Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations

Gauge-in v arian t description of sev eral (2+1)-dimens ional in tegrable nonlinear e v olution equations V. G. Dubro vsky and A. V. Gr amol in No v osibirsk State T echnical Univ ersit y , No v osibirsk, R u ssia E-mail: dubr ovsky@academ.o rg and gramolin@gmail.c om Abstract W e obtain new gauge-in v arian t forms of t w o-dimensional in tegrable sy s tems o f non- linear e quations: the Sa w ada–Kotera and Kaup–Kup er s c hmidt system, the ge neralized system of disp ersive long wa v es, and the Nizhnik–V eselo v–No vik o v sy s tem. W e show ho w these forms imply b oth n ew and well -kno wn t w o-dimensional integrable nonlin- ear equations: the Saw ada–Kotera equation, Kau p –Kup ersc hmidt equ ation, disp ersiv e long-w a v e system, Nizhnik–V eselo v–No viko v equation, and mo dified Nizhnik–V eselo v– No vik o v equation. W e consider Miura-type transformations b et we en nonlinear equ a- tions in different gauges. Keyw ords: Sa w ada–Kotera equation, Kaup–Kup ersc hmidt equation, generalized disp ersive long-w a v e equation, D a v ey–Stew art son equation, Nizhnik–V eselo v–Novik ov equation 1 In tro d uction The f undamental metho ds based on gauge t r a nsformations and the concept o f gauge in- v ariance are curren tly widely used in ph ysics a nd mathematics, in part icular, in the theory of in tegrable nonlinear equations. The first applications of these metho ds in t he theory of in tegrable nonlinear equations w ere prop osed in [1 –7] ( a lso see [8–13] and the references therein). A great man y gauge-equiv alent pairs of in tegrable mo dels ha v e b een f o und. In the one- dimensional case, these include the nonlinear Sc hr¨ odinger equation and Heise n b erg ferromag- net equation, the equations describing the ma ssiv e Thirring mo del and the tw o-dimensional relativistic field theory , and the Kortew eg–de V ries (KdV) and mo dified K dV (mKdV) equa- tions. In the t w o-dimensional case, the curren t ly most widely known gauge-equiv alen t pairs of equations are the K a dom tsev–P etviashv ili (KP) a nd mo dified KP (mKP) equations and the D a v ey–Stew ar t son and Ishimori equations (see [8–14] and the references therein). W e introduce the required terms and illustrate the unifying ro le of gauge transformations and gauge inv ariance with the well-kno wn example of t he interaction b et w een a c harged spinless pa rticle and an external electromagnetic field with the v ector and scalar p oten tials ~ A ( ~ r, t ) a nd ψ ( ~ r , t ), i ~ ψ t =  b ~ p − q ~ A  2 2 m ψ + q φψ , (1.1) 1 where b ~ p = − i ~ ~ ∇ is the particle momen tum o p erator. The part icle coupling t o external fields has the w ell-kno wn gauge-in v ariant form. F ro m the standp oint o f the in v erse scattering method, Eq. (1.1) is an auxiliary linear problem: a linear partial differen tial equation with v ariable co efficien ts fo r the w av e func- tion ψ . Under the gauge transformatio n ψ → ψ ′ = g − 1 ψ , ψ = g ψ ′ = exp  iχ ( ~ r , t ) q ~  ψ ′ , (1.2) Eq. (1.1) preserv es its form if the p oten tials ~ A and φ are transformed a s ~ A → ~ A ′ = ~ A − ~ ∇ χ, φ → φ ′ = φ + χ t . (1.3) Eliminating the gauge function χ from ( 1.3), we o btain the relations [ ~ ∇ × ~ A ′ ] = [ ~ ∇ × ~ A ] , − ~ ∇ φ ′ − ∂ ~ A ′ ∂ t = − ~ ∇ φ − ∂ ~ A ∂ t , (1.4) whic h mean that t he quan tities ~ B def = [ ~ ∇ × ~ A ] , ~ E def = − ~ ∇ φ − ∂ ~ A ∂ t , (1.5) whic h are well kno wn in electro dynamics as the mag netic field induction ~ B and electric field strength ~ E , are inv arian ts of ga uge transformatio ns. Moreo v er, definitions (1.5) for the in v ariants ~ B and ~ E imply the subsy stem of equations div ~ B = ~ ∇ · ~ B = 0 , ~ ∇ × ~ E = − ∂ ~ B ∂ t , (1.6) whic h is a (sourceless) subsys tem of the fundamen tal system of Maxw ell equations. Th us, starting from the principle of lo cal gauge inv a riance, w e can o btain the gauge- inv arian t non- stationary Schr¨ odinger equation for a c harged spinle ss particle in an external electromagnetic field, the gauge inv arian ts ~ B and ~ E , and a (sourceless) subsystem of fundamen tal Maxw ell equations. Similar considerations are applicable t o the case of in t egrable nonlinear equations. 2 Gauge-in v arian t in tegrable syste m o f KP and mKP equations W e illustrate the metho ds based on the notion o f gauge inv ariance with the w ell- known example of an integrable system of nonlinear KP and mKP equations. Auxiliary linear problems for the KP and mKP equations are particular cases of the follow ing linear problems (with the resp ectiv e gauges C ( u 0 6 = 0 , u 1 = 0) and C ( u 0 = 0 , u 1 6 = 0) for the KP a nd mKP equations): L 1 ψ =  σ ∂ y + ∂ 2 x + u 1 ∂ x + u 0  ψ = 0 , (2.1) L 2 ψ =  ∂ t + 4 ∂ 3 x + v 2 ∂ 2 x + v 1 ∂ x + v 0  ψ = 0 , (2.2) 2 where the constant v alues σ = i o r σ 2 = − 1 and σ = 1 or σ 2 = 1 resp ectiv ely cor r esp o nd to the cases of KP–I or mKP–I and KP–I I or mKP –I I equations. Because of (2.1) and (2.2 ), the compatibilit y conditio n in the Lax fo r m [ L 1 , L 2 ] = 0 leads to a system o f ev olution equations for the field v ariables u 1 and u 0 [5, 7], u 1 t + u 1 xxx − 3 2 u 2 1 u 1 x − 3 σ u 1 x ∂ − 1 x u 1 y + 3 σ 2 ∂ − 1 x u 1 yy + + 6 u 0 xx − 6 σ u 0 y + 6  u 0 u 1  x − 2 v 0 x = 0 , u 0 t + 4 σ u 0 xxx + 6 u 0 u 0 x + 6 u 1 u 0 xx + 3 2 u 2 1 u 0 x + 3 u 0 x u 1 x − − 3 σ u 0 x ∂ − 1 x u 1 y − u 1 v 0 x − v 0 xx − σ v 0 y = 0 . (2.3) F or the gauge C ( u 0 6 = 0 , u 1 = 0) with the c hoice of the v ariable v 0 = 3 u 0 x − 3 σ ∂ − 1 x u 0 y , system (2.3) implies the KP equation [1, 15] u 0 t + u 0 xxx + 6 u 0 u 0 x + 3 σ 2 ∂ − 1 x u 0 yy = 0 . (2.4) F or the gauge C ( u 0 = 0 , u 1 6 = 0) with the choice of the v aria ble v 0 = const, system (2.3) implies the mKP equation u 1 t + u 1 xxx − 3 2 u 2 1 u 1 x − 3 σ u 1 x ∂ − 1 x u 1 y + 3 σ 2 ∂ − 1 x u 1 yy = 0 . (2.5) System of equations (2.3) is called the system o f KP–mKP equations. In terms of the pure ga ug e v ariable ρ and the inv arian t w 0 of the gauge transfor mat ions ψ → ψ ′ = g − 1 ψ determined by the expressions u 1 ( x, y , t ) = 2 ρ x ρ , w 0 = u 0 − 1 2 u 1 x − 1 4 u 2 1 − σ 2 ∂ − 1 x u 1 y , (2.6) system of equations (2 .3) b ecomes ρ t + 4 ρ xxx + 6 ρ x w 0 + 3 ρw 0 x − 3 σ ∂ − 1 x w 0 y − ρv 0 = 0 , (2.7) w 0 t + w 0 xxx + 6 w 0 w 0 x + 3 σ 2 ∂ − 1 x w 0 yy = 0 . (2.8) System (2.7), (2.8) is the manifestly gauge-inv arian t fo rm of system of KP–mKP equa- tions (2 .3). T his system has the following structure. It contains gaug e- in v ariant KP equa- tion ( 2 .8) for the gauge in v ariant w 0 and Eq. (2.7) for the pure g auge v ariable ρ with ad- ditional terms containing t he gaug e in v arian t w 0 and the additio na l field v ariable v 0 . If the in v ariant is zero, w 0 = 0, then KP–mKP system (2.7), (2.8) reduces to the linear ev olution equation fo r the pure gauge v ariable ρ : ρ t + 4 ρ xxx − ρv 0 = 0 . (2.9) W e use the gauge inv arian t w 0 to obtain the Miura-type transformatio n from Eq. (2.8 ), w 0 = u 0 = − 1 2 u 1 x − 1 4 u 2 1 − σ 2 ∂ − 1 x u 1 y , (2.10) 3 whic h relates the solutions u 0 (for the gauge C ( u 0 6 = 0 , u 1 = 0)) and u 1 (for the gauge C ( u 0 = 0 , u 1 6 = 0)) of KP equations (2.4 ) and mKP equations (2 .5). This also follo ws from Eq. (2.8) written for the gauges C ( u 0 6 = 0 , u 1 = 0) and C ( u 0 = 0 , u 1 6 = 0): u 0 t + u 0 xxx + 6 u 0 u 0 x + 3 σ 2 ∂ − 1 x u 0 yy = − 1 2  ∂ x + 1 2 u 1 + σ ∂ − 1 x ∂ y  u 1 t + u 1 xxx − 3 2 u 2 1 u 1 x − 3 σ u 1 x ∂ − 1 x u 1 y + 3 σ 2 ∂ − 1 x u 1 yy  = 0 . (2.11) W e stress that t he separation o f the phy sical and the pure gauge degrees o f freedom in in tegrable nonlinear equations and t heir gauge-inv arian t fo rm ulation can b e used to study the structure of these equations and the r elat io ns b et w een their different gauge-equiv alent realizations. 3 Manifestly gauge-inv a rian t in tegrable syste m of t w o -dimensio nal Kaup–Kup ersc hmidt and Sa wa da– Kotera equations In this section, w e briefly discuss the results obtained up to no w concerning the manifestly gauge-in v aria nt formulation o f t w o- dimensional in tegrable generalizations of the Kaup–Kup ersc hmidt (2 D KK) and Saw ada–Kot era (2 D SK) nonlinear equa- tions. The auxiliary linear problems for these equations are particular cases (in the gauge C ( u 2 = 0 , u 1 6 = 0 , u 0 = u 1 x / 2) fo r the 2 D KK equation and in the g auge C ( u 2 = 0 , u 1 6 = 0 , u 0 = 0) for the 2 D SK equation) of the problems [7] L 1 ψ =  σ ∂ y + ∂ 3 x + u 2 ∂ 2 x + u 1 ∂ x + u 0  ψ = 0 , L 2 ψ =  ∂ t + κ∂ 5 x + v 4 ∂ 4 x + v 3 ∂ 3 x + v 2 ∂ 2 x + v 1 ∂ x + v 0  ψ = 0 . (3.1) In terms of t he pure gauge v ariable ρ and the in v ariants w 0 and w 1 of the gauge trans- formations ψ → ψ ′ = g − 1 ψ giv en by the express ions u 2 = 3 ρ x ρ , w 1 = u 1 − u 2 x − 1 3 u 2 2 , (3.2) w 0 = u 0 − 1 3 u 1 u 2 − 1 3 u 2 xx + 2 27 u 3 2 − σ 3 ∂ − 1 x u 2 y , (3.3) w e obtain the manifestly gauge-inv arian t inte grable system o f 2 D KK–2 D SK nonlinear equa- tions ρ t + κρ xxxxx − ρv 0 + 5 3 κ  ρ xx w 1  x + 5 3 κ  ρ x w 0  x + 5 9 κρ x w 2 1 + 10 9 κρ x w 1 xx + + 10 9 κρw 0 xx + 10 9 κρw 0 w 1 − 5 9 κσ ρ x ∂ − 1 x w 1 y − 5 9 κσ ρ∂ − 1 x w 0 y = 0 , (3.4) w 1 t − 1 9 κw 1 xxxxx − 5 9 κ  w 1 w 1 xx  x − 5 3 κ  w 0 w 1 x  x − 5 9 κw 2 1 w 1 x + 10 3 κw 0 w 0 x − − 5 9 κσ w 1 xxy − 5 9 κσ w 1 w 1 y + 5 9 κσ 2 ∂ − 1 x w 1 yy − 5 9 κσ w 1 x ∂ − 1 x w 1 y = 0 , (3.5) w 0 t − 1 9 κw 0 xxxxx − 5 9 κ  w 0 w 1  xxx − 5 9 κ  w 0 w 1 xx  x + 5 3 κ  w 0 w 0 x  x − 5 9 κ  w 0 w 2 1  x − − 5 9 κσ w 0 xxy − 10 9 κσ w 0 w 1 y − 5 9 κσ w 1 w 0 y + 5 9 κσ 2 ∂ − 1 x w 0 yy − 5 9 κσ w 0 x ∂ − 1 x w 1 y = 0 . (3.6) 4 This system consists of gauge-inv arian t system of equations (3.5), ( 3.6) fo r the gauge in v ari- an ts w 0 and w 1 and Eq. (3.4 ) for the pure ga ug e v ariable ρ with additional terms containing the gauge in v ariants w 0 and w 1 and the additional v ariable v 0 . F or the zero-v alued in v ariants w 0 = 0 and w 1 = 0, the 2 D KK– 2 D SK system giv en by (3.5) and (3.6) reduces to the linear ev olution equation for the pure gauge v ariable ρ : ρ t + κρ xxxxx − ρv 0 = 0 . (3.7) Gauge-inv a rian t system of equations (3.5), (3 .6) for the inv a rian ts w 0 and w 1 coincides in form with the system obtained in [7]. This sys tem admits t he follow ing reductions: 1. In the case w 0 = 0, w e obtain an equation for the in v ariant w 1 exactly coinciding with the 2 D SK equation [7, 16]. 2. In the case w 0 = w 1 x / 2, we o bt a in an equation fo r the in v arian t w 1 exactly coinciding with the 2 D KK equation [7, 16]. Ob viously , the 2 D KK and 2 D SK equations, as equations b elonging to differen t sets of in- v ariants , are not gauge-equiv alen t to each other. 4 A manifestly g auge-in v arian t in te grable t w o- dimension al gen eralized d isp ersiv e long -w a v e syst em The gauge-inv arian t formulation of in tegrable sy stems of nonlinear equations can b e obtained in all cases where t he gauge freedom is tak en in to accoun t in the corresp onding auxiliary linear pro blems. In [17], suc h a form ulation w as obtained for the auxiliary linear pro blems L 1 ψ =  ∂ 2 ξ η + u 1 ∂ ξ + v 1 ∂ η + u 0  ψ = 0 , (4.1) L 2 ψ =  ∂ t + κ 1 ∂ 2 ξ + κ 2 ∂ 2 η + ˜ u 1 ∂ ξ + ˜ v 1 ∂ η + v 0  ψ = 0 , (4.2) where κ 1 and κ 2 are constan ts, ξ = x + σ y and η = x − σ y ar e spatial v ariables, σ 2 = ± 1, and the deriv ativ es are ∂ ξ = ∂ /∂ ξ , ∂ η = ∂ /∂ η , ∂ 2 ξ = ∂ 2 /∂ ξ 2 , etc. Suc h a c hoice of auxiliary linear problems (4.1) a nd (4.2) leads to the w ell-kno wn tw o- dimensional generalized disp ersiv e long-w a v e (2 D gD L W) equation [18], the system of Dav ey–Stew artson (DS) equations [1 9], their reductions, and some other equations. All t he listed w ell-kno wn in tegrable nonlinear equations w ere previously obtained from t he compatibilit y condition f or auxiliary linear problems (4.1) and ( 4 .2) in the f o rm of the Manak ov triad represen tat io n [20] [ L 1 , L 2 ] = B L 1 . (4.3) T o obtain the manifestly gauge-in v arian t form ulation of the corresp o nding integrable system of nonlinear equations, it is conv enien t to use the classical gauge inv arian ts w 2 , e w 2 , and w 1 , w 2 def = u 0 − u 1 ξ − u 1 v 1 = u ′ 0 − u ′ 1 ξ − u ′ 1 v ′ 1 , e w 2 def = u 0 − v 1 η − u 1 v 1 = u ′ 0 − v ′ 1 η − u ′ 1 v ′ 1 , w 1 def = u 1 ξ − v 1 η = u ′ 1 ξ − v ′ 1 η , (4.4) 5 and the pure gauge v ariable ρ related to t he field v ariable u 1 ( ξ , η , t ) as u 1 def = (lo g ρ ) η . (4.5) The v ariable ρ corresp o nds to the pure gauge degrees of freedom and is tra nsfor med according to the simple law ρ → ρ ′ = g ρ under the gauge transformations ψ → ψ ′ = g − 1 ψ . W e note that the in v ariants w 2 and e w 2 in a uxiliary problem (4.1) are the Laplace in v ari- an ts h = w 2 and k = e w 2 for the corresp onding classical differen tial equation (see, e.g., the w ell-kno wn F orsyth mono g raph on differen tial equations [21]). In the case under study with auxiliary linear problem (4.2) determining the time ev olu- tion, the corresp onding integrable system of nonlinear equations in terms of the v ariables ρ , w 1 , and w 2 b ecomes ρ t = − κ 1 ρ ξ ξ − κ 2 ρ ηη − 2 κ 1 ρ∂ − 1 η w 2 ξ + 2 κ 2 ρ η ∂ − 1 ξ w 1 + v 0 ρ, (4.6) w 1 t = − κ 1 w 1 ξ ξ + κ 2 w 1 ηη − 2 κ 1 w 2 ξ ξ + 2 κ 2 w 2 ηη − 2 κ 1  w 1 ∂ − 1 η w 1  ξ + 2 κ 2  w 1 ∂ − 1 ξ w 1  η , (4.7) w 2 t = κ 1 w 2 ξ ξ − κ 2 w 2 ηη − 2 κ 1  w 2 ∂ − 1 η w 1  ξ + 2 κ 2  w 2 ∂ − 1 ξ w 1  η . (4.8) Gauge-inv a rian t subsy stem (4 .7), (4.8) for the inv arian ts w 1 = u 1 ξ − v 1 η and w 2 = u 0 − u 1 ξ − u 1 v 1 of system of equations ( 4 .6)–(4.8) with u 1 = 0, v 1 = − q / 2, and u 0 = (1 + r − q η ) / 4 in terms of the v ariables q and r b ecomes q t = − κ 1 ∂ − 1 η r ξ ξ + κ 2 r η − κ 1 2  q 2  ξ + κ 2 q η ∂ − 1 ξ q η , r t = − κ 1 q ξ + κ 2 ∂ − 1 ξ q ηη − κ 1 q ηξ ξ + κ 2 q ηη η − κ 1  r q  ξ + κ 2  r ∂ − 1 ξ q η  η . (4.9) In the part icular case where κ 2 = 0, system of equations (4.9) reduces to the w ell-kno wn in tegrable 2 D gDL W system [1 8] q tη = − κ 1 r ξ ξ − κ 1 2  q 2  ξ η , r tξ = − κ 1  q r + q + q ξ η  ξ ξ . (4.10) In the one-dimensional limit ξ = η , system (4.9 ) ( κ 1 − κ 2 = 1) and system (4.10) ( κ 1 = 1) reduce to the w ell-kno wn disp ersiv e long-w a v e equation (see, e.g., [22]). Therefore, integrable system of nonlinear equations (4.6)–( 4 .8) can b e called the 2 D gD L W system. In tegrable system (4.6)–( 4.8) in terms of the v ariables φ = log ρ , w 1 , and w 2 b ecomes φ t = − κ 1 φ ξ ξ − κ 2 φ ηη − κ 1 ( φ ξ ) 2 − κ 2 ( φ η ) 2 − 2 κ 1 ∂ − 1 η w 2 ξ + 2 κ 2 φ η ∂ − 1 ξ w 1 + v 0 , (4.11) w 1 t = − κ 1 w 1 ξ ξ − κ 2 w 1 ηη − 2 κ 1 w 2 ξ ξ + 2 κ 2 w 2 ηη − 2 κ 1  w 1 ∂ − 1 η w 1  ξ + 2 κ 2  w 1 ∂ − 1 ξ w 1  η , (4.12) w 2 t = κ 1 w 2 ξ ξ − κ 2 w 2 ηη − 2 κ 1  w 2 ∂ − 1 η w 1  ξ + 2 κ 2  w 2 ∂ − 1 ξ w 1  η . (4.13) In tegrable system (4 .6)–(4.8) in terms of t he v ariables φ = lo g ρ , w 2 , and e w 2 = w 2 + w 1 b ecomes more symmetric, φ t = − κ 1 φ ξ ξ − κ 2 φ ηη − κ 1 ( φ ξ ) 2 − κ 2 ( φ η ) 2 − 2 κ 1 ∂ − 1 η w 2 ξ + 2 κ 2 φ η ∂ − 1 ξ w 1 + v 0 , (4.14) w 2 t = κ 1 w 2 ξ ξ − κ 2 w 2 ηη − 2 κ 1  w 2 ∂ − 1 η ( e w 2 − w 2 )  ξ + 2 κ 2  w 2 ∂ − 1 ξ ( e w 2 − w 2 )  η , (4.15) e w 2 t = − κ 1 e w 2 ξ ξ + κ 2 e w 2 ηη − 2 κ 1  e w 2 ∂ − 1 η ( e w 2 − w 2 )  ξ + 2 κ 2  e w 2 ∂ − 1 ξ ( e w 2 − w 2 )  η . (4.16) All the mutually equiv alen t integrable systems of 2 D gDL W nonlinear equations consid- ered ab ov e, (4.6)–(4.8), (4.11)–(4.13), and (4.14)–(4.16) hav e the fo llowing common charac- teristic gauge structure: 6 a . They contain manifestly gauge-in v arian t subsystems (4.7), (4.8) and ( 4 .12), (4.1 3) of nonlinear equations for the gauge inv arian ts w 1 and w 2 (or, equiv a len tly , subsys- tem (4.15), (4.16) for the gauge in v arian ts w 2 and e w 2 ). b . They con tain Eq. (4.6 ) for the pure gauge v a r iable ρ (or Eq. (4.11) for t he v a riable φ = log ρ ), whic h satisfies a simple transformat io n la w ρ → ρ ′ = g ρ with additional terms con t aining the gauge in v ariants and t he additional field v ariable v 0 . Suc h a structure of the 2 D gDL W system reflects a significan t gauge freedom in auxiliary linear pro blems (4.1) and (4 .2 ). W e consider sev eral particular ga uges of sy stems of 2 D gD L W equations (4.6)– (4.8), (4.11)–(4.13), and (4.14)–(4.16). W e let C ( u 1 , v 1 , u 0 ) denote the gauge in general p osi- tion. In t he gaug e C ( u 1 = φ η , v 1 = φ ξ , u 0 = φ ξ η + φ ξ φ η ), which by definitions (4.4) o f the in v ariants corr esp o nds to the zero v alues of the in v aria nts w 1 and w 2 , w 1 = u 1 ξ − v 1 η = 0 , w 2 = u 0 − u 1 ξ − u 1 v 1 = 0 , e w 2 = 0 , (4.17) system of 2 D gD L W equations (4.14)–(4.16) reduces to the tw o-dimensional B ¨ urgers equation in p otential form φ t = − κ 1 φ ξ ξ − κ 2 φ ηη − κ 1 ( φ ξ ) 2 − κ 2 ( φ η ) 2 + v 0 , (4.18) or, in terms of the v ariable ρ relat ed to the Hopf– Cole t r a nsformation φ = log ρ , to the linear diffusion equation ρ t = − κ 1 ρ ξ ξ − κ 2 ρ ηη + v 0 ρ. (4.19) It fo llows from the ab o v e construction that Eq. ( 4 .18) (or Eq. (4.19)) is the compatibilit y condition f or auxiliary problems (4.1) and (4.2) in Lax fo r m, [ L 1 , L 2 ] = B ( w 1 ) L 1 ≡ 0 . (4.20) In another simple gauge, C ( u 1 = φ η , v 1 = 0 , u 0 = 0), whic h b y definitions (4.4) corre- sp onds to the inv arian ts w 1 = φ ξ η , w 2 = − φ ξ η , e w 2 = 0 , (4.21) system of 2 D gD L W equations ( 4 .14)–(4.16) again reduces to the single B¨ urgers-t yp e equation in p otential form φ t = κ 1 φ ξ ξ − κ 2 φ ηη − κ 1 ( φ ξ ) 2 + κ 2 ( φ η ) 2 + v 0 . (4.22) This equation can b e linearized b y the Hopf– Cole transformation φ = − log ρ to the corre- sp onding linear equation ρ t = κ 1 ρ ξ ξ − κ 2 ρ ηη − ρv 0 . (4.23) In the gauge C ( u 1 = 0 , v 1 = − q ξ /q , u 0 = p q ), it follows from (4.4) that the inv arian ts w 1 , w 2 , and e w 2 are give n b y the expressions w 1 =  log q  ξ η , w 2 = u 0 = p q , e w 2 = p q +  log q  ξ η , (4.24) the v ariable ρ according to (4.5) has a constan t v a lue, and the v ariable φ is equal to zero. According to (4.14), for the v ariable v 0 , we hav e v 0 = − 2 κ 1 ∂ − 1 η w 2 ξ = − 2 κ 1 ∂ − 1 η  p q  ξ . (4.25) 7 After some calculations in the case under study , system of 2 D gD L W equations (4.14)–(4.16) implies the w ell-kno wn syste m of DS equations [19] for the v ariables p and q : p t = κ 1 p ξ ξ − κ 2 p ηη + 2 κ 1 p ∂ − 1 η  p q  ξ − 2 κ 2 p ∂ − 1 ξ  p q  η , q t = − κ 1 q ξ ξ + κ 2 q ηη − 2 κ 1 q ∂ − 1 η  p q  ξ + 2 κ 2 q ∂ − 1 ξ  p q  η . (4.26) In the gauge C ( u 1 = p η , v 1 = q ξ , u 0 = p η q ξ ), it follow s f r om ( 4 .4) that the inv arian ts can b e expressed in terms of the v aria bles q and p as w 1 = p ξ η − q ξ η , w 2 = − p ξ η , e w 2 = − q ξ η . (4.27) W e substitute w 1 , w 2 , and e w 2 giv en b y (4 .2 7) in system (4.14)–(4.1 6) and obtain three equations for the v ariables p and q . Equation (4.14) for φ ≡ p implies the first equation p t = κ 1 p ξ ξ − κ 2 p ηη − κ 1 ( p ξ ) 2 + κ 2 ( p η ) 2 − 2 κ 2 p η q η + v 0 . (4.28) The other t w o equations f ollo w from Eqs. (4 .15) a nd (4.16) for the inv arian ts w 2 and e w 2 and can b e expresse d in terms of the v ariables p and q as p t = κ 1 p ξ ξ − κ 2 p ηη − κ 1 ( p ξ ) 2 + κ 2 ( p η ) 2 + 2 κ 1 ∂ − 1 η  p ξ η q ξ  − 2 κ 2 ∂ − 1 ξ  p ξ η q η  , (4.29) q t = − κ 1 q ξ ξ + κ 2 q ηη + κ 1 ( q ξ ) 2 − κ 2 ( q η ) 2 − 2 κ 1 ∂ − 1 η  q ξ η p ξ  + 2 κ 2 ∂ − 1 ξ  q ξ η p η  . (4.30) Equations (4.28 ) and (4.29) are compatible under the c hoice o f the v ariable v 0 = 2 κ 1 ∂ − 1 η  p ξ η q ξ  + 2 κ 2 ∂ − 1 ξ  q ξ η p η  . (4.31) In this case, system of three equations (4.28)–(4.30) reduces to system of t wo equations (4.29) and (4.30) , whic h contains the deriv ativ es p ξ η q ξ , p ξ η q η , etc., in nonlo cal terms. Similarly , in the gauge C ( u 1 = p η , v 1 = q ξ , u 0 = 0), it f o llo ws from (4.4) that the in v ari- an ts w 1 , w 2 , and e w 2 can b e expressed as w 1 = p ξ η − q ξ η , w 2 = − p ξ η − p η q ξ , e w 2 = − q ξ η − p η q ξ . (4.32) Equation (4 .1 1) for φ ≡ p with (4.32) taken into account b ecomes p t = κ 1 p ξ ξ − κ 2 p ηη − κ 1 ( p ξ ) 2 + κ 2 ( p η ) 2 − 2 κ 2 p η q η + 2 κ 1 ∂ − 1 η  p η q ξ  ξ + v 0 . (4.33) Substituting expressions (4.3 2) for w 1 and w 2 , we transform Eq. (4.12) as p t − q t = κ 1  p + q  ξ ξ − κ 2  p + q  ηη − κ 1  p ξ − q ξ  2 + + κ 2  p η − q η  2 + 2 κ 1 ∂ − 1 η  p η q ξ  ξ − 2 κ 2 ∂ − 1 ξ  p η q ξ  η . (4.34) Subtracting Eq. (4.34) from (4.33), w e o btain the ev olution equation for the v a riable q : q t = − κ 1 q ξ ξ + κ 2 q ηη + κ 1  q ξ  2 − κ 2  q η  2 − 2 κ 1 p ξ q ξ + 2 κ 2 ∂ − 1 ξ  p η q ξ  η + v 0 . (4.35) It follo ws from ( 4 .32) that Eq. (4.13) for the in v ariant w 2 in terms of the v ariables p and q b ecomes  p ξ η + p η q ξ  t = κ 1  p ξ η + p η q ξ  ξ ξ − κ 2  p ξ η + p η q ξ  ηη − − 2 κ 1  ( p ξ η + p η q ξ )( p ξ − q ξ )  ξ + 2 κ 2  ( p ξ η + p η q ξ )( p η − q η )  η . (4.36) 8 Equations (4.34 )–(4.36) are compatible if the v ariable v 0 satisfies the relation v 0 ξ η + p η v 0 ξ + q ξ v 0 η = 0 . (4.37) In the case where v 0 ≡ 0, system of three equations (4.3 4)–(4.36) implies the equiv a len t system of t w o equations p t = κ 1 p ξ ξ − κ 2 p ηη − κ 1  p ξ  2 + κ 2  p η  2 − 2 κ 2 p η q η + 2 κ 1 ∂ − 1 η  p η q ξ  ξ , q t = − κ 1 q ξ ξ + κ 2 q ηη + κ 1  q ξ  2 − κ 2  q η  2 − 2 κ 1 p ξ q ξ + 2 κ 2 ∂ − 1 ξ  p η q ξ  η . (4.38) This system of equations w as first obtained in a differen t con text in [23]. T o conclude this section, we discuss Miura-ty p e transformat ions b etw een differen t sys- tems of no nlinear second-order DS-type equations obtained ab o v e in differen t gauges. F or con v enience , we let P ≡ p and Q ≡ q denote the solutio ns of system of nonlinear DS-ty p e equations (4 .2 6). W e use t he in v ariants w 1 and w 2 to obt a in the relations b et w een the v ariables P and Q of D S sys tem (4.26) and the v ariables p and q of system (4 .2 9), (4.3 0): w 1 =  log Q  ξ η = p ξ η − q ξ η , w 2 = P Q = − p ξ η . (4.39) Relation (4.3 9) implies the corresp o nding Miura-t yp e transformation Q = e p − q , P = − p ξ η e q − p . (4.40) Quite similarly , for the pair of systems of DS-type equations (4 .26) and (4.38), w e obtain w 1 =  log Q  ξ η = p ξ η − q ξ η , w 2 = P Q = − p ξ η − p η q ξ . (4.41) F ro m (4.41), w e also obta in a Miura-t ype t r ansformation b et w een the corresp onding solutions Q = e p − q , P = −  p ξ η + p η q ξ  e q − p . (4.42) T ransformations ( 4 .40) and (4.4 2 ) p ermit obtaining solutions of the w ell-kno wn system of DS equations (4 .2 6) from the corresp onding solutions o f systems of Eqs. (4.29), (4.30), and (4.38). Therefore, these transformat io ns are Miura- t yp e transforma t io ns b etw een the solutions of gauge-equiv alen t systems of second-order DS-type equations. 5 Manifestly gauge-inv a rian t system of in tegrable Nizhnik–V eselo v– No vik o v equati ons W e consider the results recen tly obtained in [17] concerning the gauge-inv a rian t fo r mulation of t w o-dimensional nonlinear ev olution equations inte grable using the t w o auxiliary linear problems L 1 ψ =  ∂ 2 ξ η + u 1 ∂ ξ + v 1 ∂ η + u 0  ψ = 0 , (5.1) L 2 ψ =  ∂ t + κ 1 ∂ 3 ξ + κ 2 ∂ 3 η + u 2 ∂ 2 ξ + v 2 ∂ 2 η + ˜ u 1 ∂ ξ + ˜ v 1 ∂ η + v 0  ψ = 0 , (5.2) where κ 1 and κ 2 are constants , ξ = x + σ y and η = x − σ y are spatial co ordinates, and σ 2 = ± 1. 9 W e use the compatibility condition in the Manako v t r iad form [20] (see (4 .3)) to reduce auxiliary linear problems (5.1 ) and (5.2) to the previously obtained and w ell-known Nizhnik– V eselo v–No vik o v (NVN) equations a nd to some other equations [2 4–27] (also see [12, 13] and the references therein). In the case under study , just as for the system of 2 D gD L W equations, form ulas (4.4) and (4.5) giv e the field v ariables, the classical in v arian ts w 2 and w 1 , and the pure g auge v ariable ρ , whic h are con v enien t for the gauge-inv arian t form ulation. In the case of auxiliary linear pro blem (5.2), the first in v ariant w 1 is equal to zero ( w 1 ≡ 0 ), and the correspo nding in tegrable system of nonlinear equations [17] in terms o f the v ariables ρ and w 2 has the f o rm ρ t = − κ 1 ρ ξ ξ ξ − κ 2 ρ ηη η − 3 κ 1 ρ ξ ∂ − 1 η w 2 ξ − 3 κ 2 ρ η ∂ − 1 ξ w 2 η + v 0 ρ, (5.3) w 2 t = − κ 1 w 2 ξ ξ ξ − κ 2 w 2 ηη η − 3 κ 1  w 2 ∂ − 1 η w 2 ξ  ξ − 3 κ 2  w 2 ∂ − 1 ξ w 2 η  η . (5.4) System of equations (5.3), (5.4) in terms of the v ariables φ = log ρ and w 2 b ecomes φ t = − κ 1 φ ξ ξ ξ − κ 2 φ ηη η − κ 1 ( φ ξ ) 3 − κ 2 ( φ η ) 3 − 3 κ 1 φ ξ φ ξ ξ − 3 κ 2 φ η φ ηη − − 3 κ 1 φ ξ ∂ − 1 η w 2 ξ − 3 κ 2 φ η ∂ − 1 ξ w 2 η + v 0 , (5.5) w 2 t = − κ 1 w 2 ξ ξ ξ − κ 2 w 2 ηη η − 3 κ 1  w 2 ∂ − 1 η w 2 ξ  ξ − 3 κ 2  w 2 ∂ − 1 ξ w 2 η  η . (5.6) Equation (5.4) (o r (5.6)) for the g a uge inv a r ian t w 2 = u 0 − u 1 ξ − u 1 v 1 of the last t w o systems exactly coincides in form with the w ell-kno wn NVN equation [24, 25] u t = − κ 1 u ξ ξ ξ − κ 2 u ηη η − 3 κ 1  u∂ − 1 η u ξ  ξ − 3 κ 2  u∂ − 1 ξ u η  η . (5.7) In tegrable system (5.3), (5.4) (or (5.5), (5.6)) can t herefore b e called the system o f NVN equations. In tegrable system o f NVN equations ( 5.3), (5.4) (or ( 5.5), (5.6)) has the f ollo wing gauge structure: a . It con tains manifestly gauge- inv arian t equation (5.4) (or (5.6)) for the in v ariant w 2 . b . It con tains Eq. (5 .3 ) (or (5.5)) for the pure gauge v ariable ρ (or φ ) with some additional terms containing the g a uge inv a r ian t w 2 and the additional field v aria ble v 0 used in auxiliary linear problem (5.2). F or w 2 = 0, system of NVN equations (5 .3), (5.4) (or (5.5), (5.6)) reduces to t he single linear equation ρ t = − κ 1 ρ ξ ξ ξ − κ 2 ρ ηη η + v 0 ρ. (5.8) In terms of t he v ariable φ = log ρ , linear equation (5.8) lo oks lik e the third- order B ¨ urgers equation φ t = − κ 1 φ ξ ξ ξ − κ 2 φ ηη η − κ 1 ( φ ξ ) 3 − κ 2 ( φ η ) 3 − 3 κ 1 φ ξ φ ξ ξ − 3 κ 2 φ η φ ηη + v 0 , (5.9) whic h is linearized b y the substitution φ = lo g ρ and is hence C -integrable. W e let C ( φ, u 0 , v 0 ) denote the gauge corres p onding to the no nzero field v ariables u 1 = φ η , v 1 = φ ξ , u 0 , and v 0 of auxiliary linear problems (5.1) and (5.2) and hence to system o f NVN equations (5.5), (5.6) in general p osition. In differen t gauges, the NVN system implies 10 differen t gauge-equiv alen t in tegrable nonlinear equations. The solutions of these equations are related b y Miura-t yp e tra nsfor ma t ions. F or example, in the g auge C (0 , u 0 , 0), system of NVN equations (5.5), (5.6) reduces to the we ll-know n NVN equation [24 , 25] for the field v ariable u 0 : u 0 t = − κ 1 u 0 ξ ξ ξ − κ 2 u 0 ηη η − 3 κ 1  u 0 ∂ − 1 η u 0 ξ  ξ − 3 κ 2  u 0 ∂ − 1 ξ u 0 η  η . (5.10) In ano t her gauge, C ( φ, 0 , v 0 ), NVN system (5 .5), (5.6 ) b ecomes φ t = − κ 1 φ ξ ξ ξ − κ 2 φ ηη η − κ 1 ( φ ξ ) 3 − κ 2 ( φ η ) 3 + + 3 κ 1 φ ξ ∂ − 1 η  φ ξ φ η  ξ + 3 κ 2 φ η ∂ − 1 ξ  φ ξ φ η  η + v 0 , (5.11)  ∂ 2 ξ η + φ η ∂ ξ + φ ξ ∂ η  φ t =  ∂ 2 ξ η + φ η ∂ ξ + φ ξ ∂ η  h − κ 1 φ ξ ξ ξ − κ 2 φ ηη η − − κ 1 ( φ ξ ) 3 − κ 2 ( φ η ) 3 + 3 κ 1 φ ξ ∂ − 1 η  φ ξ φ η  ξ + 3 κ 2 φ η ∂ − 1 ξ  φ ξ φ η  η i . (5.12) By (5.11) and (5 .1 2), NVN system (5.5) , (5.6) in the gaug e C ( φ, 0 , v 0 ) reduces to t he system of equations φ t = − κ 1 φ ξ ξ ξ − κ 2 φ ηη η − κ 1 ( φ ξ ) 3 − κ 2 ( φ η ) 3 + + 3 κ 1 φ ξ ∂ − 1 η  φ ξ φ η  ξ + 3 κ 2 φ η ∂ − 1 ξ  φ ξ φ η  η + v 0 ,  ∂ 2 ξ η + φ η ∂ ξ + φ ξ ∂ η  v 0 = 0 . (5.13) F or v 0 = 0, system of equations (5.13) reduces to the w ell-kno wn mo dified NVN (mNVN) equation φ t = − κ 1 φ ξ ξ ξ − κ 2 φ ηη η − κ 1 ( φ ξ ) 3 − κ 2 ( φ η ) 3 + 3 κ 1 φ ξ ∂ − 1 η  φ ξ φ η  ξ + 3 κ 2 φ η ∂ − 1 ξ  φ ξ φ η  η , (5.14) whic h w as first obtained in a differen t context b y K onop elc henk o [26 ]. W e note that mNVN equation (5.14 ) differs from t he mNVN equation o bt a ined b y Bogdano v [27]. The new sys tem of equations (5.13) can also b e called a system of mNVN equations. Ob viously , the solutions u 0 and φ of NVN equations (5.10) and mNVN equations (5.14) are related b y the Miura-type transformation u 0 = − φ ξ η − φ ξ φ η (5.15) through the gauge in v arian t w 2 = u 0 = − φ ξ η − φ ξ φ η (calculated in the differen t gauges C (0 , u 0 , 0) and C ( φ, 0 , 0)). In the one-dimensional limit with coinciding deriv ativ es ∂ ξ = ∂ η , mNVN equation ( 5.14) reduces to mKdV equation in p oten tial form φ t = − κ φ ξ ξ ξ + 2 κ ( φ ξ ) 3 , (5.16) where κ = κ 1 + κ 2 . In terms of the v ariable v 1 = φ ξ , t his is the mKdV equation v 1 t = − κ v 1 ξ ξ ξ + 6 κ v 2 1 v 1 ξ . (5.17) 11 6 Conclus ion The ideas of gauge in v ariance are curren tly widely used in the theory of integrable nonlin- ear equations. A manifestly gaug e-in v ariant form ulation of in tegrable systems of nonlinear equations can b e give n in all cases where the gauge freedom is prop erly t a k en in to accoun t in the correspo nding a uxiliary linear problems. Ac kno wledgmen ts. 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