + 2 √ 3 c 1 v 1 e − v − 1 2 e 2 v ( u 2 + c 2 ) − c 2 1 e − 2 v ; (3.1) → (3.27) u ′ = 1 3 √ 2 e v (3 u 1 + c 1 e v + 2 u 2 e v ) , v ′ = 3 2 v 2 − 3 4 v 2 1 − u 1 e v + 1 3 e 2 v ( c 1 − 2 u 2 ); (3.1) → (3.28) u ′ = 3 2 √ 2 e v ( u 1 + 2 c 2 u + c 1 u 2 e v ) , v ′ = 3 2 v 2 − 3 4 v 2 1 − 3 2 c 1 u 1 e v + 3 c 2 v 1 − 3 4 (2 c 2 + c 1 ue v ) 2 − 3 4 u 2 e 2 v ; (3.1) → (3.29) u ′ = 3 2 √ 2 e v  u 1 + ( c + u 2 ) e v  , v ′ = 3 2 v 2 − 3 4 v 2 1 − 3 2 u 1 e v + 3 2 ( c − u 2 ) e 2 v ; (3.1) → (3.30) u ′ = 3 2 √ 2 ( u 2 − u 1 v 1 ) + √ 2  c 2 e u − c 3 e − u − 3 c 1 u 1 e − v  , v ′ = 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) + c 2 e u + c 3 e − u − 6 c 1 v 1 e − v − 3 c 2 1 e − 2 v ; (3.1) → (3.32) u ′ = 3 2 √ 2 ( u 2 − u 1 v 1 ) − 3 √ 2( cu 1 + 2 k v 1 ) e − v − 6 ck √ 2 e − 2 v , v ′ = 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) − 3( k u 1 + 2 cv 1 ) e − v − 3( c 2 + k 2 ) e − 2 v ; (3.1) → (3.33) u ′ = 3 √ 2 ( u 2 − u 1 v 1 ) − √ 2  c 2 e − u + 6 c 1 c 3 e u − 3 c 1 u 1 e u + v + 3 c 3 u 1 e − v  , (3.1) → (3.34) v ′ = 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) + c 2 e − u − 3 c 2 1 e 2( u + v ) − 3 c 1 u 1 e u + v − 6 c 3 v 1 e − v − 3 c 2 3 e − 2 v ; u ′ = 3 √ 2 ( u 2 − u 1 v 1 ) + 3 √ 2  c 1 e 2( u + v ) + c 2 u 1 e u + v  , v ′ = 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) + 3( c 1 − c 2 2 ) e 2( u + v ) − 3 c 2 u 1 e u + v ; (3.1) → (3.35 ) u ′ = √ 2  3 2 ( u 2 − u 1 v 1 ) − 2 3 k c 2 1 e − 2 v + 3 c 2 u 1 e u + v + 2 k c 1 c 2 e u  + c 1 √ 2 ( k u 1 + 2 v 1 ) e − v , k 2 = 1 , v ′ = 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) − 2 3 c 2 1 e − 2 v − 3 c 2 2 e 2( u + v ) − 3 c 2 u 1 e u + v + 2 c 1 c 2 e u + c 1 ( u 1 + 2 k v 1 ) e − v ; (3.1) → (3.36) u ′ = 3 2 √ 2 ( u 2 − u 1 v 1 ) + 3 √ 2  c 1 e u + v + c 2 e v − u − c 3 e − v  u 1 + 6 c 3 √ 2 ( c 2 e − u − c 1 e u ) , v ′ = 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) − 3 c 2 1 e 2( u + v ) − 3 c 2 2 e 2( v − u ) + 3 u 1 ( c 2 e v − u − c 1 e u + v ) − 6 c 3 v 1 e − v − 6 c 1 c 2 e 2 v − 3 c 2 3 e − 2 v ; (3.1) → (3.37) u ′ = 3 2 √ 2 ( u 2 − u 1 v 1 ) + 1 3 √ 2 c 2 2 e − 2 v − 1 2 √ 2 c 2 ( u 1 − 4 v 1 ) e − v − 2 √ 2 c 1 c 2 e u + 3 2 √ 2 c 1 (2 c 1 e 2( u + v ) + 3 u 1 e u + v ) , v ′ = 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) − 15 4 c 2 1 e 2( u + v ) − 9 2 c 1 u 1 e u + v + 5 2 c 1 c 2 e u − 5 12 c 2 2 e − 2 v + c 2 ( u 1 − v 1 ) e − v ; (3.1) → (3.38) u ′ = 3 2 √ 2 ( u 2 − u 1 v 1 ) + 3 √ 2 u 1  c 1 e u + v + c 2 e v − u  − 2 3 √ 2 c 2 3 e − 2 v + 2 √ 2 c 3 ( c 2 e − u − c 1 e u ) − √ 2 c 3 ( u 1 + 2 v 1 ) e − v , v ′ = 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) − 3 e 2 v ( c 1 e u + c 2 e − u ) 2 − c 3 ( u 1 + 2 v 1 ) e − v + 2 c 3 ( c 2 e − u − c 1 e u ) − 2 3 c 2 3 e − 2 v + 3 u 1  c 2 e v − u − c 1 e u + v  ; (3.1) → (3.39) u ′ = 3 √ 2 u 2 − 3 g v 1 − 2 g 3 p u 1 − g 2 − 3 √ 2 (3 g + 2 c ) p u 1 − g 2 , v ′ = 3( v 1 − 2 cg − c 2 ) , g = u + v ; (3.1) → (3.40) u ′ = u, v ′ = v 1 ; (3.3) → (3.5) u ′ = 3 2 u 2 + u 2 + v , v ′ = u, (3.3) → (3.6) this sub s titution is inv ertible, see b elo w (3.6) → (3.3); u ′ = u 1 + 1 2 v 2 , v ′ = v − k ; (3.3) → (3.8) In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 19 u ′ = 4 u 2 + u 1  16 3 √ u + v 1 − 2 c 1  + 16 9 u 2 + 1 2 c 2 1 u, v ′ = − u 1 + v 2 √ u + v 1 − 4 3 v 1 + c 1 √ u + v 1 ; (3.3) → (3.18) u ′ = u 1 , v ′ = v ; (3.4) → (3.3) u ′ = u 1 , v ′ = v 1 ; (3.4) → (3.5) u ′ = 3 2 u 3 + 2 uu 1 + v 1 , v ′ = u, (3.4) → (3.6) this sub s titution has the quasi-lo cal inv erse s u bstitution, see b elo w (3.6) → (3.4); u ′ = u 2 + v v 1 , v ′ = v − k ; (3.4) → (3.8) u ′ = 3 u 3 − 6( u 1 v 1 + v u 2 ) + 2 uu 1 , v ′ = 3 v 1 − 3 v 2 + u ; ( 3.4) → (3.14) u ′ = e − v , v ′ = 3 2 v 2 − 3 4 v 2 1 − ue 2 v ; (3.4) → (3.25) u ′ = u 1 + √ 3 e v ( u 2 + u 1 v 1 ) , v ′ = 3 2 v 2 − 3 4 v 2 1 − ue 2 v + √ 3 v 1 e − v − 1 4 e − 2 v ; (3 .4) → (3.26) u ′ = v , v ′ = − 3 2 v 2 − v 2 + u ; (3.6) → (3.3) u ′ = v , v ′ = − 3 2 v 2 − v 2 + w, w x = u, w t = u 2 + uv ; ( 3.6) → (3.4) u ′ = v 1 , v ′ = − 3 2 v 3 − v 2 1 + u ; (3.6) → (3.5) u ′ = u + k , v ′ = v 1 − 1 2 v 2 ; (3.6) → (3.7) u ′ = v − k , v ′ = u 1 − 3 2 v 2 − 1 2 ( v 2 + 3 k 2 ) + 2 k v ; (3.6) → (3.8) u ′ = 3 u 1 − 3 2 v 1 − 3 4 ( u 2 + v 2 ) , v ′ = 9 4 v 3 − 9 4 u ( u 2 + 2 v 2 ) + 9 4 v v 2 − 9 2 u 2 1 + 9 4 u 2 (2 u 1 + v 1 ) − 9 4 v v 1 (2 u + v ) − 9 16 ( u 2 − v 2 ) 2 ; (3.6) → (3.10) u ′ = − 3 u 2 + 3 2 v 1 , v ′ = 9 4 v 3 + 9 2 u 1 v 2 − 9 16 ( u 2 1 − 2 u 2 + 2 v 1 ) 2 + 9 4 ( u 2 1 + v 1 ) 2 − 9 8 u 4 1 ; (3.6) → (3.11) u ′ = − 3 u 1 + 3 2 v , v ′ = 9 4 v 2 + 9 2 uv 1 − 9 16 ( u 2 − 2 u 1 + 2 v ) 2 + 9 4 ( u 2 + v ) 2 − 9 8 u 4 ; (3.6) → (3.12) u ′ = − 3( v 1 + 2 u 1 ) + 3 2 c 1 g , g = u + v , v ′ = 9 2 v 3 + 9 4 c 1 ( u 2 − v 2 ) + 18 g v 2 − 9 u 2 1 + 9 v 2 1 + 9( c 1 g + 2 g 2 ) u 1 − 9 8 c 2 1 ( u 1 + g 2 ) + 36 g 2 v 1 + 9 g 4 ; (3.6) → (3.13) u ′ = 3 v 1 + u − 3 v 2 , v ′ = − 9 2 v 3 + 3 2 u 2 + 9 v v 2 − 6 u 1 v + 6(3 v 2 − u ) v 1 + 3 v 2 (2 u − 3 v 2 ); (3.6) → (3.14) u ′ = − u 1 + v 2 √ u + v 1 + c 1 √ u + v 1 − 4 3 v 1 , v ′ = 3 2 u 3 + v 4 √ u + v 1 − 3 4 c 1 u 2 + v 3 √ u + v 1 + 8 3 u u 1 + v 2 √ u + v 1 + 9 8 ( u 1 + v 2 ) 3 ( u + v 1 ) 5 / 2 − ( u 1 + v 2 ) 2 u + v 1 − 3 8 u 1 + v 2 ( u + v 1 ) 3 / 2  6( u 2 + v 3 ) − c 1 ( u 1 + v 2 )  + 2( v 3 + 2 u 2 + c 1 v 2 ) + 16 9 ( u 2 − v 2 1 ) − 8 3 ( v 2 − u 1 − c 1 v 1 ) √ u + v 1 − 1 2 c 2 1 ( u + 2 v 1 ); (3.6) → (3.18) u ′ = √ u + v 1 − k , v ′ = − 3 4 u 2 + v 3 √ u + v 1 + 3 8 ( u 1 + v 2 ) 2 ( u + v 1 ) 3 / 2 + 2 k √ u + v 1 − v 1 − u 2 ; (3.6) → (3.19) u ′ = i √ 6 u 1 + u 2 + v , v ′ = 3 2 ( u 2 1 − v 2 ) + i √ 6 uv 1 − v 2 ; (3.6) → (3.20) 20 A.G. Meshko v and M.Ju . Balakhnev u ′ = 3( u 1 + v 1 − k u ) , v ′ = − 9 2 v 3 − 9 2 uu 2 + 9( u + k ) v 2 − 27 4 u 2 1 − 9 v 2 1 + 9 u ( u + k ) u 1 − 9( u 2 + k 2 ) v 1 − 9 4 u 2 ( u 2 + 2 k 2 ); (3.6) → (3.21) u ′ = − 3 u 2 + 3 2 v 1 + 3 2 e u ( c 1 + 4 √ c 2 u 1 ) , v ′ = 9 4 v 3 − 9 4  1 2 u 2 1 − u 2 + v 1 + c 1 e u + 2 c 2 e 2 u  2 + 9 4 c 1 u 2 e u + 9 2 u 1 v 2 + 9 4 v 2 1 + 9 8 u 2 1 ( u 2 1 + 4 v 1 ) + 9 4 e u  c 1 (5 u 2 1 + 2 v 1 ) − 4 √ c 2 ( v 2 + 2 u 1 v 1 + u 3 1 )  + 18 c 2 2 e 4 u + 9 4 e 2 u  c 1 ( c 1 − 12 √ c 2 u 1 ) + 4 c 2 (3 u 2 1 + 2 v 1 )  + 18 c 2 e 3 u ( c 1 − 2 √ c 2 u 1 ); (3.6) → (3.22) u ′ = − 3 u 2 − 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) − 2 √ 6 c 2 ( u 1 + v 1 ) e u + v − c 1 e − 2 v − c 2 e 2( u + v ) + c 3 e 2( v − u ) , v ′ = 9 4 v 4 − 9 4 u 1 u 3 + 9 4 v 3 (2 u 1 + v 1 ) − 9 2 u 2 ( u 2 + u 2 1 ) − 9 16 ( u 2 1 − v 2 1 ) 2 + 9 4 v 2 ( u 2 1 + 2 u 1 v 1 − v 2 1 ) + 4 √ 6 c 2 v 1  2 c 3 e 3 v − u − c 1 e u − v  + c + 3 √ 6 c 2  v 3 − u 1 u 2 + v 2 (2 u 1 + v 1 ) + 2 u 1 v 1 ( u 1 + v 1 )  e u + v + 6 v 2  c 2 e 2( u + v ) + c 3 e 2( v − u ) − c 1 e − 2 v  + 3 2 c 1 ( u 1 − v 1 )( u 1 − 3 v 1 ) e − 2 v − c 2 1 e − 4 v + 4 c 2 c 3 e 4 v − 3 2 c 2 ( u 1 − v 1 ) 2 e 2( u + v ) + 3 2 c 3 ( u 1 − 3 v 1 ) 2 e 2( v − u ) ; (3.6) → (3.31) u ′ = 3 u 2 − 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) − 3 k e − v ( u 1 + 2 v 1 ) − 3( c 1 + 2 k 2 ) e − 2 v , v ′ = 9 4 v 4 − 9 4 u 1 u 3 + 9 4 v 3 ( v 1 − 2 u 1 ) + 9 2 u 2 ( u 2 1 − u 2 ) − 9 16 ( u 2 1 − v 2 1 ) 2 + 9 4 v 2 ( u 2 1 − 2 u 1 v 1 − v 2 1 ) − 9 c 2 1 e − 4 v + 18 kc 1 e − 3 v ( u 1 + 2 v 1 ) + 9 2 e − 2 v  4 k 2 u 2 − 4 c 1 v 2 − 2 k 2 ( u 1 + v 1 ) 2 + c 1 ( u 1 + v 1 )( u 1 + 3 v 1 )  − 9 2 k e − v  u 3 + 2 v 3 − 4 u 2 ( u 1 + v 1 ) + v 2 (2 v 1 − 3 u 1 ) + u 2 1 ( u 1 + 2 v 1 )  , c 1 = c 2 − k 2 ; (3.6) → (3.33) u ′ = − 3 u 2 − 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) − 3 c 2 1 e 2( u + v ) − 3 c 1 (3 u 1 + 2 v 1 ) e u + v + c 2 e − u − k e − 2 u , k = 3 c 2 3 , v ′ = 9 4 ( v 4 − u 1 u 3 ) + 9 4 v 3 (2 u 1 + v 1 ) − 9 2 u 2 ( u 2 + u 2 1 ) + 9 4 v 2 ( u 2 1 + 2 u 1 v 1 − v 2 1 ) − 9 16 ( u 2 1 − v 2 1 ) 2 − 9 c 4 1 e 4( u + v ) − 18 c 3 1 (3 u 1 + 2 v 1 ) e 3( u + v ) + 6 c 2 1 c 2 e u +2 v + 6 k c 2 1 e 2 u + 2 k c 2 e − u − 2 v + 6 k c 1 ( u 1 − 2 v 1 ) e u − v + 3 2 c 2 e − u ( u 2 + 2 v 2 + v 2 1 ) − 9 2 c 1 e u + v  u 3 − 2 v 3 + u 2 (9 u 1 + 4 v 1 ) − v 2 ( u 1 + 2 v 1 ) + 4 u 2 1 ( u 1 + v 1 )  − 3 2 k e − 2 v  4 v 2 + u 1 (4 v 1 − u 1 ) − 3 v 2 1  − k 2 e − 4 v + 6 c 1 c 2 e v ( u 1 + 2 v 1 ); (3.6) → (3.34) u ′ = 3 u 2 − 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) + 3( c 1 − c 2 2 ) e 2( u + v ) + 3 c 2 ( u 1 + 2 v 1 ) e u + v , v ′ = 9 4 v 4 − 9 4 u 1 u 3 + 9 4 v 3 ( v 1 − 2 u 1 ) + 9 2 u 2 ( u 2 1 − u 2 ) − 9 16 ( u 2 1 − v 2 1 ) 2 + 9 4 v 2 ( u 2 1 − 2 u 1 v 1 − v 2 1 ) − 9 k c 2 2 e 4( u + v ) + 18 kc 2 e 3( u + v ) ( u 1 + 2 v 1 ) + 9 2 e 2( u + v )  2 c 2 2 u 2 + 4 c 1 v 2 + c 1 ( u 1 + 3 v 1 ) 2 − c 2 2 ( u 2 1 + 4 u 1 v 1 + 5 v 2 1 )  − 9 2 c 2 e u + v  u 3 + 2 v 3 + u 2 ( u 1 + 4 v 1 ) + v 2 (2 v 1 − u 1 )  , k = c 2 2 − 2 c 1 ; (3.6) → (3.35) u ′ = 3 u 2 − 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) − 3 c 2 2 e 2( u + v ) + 3 c 2 ( u 1 + 2 v 1 ) e u + v + 2 c 1 c 2 e u + c 1 e − v ( u 1 + 2 v 1 ) − 2 3 c 2 1 e − 2 v , v ′ = 9 4 v 4 − 9 4 u 1 u 3 + 9 4 v 3 ( v 1 − 2 u 1 ) + 9 2 u 2 ( u 2 1 − u 2 ) − 9 16 ( u 2 1 − v 2 1 ) 2 + 9 4 v 2 ( u 2 1 − 2 u 1 v 1 − v 2 1 ) − 9 c 4 2 e 4( u + v ) + 18 c 3 2 e 3( u + v ) ( u 1 + 2 v 1 ) + 12 c 1 c 3 2 e 3 u +2 v + 9 2 c 2 2 e 2( u + v ) (2 u 2 − u 2 1 − 4 u 1 v 1 − 5 v 2 1 ) − 4 c 2 1 c 2 2 e 2 u + c 2 1 e − 2 v  2 u 2 − ( u 1 + v 1 ) 2  In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 21 − 9 2 c 2 e u + v  u 3 + 2 v 3 + u 2 ( u 1 + 4 v 1 ) + v 2 (2 v 1 − u 1 )  − 6 c 1 c 2 2 e 2 u + v ( u 1 + 2 v 1 ) − 3 c 1 c 2 e u (3 u 2 + 2 v 2 + 2 u 1 v 1 + 3 v 2 1 ) + 3 2 c 1 e − v  u 3 + 2 v 3 − 4 u 2 ( u 1 + v 1 ) + v 2 (2 v 1 − 3 u 1 ) + u 2 1 ( u 1 + 2 v 1 )  ; (3.6) → (3.36) u ′ = 3 u 2 − 3 2 v 2 − 3 4 ( u 2 1 + v 2 1 ) − 15 4 c 2 1 e 2( u + v ) + 9 2 c 1 ( u 1 + 2 v 1 ) e u + v + 5 2 c 1 c 2 e u + c 2 e − v ( u 1 + 2 v 1 ) − 5 12 c 2 2 e − 2 v , v ′ = 9 4 v 4 − 9 4 u 1 u 3 + 9 4 v 3 ( v 1 − 2 u 1 ) + 9 2 u 2 ( u 2 1 − u 2 ) − 9 16 ( u 2 1 − v 2 1 ) 2 + 9 4 v 2 ( u 2 1 − 2 u 1 v 1 − v 2 1 ) − 81 16 c 4 1 e 4( u + v ) + 27 4 c 3 1 e 3( u + v ) ( u 1 + 2 v 1 ) + 27 4 c 3 1 c 2 e 3 u +2 v + 9 8 c 2 1 e 2( u + v ) (18 u 2 + 16 v 2 − 5 u 2 1 − 12 u 1 v 1 − 9 v 2 1 ) − 27 8 c 2 1 c 2 2 e 2 u + 3 4 c 1 c 3 2 e u − 2 v − 9 4 c 1 c 2 2 e u − v ( u 1 + 2 v 1 ) − 27 4 c 1 e u + v  u 3 + 2 v 3 + u 2 ( u 1 + 4 v 1 ) + v 2 (2 v 1 − u 1 )  − 3 4 c 1 c 2 e u (17 u 2 + 14 v 2 + 12 u 1 v 1 + 19 v 2 1 ) + 1 2 c 3 2 e − 3 v ( u 1 + 2 v 1 ) + 1 8 c 2 2 e − 2 v  16 u 2 + 12 v 2 − 11 u 2 1 − 28 u 1 v 1 − 17 v 2 1  − 1 16 c 4 2 e − 4 v + 3 2 c 2 e − v  u 3 + 2 v 3 − 4 u 2 ( u 1 + v 1 ) + v 2 (2 v 1 − 3 u 1 ) + u 2 1 ( u 1 + 2 v 1 )  ; (3.6) → (3.38) u ′ = 3 2 g 3 − u 2 + 2 g v 1 p u 1 − g 2 + 6 g p u 1 − g 2 − 6 u 1 − 3 v 1 − 3 c (2 g + c ) , g = u + v , v ′ = 9 2 v 3 + 9(3 g + c ) u 2 + 9(2 g + c ) v 2 − 9 4 (2 g 3 − u 2 + 2 g v 1 ) 2 u 1 − g 2 + 9(2 u 1 + v 1 ) 2 − 18 u 2 1 + 18 p u 1 − g 2  v 2 + 2(2 g + c )( v 1 + g 2 )  + 36(2 g + c )( u 1 − g 2 ) 3 / 2 − 9 g 2 u 1 − 18  ( g + c ) 2 + 2 g 2  v 1 − 36 cg  ( g + c ) 2 + g 2  − 45 g 4 ; (3.6) → (3.40) u ′ = v − 2 k , v ′ = u − 3 2 v 1 + 4 3 k 3 t ; (3.7) → (3.8) this sub s titution is inv ertible: u ′ = 3 2 u 1 + v − 4 3 k 3 t, v ′ = u + 2 k ; (3.8) → (3.7) u ′ = − 1 2 v , v ′ = √ u + v 1 ; (3.8) → (3.19) u ′ = c 1 √ 2 e − v , v ′ = − v 1 + 2 3 u e v ; (3.10) → (3.24) u ′ = ue v , v ′ = v 1 − c 1 ue v − 2 c 2 ; (3.10) → (3.29) u ′ = ( u + √ − c ) e v , v ′ = v 1 + ( √ − c − u ) e v ; (3.10) → (3.30) u ′ = u 1 + 2 k e − v , v ′ = v 1 − 2 c e − v ; (3.10) → (3.33) u ′ = u 1 + ce u + v , v ′ = v 1 − k e u + v , where c and k are ro ots of z 2 − 2 c 2 z + 2 c 1 = 0; (3.10) → (3.35) u ′ = u 1 − 2 3 c 1 e − v , v ′ = v 1 − 2 c 2 e u + v + 2 3 c 1 e − v ; (3.10) → (3.36) u ′ = u 1 + 2 c 1 e u + v − 2 3 c 2 e − v , v ′ = v 1 − c 1 e u + v + 1 3 c 2 e − v ; ( 3.10) → (3.38) u ′ = 2 p u 1 − ( u + v ) 2 , v ′ = 2( u + v + c ); (3.10) → (3.40) u ′ = − 2 v , v ′ = 2 3 u − 2( v 1 + v 2 ); (3.12) → (3.14) this sub s titution is inv ertible: u ′ = 3 2 ( v − u 1 ) + 3 4 u 2 , v ′ = − 1 2 u ; (3.14) → (3.12) u ′ = 3 2 ( u 1 − uv ) , v ′ = 1 2 ( u − v ); (3.14) → (3.10) u ′ = 3( u + v ) 2 − 3 u 1 + 3 16 c 2 1 , v ′ = 1 4 c 1 − u − v ; (3.14) → (3.13) u ′ = u, v ′ = v 1 ; (3.14) → (3.15) u ′ = 4 3 u + 3 16 c 2 1 , v ′ = 1 4 c 1 − 2 3 √ u + v 1 ; (3.14) → (3.18) 22 A.G. Meshko v and M.Ju . Balakhnev u ′ = u, v ′ = 1 2 v 1 + 1 √ 3 ce − v ; (3.14) → (3.23) u ′ = c 1 √ 2 u, v ′ = 1 2 v 1 − 1 3 u e v − √ 2 2 c 1 e − v ; (3.14) → (3.24) u ′ = − u 1 e v − 1 3 e 2 v  2 u 2 + c 1  , v ′ = − 1 2 v 1 + 1 3 √ − 2 c 1 e v ; ( 3.14) → (3.28) u ′ = u 1 e v + 1 3 e 2 v  2 u 2 + c 1  , v ′ = − 1 2 v 1 + 2 3 u e v ; (3.14) → (3.28) u ′ = 3 2 e v ( u 1 + 2 c 2 u + c 1 u 2 e v ) , v ′ = − 1 2 v 1 + 1 2 ( c 1 + 1) u e v + c 2 ; (3 .14) → (3.29) u ′ = 9 4 ( u + v ) 2 − 9 4 u 1 , v ′ = 9 4 u ; (3.18) → (3.13) this sub s titution is inv ertible: u ′ = 4 9 v , v ′ = 2 3 √ u + v 1 − 4 9 v . (3.13) → (3.18) The graph of th e subs titutions is v ery cumbers ome, therefore we sho w the most interesting subgraph only . ✍✌ ✎☞ 3.1 ✍✌ ✎☞ 3.13 ✍✌ ✎☞ 3.18     ✠ ❅ ❅ ❅ ❅ ❘ ✛ ✲ ✍✌ ✎☞ ✍✌ ✎☞ 3.6 3.3     ✒ ❅ ❅ ❅ ❅ ■ ✲ ✛ ◗ ◗ ◗ ◗ s ✍✌ ✎☞ ✑ ✑ ✑ ✑ ✰ ✍✌ ✎☞ ✻ 3.5 3.4 ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ☛ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❯ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✌ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✕ ✍✌ ✎☞ 3.8 ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ◆ ✍✌ ✎☞ 3.7 ✛ ✲ ✍✌ ✎☞ 3.26 ✍✌ ✎☞ ✛ ✲ 3.25 Fig. 1. A subgraph of th e dif ferential substitutions. Commen ts. (1) System (3.1) coincides with (1.2) and (3.3) coincides w ith (1.3 ). (2) It w as simpler to obtain some systems from (3.6 ) than (3.3 ) or vice v ersa. These systems are connected by the second order inv ertible sub stitution, see (3.6) → (3.3) and (3.3) → (3.6). Hence, eac h system obtained fr om (3.6) can b e obtained fr om (3.3) and vice v ersa. (3) Fifteen systems (3.9)–(3.15), (3.18), (3.22), (3.33)–(3.3 6 ), (3.38) and (3.40) can b e obtained from b oth (3.1) and (3.6) b y the presente d dif feren tial s ubstitutions. T w elv e systems (3.2), (3.16), (3.17), (3.23), (3.24), (3.27)–(3.30), (3.32), (3.37) and (3.39) can b e obtained from (3.1). The remaining elev en systems (3.3)–(3.5), (3.7), (3.8), (3.19)–(3.21), (3.25), (3.26) and (3.31) can b e ob tained from (3.6) or (3.3). Remark 7. As the systems (3.1) and (3.3) h av e the Lax r epresen tations, then all systems from the list ha v e the Lax represent ations in a generalized meaning (see Section 5). Remark 8. Some of the presented sub stitutions are sup erp osit ions of low er order substitutions, other substitutions are irreducible. Remark 9. S ystem (3.1) admits the f ir st and second order substitutions and do es not admit the third and four th ord er substitutions. Probably it do es not admit any h igher order su bstitutions, In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 23 either. Systems (3.4) and (3.6) admit th e substitutions from th e f irst till four th orders. W e do n ot present higher ord er sub stitutions for (3.4 ) b eca use simpler substitutions exist for (3.6). Fifth and higher order su bstitutions f or systems (3.4) and (3.6) ha v e n ot b een computed b ec ause the computations are extremely cumbers ome. Remark 10. Th ere are some additional d if ferenti al s ubstitutions un der the constraints for constan ts in the systems. F or example, there are substitutions (3.1) → (3.31) f or c 1 = 0 and (3.6) → (3.32) for c 3 = 0 or (3.6) → (3.39) for c 2 = 0 and so on. These su bstitutions are not so imp ortant and w e do not present them here. Remark 11. Unexp ectedly , the well kn own systems (3.1) and (3.3) are implicitly connected as it is s h o wn in Fig. 1. 5 Examples of zero cu rv ature represen tatio ns The IST metho d for nonlinear equations w ith tw o indep enden t v ariables is based on inv estig ation of a linear o v erdetermined system L ( u , λ, ∂ x ) ψ = 0 , ψ t = A ( u , λ, ∂ x ) ψ , (5.1) where L an d A are ord in ary lin ear op e rators, u is a smo oth (vect or) fun ction satisfying a n on- linear partial dif fer ential equ ation E ( u ) = 0 and λ is a p arameter. The op erators L , A may b e b oth scalar and matrix. Th e op erators L , A and E ma y b e also pseu do dif feren tial or in tegro- dif ferential . The compatibilit y condition of system (5.1 ) r eads  ∂ L ∂ t + LA  ψ Lψ = 0 = 0 . There are t w o wa ys to satisfy this condition. The f irst op erat or condition w as introduced b y P .D. Lax [37]: ∂ L ∂ t + LA = AL, or ∂ L ∂ t = [ A, L ] . (5.2) The second m ore general op erator condition was introdu ced in [38], s ee also [39]: ∂ L ∂ t + LA = B L. (5.3) If an equation E ( u ) = 0 is equiv alen t to equation (5.2), then (5.2) is said to b e the Lax represent ation of the equation E ( u ) = 0. Th e pair of op erators ( L, A ) is said to b e the ( L, A )- pair or the Lax pair. If an equation E ( u ) = 0 is equiv alent to equ ation (5.3), th en (5.3) is said to b e the ( L, A, B ) represent ation of the equation E ( u ) = 0 or the triad r ep resen tation. In all cases, op erato r L must essen tially dep e nd on th e parameter λ . This p arameter cannot b e remo v ed by a gauge transf ormation L → f − 1 Lf with some smo oth f unction f , in p articular. If system (5.1) is d if ferentia l, then the standard substitution ψ = Ψ 1 , ψ x = Ψ 2 and so on, pro vides th e follo wing f irs t order sys tem: Ψ x = U Ψ , Ψ t = V Ψ , (5.4) where U and V are square m atrices dep ending on u and λ . The compatibilit y condition of linear system (5.4) reads U t − V x + [ U, V ] = 0 (5.5) 24 A.G. Meshko v and M.Ju . Balakhnev if E ( u ) = 0. Usually a stronger condition is requ ir ed: (5. 5) is v ali d if f E ( u ) = 0. In this case equation (5.5) is s aid to b e the zero curv ature repr esen tation. F or an ev olutionary system u t = K ( u , u x , . . . , u n ) , u = { u α } the matrix U usually dep end on u only , bu t it ma y dep end on u , u x , u xx , and so on. L et u s consider the general case. If s ome smo oth f unctions F = F ( u , u x , . . . , u r ) and Φ = Φ( u , u x , . . . , u p ) satisfy the condi- tion  ∂ t F + Φ  u t = K = 0 , (5.6) then one obtains Φ u t = K = Φ , ∂ t F u t = K = ∂ F ∂ u α i ∂ t u α i u t = K = ∂ F ∂ u α i ∂ i x u α t u t = K = ∂ F ∂ u α i D i x K α , where the summation o v er i = 0 , . . . , r and α = 1 , . . . , m is implied. This implies Φ = − ∂ F ∂ u α i D i x K α according to (5.6). Using this result one obtains th e follo wing id en tit y: ∂ t F + Φ ≡ ∂ F ∂ u α i D i x ( u α t − K α ) , ∀ u (5.7) for any F and Φ s atisfying (5.6). Let us apply identi t y (5.7) to equation (5.5). If th e matrix U d ep ends on u only then U t − V x + [ U, V ] = ∂ U ∂ u α ( u α t − K α ) , ∀ u . (5.8) It is ob vious n o w that equation (5.5) is equ iv alen t to u t = K if f the matrices ∂ U /∂ u α , α = 1 , . . . , m are linearly indep e ndent. Su pp o se no w th at th e matrix U dep ends on u and u x then one obtains U t − V x + [ U, V ] = ∂ U ∂ u α ( u α t − K α ) + ∂ U ∂ u α x D x ( u α t − K α ) , ∀ u . (5.9) If the matrices ∂ U /∂ u α , ∂ U / ∂ u β x , α, β = 1 , . . . , m are linearly indep enden t, then equation (5.5) is equiv alent to u t = K again. Otherwise, equation (5.5) w ould b e equiv alent to some dif ferential consequence of the system u t = K that is a more general system than the original one. In this case we call equation (5.5) the gener alize d zero curv a ture representa tion. It is well kno wn that equations (5.4) and (5.5) are co v ariant und er the follo wing tran s forma- tion ¯ Ψ = S − 1 Ψ , ¯ U = S − 1 ( U S − S x ) , ¯ V = S − 1 ( V S − S t ) , (5.10) where S is an y non-d egenerate matrix. This transformation is called a gauge one. Any gauge transformation is in v ertible and pr eserv es compatibilit y of system (5.4). Tw o ( L, A )-pairs w ere p rop osed for system (1.2) in [3]. On e of these ( L, A )-pairs coincides with the ( L, A )-pair that was present ed in [4]. Th e L -op erator of the common ( L, A )-pair tak es the form L = ( ∂ 2 x + f )( ∂ 2 x − g ), where f = 1 6 ( u √ 2 − 2 v ) , g = 1 6 ( u √ 2 + 2 v ) . In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 25 The temp oral Lax equation reads ψ t = Aψ , w h ere A is a fr actional degree of L . The spatial Lax equation Lψ = λ 2 ψ can b e transformed int o the system ( ∂ 2 x − g ) ψ = λϕ , ( ∂ 2 x + f ) ϕ = λψ and then in to the normal form (5.4), were U =     0 1 0 0 ( u √ 2 − 2 v ) / 6 0 λ 0 0 0 0 1 λ 0 − ( u √ 2 + 2 v ) / 6 0     , V =       ( u 1 √ 2 + v 1 ) / 6 − ( u 1 √ 2 + v 1 ) / 3 0 − 2 λ f 1 + f 2 − ( u 1 √ 2 + v 1 ) / 6 λv / 3 0 0 − 2 λ − ( u 1 √ 2 − v 1 ) / 6 ( u 1 √ 2 − v 1 ) / 3 λv / 3 0 f 1 − f 2 ( u 1 √ 2 − v 1 ) / 6       . (5.11) Here f 1 and f 2 tak e the follo wing form: f 1 = 1 18 (3 v 2 − 2 u 2 + 2 v 2 ) − 2 λ 2 , f 2 = √ 2 18 (3 u 2 + uv ) . Matrices (5.11) realize the zero curv ature repr esen tation of sys tem (1.2). System (1.3) also has t w o Lax represen tations (see [3]). Using th e simpler L -op erator, we ha v e f ound, similarly to the previous case, the follo wing matrices that realize the zero curv ature represent ation of system (1.3): ˜ U =     0 1 0 0 − v / 3 − λ 0 1 0 0 0 0 1 u/ 9 0 λ − v / 3 0     , ˜ V =      v 1 / 6 2 λ − v / 3 0 − 2 h − λv / 3 − v 1 / 6 v / 3 0 u 1 / 9 − 2 u/ 9 v 1 / 6 − 2 λ − v / 3 uv / 27 + u 2 / 9 − u 1 / 9 h + λv / 3 − v 1 / 6      , (5.12) where h = 1 6 v 2 + 1 9 ( v 2 − 2 u ) − 2 λ 2 . Let us consider an admissible dif feren tial substitution u = f ( ˜ u i , ˜ v j ), v = g ( ˜ u i , ˜ v j ) of sys- tem (1.1). Sub s tituting u and v in the matrices U ( u, v ) and V ( u i , v j ) one obtains ˆ U ( ˜ u i , ˜ v j ) = U ( f , g ) , ˆ V ( ˜ u i , ˜ v j ) = V ( D k x f , D l x g ) . As ˆ U dep ends on deriv a tiv es of ˜ u or ˜ v , then one has a generalized zero cur v ature rep resen tation. T o obtain an ordinary zero curv atur e r epresent ation one can tr y to remo v e higher order deriv ative s fr om th e m atrix ˆ U using the gauge tr ansformation (5.10). But this is not alw a y s p ossible (see example B b e lo w). A. P erforming the sub stitution (3.1) → (3.9) into the matrix U fr om (5.11) one obtains ˆ U =     0 1 0 0 ( u 2 − v 2 ) / 2 + h 1 0 λ 0 0 0 0 1 λ 0 − ( u 2 + v 2 ) / 2 + h 2 0     , 26 A.G. Meshko v and M.Ju . Balakhnev where h 1 = 1 4 ( u 1 − v 1 ) 2 , h 2 = 1 4 ( u 1 + v 1 ) 2 . One can easily v erify that matrices A = ˆ U u 2 and B = ˆ U v 2 are comm utativ e, hence the system S x = ( Au xx + B v xx ) S has the follo wing solution S = exp ( Au x + B v x ). Th e m atrix U 1 = ¯ ˆ U ev aluated according to (5.10) tak es the f ollo wing form U 1 = 1 2     u x − v x 2 0 0 0 v x − u x 2 λ 0 0 0 − u x − v x 2 2 λ 0 0 u x + v x     . No w another gauge tran s formation is p ossible with the follo wing d iagonal matrix: S 1 = exp  Z diag ( U 1 ) dx  , where diag ( U 1 ) is th e main diagonal of U 1 . This gauge transf ormation p ro vides the f ollo wing matrix U 2 =     0 e v − u 0 0 0 0 λe − v 0 0 0 0 e u + v λe − v 0 0 0     . A corresp o nding V -matrix can b e obtained by solving equation (5.5) directly or by the p revious t w ofold gauge tr an s formation. This matrix take s th e follo wing f orm V 2 =         0 e v − u f 1 λe − u ( u 1 + v 1 ) − 2 λe v − 2 λ 2 e u − v 0 λ 4 e − v (2 v 2 − 3 u 2 1 + v 2 1 ) λe u ( u 1 − v 1 ) λe u ( v 1 − u 1 ) − 2 λe v 0 e u + v f 2 λ 4 e − v (2 v 2 − 3 u 2 1 + v 2 1 ) − λe − u ( u 1 + v 1 ) − 2 λ 2 e − u − v 0         , where f 1 = − u 2 − 1 2 v 2 + u 1 v 1 + 1 4 ( u 2 1 + v 2 1 ) , f 2 = u 2 − 1 2 v 2 − u 1 v 1 + 1 4 ( u 2 1 + v 2 1 ) . The matrices U 2 and V 2 realize the zero curv at ure representat ion of system (3.9). B. Substitution (3.1) → (3.17) redu ces matrix U from (5.11) to the follo wing f orm ˆ U =     0 1 0 0 ( k − v ) / 3 + R 0 λ 0 0 0 0 1 λ 0 ( k − v ) / 3 − R 0     , where R = p 2( u 1 + v 2 ) / 6. It is ob vious that one cannot remov e u 1 from ˆ U by a gauge trans- formation. It is clear fr om the structur e of th e matrix ˆ U that U t − V x + [ U, V ] = A  k v 1 + 1 2 u 2 − 1 2 v 3 − v t  + B D x  u 3 − 3 4 (2 v v 1 + u 2 ) 2 v 2 + u 1 + 3 v v 2 + 3 2 v 2 1 + 2 3 v 3 − k (2 v 2 + u 1 ) − u t  , In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 27 where A and B are some linearly ind ep enden t matrices. T h us, this zero cur v ature r epresen tation for s ystem (3.17) is generalized. Of cour se, one may introd u ce here the n ew v ariable u ′ = u 1 to obtain an ordinary zero curv ature representa tion. But we do n ot know if it is alw a ys p o ssible. C. P erforming the substitution (3.3) → (3.6) into the matrix U from (5.12) one obtains ˆ ˜ U =     0 1 0 0 − u/ 3 − λ 0 1 0 0 0 0 1 u 2 / 6 + ( u 2 + v ) / 9 0 λ − u/ 3 0     . The f irst gauge transform ation is p erformed using S 1 = exp( u 1 ( ∂ ˆ ˜ U /∂ u 2 )): S 1 =     1 0 0 0 0 1 0 0 0 0 1 0 u 1 / 6 0 0 1     . The transformed U -mat rix is ˜ U 1 =     0 1 0 0 − u/ 3 − λ 0 1 0 u 1 / 6 0 0 1 ( u 2 + v ) / 9 − u 1 / 6 λ − u/ 3 0     . The second gauge transformation is p erf ormed usin g S 2 = exp( u ( ∂ ˜ U 1 /∂ u 1 )): S 2 =     1 0 0 0 0 1 0 0 u/ 6 0 1 0 0 − u/ 6 0 1     . The result of the t w ofold gauge transformation is ˜ U 2 =     0 1 0 0 − u/ 6 − λ 0 1 0 0 − u/ 3 0 1 u 2 / 36 + v / 9 0 λ − u/ 6 0     , ˜ V 2 =      − u 1 / 6 2 λ 0 − 2 f 3 − λu/ 3 − u 1 / 6 u/ 3 0 ( uu 1 − v 1 ) / 18 − λu 1 / 3 − u 2 / 3 − u 2 / 6 − 2 v / 9 u 1 / 6 − 2 λ f 4 ( v 1 − uu 1 ) / 18 − λu 1 / 3 f 3 + λu/ 3 u 1 / 6      , where f 3 = − 1 6 u 2 − 1 18 u 2 − 2 9 v − 2 λ 2 , f 4 = 1 18 ( uu 2 − v 2 + u 2 1 ) + 1 108 u 3 − 1 27 uv − 2 3 uλ 2 . Matrices ˜ U 2 and ˜ V 2 realize the zero curv ature representa tion of system (3.6). 6 Conclusion The examples in Section 5 illustrate the fact that some systems p ossess ordinary zero curv a- ture representa tion w hile others p ossess generalized zero curv atur e r epresen tation. All these 28 A.G. Meshko v and M.Ju . Balakhnev represent ations are obtained fr om the Drinfeld–Sok olo v L , A op erators by usin g corresp ondin g dif ferential sub stitutions listed in Section 4. Matrice s U and V that realize all zero curv atur e represent ations h a v e the size 4 × 4. Thus, the t w o-f ield ev olutionary systems presente d ab o ve are int egrable in p rinciple by the inv ers e sp e ctral transform metho d. But the fact is that the in v erse scattering p roblem for dif ferentia l equations with order more than t w o is extremely dif f icult. Th at is why other metho ds for solution of equations may b e usefu l [40]. T hey may b e B¨ ac klun d tr an s formations [41], Darb oux transformations [42, 43], Hirota metho d [44] or n umeric sim ulating (see [45], for example). Ac kno wledgmen ts W e are grateful to Professor V.V. S ok olo v for helpf ul discussions. Th is work wa s sup p orted by F ederal Agency for Edu cation of Russ ian F ederation, pro ject # 1.5.07. References [1] Ablowitz M.J., Segur M., Solitons and the inv erse scattering transform, SI AM, Philadelphia, 1981. [2] Zakharov V.E., Manako v S.V., Novik o v S.P ., Pitaevsky L.P ., Theory of solitons: inv erse problem method , Nauk a, Mosco w, 1980. [3] Drinfeld V .G., Sokolo v V.V., New evolution equations h a ving (L-A) pairs, T rudy Sem. S.L. Sob oleva, Inst. Mat., Novosibirsk 2 (1981), 5–9 (in Russian). [4] Hirota R., S atsuma J., Soliton solutions of a coupled Korteweg–de V ries equ ation, Phys. L ett . A 85 (1981), 407–408 . [5] Antono wicz M., F ordy A., Coupled KdV eq u ations with multi-Hamiltonian structures, Phys. D 28 (1987), 345–357 . [6] Ma W.-X. A class of coupled KdV sy stems and their bi-Hamiltonian form ulation, J. Phys. A: Math. Gen. 31 (1998), 7585–75 91, solv-int/9803 009 . [7] Ma W.-X ., P a vlov M., Ex t ending Hamiltonian op erators to get bi-Hamiltonian of coupled KdV systems, Phys. L ett. A 246 (1998), 511–522, solv-int/980700 2 . [8] F uchssteiner B., The Lie algebra structure of d egenerate Hamiltonian and b i-Hamiltonian systems, Pr o gr. The or et. Phys. 68 (1982), 1082–1104 . [9] Nutk u Y., Oˇ guz ¨ O., Bi-Hamiltonian structu re of a pair of coupled K d V equations, Nuovo Cimento 105 (1990), 1381–13 83. [10] Gerdt V .P ., Zh arko v A.Y., Computer classif i cation of integrable coup led KdV-like sy stems, J. Symb oli c Comput. 10 (1990), 203–207. [11] F oursov M.V., On integrable coupled KdV-typ e systems, Inverse Pr oblems 16 (2000), 259–274. [12] Zharko v A.Y., Computer classif ica tion of th e integrable coupled KdV -like systems with unit main matrix, J. Symb olic Comput. 15 (1993), 85–90. [13] Karasu A., P ainlev´ e classif icatio n of coup led Korteweg–de V ries-type systems, J. Math. Phys. 38 (1997), 3616–36 22. [14] F oursov M.V., T ow ards the complete classif ication of homogeneous tw o-comp on ent integrable eq uations, J. Math. Phys. 44 (2003), 3088–3096. [15] Meshko v A.G., S okol o v V.V., Integrable evolution eq uations on the N -dimensional sphere, Comm. Math. Phys. 232 (2002), 1–18. [16] Meshko v A.G., Sokolo v V.V., Classif ica tion of integrable div ergen t N -comp onent ev olution systems, The or et. and M ath. Phys. 139 (2004), 609–622. [17] Balakhnev M.Ju., A class of integrable evolutionary vector equations. The or et. and Math. Phys. 142 (2005), 8–14. [18] Balakhnev M.Ju., Meshko v A.G., Integrable anisotropic evolution equ ations on a sphere, SIGMA 1 (2005), 027, 11 pages, nlin.SI/0512032. [19] Balakhnev M.Ju., Meshko v A.G., On a classif i cation of integrable vectorial evolutionary equations, submit- ted. In tegrable Evo lutionary Systems and Th eir Dif f eren tial Subs titutions 29 [20] Kulemin I.V., Meshk o v A.G., T o th e classif ication of integrable systems in 1+1 dimensions, in Proceedings Second International Conference “Sy mmetry in Nonlinear Mathematical Physics” (July 7–13, 1997, Kyiv), Editors M. Shkil, A. Nikitin and V. Boyk o, Institute of Mathematics, Kyiv, 1997, 115–121. [21] Meshko v A.G., On symmetry classif ication classif ication of third order evolutionary systems of divergen t type, F undam. Prikl. Mat. 12 ( 2006), n o. 7, 141–161 (in Russian). [22] Ibragimo v N.Kh., Shabat A.B., On inf inite Lie–B¨ a cklund algebras, F unktsional. A nal. i Prilozhen. 14 (1980), no. 4, 79–80 (in Russian). [23] Svinolup ov S.I., Sokolo v V.V., Evolution equations with nontrivial conserv ation la ws, F unktsional. Anal. i Prilozhen. 16 (1982), no. 4, 86–87. [24] Sokolo v V.V., Shabat A .B., Classif ication of integrable evo lution equations, Soviet Scientific Re views, Se c- tion C 4 ( 1984), 221–28 0. [25] Mikhailo v A.V., Shabat A.B., Y amilov R.I., The symmetry approach to the classif icati on of integrable equations. Complete lists of integrable systems, Russian M ath. Surveys 42 (1987), n o. 4, 1–63. [26] Mikhailo v A.V., S h abat A.B., Soko lo v V.V., The sym met ry approach to the classif icati on of integrable equations, in I ntegrabil ity and Kin et ic Equations for S olitons, Editors V.G. Bahta riar, V.E. Zak haro v an d V.M. Chernousenko, Nauko v a Dumka, Kyiv, 1990, 213–279 (English transl.: in What is Integra bilit y?, Springer-V erlag, New Y ork, 1991, 115–184 ). [27] F ok as A .S., S ymmetries and integ rabilit y , Stud. Appl. Math. 77 (1987), 253–299. [28] Galindo A ., Martinez L., Kernels and ranges in the v aria tional formalis m, L ett. Math. Phys. 2 (1978), 385–390 . [29] Drinfeld V.G., S okol ov V.V., Lie algebras and equations of Kortew eg–de V rie s typ e, Curr ent Pr oblems in Mathematics , V ol. 24, Itogi Nauki i T ekhniki, VINITI, Mosco w, 1984, 81–180 (English transl.: J. Sov. Math. 30 (1985), 1975–20 35). [30] Balakhnev M.Y u., Kulemin I .V., Dif feren tial substitutions for third-order evolution systems, Differ. Ur avn. Pr otsessy Upr. (2002), no. 1, 7 pages (in Russian), a v aila ble at http://www.neva.r u/journal/j / . [31] Zakharov V.E., Shabat A.B., Exact t heory of t w o-dimensional self-focusing and one-dimensional self- mod u lation of wa ves in nonlinear media, ˇ Z. Eksp er. T e or et. Fiz. 61 (1971), no. 1, 118–134 (English transl.: Soviet Physics JETP 34 (1972), n o. 1, 62–69 ). [32] Chen H.H., Lee Y.C., Liu C.S., Integrabilit y of n onlinear Hamiltonian systems by inv erse scattering meth od , Phys. Scripta 20 (1979), 490–492 . [33] Meshko v A.G., N ecessary conditions of the integra bility , Inverse Pr oblems 10 (1994), 635–653 . [34] F uchssteiner B., F ok as A.S., Symplectic structures, th eir B¨ ac klund transformations and hered itary symme- tries, Phys. D 4 (1981/8 2), 47–66. [35] Drinfeld V.G., Sokolo v V .V., Equations that are related to the Kortew eg–de V ries equ ation, Dokl. Akad. Nauk SSSR 284 (1985), n o. 1, 29–33 (in Russian). [36] Meshko v A .G., T o ols for symmetry analysis of PDEs, Di ffer. Ur avn. Pr otsessy Upr. (2002), no. 1, 53 pages, a v aila ble at http://www.neva.r u/journal/j / . [37] Lax P .D., Integrals of nonlinear equations of evo lution and solitary wa v es, Comm. Pur e Appl. Math. 21 (1968), 467–490 . [38] Manako v S.V ., The metho d of the inv erse scattering problem, and tw o-dimensional evolution equations, Usp ehi Mat. Nauk 31 (1976), no. 5, 245–246 (in Russian). [39] Zakharov V.E., I nv erse scattering metho d, in Solitons, Editors R.K. Bullough and P .J. Caudrey , S pringer- V erlag, New Y ork, 1980, 270–309. [40] Do dd R .K., Eilbeck J.C., Gibb on J.D., Morris H.C., Solitons and nonlinear wa v e equations, A cademic Press Inc., London, 1984. [41] Miura R.M. (Editors), B¨ ac klund transformations, the inv erse scattering metho d, solitons, and their appli- cations, N SF R esearc h W orkshop on Contact T ransformations, L e ctur e Notes in Mathematics , V ol. 515, 1976. [42] Matveev V.B., Salle M.A., Darb oux transformations and solitons, Springer, Berlin, 1991. [43] Gu C.H., Hu H.S., Zhou Z.X., D arbou x transformations in soliton theory and its geometric ap plications, Shanghai Scientif i c and T ec hnical Publishers, Shanghai, 1999. [44] Hirota R., The direct metho d in soliton th eory , Cam bridge Universit y Press, Cam bridge, 2004. [45] Zienkiewcz O.C., Morgan K., Finite elements and approximation, Wiley-Interscience Publication, New Y ork, 1983.