Dispersionless Davey-Stewartson system: Lie symmetry algebra, symmetry group and exact solutions
aa r X i v : . [ n li n . S I] F e b Dispersionless Davey–Stewartson system: Lie symmetry algebra,symmetry group and exact solutions
Faruk G¨ung¨or ∗ Cihangir ¨Ozemir † Department of Mathematics, Faculty of Science and Letters,Istanbul Technical University, 34469 Istanbul, TurkeyIn memory of Prof. Pavel Winternitz: A great mentor, collaborator and friend.
Abstract
Lie symmetry algebra of the dispersionless Davey–Stewartson (dDS) system is shown tobe infinite-dimensional. The structure of the algebra turns out to be Kac–Moody–Virasoroone when the system is of type dDS-I. For the type dDS-II system this rare structure isspoilt. Symmetry group transformations are constructed using a global approach. Theyare split into both connected and discrete ones. Several exact solutions are obtained asan application of the symmetry properties.
Keywords:
Dispersionless Davey–Stewartson system, Lie symmetry algebra and group,Exact solutions.
The aim of this work is to study the semi-classical or dispersionless limit of the Davey-Stewartson system in (2+1) dimensions from a group-theoretical point of view. A very recentwork [1], followed by [2], considers the following ε -dependent form of the Davey–Stewartson(DS) system parametrized by ε iεq t + ε q xx + σq yy ) + δqφ = 0 , (1.1a) σφ yy − φ xx + ( | q | ) xx + σ ( | q | ) yy = 0 , (1.1b)where q ( x, y, t ) is a complex wave function and φ ( x, y, t ) is a real (mean-flow) function. Eq.(1.1) with σ = 1 is called the DS-I system and with σ = − δ = 1 is the focusing case and δ = − φ → φ + | q | in(1.1), we obtain iεq t + ε q xx + ε σ q yy = − δq | q | − δqφ, (1.2a) φ xx − σφ yy = 2 σ ( | q | ) yy (1.2b) ∗ e-mail: [email protected] † e-mail: [email protected] iψ t + ψ xx + ǫ ψ yy = ǫ | ψ | ψ + ψw, (1.3a) w xx + δ w yy = δ ( | ψ | ) yy , (1.3b)where δ , δ , ǫ and ǫ are real constants, ǫ = ∓ ǫ = ∓
1. This system is integrable onlywhen δ = − ǫ .The author of [1] applies the Madelung type transformation q = √ u exp( i sε ) , u > , s ∈ R (1.4)in (1.1), where u ( x, y, t ) > s ( x, y, t ) is the real phase function,and obtains the following two different systems in the dispersionless (semiclassical) limit ε → u t + ( us x ) x + ( us y ) y = 0 , (1.5a) s t + 12 ( s x + σs y ) − δφ = 0 , (1.5b) σφ yy − φ xx + u xx + σu yy = 0 . (1.5c)In (1.5), σ = 1 corresponds to the dispersionless DS-I (dDS-I) system and σ = − i ~ Ψ ~ t + ~ ~ + f ( | Ψ ~ | )Ψ ~ = 0 , x ∈ R d , t ∈ R + , (1.6a)Ψ ~ ( x,
0) = A ( x ) exp (cid:18) i ~ S ( x ) (cid:19) (1.6b)which are parametrized by ~ and supposedly have solutions Ψ ~ ( x, t ). Investigating the semi-classical limit of this problem is to ”determine the limiting behaviour of any Ψ ~ ( x, t ) as ~ → ” [4, 5]. Also of both mathematical and physical interest is the existence of the limiting behaviourof the conserved densities of mass, momentum and energy. Refs. [4, 5, 6] focus on a query onvalidity of this limiting case before any breakdown occurs in the solutions.We would like to mention some of the discussions reported in [7] regarding nonlinearSchr¨odinger equation, Davey–Stewartson system and their semi-classical limit as we find themuseful for an understanding of the results available in the current manuscript. Ref. [7] notesthe nonlinear Schr¨odinger equation iǫ Ψ t + ǫ − ρν | Ψ | ν Ψ = 0 , x ∈ R d , t ∈ R (1.7)in which Ψ is a complex-valued function, ∆ is the Laplace operator in d dimensions, and ǫ << < ν < ∞ . The case ρ = 1 is called the defocusing case and ρ = − ”It is a fundamental principle in quantum mechanics that,when the time and distance scales are large enough relative to the Planck constant ~ , the systemwill approximately obey the laws of classical, Newtonian mechanics. This is usually rephrased ina colloquial form as: in the limit as ~ → quantum mechanics becomes Newtonian mechanics.The asymptotics of quantum variables as ~ → are known as ’semiclassical’, which expressesthis limiting behaviour.” Since the constant ǫ in (1.7) is located at the same places with the Planck constant ~ in linearSchr¨odinger equation of quantum mechanics, the limit ǫ → √ ue iS/ǫ in (1.7), thesystem u t + ∇ · ( u ∇ S ) = 0 , (1.8a) S t + 12 |∇ S | + ρν u ν = ǫ √ u ) √ u . (1.8b)is obtained. Letting ǫ →
0, the semi-classical limit of the NLS equation (1.7) is obtained [7].Ref. [7] considers the Davey–Stewartson system iǫ Ψ t + ǫ Ψ xx − αǫ Ψ yy + 2 ρ (Φ + | Ψ | )Ψ = 0 , (1.9a)Φ xx + β Φ yy + 2( | Ψ | ) xx = 0 (1.9b)in which α , β and ρ assume the values ∓
1, and ǫ is the dispersion parameter which is small. For α = β = − α = β = 1 it is called DS-II equation. DS-Iequation is integrable. The main focus of Ref. [7] is to study numerically the dispersionlesslimit of the DS II system which appears nonlocally due to the operator D − u t + 2( uS x ) x − uS y ) y = 0 , (1.10a) S t + S x − S y + 2 ρ D − D − ( u ) = 0 (1.10b)with D ± = ∂ x ± ∂ y .Ref. [9] performs numerical experiments to study the semi-classical limit for the focusingcubic nonlinear Schr¨odinger equation with analytic initial data. Ref. [10] also carried out nu-merical analysis which were in agreement with the results of [9] and obtained a train of solitons.[11] considers the NLS equation in two-dimensions with a general nonlinearity including a radi-ally symmetric potential and studies the semi-classical limit. We see further treatments in [12],[13]; and we also would like to mention the book [14] by the same author on NLS equations.The Benney system of equations, originally introduced in [15], has been obtained in [16, 17]as the quasiclassical limit of NLS equations, which are integrable by the inverse scatteringmethod. Dispersionless limits of several equations in (1+1), (2+1) and (3+1) dimensions in-cluding Heisenberg ferromagnet equation, Landau–Lifshitz equation, KdV, NLS and DS equa-tions are listed in [18]. A discussion of dispersionless limits of integrable systems can be found in[19], which involves Kadomtsev–Petviashvili (KP), generalized Benney and Davey–Stewartson(DS) equations. As we mentioned in Introduction, the dDS system under investigation in thismanuscript is introduced in [1]. As a continuation of this work, a new hierarchy of compatiblePDEs defining infinitely many symmetries, which is associated with the dDS system is presentedin [2]. Ref. [20] presents a generalized Weierstrass-type representation for highly corrugated3urfaces with a slow modulation in the four- and three-dimensional Euclidean spaces, whichfind applications in applied mathematics, string theory, membrane theory and various otherfields [20]. It is stated in [20] that integrable deformations of such corrugated surfaces in R and R are induced by the hierarchy of dispersionless DS (dDS) equations and dispersionlessmodified Veselov–Novikov equations. In this context, the authors obtain the quasi-classical(i.e., dispersionless) limit of the Davey–Stewartson systems.Davey–Stewartson system (1.3) was obtained in [21] in two space dimensions by a perturba-tion expansion for the evolution of a wave packet a water of finite depth, ψ and w standing forshort wave and long wave modulation amplitude functions. It is a special case of the Benney–Roskes system obtained in [22]. The generalized Davey–Stewartson (GDS) system in (2+1)dimensions which governs the dynamics of ”a short transverse wave, a long transverse waveand a long longitudinal wave” was derived in [23]. In (3+1) dimensions, we can cite [24] as thederivation of a DS system in a collisionless unmagnetized plasma via multiple-scale asymptoticexpansion method for three-dimensional modulation of an electron-acoustic wave.Lie symmetry algebra of the DS equations was shown in [3] to be an infinite-dimensional Liealgebra isomorphic to the Kac–Moody–Virasoro (KMV) algebra exactly in the integrable case.The variable-coefficient DS (VCDS) system has been considered in [25] for a Lie symmetryanalysis and in particular conditions for the VCDS system to be transformable to the standardintegrable DS system. Lie algebra of the GDS system is also infinite-dimensional, as studiedin [26, 27]. In the (3+1)-dimensional case, Virasoro part of the Lie algebra degenerates to afinite-dimensional subalgebra and hence the full invariance algebra is an indecomposable oneas semi-direct sum of this subalgebra with an infinite-dimensional ideal [28]. It is also ourintention here to analyse system (1.5) from a group-theoretical perspective by exploring its Liesymmetry algebra and group.The paper is organized as follows. In Section 2 we find the Lie symmetry algebra of thesystem using the infinitesimal approach. Section 3 is devoted to the derivation of symmetrygroup transformations of the dDS system (1.5) by the global (not infinitesimal) method. Section4 provides reductions via a one-dimensional and two-dimensional subalgebras and their analysisto obtain exact solutions in some cases. The Lie symmetry algebra L σ of the dDS system (1.5) will be realized in terms of vector fieldsof the form V = τ ∂ t + ξ∂ x + η∂ y + ζ ∂ u + ζ ∂ s + ζ ∂ φ , (2.1)where τ , ξ , η , ζ , ζ and ζ are functions of t, x, y, u, s, φ to be determined from the infinitesimalinvariance condition. Applying this condition and solving an overdetermined system of linearPDEs we find that a general element of L σ can be written in the form V = X ( τ ) + Y ( g ) + Z ( h ) + W ( m ) + µ D , (2.2)4here µ is an arbitrary constant and D = x∂ x + y∂ y + 2 u∂ u + 2 s∂ s + 2 φ∂ φ , (2.3a) X ( τ ) = τ ( t ) ∂ t + ˙ τ ( t )2 (cid:16) x∂ x + y∂ y (cid:17) − ˙ τ ( t ) u∂ u + ¨ τ ( t )4 (cid:16) x + σy (cid:17) ∂ s + h ... τ ( t )4 δ (cid:16) x + σy (cid:17) − ˙ τ ( t ) φ i ∂ φ , (2.3b) Y ( g ) = g ( t ) ∂ x + x ˙ g ( t ) ∂ s + xδ ¨ g ( t ) ∂ φ , (2.3c) Z ( h ) = h ( t ) ∂ y + σy ˙ h ( t ) ∂ s + yσδ ¨ h ( t ) ∂ φ , (2.3d) W ( m ) = m ( t ) ∂ s + 1 δ ˙ m ( t ) ∂ φ . (2.3e)The functions g ( t ) and m ( t ) are arbitrary, but τ ( t ) and h ( t ) are split into two classes dependingon whether σ = 1 or σ = − τ ( t ) = arbitrary, σ = 1 ,τ t + τ , σ = − . (2.4)and h ( t ) = arbitrary, σ = 1 ,h = const. , σ = − . (2.5)The commutation relations are given as follows[ Y ( g ) , D ] = Y ( g ) , [ Z ( h ) , D ] = Z ( h ) , [ W ( m ) , D ] = 2 W ( m ) (2.6)and [ X ( τ ) , X ( τ )] = X ( τ ˙ τ − τ ˙ τ ) , (2.7a)[ Y ( g ) , Y ( g )] = W ( g ˙ g − g ˙ g ) , (2.7b)[ Z ( h ) , Z ( h )] = W (cid:16) σ ( h ˙ h − h ˙ h ) (cid:17) , (2.7c)[ X ( τ ) , Y ( g )] = Y ( τ ˙ g − g ˙ τ ) , (2.7d)[ X ( τ ) , Z ( h )] = Z ( τ ˙ h − h ˙ τ ) , (2.7e)[ X ( τ ) , W ( m )] = W ( τ ˙ m ) (2.7f)and the vanishing ones are, just to note,[ D , X ( τ )] = [ W ( m ) , W ( m )] = [ Y ( g ) , Z ( h )] = [ Y ( g ) , W ( m )] = [ Z ( h ) , W ( m )] = 0 . (2.8)(a) σ =
1, dDS-I:
The symmetry algebra represented by (2.2) depends on four arbitraryfunctions τ ( t ), g ( t ), h ( t ), m ( t ).(b) σ = −
1, dDS-II:
The general element of the symmetry algebra is given by V = X ( τ t + τ ) + Y ( g ) + Z ( h ) + W ( m ) + µ D (2.9)5ith g ( t ) and m ( t ) being arbitrary. In this case, though the symmetry algebra is still infinite-dimensional, it gets reduced to a large degree. Indeed, the number of arbitrary functions getnarrowed down from four to two. In terms of vector fields T = X (1) = ∂ t , (2.10a) Q = ∂ y , (2.10b) D = x∂ x + y∂ y + 2 u∂ u + 2 s∂ s + 2 φ∂ φ , (2.10c) D = X ( t ) = t∂ t + x ∂ x + y ∂ y − u∂ u − φ∂ φ , (2.10d)(2.9) can be written in the form V = τ T + τ D + h Q + µ D + Y ( g ) + W ( m ) . (2.11)By taking a linear combination of D and D D ′ = D − D = t∂ t − u∂ u − s∂ s − φ∂ φ (2.12)we see that the solvable finite-dimensional subalgebra L = {T , D ′ , Q , D } is a direct sum oftwo nonabelian algebras L = {T , D ′ } ⊕ {Q , D } (2.13)with nonzero commutation relations [ T , D ′ ] = T and [ Q , D ] = Q . Theorem 2.1 (i) System (1.5) with σ = 1 (dispersionless DS-I equation) admits an infinitedimensional Lie symmetry algebra L with the infinitesimal generator V = X ( τ ) + Y ( g ) + Z ( h ) + W ( m ) + µ D including four arbitrary smooth functions τ ( t ) , g ( t ) , h ( t ) and m ( t ) .The algebra has the Levi decomposition of a simple algebra and its nilradical L = {X ( τ ) } ⊃ + {Y ( g ) , Z ( h ) , W ( m ) , D } . (2.14) The algebra {X ( τ ) } ⊃ + {Y ( g ) , Z ( h ) , W ( m ) } is isomorphic to the DS symmetry algebra of (1.3) , which has a Kac–Moody–Virasoro (KMV) structure [3].(ii) System (1.5) with σ = − (dispersionless DS-II equation) admits an infinite dimensionalLie symmetry algebra L − with the infinitesimal generator (2.11) involving two arbitrarysmooth functions g ( t ) and m ( t ) . The Lie algebra has the semi-direct sum structure of twoLie algebras L − = {T , Q , D , D} ⊃ + {Y ( g ) , W ( m ) } . (2.15) Remark 2.1
The absence of KMV loop structure for σ = − is atypical of many integrablePDEs in 2+1-dimensions like KP equation and its dispersionless variant and integrable DSsystem. Symmetry group transformations
In this Section we will find the symmetry transformation that maps (1.5) to itself. One param-eter subgroups of the full symmetry group can be obtained by integrating the vector fields inthe symmetry algebra. When the symmetry algebra is infinite-dimensional then the integrationprocedure can be quite tricky. Here we shall pursue an alternative global approach to find thefull symmetry group. An added bonus is that all possible discrete symmetry transformations,not possible in infinitesimal approach, can be caught.Inspection of the symmetry algebra (2.1)-(2.3) suggests that we can restrict the symmetrytransformations to the fiber-preserving ones of the form u ( x, y, t ) = A ( x, y, t ) ˜ u (˜ x, ˜ y, ˜ t ) + B ( x, y, t ) , (3.1a) s ( x, y, t ) = A ( x, y, t ) ˜ s (˜ x, ˜ y, ˜ t ) + B ( x, y, t ) , (3.1b) φ ( x, y, t ) = A ( x, y, t ) ˜ φ (˜ x, ˜ y, ˜ t ) + B ( x, y, t ) , (3.1c)where ˜ x = X ( x, y, t ), ˜ y = Y ( x, y, t ), ˜ t = T ( t ), under the invertibility condition A A A = 0 , ∂ (˜ x, ˜ y ) ∂ ( x, y ) ˙ T = 0 . (3.2)We determine the functions X , Y , T , A i , B i , i = 1 , , u ˜ t + (˜ u ˜ s ˜ x ) ˜ x + (˜ u ˜ s ˜ y ) ˜ y = 0 , (3.3a)˜ s ˜ t + 12 (˜ s x + σ ˜ s y ) − δ ˜ φ = 0 , (3.3b) σ ˜ φ ˜ y ˜ y − ˜ φ ˜ x ˜ x + ˜ u ˜ x ˜ x + σ ˜ u ˜ y ˜ y = 0 . (3.3c)When we plug the transformation (3.1) in (1.5), we obtain the transformed system (3.3), whereeach equation includes several extra partial derivatives of the dependent variables. Eq. (3.3a)must not include the term ˜ s ˜ y ˜ y . This requires A B ( Y x + Y y ) = 0, which gives B = 0. Requiringthat no terms ˜ u ˜ s ˜ x ˜ y in (3.3a) and ˜ u ˜ x ˜ y and ˜ φ ˜ x ˜ y in (3.3c) occur, we find the constraints A A ( X x Y x + X y Y y ) = 0 , (3.4a) A ( X x Y x + σX y Y y ) = 0 , (3.4b) A ( X x Y x − σX y Y y ) = 0 . (3.4c)Since A A = 0, we obtain that X x Y x = 0, which gives two possibilities: either we have Y x = X y = 0 or X x = Y y = 0.a) X = X ( x, t ), Y = Y ( y, t ):In this case, in Eq. (3.3b) there is a term that does not include any of the dependentvariables, which must vanish: B ,t + 12 B ,x + σ B ,y − δB = 0 (3.5)so we have B ( x, y, t ) = 1 δ (cid:18) B ,t + 12 B ,x + σ B ,y (cid:19) . (3.6)7oreover, eliminating the terms ˜ s , ˜ s ˜ s ˜ x and ˜ s ˜ s ˜ y that appear in (3.3b) results in A ,t + A ,x B ,x + σA ,y B ,y = 0 , (3.7a) A A ,x X x = 0 , (3.7b) A A ,y Y y = 0 , (3.7c)therefore A is a constant, say A ( x, y, t ) = 1 /µ. After this, (3.3b) takes the form˜ s ˜ t + X x µ ˙ T ˜ s x + σY y µ ˙ T ˜ s y − δµA ˙ T ˜ φ + 1˙ T ( X t + X x B ,x )˜ s ˜ x + 1˙ T ( Y t + σY y B ,y )˜ s ˜ y = 0 . (3.8)If we compare (3.8) with (3.3a), we arrive at X ( x, t ) = ǫ q µ ˙ T x + α ( t ) , Y ( y, t ) = ǫ q µ ˙ T y + β ( t ) , A ( x, y, t ) = 1 µ ˙ T ( t ) (3.9)with ǫ = 1 and ǫ = 1. Furthermore, solving X t + X x B ,x = 0 and Y t + σY y B ,y = 0 we findthat B ( x, y, t ) = −
14 ¨ T ˙ T ( x + σy ) − ǫ ˙ α q µ ˙ T x − ǫ σ ˙ β q µ ˙ T y + γ ( t ) , (3.10)after which transformation of (1.5b) to (3.3b) is achieved successfully. Now the transformedform of (1.5c) becomes σ ˙ TµA ˜ φ ˜ y ˜ y − ˙ TµA ˜ φ ˜ x ˜ x + ˜ u ˜ x ˜ x + σ ˜ u ˜ y ˜ y + 2 ǫ A ,x A q µ ˙ T ˜ u ˜ x + 2 ǫ σA ,y A q µ ˙ T ˜ u ˜ y + A ,xx + σA ,yy µA ˙ T ˜ u = 0 . (3.11)We see that we must have A ( x, y, t ) = ˙ T /µ . Then, (1.5c) remains invariant as (3.3c). Thereremains only to check the transformation of (1.5a). Now let us give the transformation explicitly X = ǫ q µ ˙ T x + α ( t ) , Y = ǫ q µ ˙ T y + β ( t ) , T = T ( t ) , ˙ T > , µ > ,u = 1 µ ˙ T ˜ u ( X, Y, T ) ,s = 1 µ ˜ s ( X, Y, T ) − q µ ˙ T ( ǫ ˙ αx + ǫ σ ˙ βy ) − ¨ T T ( x + σy ) + γ ( t ) ,φ = 1 µ ˙ T ˜ φ ( X, Y, T ) + ǫ σδ q µ ˙ T (cid:18) ˙ α ¨ T ˙ T − ¨ α (cid:19) x + ǫ σδ q µ ˙ T (cid:18) ˙ β ¨ T ˙ T − ¨ β (cid:19) y − δ S { T, t } ( x + σy ) + 12 δµ ˙ T ( ˙ α + σ ˙ β ) + 1 δ ˙ γ, (3.12)where ǫ = 1 and ǫ = 1 and S { T, t } = ... T ˙ T −
32 ¨ T ˙ T (3.13)is the Schwarzian derivative of T ( t ) with respect to t . The transformed system becomes˜ u ˜ t + (˜ u ˜ s ˜ x ) ˜ x + (˜ u ˜ s ˜ y ) ˜ y + (1 − σ ) ¨ T T ˜ u + (1 − σ ) (cid:18) ˙ β ˙ T + ǫ q µ ˙ T ¨ T T y (cid:19) ˜ u ¯ y = 0 , (3.14a)˜ s ˜ t + 12 (˜ s x + σ ˜ s y ) − δ ˜ φ = 0 , (3.14b) σ ˜ φ ˜ y ˜ y − ˜ φ ˜ x ˜ x + ˜ u ˜ x ˜ x + σ ˜ u ˜ y ˜ y = 0 . (3.14c)8t is obvious that when σ = 1, ǫ = ǫ = 1, the system remains invariant under the Lie grouptransformation (3.12) involving four arbitrary functions T ( t ), α ( t ), β ( t ) and γ ( t ).When σ = −
1, we must have ¨ T = 0 and ˙ β = 0 for invariance of the system, hencethe transformation includes two arbitrary functions α ( t ) and γ ( t ), ǫ , ǫ and four arbitraryparameters a, b, µ, β ( µa > T ( t ) = at + b, X ( x, t ) = ǫ √ µa x + α ( t ) , Y ( y, t ) = ǫ √ µa y + β ,u = aµ ˜ u ( X, Y, T ) ,s = 1 µ ˜ s ( X, Y, T ) − ǫ ˙ α √ µa x + γ ( t ) ,φ = aµ ˜ φ ( X, Y, T ) − ǫ ¨ αδ √ µa x + ˙ α δµa + ˙ γδ , (3.15)where ǫ = 1, ǫ = 1 as above.Transformations (3.12) and (3.15) include both continuous and discrete ones. In the firstcase, the continuous one corresponds to ( ǫ , ǫ ) = (1 , ǫ , ǫ ) = ( − , ǫ , ǫ ) = (1 , − ǫ , ǫ ) = ( − , −
1) with particular values assigned to all remaining parametersand functions.In the continuous case, they can be written in the infinitesimal form( u, s, φ ) = (˜ u, ˜ s, ˜ φ ) + ε ( Q [˜ u ] , Q [˜ s ] , Q [ ˜ φ ]) + O ( ε ) (3.16)by introducing a group parameter ε into µ and arbitrary functions T, α, β, γ through T ( t ) = t + ετ ( t ) + O ( ε ) , µ = 1 + 2 µ ε + O ( ε ) ,α ( t ) = εg ( t ) + O ( ε ) , β ( t ) = εh ( t ) + O ( ε ) , γ ( t ) = εm ( t ) + O ( ε ) (3.17)and expanding (3.12) into Taylor series near ε = 0, keeping only terms linear in ε . The functions Q , Q , Q will appear precisely as characteristic functions defined by Q α [ u ] = Q ( x, y, t, u, u x , u y , u t ) = τ u t + ξu x + ηu y − ζ α , α = 1 , , ǫ , ǫ ) with other parameters set equal to zeroinduce reflection symmetries. For instance, ( ǫ , ǫ ) = ( − , −
1) means invariance under thediscrete group ( t, x, y, u, s, φ ) → ( t, − x, − y, u, s, φ ) . The same reasoning holds for (3.15). In the continuous (connected) case we put ( ǫ , ǫ ) =(1 ,
1) and if we write a = 1 + ετ , b = ετ , β = εh , µ = 1 + 2 µ ε, (3.18)then we recover the infinitesimal generator (2.9). For other choices of ( ǫ , ǫ ) discrete transfor-mations are extracted.When σ = 1 we can readily obtain the SL(2 , R ) symmetry group of the subalgebrasl(2 , R ) ≃ n X (1) , X ( t ) , X ( t ) o
9y restricting the parameters and arbitrary functions in (3.12) with ( ǫ , ǫ ) = (1 ,
1) as µ = 1 , α ( t ) = β ( t ) = γ ( t ) = 0 , T ( t ) = at + bct + d , ad − bc = 1 . (3.19)The condition ¨ T = 0 ( c = 0) is true when σ = 1 so that t transforms via M¨obius transfor-mations. The associated symmetry group transformations in this case depend on three groupparameters a, b, c X = xct + d , Y = yct + d , T = at + bct + d ,u = 1( ct + d ) ˜ u ( X, Y, T ) ,s = ˜ s ( X, Y, T ) + c ct + d ) ( x + y ) ,φ = 1( ct + d ) ˜ φ ( X, Y, T ) . (3.20)On the other hand, if we restrict T ( t ) to the solution of the Schwarzian equation S { T, t } = 2 ρ ( t ) (3.21)for some ρ , then T can be expressed as the three-parameter ratio of the linear combinationof two linearly independent solutions λ ( t ) , ν ( t ) of the second order linear ordinary differentialequation ¨Ω + ρ ( t )Ω = 0. In this case the symmetry group is somewhat more general (letting α, β, γ remain free) X = λ − x, Y = λ − y, T = λ λ = aλ + bνcλ + dν , ( ad − bc ) W ( λ, ν ) = 1 ,u = λ − ˜ u ( X, Y, T ) ,s = ˜ s ( X, Y, T ) + ˙ λ λ ( x + y ) − λ ( ˙ αx + ˙ βy ) + γ,φ = λ − ˜ φ ( X, Y, T ) − ρ δ ( x + y ) − δλ [(2 ˙ α ˙ λ + λ ¨ α ) x + (2 ˙ β ˙ λ + λ ¨ β ) y ]++ λ δ ( ˙ α + ˙ β ) + 1 δ ˙ γ. (3.22)We note that the Wronskian W ( λ , λ ) of the set { λ , λ } is unity and ˙ T = λ − .Transformation formula (3.20) or (3.22) states that if the functions (˜ u, ˜ s, ˜ φ ) solve the originalsystem with σ = 1 then do also ( u, s, φ ). A class of seed solutions can be chosen in the form˜ u = U ( x, y ) , ˜ s = F ( t ) , ˜ φ = δ − ˙ F ( t ) , (3.23)where U is a solution to the Laplace equation ∆ U = 0 in the plane and F is an arbitraryfunction. Applying this solution to transformation (3.22) generates solutions in which all inde-pendent variables are involved. 10) X = X ( y, t ), Y = Y ( x, t ):Similar to the previous case, setting equal to zero a term added in (3.3b) we have B ( x, y, t ) = 1 δ (cid:18) B ,t + 12 B ,x + σ B ,y (cid:19) . (3.24)The terms ˜ s , ˜ s ˜ s ˜ x and ˜ s ˜ s ˜ y that appear in (3.3b) are eliminated if we put the constraints A ,t + A ,x B ,x + σA ,y B ,y = 0 , (3.25a) A A ,y X y = 0 , (3.25b) A A ,x Y x = 0 , (3.25c)from which it follows that A should be a constant, say A ( x, y, t ) = 1 /µ. With this information,(3.3b) takes the form˜ s ˜ t + σX y µ ˙ T ˜ s x + Y x µ ˙ T ˜ s y − δµA ˙ T ˜ φ + 1˙ T ( X t + σX y B ,y )˜ s ˜ x + 1˙ T ( Y t + Y x B ,x )˜ s ˜ y = 0 . (3.26)We find X ( y, t ) = ǫ q σµ ˙ T y + α ( t ) , Y ( x, t ) = ǫ q σµ ˙ T x + β ( t ) , A ( x, y, t ) = 1 µ ˙ T ( t ) (3.27)where ǫ = 1 and ǫ = 1. Furthermore, solving X t + σX y B ,y = 0 and Y t + Y x B ,x = 0 we find B ( x, y, t ) = −
14 ¨ T ˙ T ( x + σy ) − ǫ σ ˙ α q σµ ˙ T y − ǫ ˙ β q σµ ˙ T x + γ ( t ) , (3.28)after which transformation of (1.5b) to (3.3b) is achieved successfully. Now let us have a lookat how (1.5c) has transformed: − σ ˙ TµA ˜ φ ˜ y ˜ y + ˙ TµA ˜ φ ˜ x ˜ x + ˜ u ˜ x ˜ x + σ ˜ u ˜ y ˜ y + 2 ǫ σ q σµ ˙ T A ,y µA ˙ T ˜ u ˜ x + 2 ǫ q σµ ˙ T A ,x µA ˙ T ˜ u ˜ y + A ,xx + σA ,yy µA ˙ T ˜ u = 0 . (3.29)This gives us A ( x, y, t ) = − ˙ T ( t ) /µ , and the invariance of (1.5c) in the form (3.3c) is achieved.In summary, we have obtained˜ u ˜ t + σ (cid:20) (˜ u ˜ s ˜ x ) ˜ x + (˜ u ˜ s ˜ y ) ˜ y (cid:21) + (1 − σ ) ¨ T T ˜ u + (1 − σ ) (cid:18) ˙ α ˙ T + ǫ q µσ ˙ T ¨ T T y (cid:19) ˜ u ¯ x = 0 , (3.30a)˜ s ˜ t + 12 (˜ s x + σ ˜ s y ) − δ ˜ φ = 0 , (3.30b) σ ˜ φ ˜ y ˜ y − φ ˜ x ˜ x + ˜ u ˜ x ˜ x + σ ˜ u ˜ y ˜ y = 0 . (3.30c)Demanding invariance of the original system forces to put σ = 1 in the transformed one (3.30).Then, the associated symmetry transformation involves four arbitrary functions T ( t ), α ( t ), β ( t )11nd γ ( t ) X = ǫ q µ ˙ T y + α ( t ) , Y = ǫ q µ ˙ T x + β ( t ) , T = T ( t ) ,u = − µ ˙ T ˜ u ( X, Y, T ) ,s = 1 µ ˜ s ( X, Y, T ) − q µ ˙ T ( ǫ ˙ αy + ǫ ˙ βx ) − ¨ T T ( x + y ) + γ ( t ) ,φ = 1 µ ˙ T ˜ φ ( X, Y, T ) + ǫ δµ q µ ˙ T (cid:18) ˙ α ¨ T ˙ T − ¨ α ˙ T (cid:19) y + ǫ δµ q µ ˙ T (cid:18) ˙ β ¨ T ˙ T − ¨ β ˙ T (cid:19) x − δ S { T, t } ( x + y ) + 12 δµ ˙ T ( ˙ α + ˙ β ) + 1 δ ˙ γ. (3.31)We note that transformations (3.31) will only induce discrete ones as they cannot be contin-uously connected to the identity transformation. For instance, system (1.5) is invariant underthe group of discrete transformations( x, y, t, u, s, φ ) → ( y, x, t, − u, s, φ ) . In this Section we construct some exact solutions invariant under the one-dimensional subal-gebra Y ( g ) + W ( m ) and also two-dimensional subalgebras obtained by embedding X (1) and X ( t ) into Y ( g ) + W ( m ). Y ( g ) + W ( m ) First we perform reduction by the subalgebra b V = Y ( g ) + W ( m ), g = 0. We observe that g ( t )can be removed from b V by an application of the symmetry group generated by exp { ε Y ( G ) } ˜ x = x + εG ( t ) , ˜ y = y, ˜ t = t, ˜ u = u, ˜ s = s + ε (cid:18) x + ε (cid:19) ˙ G ( t ) , ˜ φ = φ + εδ (cid:18) x + ε (cid:19) ¨ G ( t ) (4.1)with G defined by G ( t ) = gε Z tt ( g − m )( z ) dz + cg ( t ) , where c is a constant. Thus b V is conjugate to the gauge subgroup W ( m ). But this does notlead to any reduction so we prefer to keep g ( t ) as it is.Solutions of (1.5) invariant under the group of transformations generated by Y ( g ) + W ( m )will have the form u = U ( y, t ) , (4.2a) s = S ( y, t ) + ˙ g ( t )2 g ( t ) x + m ( t ) g ( t ) x, (4.2b) φ = Φ( y, t ) + ¨ g ( t )2 δg ( t ) x + ˙ m ( t ) δg ( t ) x. (4.2c)12ubstituting this in (1.5) results in U t + ( U S y ) y + ˙ gg U = 0 , (4.3a) S t + σ S y − δ Φ = − m g , (4.3b) U yy + Φ yy = g ′′ σδg . (4.3c)From (4.3c) and (4.3b) we findΦ( y, t ) = − U ( y, t ) + ¨ g ( t )2 σδg ( t ) y + f ( t ) y + f ( t ) , (4.4a) U ( y, t ) = − δ S t − σ δ S y + ¨ g ( t )2 σδg ( t ) y + f ( t ) y + f ( t ) − m ( t ) δg ( t ) . (4.4b)Substituting (4.4b) into (4.3a), it follows that S ( y, t ) should satisfy the PDE (cid:18) σ ¨ g g y + δf y + δf − m g (cid:19) S yy − (cid:18) S t + 32 σS y (cid:19) S yy − S tt − (1 + σ ) S y S yt − σ ˙ g g S y + (cid:18) σ ¨ gg y + δf (cid:19) S y − ˙ gg S t + σ ... g g y + δ ( ˙ f + f ˙ gg ) y + δ ( ˙ f + f ˙ gg ) + m ˙ g g − m ˙ mg = 0 . (4.5)In order to find an exact solution of the highly nonlinear equation (4.5) we introduce anotherequation S t + 32 σS y = 0 , (4.6)which is compatible with the given one. In other words, we look at the possibility of a nontrivialcommon solution of (4.5) and (4.6). Eq. (4.6) admits the separable solution S ( y, t ) = ( y + ℓ ) σt + ℓ (4.7)where ℓ , ℓ are integration constants. When we substitute (4.7) in (4.5) we obtain a seconddegree polynomial in y with coefficients in t as a compatibility condition, which has to vanishidentically so that (4.5) holds for (4.7). The coefficient of y is zero if g ( t ) satisfies the equation(6 σt + ℓ ) ... g + 6(6 σt + ℓ ) ¨ g + 8(6 σt + ℓ ) ˙ g + 48(1 − σ ) g = 0 . (4.8)The general solution of this Euler type equation is g ( t ) = K Q / + K Q / + K Q − σ , Q ( t ) = 6 σt + ℓ , (4.9)where K , K , K are arbitrary constants. The coefficients of y and y vanish when f ( t ) = − ℓ σδQ + 1 g (cid:20) C Q − σ + C ℓ Q − σ + m δg − δ K ℓ σ (27 σ − Q − σ (cid:21) , (4.10) f ( t ) = − ℓ σδQ (cid:20) σ − g K Q − σ (cid:21) + C g Q − σ , (4.11)13ith the arbitrary constants C and C . Therefore we are able to find an exact solution of (1.5)depending on one arbitrary function m ( t ). Explicitly, solution to (1.5) is u = U ( y, t ) = 4 σ ( y + ℓ ) δ (6 σt + ℓ ) + ¨ g ( t )2 σδg ( t ) y + f ( t ) y + f ( t ) − m ( t ) δg ( t ) , (4.12a) s = ( y + ℓ ) σt + ℓ + ˙ g ( t )2 g ( t ) x + m ( t ) g ( t ) x, (4.12b) φ = − U ( y, t ) + ¨ g ( t )2 δg ( t ) x + ˙ m ( t ) δg ( t ) x + ¨ g ( t )2 σδg ( t ) y + f ( t ) y + f ( t ) . (4.12c)The symmetry algebra of u t = u y is ten-dimensional. One of its elements is X = 2 t∂ y − y∂ u and generates a Galilei transformation. The action of X on U ( y, t ) gives the solution u ( y, t ) = U (˜ y, ˜ t ) − ǫy ǫt ) , ˜ y = y − ǫt , ˜ t = t − ǫt , (4.13)and in particular on the separable solution u = ( y + c ) c − t . (4.14)gives the less trivial solution u = (4 − ǫc ) y + 8 c y + 4 c (1 + ǫt )4[ c + ( ǫc − t . (4.15)By scaling with respect to t or y , we can write S ( y, t ) = (4 − ǫc ) y + 8 c y + 2 c (2 − ǫσt )2[2 c + 3 σ (4 − ǫc ) t ] . (4.16)as solution to (4.6). As with (4.7), this solution can be used to find other compatible solutionsof (4.5). Now we will perform symmetry reductions to ordinary differential equations (ODEs) usingthe two-dimensional subalgebras {X ( t ) , Y ( g ) + W ( m ) } and {X (1) , Y ( g ) + W ( m ) } . A two-dimensional subalgebra can be either abelian or non-abelian. We determine the functions g ( t )and m ( t ) so that these two canonical forms are realized. Using the commutation relationsfor the generators given above in (2.7d) and (2.7f) we find the following four two-dimensionalsubalgebras L . = {X ( t ) , a Y ( t / ) + b W ( t ) } , L . = {X (1) , a Y ( e t ) + b W ( e t ) } ,L . = {X ( t ) , a Y ( t / ) + b W (1) } , L . = {X (1) , a Y (1) + b W (1) } , (4.17)with L . , L . being non-abelian and L . , L . abelian. These subalgebras are common to L and L − , that is, they are admitted by (1.5) both when σ = 1 and σ = − eduction with L . :By finding invariants, we see that the ansatz u = 1 t U ( r ) ,s = 3 x t + b xa √ t + S ( r ) ,φ = 3 x δt + b xa δt / + 1 t Φ( r ) (4.18)with r = y √ t provides reduction to a system of ODEs2( U S ′ ) ′ − rU ′ + U = 0 ,σ S ′ ) − r S ′ − δ Φ + b a = 0 ,U ′′ + Φ ′′ = 3 σ δ . (4.19)Solving this system, we get U ( r ) = − σ δ ( S ′ ) + r δ S ′ + 3 σ δ r + c r + c − b δa , Φ( r ) = σ δ ( S ′ ) − r δ S ′ + b δa , (4.20)where S ( r ) satisfies (cid:18) σ − r + 2 δc r + 2 c δ − b a (cid:19) S ′′ + ( σ + 2) rS ′ S ′′ − σ ( S ′ ) S ′′ + (1 − σ S ′ ) + ( 3 σ r + 2 c δ ) S ′ − σ r + c δ − b a = 0 . (4.21)Eq. (4.21) equation admits a quadratic solution S ( r ) = ar + br + c . This occurs under thefollowing conditions between the coefficients: k = (6 σa + σ − b − c δ (4 a + 1) − bc δ + (4 a + 1) b a = 0 , (4.22a) k = 24 σba − b + c δ ) a − σb , (4.22b) k = 24 σa − σ + 6) a + (1 − σ a + 3 σ . (4.22c)Solving this system we find a = {− σ , , σ } , b = 16 σac δ a − σa − c = b δa + 96 ac δ (48 a − a − , σ = 1 ,b δa + 32 a (4 a − c δ (1 + 4 a )(48 a + 16 a − , σ = − . (4.24)15he quadratic 48 a − σa − { , − (4 / σ ) , } . Finally we find thesolution of (1.5) as u = 1 δ ( a − σa + 3 σ y t + ( c + b δ − abσδ ) yt / + ( c − b δa − σb δ ) 1 t ,s = 3 x t + b xa √ t + a y t + b y √ t + c,φ = 3 x δt + b xa δt / + aδ (2 aσ − y t + b δ (4 aσ − yt / + ( b a + b σ δt . (4.25) Reduction with L . :Invariance under L . leads to u = U ( y ) ,s = b a x + x S ( y ) ,φ = b δa x + 12 δ x + Φ( y ) (4.26)and to the reduced system ( U S ′ ) ′ + U = 0 ,σ S ′ ) − δ Φ + b a = 0 ,U ′′ + Φ ′′ = σδ . (4.27)Some work on these equations results in U ( y ) = − σ δ ( S ′ ) + σ δ y + c y + c , Φ( y ) = σ δ ( S ′ ) + b δa , (4.28)where S ( y ) satisfies the ODE( σy + 2 c δy + 2 c δ ) S ′′ − σ ( S ′ ) S ′′ + 2( σy + c δ ) S ′ − σ ( S ′ ) + σy + 2 c δy + 2 c δ = 0 . (4.29)Again, a polynomial solution to this equation in the form S ( y ) = ay + by + c is possible. Thisimplies an expression of the form k y + k y + k ≡
0, where k = σb (1 + 6 a ) − c δ (1 + 2 a ) − bc δ, (4.30a) k = 2 σb (12 a + 2 a − − c δ (1 + 4 a ) , (4.30b) k = σ (24 a + 4 a − a − . (4.30c)Solving k = 0 we find three possible values a = {− , − , } . k = 0 and k = 0 are solved togive b = c σδ (1 + 4 a )12 a + 2 a − , c = 3 σδc (1 + 4 a )2(12 a + 2 a − (4.31)16nd c remains arbitrary. Therefore we arrive at the solution u = − σ δ (2 ay + b ) + σ δ y + c y + c ,s = b a x + x ay + by + c,φ = b δa x + 12 δ x + σ δ (2 ay + b ) + b δa . (4.32) Reduction with L . :Reduction formula u = 1 t U ( r ) ,s = x t + b xa √ t + S ( r ) ,φ = − x δt + 1 t Φ( r ) (4.33)with r = y √ t leads to the reduced system of ODEs2( U S ′ ) ′ − ( rU ) ′ = 0 ,σ S ′ ) − r S ′ − δ Φ + b a = 0 ,U ′′ + Φ ′′ = − σ δ . (4.34)After some calculations, we see that U ( r ) is a solution of the cubic equation4 δU + [( σ − r − δ ( c r + c ) + 2 b a ] U + c ( σ − rU + 12 σc = 0 (4.35)and Φ( r ) = − U ( r ) − σ δ r + c r + c ,S ( r ) = c Z drU ( r ) + r c . (4.36)When σ = 1, c = 0, we find U ( r ) = U , a constant, which is a root of the cubic equation4 δU + (cid:16) b a − c δ (cid:17) U + 12 c = 0 . (4.37)Therefore, a solution to (1.5) when σ = 1 is obtained as u = U t ,s = x t + b xa √ t + y t + c y U √ t + c ,φ = − δt ( x + y ) + c − U t . (4.38)17 eduction with L . :For this subalgebra we have u = U ( y ) , s = b a x + S ( y ) , φ = Φ( y ) (4.39)and the reduced system is ( U S ′ ) ′ = 0 ,σ S ′ ) − δ Φ + b a = 0 ,U ′′ + Φ ′′ = 0 . (4.40)We see that U is a solution to the algebraic equation2 δU + [ b a − δ ( c y + c )] U + σc = 0 (4.41)and Φ( y ) = − U ( y ) + c y + c ,S ( y ) = c Z dyU ( y ) + c . (4.42)Finally, we comment that most of the solutions obtained in this Section are polynomials in x , and y , but with algebraic coefficients in t . Symmetry group transformations (3.12), (3.15)or (3.22) can be applied to all of these solutions to construct additional solutions. References [1] G. Yi. On the dispersionless Davey–Stewartson system: Hamiltonian vector field Lax pairand relevant nonlinear Riemann–Hilbert problem for dDS-II system.
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