About a gauge-invariant description of some 2+1-dimensional integrable nonlinear evolution equations

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1 About a gauge-invariant description of some +-dimensional integrable nonlinear evolution equations V.G. Dubrovsky, M. Nahrgang Novosibirsk State Technical University, Novosibirsk, prosp. Karl Marx 0, 63009, Russia. Abstract A mathematical two-dimensional generalization of the Korteweg-de-Vries equation, the Nijnik-Veselov-Novikov (NVN) equation is derived within a gauge-invariant formulation. This modified Nijnik-Veselov-Novikov (mnvn) equation can be used as a basis for finding exact solutions of the NVN equation. NVN-equation, its triad representation The well known +-dimensional integrable nonlinear evolution Nijnik-Veselov-Novikov (NVN) equation [,]: U t + κ U ξξξ + κ U ηηη + 3κ (U η U ξ ) ξ + 3κ (U ξ U η ) η = 0 () has the following Manakov triad (L, L, B) representation [L, L ] = BL, () i.e. this equation can be represented as a compatibility condition (weaker than the Lax form [L, L ] = 0) of two linear auxiliary problems L ψ = ψ ξη + Uψ = 0; (3) L ψ = ψ t + κ ψ ξξξ + κ ψ ηηη + 3κ ( η U ξ )ψ ξ + 3κ ( ξ U η )ψ η = 0 (4) with B in () given by the expression B = 3κ η U ξξ + 3κ ξ U ηη. (5) It is instructive to re-derive the representation () starting from more general auxiliary linear problems in general position: L ψ = ψ ξη + U ψ ξ + U ψ η + Uψ = 0; (6) L ψ = ψ t + U 3 ψ ξξξ + V 3 ψ ηηη + U ψ ξξ + V ψ ηη + Ũψ ξ + Ṽψ η + Ũψ = 0. (7) Permanent address for correspondence: AG Vielteilchenphysik, Fachbereich Physik, Philipps- University Marburg, Renthof 5, Marburg.

2 After some calculations one obtains the following expressions for the coefficients at the degrees of derivatives n ξ m η of the commutator [L, L ]: 4 ξ : U 3η ; 4 η : V 3ξ ; (8) 3 ξ η : U 3ξ ; 3 η ξ : V 3η ; (9) 3 ξ : U 3ξη + U η + U U 3ξ 3U 3 U ξ + V U 3η ; (0) 3 η : V 3ξη + V ξ + V V 3η 3V 3 V η + U V 3ξ ; () ξ η : U ξ 3U 3 V ξ ; η ξ : V η 3V 3 U η ; () ξ : U ξη + Ũη 3U 3 U ξξ + U U ξ U U ξ + V U η 3U 3 U ξ ; (3) η : V ξη + Ṽξ 3V 3 V ηη + V V η V V η + U V ξ 3V 3 V η ; (4) ξη : Ũξ + Ṽη 3V 3 U ηη V U η 3U 3 V ξξ U V ξ ; (5) ξ : U t + Ũη + Ũξη U 3 U ξξξ V 3 U ηηη U U ξξ V U ηη + +U Ũ ξ ŨU ξ ṼU η + V Ũ η 3U 3 U ξξ U U ξ ; (6) η : V t + Ũξ + Ṽξη U 3 V ξξξ V 3 V ηηη U V ξξ V V ηη + +V Ṽ η ṼV η ŨV ξ + U Ṽ ξ 3V 3 U ηη V U η ; (7) 0 : U t + Ũξη + U Ũ ξ + V Ũ η U 3 U ξξξ V 3 U ηηη V U ηη U U ξξ ŨU ξ ṼU η. (8) Setting the coefficients at the highest degrees of derivatives n ξ m η of the commutator [L, L ] to zero one finds from (8) and (9), that the coefficients U 3 and V 3 are constants: U 3 = κ = const., V 3 = κ = const.. (9) Taking into account (9) and setting to zero the coefficients at 3 ξ, ξ η, ξ η and 3 η one obtains from (0)-() the relations: Integration of (0) yields: U ξ = 3κ V ξ, V η = 3κ U η ; (0) U η = 3κ U ξ, V ξ = 3κ V η. () U = 3κ V + f(η, t), V = 3κ U + g(ξ, t) () with the arbitrary functions f(η, t) and g(ξ, t). relations: Insertion of () into () leads to the U η = 3κ V η + f η (η, t) = 3κ U ξ ; (3) V ξ = 3κ U ξ + (ξ, t) = 3κ V η. (4) The relations (3) and (4) are compatible with each other if f = 0 and thus from ()-(4) important relations for the field variables U, V, U, V can be obtained: U ξ = V η ; (5) U = 3κ V, V = 3κ U. (6)

3 Setting the coefficients (3) and (4) at ξ and η to zero one obtains the following expressions for Ũη and Ṽξ: Integration of (7) and (8) yields Ũ η = 3κ U ξ 3κ U V ξ + 3κ V V η ; (7) Ṽ ξ = 3κ U η 3κ V U η + 3κ U U ξ. (8) Ũ = 3κ η U ξ 3κ η U V ξ + 3 κ V + f (ξ, t); (9) Ṽ = 3κ ξ U η 3κ ξ V U η + 3 κ U + g (η, t) (30) including as constants of integration the arbitrary functions f (ξ, t), g (η, t). In further calculations we choose these functions being equal to zero: f (ξ, t) = g (η, t) = 0. Inserting the expressions (9) and (30) for Ũ and Ṽ into (5) and taking into account (5),(6) the following expression for the coefficient at the derivative ξη can be obtained: B(W ) = 3κ η W ξξ + 3κ ξ W ηη, W := U U ξ U V. (3) This coefficient B will be equal to zero if, for example, the following relation is satisfied: κ W ξξξ + κ W ηηη = 0. (3) Evidently it is difficult to satisfy this relation between U, U and V nontrivially. For example, one can show that the choice W = 0, i. e. U = U ξ + U V, does not lead to any nonlinear equations for the field variables U, U and V. According to the choice made in the papers [,,3] we will consider the case in which B(W ) 0 and require that the commutator [L, L ] is proportional to L with B(W ) from (3): [L, L ] = B(W )L. (33) Using (33) and (6)-(8) and setting the coefficients at the derivatives ξ, η and 0 to zero one derives the undetermined system of three nonlinear equations for the four field variables U, U, V and Ũ: ξ : U t + Ũη + Ũξη U 3 U ξξξ V 3 U ηηη U U ξξ V U ηη + +U Ũ ξ + V Ũ η ŨU ξ ṼU η 3U 3 U ξξ U U ξ B(W )U = 0; (34) η : V t + Ũξ + Ṽξη U 3 V ξξξ V 3 V ηηη U V ξξ V V ηη + +V Ṽ η + U Ṽ ξ ṼV η ŨV ξ 3V 3 U ηη V U η B(W )V = 0; (35) 0 : U t + Ũξη + U Ũ ξ + V Ũ η U 3 U ξξξ V 3 U ηηη V U ηη U U ξξ ŨU ξ ṼU η B(W )U = 0. (36) Indeterminacy of the obtained system of nonlinear equations (as will be shown below) is connected with gauge freedom of the linear auxiliary problems (6) and (7). 3

4 Using formulas (5), (6) and (9)-(3) one can rewrite this system of equations (34)-(36) in the following more explicit form: U t + κ U ξξξ + κ U ηηη 3 κ V V η 3 κ U U η +3κ (V η (U ξ U V ξ )) η + 3κ (U ξ (U η U η V )) η Ũη = 0; (37) V t + κ V ξξξ + κ V ηηη 3 κ V V ξ 3 κ U U ξ +3κ (V η (U ξ U V ξ )) ξ + 3κ (U ξ (U η U η V )) ξ Ũξ = 0; (38) U t + κ U ξξξ + κ U ηηη + 3κ V U ξξ + 3κ U U ηη κ V U ξ + 3 κ U U η 3κ (V V ξ + V ξξ )U 3κ (U U η + U ηη )U + +3κ (U η (U ξ U V ξ )) ξ + 3κ (U ξ (U η U η V )) η Ũξη U Ũ ξ V Ũ η = 0. (39) The undeterminate system of equations (37)-(39) for the four field variables U, U, U and Ũ is integrable by the linear auxiliary problems (6) and (7) and can be represented as a compatibility condition of (6) and (7) in the form (33) with B(W ) from (3). Let us mention that due to (5), (6) and (9)-(3) the system of nonlinear integrable equations (37)-(39) can also be rewritten to the following convenient form: V t + κ V ξξξ + κ V ηηη + κ (V 3 + 3V V ξ ) ξ + κ (U 3 + 3U U η ) ξ + +3κ (V η W ξ ) ξ + 3κ (U ξ W η ) ξ Ũξ = 0; (40) U t + κ U ξξξ + κ U ηηη + κ (V 3 + 3V V ξ ) η + κ (U 3 + 3U U η ) η + +3κ (V η W ξ ) η + 3κ (U ξ W η ) η Ũη = 0; (4) U t + κ U ξξξ + κ U ηηη + 3κ V U ξξ + 3κ U U ηη + +3κ (V ξ + V )U ξ + 3κ (U η + U )U η + +3κ (U η W ξ ) ξ + 3κ (U ξ W η ) η Ũξη U Ũ ξ V Ũ η = 0. (4) By the use of (40) and (4) one obtains the following expressions for Ũξ and Ũη: Ũ ξ = V t + κ V ξξξ + κ V ηηη + κ (V 3 + 3V V ξ ) ξ + κ (U 3 + 3U U η ) ξ + +3κ (V η W ξ ) ξ + 3κ (U ξ W η ) ξ ; (43) Ũ η = U t + κ U ξξξ + κ U ηηη κ (V 3 + 3V V ξ ) η + κ (U 3 + 3U U η ) η + +3κ (V η W ξ ) η + 3κ (U ξ W η ) η. (44) It is evident from (5), (43) and (44) that Ũξη = Ũηξ, i.e. the expressions (43) and (44) are compatible with each other. Finally inserting (43) and (44) into (39) and using (5), (6), (9)-(3) one obtains Nijnik-Veselov-Novikov equation for the single field variable W : W t + κ W ξξξ + κ W ηηη + 3κ (W η W ξ ) ξ + 3κ (W ξ W η ) η = 0; (45) W := U U ξ U V. (46) 4

5 Gauge transformations of linear auxiliary problems and their invariants Let us investigate the gauge properties of the linear auxiliary problems (6) and (7) under gauge transformations ψ ψ = g ψ (47) of the wave function ψ with an arbitrary function g(ξ, η, t). Inserting ψ = gψ into (6) we find L ψ := g L gψ = [ ξη + (U + g ) ξ + (V + g ) η + U + η g + U g + V g ]ψ = 0. (48) From (48) one achieves the rules of gauge transformations of the field variables U, V and U: U = U + g, V = V + U = U + η g + U g + V g ; (49) g. (50) The exclusion of the gauge function g(ξ, η, t) from the relations (49) and (50) leads to the following invariants W := U ξ V η = U ξ V η ; (5) W := U U ξ U V = U U ξ U V (5) of the gauge transformations (47). Due to the relation (5) the first invariant is equal to zero W = U ξ V η = 0. (53) The insertion of ψ = gψ into (7) leads to the transformed second linear auxiliary problem L ψ := g L gψ = = [ t + κ 3 ξ + κ 3 η + (U + 3κ g ) ξ + (V + 3κ +(Ũ + U g + 3κ ξ +Ũ + g t g + κ g ) η + η g ) ξ + (Ṽ + V g + 3κ g ) η + ξξ g + κ ηη g + U ξ g + V η g + +Ũ g + Ṽ g ]ψ = 0. (54) The rules of transformations of the field variables of the second auxiliary linear problem (7) U, V, Ũ, Ṽ and Ũ under the gauge transformations (47) follow from (54): Ũ = Ũ + U g + 3κ ξ g, V Ũ = Ũ + g t g + κ U = U + 3κ g, V = V + 3κ g ; (55) ξξ g + κ η = Ṽ + V g + 3κ g ; (56) ηη g + U ξ g + V η g + 5 +Ũ g + Ṽ g. (57)

6 For further calculations let us present some useful formulas. It follows from (49) and (53): ξ V ξ V ξ + (U U ), ξξ V V, U U ; (58) g t ξ (V t V t ) = η (U t U t ); (59) η U η U η + (U U ) ; (60) V ξξ V ξξ + 3(V V )(V ξ V ξ ) + (V V ) 3 ; (6) ηη = U ηη U ηη + 3(U U )(U η U η ) + (U U ) 3. (6) g By the use of (9)-(3), (43), (44) and (5) one can also obtain convenient formulas for the field variables Ũ, Ṽ, and Ũ: Ũ = 3κ η W ξ + 3κ (V ξ + V ); (63) Ṽ = 3κ ξ W η + 3κ (U η + U ); (64) Ũ = ξ V t + κ V ξξ + κ U ηη + 3κ V V ξ + 3κ U U η + +κ V 3 + κ U 3 + 3κ V η W ξ + 3κ U ξ W η. (65) Using the laws of transformation (55)-(57) of the field variables of the second linear problem U, V, Ũ, and Ṽ and the formulas (58)-(65) one derives: U = U + 3κ 3κ (V + g ) = 3κ V ; (66) V = V + 3κ 3κ (U + g ) = 3κ U ; (67) Ũ = Ũ + U g + 3κ ξ = 3κ η W ξ + 3κ V + 3κ V ξ + 3κ (V V ) + 6κ V (V V ) = = 3κ η W ξ + 3κ (V ξ + V ); (68) η V = Ṽ + V g + 3κ = 3κ ξ W η + 3κ U + 3κ U η + 3κ (U U ) + 6κ U (U U ) = = 3κ ξ W η + 3κ (U η + U ); (69) Ũ = Ũ + g t g + κ ξξ g + κ ηη g + U ξ g + V η g + Ũ g + Ṽ = ξ V t + κ V ξξ + κ U ηη + 3κ V V ξ + 3κ U U η + κ V 3 + κ U κ V η W ξ + 3κ U ξ W η + ξ (V t V t ) + κ (V ξξ V ξξ) + +3κ (V ξ V ξ )(V V )) + κ (V V ) 3 + 3κ V (V ξ V ξ ) + +3κ (U ηη U ηη ) + 3κ (U η U η )(U U )) + κ (U U ) κ V (V V ) + 3κ U (U η U η ) + 3κ U (U U ) + 3κ (V V ) η W ξ + +3κ (V V )(V ξ + V ) + 3κ (U U ) ξ W η + 3κ (U U )(U η + U ) = = ξ V t + κ V ξξ + κ U ηη + 3κ V V ξ + 3κ U U η + +κ V 3 + κ U 3 + 3κ V 6 η W ξ + 3κ U ξ W η. (70)

7 These last calculations show that the expressions for the field variables of the transformed second linear problem U, V, Ũ, Ṽ and Ũ through the field variables of the first linear problem U, V and U are of the same form as for the corresponding field variables (6) and (63)-(65) of the non-transformed (initial) linear problems. This means that the auxiliary linear problems are invariant under gauge transformations: ψ ψ = g ψ; U U = U + g, V V = V + g ; U U = U + η g + U g + V U U = U + 3κ g, V V = V + 3κ g ; (7) ξ Ũ Ũ = Ũ + U g + 3κ g, Ṽ V = Ṽ + V g + 3κ g ; Ũ Ũ = Ũ + g t g + κ ξξ g + κ ηη g + U ξ g + V η g + Ũ g + Ṽ g. Starting from the linear auxiliary problems (6) and (7) in general position one can obtain them in a gauge invariant form by choosing a certain gauge function g. Indeed for the special choice of g such, that i.e. U, one obtains for the field variables: U = U + 0, V = V + 0 (7) V, η ξ V η + U V, (73) U = U V η U V = U U ξ + U V = W ; (74) U = U + 3κ 3κ (V + ) = 0; (75) g V = V + 3κ 3κ (U + ) = 0; (76) g Ũ = Ũ + U g + 3κ ξ = 3κ η W ξ + 3κ (V ξ + V ) + 3κ ( V ξ + V ) + 6κ V ( V ) = = 3κ η W ξ ; (77) V = Ṽ + V g + 3κ η = 3κ ξ W η + 3κ (U η + U ) + 3κ ( U η + U ) + 6κ U ( U ) = = 3κ ξ W η. (78) Ũ = Ũ + g t g + κ ξξ g + κ ηη g + U ξ g + V η g + Ũ g + Ṽ = ξ V t + κ V ξξ + κ U ηη + 3κ V V ξ + 3κ U U η + κ V 3 + κ U κ V η W ξ + 3κ U ξ W η ξ V t κ V ξξ + 3κ V V ξ κ U ηη κ U 3 + 3κ V ( V ξ + V ) + 3κ U ( U η + U ) 3κ V η W ξ 3κ U ξ W η 3κ V (V ξ + V ) 3κ U ( U η + U ) = 0. (79) 7 g

8 3 NVN and new modified NVN equations as realizations of different gauges. Miura type transformation between solutions of NVN and mnvn equations According to the results of the previous sections it exists a gauge-invariant formulation of the NVN equation W t + κ W ξξξ + κ W ηηη + 3κ (W η W ξ ) ξ + 3κ (W ξ W η ) η = 0; (80) W := U U ξ U V. (8) This equation can be represented as a compatibility condition of two linear auxiliary problems L (W ) in the form L (W ) ψ := ψ ξη + W ψ = 0; (8) ψ := ψ t + κ ψ ξξξ + κ ψ ηηη + 3κ ( η W ξ )ψ ξ + 3κ ( ξ W η )ψ η = 0 (83) [L (W ), L (W ) ] = B(W )L (W ) (84) of Manakov s triad (L (W ), L (W ), B(W )) with B(W ) given by the formula B = 3κ η W ξξ + 3κ ξ W ηη. (85) The representation (84) can be obtained from the analogous given by (33) in general position by an appropriately chosen gauge transformation g [L, L ] [L (W ), L (W ) ] = B(W )g L B(W )L (W ). (86) Let us denote by C(U, U, V ) the gauge which corresponds to the general position with nonzero field variables U, U, V. The fields of different gauges C(U, U, V ) and C(U, U, V ) are connected through the gauge invariant W : U U ξ U V = U U ξ U V. (87) So, after the gauge transformation of the linear auxiliary problems (3) and (4) with the gauge function g given by (7) and (73) one obtains ψ ψ (W ) = g ψ (88) L (W ) ψ (W ) := g L gψ (W ) = ψ (W ) ξη + W ψ (W ) = 0; (89) L (W ) ψ (W ) := g L gψ (W ) = = ψ (W ) t + κ ψ (W ) ξξξ + κ ψ ηηη (W ) + 3κ ( η W ξ )ψ (W ) ξ + 3κ ( ξ W η )ψ η (W ) = 0 (90) 8

9 linear auxiliary problems depending on the gauge invariant field variable W = U U ξ U V. One can also apply the gauge transformation of the compatibility condition of the auxiliary linear problems (3) and (4) [L (W ), L (W ) ] := g [L, L ] B(W )g L B(W )L (W ). (9) As result one obtains a compatibility condition of auxiliary linear problems (8) and (83). It is evident that this condition exactly yields the Nijnick-Veselov-Novikov equation: W t + κ W ξξξ + κ W ηηη + κ (W η W ξ ) ξ + κ (W ξ W η ) η = 0 (9) for the gauge invariant W. Due to the relation U ξ = V η one can express the field variables U and V through the potential φ(ξ, η, t): (V, U ) := (φ ξ, φ η ). (93) By such a definition of U and V the invariant W of the gauge transformation will be of the form W (φ, U) = U U ξ U V = U φ ξη φ ξ φ η. (94) The equation (93) gives a manifestly gauge-invariant description of the nonlinear equations connected with the problems (3) and (4). If one fixes the gauge one obtains concrete integrable nonlinear equations. In the different gauges we obtain, at first sight, different nonlinear equations. They are, however, gauge-equivalent to each other. Let us denote by C(φ, U) the gauge in which the field variables U, V and U of the first auxiliary linear problem (3) are given via the functions U = φ η (ξ, η, t), V = φ ξ (ξ, η, t), U(ξ, η, t). (95) The function W (φ, U) given by (94) is invariant under the gauge transformation i.e. φ φ = φ + lng, (96) U U = U = η g + φ η g + φ ξ g, (97) W (φ, U ) = U φ ξη φ ξφ η = U φ ξη φ ξ φ η = W (φ, U). (98) The variables φ and U in different gauges are connected by certain relations. For example, the field variable U in the gauge C(U, 0) is connected due to (79) with the field φ in the gauge C(0, φ) by the relation: U = φ ξη φ ξ φ η = ( ξη + φ ξ η + φ η η )φ. (99) In the gauge C(0, φ) we have the Nijnik-Veselov-Novikov equation for U: U t + κ U ξξξ + κ U ηηη + κ (U η U ξ ) ξ + κ (U ξ U η ) η = 0 (00) but in the gauge C(0, φ) one obtains an equation for the field φ: (φ ξη + φ ξ φ η ) t + κ (φ ξη + φ ξ φ η ) ξξξ + κ (φ ξη + φ ξ φ η ) ηηη + +κ ((φ ξη + φ ξ φ η ) η (φ ξη + φ ξ φ η ) ξ ) ξ + κ ((φ ξη + φ ξ φ η ) ξ (φ ξη + φ ξ φ η ) η ) η = = ( ξη + φ ξ η + φ η η )(φ t + κ φ ξξξ + κ φ ηηη κ φ3 ξ κ 3 φ3 η 3κ φ ξ η (φ η φ ξξ ) 3κ φ η ξ (φ ξ φ ηη )) = 0 (0) 9

10 This last result shows that one can introduce a new version of modified NVN equation, mnvn-equation: φ t + κ φ ξξξ + κ φ ηηη κ φ3 ξ κ 3 φ3 η 3κ φ ξ η (φ η φ ξξ ) 3κ φ η ξ (φ ξ φ ηη ) = 0 (0) By a Miura-type transformation U = φ ξη φ ξ φ η (03) one can construct from the solution φ(ξ, η, t) of the mnvn-equation (0) the solution of th NVN-equation (00)! Acknowledgements Marlene Nahrgang is grateful for receiving financial and ideal support from DAAD (German Academic Exchange Service), the German Academic Foundation and Alfried Krupp von Bohlen und Halbach-Stiftung. She expresses her thankfulness towards these organizations as well as to Professor V.G. Dubrovsky, Departement of Applied and Theoretical Physics, and the heads of the Center for International Relations of Novosibirsk State Technical University for the received support. References [] S.P. Novikov, V.E. Zakharov, S.V. Manakov, L.V. Pitaevski: Solition Theory: The inverse Seattering Method (Plenum, New York, 984) [the Russian original appeared in 980 with Nauka (Moskow)]. [] M.J. Ablowitz, P.A. Clarkson: Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Notes Series (Cambridge University Press, Cambridge 99). 0