Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy

Non-Hamiltonian generalizations of the disp ersionless 2DTL hierarc h y L.V. Bogdano v ∗ Marc h 9, 2022 Abstract W e consider tw o-comp onen t in tegr able generalizatio ns of the dis - per sionless 2DTL hierarch y connected with non- Ha miltonian vector fields, similar to the Manako v-Santini hierarch y generalizing the dKP hierarch y . They form a one-para metric family connected b y ho do graph t yp e tra ns formations. Generating equatio ns and L ax-Sato equations are introduced, a dressing scheme based on the vector nonlinear Rie- mann pr oblem is formulated. The simplest tw o-c omp onent general- ization of the disp ersionless 2 DTL equation is derived, its differential reduction analogous to the Duna jski interpolating system is presented. A symmetric t wo-compo ne nt genera liz ation of the disp ersio nless ellip- tic 2DTL equation is also constructed. 1 In tro duc tion Recen tly S.V. Manako v and P .M. S an tini introd uced a t w o-comp onent sys- tem generalizing the disp ersionless KP equation to th e case of non-Hamiltonian v ector fields in the Lax p air [1, 2], u xt = u y y + ( uu x ) x + v x u xy − u xx v y , v xt = v y y + uv xx + v x v xy − v xx v y , (1) and the Lax pair is ∂ y Ψ = (( λ − v x ) ∂ x − u x ∂ λ ) Ψ , ∂ t Ψ = (( λ 2 − v x λ + u − v y ) ∂ x − ( u x λ + u y ) ∂ λ ) Ψ , (2) ∗ L.D. Land au ITP RAS , Moscow, Ru ssia, e-mail leonid@landau.ac.ru 1 where u , v are functions of x , y , t , and λ pla ys a role of a sp ectral v ari- able. F or v = 0 the sys tem (1) reduces to the dKP (Kh ohlo v-Zab olotsk a y a) equation u xt = u y y + ( uu x ) x . (3) Resp ectiv ely , the r eduction u = 0 gives an equ ation [3] v xt = v y y + v x v xy − v xx v y . (4) The hierarc h y r elated to this sys tem w as stu died in [4, 5]. It w as demon- strated that the Manak o v-Sant ini hierarc hy represents a case N=1 of a gen- eral (N+1)-comp onent hierarch y . This general hierarc hy is connected with comm utativit y of (N+1)-dimensional vect or vields con taining a deriv ative with resp ect to the sp ectral v ariable, w ith the coefficient s of v ector fi elds meromorphic in th e complex plane of the sp ectral v ariable and ha ving a p ole only at one p oint (e.g., infinity , compare the Lax pair (2)). In this sense the Manak o v-Santi ni hierarc h y i s a t wo- comp onen t o ne-p oin t hierar- c h y , and generalizations of the disp ersionless 2DTL h ierarch y we are going to consider i n this pap er represent a t wo-c omp onen t t w o-p oint case, when v ector fields hav e p oles at tw o p oin ts (say , zero and in finit y). Our starting p oin t is th e formalism of the w orks [6, 4, 5], whic h we transfer to the t wo -p oin t case ha ving in m ind the represen tation of the dis- p ersionless 2DTL hierarc hy giv en in [7, 8] and the results of the recen t w ork [9], in wh ic h the dressing, th e Cauc h y prob lem and the behavior of solutions of the disp ersionless 2D T o d a equation w ere studied. W e in tro du ce generat- ing equations an d Lax-Sato equations and d ev elop a dressing sc heme based on vect or nonlinear Riemann problem. W e disco v er o ne-parametric freedom in generalizing the disp ersionless 2DTL hierarch y , and describ e ho dograph t yp e transformations connecting different generalizat ions. The simplest t wo -comp onent generalizatio n of the disp ersionless 2DTL equation reads (e − φ ) tt = m t φ xy − m x φ ty , m tt e − φ = m ty m x − m xy m t , (5) and the Lax pair is ∂ x Ψ =  ( λ + m x m t ) ∂ t − λ ( φ t m x m t − φ x ) ∂ λ  Ψ , ∂ y Ψ =  1 λ e − φ m t ∂ t + (e − φ ) t m t ∂ λ  Ψ (6) 2 (the deriv ation is giv en b elo w). F or m = t the system ( 5) reduces to the disp ers ionless 2DTL equation (e − φ ) tt = φ xy , (7) Resp ectiv ely , the reduction φ = 0 giv es an equation [3] m tt = m ty m x − m xy m t . System ( 5) do esn’t p reserv e the symmetry of the d isp ersionless 2 DTL equa- tion with resp ect to x , y v ariables, ho w ev er, we also introdu ce a sym m etric generalizat ion of the d2DTL equation. 2 Generalized disp ersionless 2DTL hierarc h y W e generalize a picture of the disp ersionless 2DTL hierarch y giv en by T ak asaki and T ake b e [7, 8], taking into account th e results of the recen t w ork [9], to the case of non-Hamilto nian v ector fields, similar to the Manak o v-San tini hierarc hy , wh ic h generalizes the disp ersionless KP h ierarc hy [1, 2, 4 , 5]. W e consider formal series Λ out = ln λ + ∞ X k =1 l + k λ − k , Λ in = ln λ + φ + ∞ X k =1 l − k λ k , (8) M out = M out 0 + ∞ X k =1 m + k e − k Λ + , M in = M in 0 + m 0 + ∞ X k =1 m − k e k Λ − , (9) M 0 = t + x e Λ + y e − Λ + ∞ X k =1 x k e ( k +1)Λ + ∞ X k =1 y k e − ( k + 1)Λ , where λ is a sp ectral v ariable. Usually w e su ggest that ‘out’ and ‘in’ comp o- nen ts of the series define the functions outside and in side the unit circle in the complex p lane of th e v ariable λ resp ectiv ely , with Λ in − ln λ , M in − M in 0 analytic in the unit disc, and Λ out − ln λ , M out − M out 0 analytic outside the unit disc and decreasing at in finit y . F or a fun ction on the complex plane, ha ving a discon tinuit y o n the un it c ircle, b y ‘in’ and ‘o ut’ comp onent s w e mean the fu nction in s ide and outside the unit disc. F or t wo -comp onent se- ries we observe a natur al con v en tion ( AB ) in = A in B in , ( AB ) out = A out B out , whic h corresp ond s to m ultiplication of resp ectiv e fun ctions on the complex plane. The co efficien ts of the series φ , m 0 , l ± k , m ± k are functions of times t , x n , y n . Usually f or simplicit y we s u ggest that only fin ite n u mb er of x k , y k are not equal to zero. 3 Generalized d isp ersionless 2DTL hierarc hy is defined by the generating relation (( J 0 ) − 1 dΛ ∧ d M ) out = (( J 0 ) − 1 dΛ ∧ d M ) in , (10) whic h m a y b e considered as a con tinuit y condition on the unit circle for the differen tial t wo -form (or just in terms of formal series), where J 0 is a determinan t of Jacobi t yp e matrix J , J =  λ∂ λ Λ ∂ t Λ λ∂ λ M ∂ t M  , J out 0 = 1 + O ( λ − 1 ), J in 0 = 1 + ∂ t m 0 + O ( λ ), and we suggest that J 0 6 = 0; the differen tial d is giv en b y d f = ∂ λ f d λ + ∂ t f d t + ∞ X k =1 ∂ f ∂ x k d x k + ∞ X k =1 ∂ f ∂ y k d y k . (11) As a result of a con tinuit y cond ition, the co efficien ts of the differentia l tw o- form in the generating relation (10) are mer omor phic . First w e will gi ve a direct deriv atio n of the Lax-Sato equat ions of gener- alized tw o-component d2DTL hierarc hy from the generating relation (10 ). It is also p ossible to giv e a deriv ation based on an in termediate general state- men t ab out linear op erators of the h ierarch y , sim ilar to the w orks [6 ], [4], but here w e p refer to demonstrate a more straigh tforw ard w ay of exploiting the generating relation (10). T aking a term of the generating relation con taining d λ ∧ d x n , we get (( J 0 ) − 1 ( ∂ λ Λ ∂ + n M − ∂ λ M ∂ + n Λ)d λ ∧ d x n ) out = (( J 0 ) − 1 ( ∂ λ Λ ∂ + n M − ∂ λ M ∂ + n Λ)d λ ∧ d x n ) in , where we int ro duce a notation ∂ + n = ∂ ∂ x n , ∂ − n = ∂ ∂ y n . Thus, taking into accoun t (8), (9), w e come to the conclusion that the fu n ctions A + n = λ ( J 0 ) − 1 ( ∂ λ Λ ∂ + n M − ∂ λ M ∂ + n Λ) (12) are p olynomials, and they can b e expr essed b y the formula A + n = (( J 0 ) − 1 ( λ∂ λ Λ)e ( n +1)Λ ) out + , (13) where the sub scripts + , − denote pro j ection op erators, ( P ∞ −∞ u n p n ) + = P ∞ n =0 u n p n , ( P ∞ −∞ u n p n ) − = P n = − 1 −∞ u n p n . I n a similar wa y , taking a term of the generating relation con taining d t ∧ d x n , we conclude that the functions B + n = ( J 0 ) − 1 ( ∂ t Λ ∂ + n M − ∂ t M ∂ + n Λ) (14) 4 are also p olynomials, and they can b e expressed b y the formula B + n = (( J 0 ) − 1 ( ∂ t Λ)e ( n +1)Λ ) out + , (15) Resolving (12), (14) as linear equations with resp ect to ∂ + n Λ, ∂ + n M , we obtain Lax-Sato equations for the times x n , ∂ + n  Λ M  = ( A + n ∂ t − B + n λ∂ λ )  Λ M  T aking t he terms of the generating relation con taining d λ ∧ d y n , d t ∧ d y n , w e obtain Lax-Sato equations for the times y n , ∂ − n  Λ M  = ( A − n ∂ t − B − n λ∂ λ )  Λ M  , where A − n = (( J 0 ) − 1 ( λ∂ λ Λ)e − ( n +1)Λ ) in − , B − n = (( J 0 ) − 1 ( ∂ t Λ)e − ( n +1)Λ ) in − . The compatibilit y of the fl ows defined by the Lax-Sato equations can b e pro ve d similar to the case of Duna j ski hierarch y [6], see also [5]. In explicit form, a complete set of Lax-Sato equ ations reads ∂ + n n + 1 − λ (e ( n +1)Λ ) λ { Λ , M } ! out + ∂ t + (e ( n +1)Λ ) t { Λ , M } ! out + λ∂ λ !  Λ M  = 0 , (16)   ∂ − n n + 1 + λ (e − ( n +1)Λ ) λ { Λ , M } ! in − ∂ t − (e − ( n +1)Λ − ) t { Λ , M } ! in − λ∂ λ    Λ M  = 0 , (17) where the defin ition of the Poisson b rac k et is { f , g } = λ ( f λ g t − f t g λ ). L ax- Sato equations for the times x = x 1 , y = y 1 , ∂ + 1 = ∂ x , ∂ − 1 = ∂ y , ∂ x Ψ =  ( λ + ( m + 1 ) t − l + 1 ) ∂ t − λl + 1 ∂ λ  Ψ , ∂ y Ψ =  1 λ e − φ m t ∂ t + (e − φ ) t m t ∂ λ  Ψ , where Ψ =  Λ M  , m = m 0 + t , corresp ond to th e Lax pair (6), where the co efficien ts in the first Lax-Sato equ ation can b e transformed to th e form (6) 5 b y taking its expansion at λ = 0, and the system (5) arises as a compatibilit y condition. Lax-Sato equations (16,17 ) defin e the e vol ution of the series Λ in , Λ out , M in , M out . Th e only term con taining an i nteract ion b et wee n Λ a nd M is { Λ , M } . The condition { Λ , M } = 1 s plits out equations for Λ and red uces the h ierarc h y (16,17) to th e d2DTL h ierarc h y , while the condition Λ = ln λ – to the h ierarc h y , considered by Mart ´ ınez Alonso and Shabat [10, 11], see also Pa vlo v [3]. 2.1 The dressi ng scheme A dressing scheme for the g eneralized t wo -comp onent d2DTL h ierarch y can b e formula ted in terms of the t wo- comp onen t n onlinear Riemann-Hilb ert problem on the un it circle S in the complex p lane of the v ariable λ , Λ out = F 1 (Λ in , M in ) , M out = F 2 (Λ in , M in ) , (18) where the functions Λ out ( λ, x , y , t ), M out ( λ, x , y , t ) are defin ed in side the unit circle, the functions Λ in ( λ, x , y , t ), M in ( λ, x , y , t ) outside the u nit circle b y the series of the form (8), (9), with Λ in − ln λ , M in − M in 0 analytic in the unit disc, and Λ out − ln λ , M out − M out 0 analytic outsid e the un it disc and decreasing at infinit y . The functions F 1 , F 2 are suggested to define (at least lo cally) a diffeomorphism of th e plane, F ∈ Diff(2), and w e call them th e dressing d ata. Let us consider a differentia l form Ω = dΛ ∧ d M . The c ond ition for this f orm on the unit circle is determined b y the Jacobian of the diffeomorphism defi n ed b y F 1 , F 2 , Ω out =     D ( F 1 , F 2 ) D (Λ in , M in )     Ω − , (19) where for the Jacobian w e use a n otation     D ( f , g ) D ( x, y )     = det D ( f , g ) D ( x, y ) = det ∂ f ∂ x ∂ f ∂ y ∂ g ∂ x ∂ g ∂ y ! . Expressing the d ifferen tial d in terms of ind ep endent v ariables λ , x , y , t (11), w e come to the conclusion that a ll the co efficien ts of the differenti al tw o- form Ω in terms of these v ariables trans f orm according to the cond ition (19). 6 Normalizing the form b y o ne of the co efficien ts, we obtain the differen tial form con tinuous on the unit circle. T aking the coefficient corresp ond in g to dλ ∧ d t , we o btain the relation     D (Λ , M ) D ( λ, t )     − 1 Ω ! out =     D (Λ , M ) D ( λ, t )     − 1 Ω ! in , (20) whic h is equiv alen t to the ge nerating r elation (10) afte r multiplicat ion by λ − 1 , J 0 = λ     D (Λ , M ) D ( λ, t )     = det  λ∂ λ Λ ∂ t Λ λ∂ λ M ∂ t M  . 2.2 Differen tial reductions Recen tly w e ha ve introdu ced a class of reductions of the Manako v-San tini hierarc hy [5] connected wit h the in terp olating system [12]. Similar r educ- tions can b e constructed f or the generalized tw o-c omp onen t d2DTL hier- arc h y , we are going to stud y them in detail else wh ere. Here w e will only present the simplest redu ction, w hic h is analogo us to the reduction of th e Manak o v-San tini sy s tem leading to the in terp olating system [12]. The re- duced h ierarc h y is defin ed b y the relation (exp( − α Λ)dΛ ∧ d M ) out = (exp( − α Λ)dΛ ∧ d M ) in , where α is a p arameter ( α = 0 corresp onds to the ca se of d2DTL hierarch y), whic h implies that J 0 = λ − α exp( α Λ) , and in terms of the system (5) we ge t a reduction e αφ = m t . This redu ction mak es it p ossible to r ewrite the sy s tem (5) as one equ ation for m , m tt = ( m t ) 1 α ( m ty m x − m xy m t ) , or in the form of deformed d2DTL equation, (e − φ ) tt = m t φ xy − m x φ ty , m t = e αφ . 7 3 T ransformations and a symmetric generalization The system (5) w e ha v e in tro d uced ab o v e do esn’t preserv e the symmetry of th e disp ersionless 2DTL equation (7) w ith resp ect to th e v ariables x , y . T o in tro d uce a symmetric generaliza tion of the equation (7) and its elliptic v ersion (e − φ ) tt = φ z ¯ z , (21) it is p ossible to c hange th e form of the series f or Λ, M to h a v e an explicit symmetry b et w een zero and infin ity in the complex p lane of th e sp ectral v ariable λ , then th e generating relation (10) will lead to symmetric Lax- Sato equations. How ev er, we pr efer to consider fi r st the transformations of the hierac h y th at will allo w us to transfer to the symmetric case and w ill giv e the connection b et wee n different generalizati ons of th e d2DTL equation. First, there is a ga uge tran s formation, pr esent already in d2DTL case [7, 8], whic h c hanges the Lax pair, but preserves the equations λ → λ exp ( − ǫφ ) , where ǫ is a parameter. Af ter this tran s formation w e get Λ of the f orm Λ out = ln λ − ǫφ + ∞ X k =1 l + k λ − k , Λ in = ln λ + (1 − ǫ ) φ + ∞ X k =1 l − k λ k . In the Lax pair on e should p erform a substitution λ → λ exp( − ǫφ ) , ∂ λ → exp( ǫφ ) ∂ λ . ∂ x → ∂ x + ǫλφ x ∂ λ , ∂ y → ∂ y + ǫλφ y ∂ λ , ∂ t → ∂ t + ǫλφ t ∂ λ , In the elliptic d 2DTL case (21) for ǫ = 1 2 w e get a sym m etric Lax p air ∂ z Ψ = L 1 Ψ =  ( λ e − 1 2 φ ) ∂ t + 1 2 ( φ z + λ e − 1 2 φ φ t ) λ∂ λ  Ψ , ∂ ¯ z Ψ = L 2 Ψ =  ( 1 λ e − 1 2 φ ) ∂ t − 1 2 ( φ ¯ z + λ e − 1 2 φ φ t ) λ∂ λ  Ψ , and on the un it circle L 1 = ¯ L 2 . 8 T o get a symmetric tw o-component generalization of the ellyptic d 2DTL equation and a symmetric Lax pair for it, w e should also use a ho d ograph t yp e transformation t = τ − αm 0 (where τ is a new ‘time’, α is a p arameter), whic h giv es M of the form M out = M out 0 + (1 − α ) m 0 + ∞ X k =1 m + k e − k Λ + , M in = M in 0 − αm 0 + ∞ X k =1 m − k e k Λ + , M 0 = τ + x e Λ + y e − Λ + . . . Deriv ative s transform as follo ws, ∂ x → ∂ x + αm 0 x 1 − αm 0 τ ∂ τ , ∂ y → ∂ y + αm 0 y 1 − αm 0 τ ∂ τ , ∂ t → ∂ τ + αm 0 τ 1 − αm 0 τ ∂ τ . Applying these tr an s formations to the system (5), where m = m 0 + t , w e ob- tain a one-parametric family of t wo-c omp onen t generalizatio n s of th e d2DTL equation. T aking x = z , y = ¯ z , ǫ = 1 2 , φ → − 2 ϕ , α = 1 2 , m 0 = − 2i µ , we get Λ out = ln λ + ϕ + ∞ X k =1 l + k λ − k , Λ in = ln λ − ϕ + ∞ X k =1 l − k λ k , M out = M out 0 + i µ + ∞ X k =1 m + k e − k Λ , M in = M in 0 − i µ + ∞ X k =1 m − k e k λ , M 0 = τ + z e Λ + ¯ z e − Λ + . . . where for the case of ellyptic d2DTL we suggest that µ , ϕ are real, and on the u nit circle λ ¯ λ = 1 M out = ¯ M in , Λ out = − ¯ Λ in . F rom the Lax pair (6) w e obtain a symm etric Lax pair ∂ z Ψ = L 1 Ψ , L 1 = ( λ e ϕ u + v ) ∂ τ + (( ϕ τ v − ϕ z ) − λu e ϕ ϕ τ ) λ∂ λ , ∂ ¯ z Ψ = L 2 Ψ , L 2 = ( 1 λ e ϕ ¯ u + ¯ v ) ∂ τ − (( ϕ τ ¯ v − ϕ ¯ z ) − 1 λ ¯ u e ϕ ϕ τ ) λ∂ λ , 9 on th e unit circle L 1 = ¯ L 2 , u = 1 1 + i µ τ , v = − i µ z 1 − i µ τ . Equation (5) transf orms to the sym metric t wo -comp onent generalizatio n of the ellyptic d2DTL equation (21), ( v ¯ z + e ϕ u∂ τ (e ϕ ¯ u ) + v ∂ τ ¯ v ) − c.c. = 0 , ( ∂ ¯ z ( ϕ τ v − ϕ z ) + e ϕ u∂ τ ( ¯ u e ϕ ϕ τ ) − v ∂ τ ( ϕ τ ¯ v − ϕ ¯ z ) + u ¯ u e 2 ϕ ϕ τ ϕ τ ) +c.c. = 0 If µ = 0 ( u = 1, v = 0), the first e qu ation v anishes, the second giv es the d2DTL equ ation for φ = ( − 2 ϕ ). If ϕ = 0, the second equation v anishes, the fi rst give s ( v ¯ z + u∂ τ ( ¯ u ) + v ∂ τ ¯ v ) − c.c. = 0 , or, in explicit form , µ τ τ = 1 2 ( µ z µ ¯ z − (1 + µ 2 τ )) − 1 ( µ 2 τ ( ∂ τ ( µ z µ ¯ z ) − i( µ z τ µ ¯ z − µ z µ ¯ z τ )) − µ z ¯ z (1 + µ 2 τ )) . Ac kno wledgmen ts The author is grateful to S.V. Manak o v and P .M. S an tini for useful dis- cussions. This researc h w as p artially sup p orted by the Russian F oundation for Basic Researc h u nder gran ts no. 10-01 -00787, 09-01-9 2439, and by the President of Russia gran t 4887.2008 .2 (scient ific schools). References [1] S. V. 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