Symmetry computation and reduction of a wave model in dimensions

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1 Nonlinear Dyn 017) 87:13 3 DOI /s ORIGINAL PAPER Symmetry computation and reduction of a wave model in 1 dimensions P. G. Estévez J. D. Lejarreta C. Sardón Received: 6 January 016 / Accepted: 30 July 016 / Published online: 1 August 016 Springer ScienceBusiness Media Dordrecht 016 Abstract We present the iterative classical point symmetry analysis of a shallow water wave equation in 1 dimensions and that of its corresponding nonisospectral, two-component Lax pair. A few reductions arise and are identified with celebrate equations in the Physics and Mathematics literature of nonlinear waves. We pay particular attention to the isospectral or nonisospectral nature of the reduced spectral problems. Keywords Lie symmetries Reduction Nonlinear Soliton KdV equation Painlevé test 1 Introduction Invariance of a differential equation under a group of transformations is synonymous of existence of symmetry and, consequently, of conserved quantities 5]. Such invariance helps us to achieve partial or complete integration of the equation. P. G. Estévez C. Sardón B) Department of Fundamental Physics, Universidad de Salamanca, Plz de la Merced s/n, Salamanca, Spain cristinasardon@usal.es P. G. Estévez pilar@usal.es J. D. Lejarreta Department of Applied Physics, Universidad de Salamanca, Plz de la Merced s/n, Salamanca, Spain leja@usal.es A conserved quantity for a first-order differential equation can lead to its integration by quadrature, whilst for higher-order ones, it leads to a reduction of their order 8,31]. Many of the existing solutions to physical phenomena described by differential equations have been obtained through symmetry arguments 8,9,31]. Nevertheless, finding conserved quantities is a nontrivial task. The most famous and established method for finding point symmetries is the classical Lie symmetry method CLS) developed by Lie in ,8,31]. Although the CLS represents a very powerful tool, it yields cumbersome calculations to be solved by hand. Notwithstanding, the increased number of available software packages for symbolic calculus has made of generalizations of the CLS analysis very tractable approaches to find conservation laws, explicit solutions, etc. If we impose that symmetries leave certain submanifold invariant, we find the class of conditional symmetries or the so-called nonclassical symmetry method NSM) introduced by Bluman and Cole 4] and later applied by many authors 3,9,15,6,7]. A remarkable difference between the CLS and NSM is that the latter provides us with no longer linear systems of differential equations from which to obtain the symmetries 5,36,37]. In this paper, we aim to perform the CLS on the socalled Bogoyavlenskii Kadomtsev Petviashvili equation denoted as 1)-BKP equation, henceforth] which takes the form 13,14]

2 14 P. G. Estévez et al. ) uxt u xxxy 8u x u xy 4u xx u y x σ u yyy = 0, σ =±1. 1) This equation is a model for evolutionary shallow water waves represented by the scalar field ux, y, t) as the height of the wave. In 38], it was proved that this equation is a reduction of the well-known 3 1)- Kadomtsev Petviashvili equation for the description of wave motion 1,18]. Notice that the 1)-BKP equation is also a modified version of the Calogero Bogoyanlevski Schiff equation 6,7,30]. Equation 1) is integrable according to the Painleve property PP) and admits a Lax pair 14]. Results on solutions of solitonic nature for 1) were pursued through the singular manifold method 14,16,34,35]. In particular, lump solitons, or solutions which decay polynomially in all directions, were found. The interest of lump solutions roots in their nontrivial dynamics and interactions 13,17,4]. In particular, we focus on the study of its corresponding Lax pair in 1 dimensions 13]. It is a complex, two-component linear problem 14] ψ xx = iψ y u x ψ, ψ t = iψ yy 4u y ψ x u xy iω y )ψ ) and its complex conjugate χ xx = iχ y u x χ, χ t = iχ yy 4u y χ x u xy iω y )χ 3) with ψx, y, t) and χx, y, t) being the eigenfunctions. The compatibility condition of the cross-derivatives ψ xxt = ψ txx )in) retrieves 1) in the form. u xt u xxxy 8u x u xy 4u xx u y = ω yy, u yy = ω xy 4) where we have introduced the auxiliary scalar field ωx, y, t). We restrict ourselves to the case σ = 1. Results for σ = 1 can be obtained by considering that if ux, y, t) is a solution for σ = 1, then ux, iy, it) is a solution for σ = 1. The compatibility condition for χ xxt = χ txx )in3) yields 4) too. Notice that the spectral parameter is not present in the Lax pair. This does not necessarily imply that it is an isospectral Lax pair. Indeed, there exists a gauge transformation which allows us to express the nonisospectral spectral linear problem as a spectral parameter-free linear problem. The converse is possible by introducing λ orspectral parameter, conveniently. The importance of reduced spectral problems and reduced spectral parameters resides in the fact that in 1 1 dimensions, it is not usual to find nonisospectral versions. The nonisospectrality of 1 1-dimensional Lax pairs gives rise to some inconveniency, since the inverse scattering transform IST) 1,] can no longer be worked upon them, for example. If the IST cannot be used, the possibilities of solving the nonlinear equation through an associated spectral problem are diminished. On the other hand, many of the Lax pairs found in 1 dimensions are nonisospectral 15,16]. For this matter, we pay closer attention at the reductions of the associated spectral problem. In this way, we stress out the importance of finding isospectral Lax pairs in 1 1 dimensions and the surprising nature of nonisospectral ones. In Sect., we introduce the CLS for partial differential equations PDEs) and apply it to the spectral problem ) and its corresponding compatibility condition 4). In Sect. 3, we will obtain a classification of its possible reductions to 11 dimensions of the equation and the Lax pair, depending on different values of the arbitrary functions appearing in the obtained symmetries. We will identify six interesting reductions. Two of them will be nontrivial. One of such nontrivial reduction corresponds with the celebrated Korteweg de Vries equation KdV equation) 10,19,3]. The second reduction found will be submitted to a second CLS calculation. So, Sect. 4 shall be devoted to the classical symmetry computation of the aforementioned second nontrivial reduction in 1 1 dimensions. A list of four reductions to ODEs will be displayed, considering different values for the constants of integration appearing in the symmetry computation. To conclude, we shall enclose a summary of the most relevant results found throughout the iterative symmetry search and reduction procedure. The classical symmetry approach A priori, we propose the most general Lie point symmetry transformation in which the coefficients of the infinitesimal generator can depend on any dependent or independent variable

3 Symmetry computation and reduction of a wave model 15 x x ɛξ 1 x, y, t, u,ω) Oɛ ), y y ɛξ x, y, t, u,ω) Oɛ ), t t ɛξ 3 x, y, t, u,ω) Oɛ ), u u ɛη u x, y, t, u,ω) Oɛ ), ω ω ɛη ω x, y, t, u,ω) Oɛ ). 5) If we search for general Lie point transformation of the Lax pair too, the spectral functions must be transformed accordingly with ψ ψ ɛη ψ x, y, t, u,ω,ψ,χ) Oɛ ), χ χ ɛη χ x, y, t, u,ω,ψ,χ) Oɛ ). 6) Associated with this transformation, there exists an infinitesimal generator X = ξ 1 x ξ y ξ 3 t η u u η ω ω η ψ ψ η χ χ, 7) where the subscripts in η p have been added accordingtothefield p to which each η is associated. This transformation must leave ), 3) and 4), invariant. From now, since Eqs. ) and 3) are equivalent, we shall exclude 3) from our calculus as a matter of simplification. Our results obtained for ψ can be similarly extrapolated for χ. To proceed with CLS, we follow the next steps 1. Introduce the infinitesimal transformation 5) and its further derivatives in Eqs. ) and 4).. Select the linear terms in ɛ and set them equal to zero. The zero order in ɛ retrieves the original equations. 3. Substitute the values of ψ xx,ψ t, u xt, u y from ) and 4), correspondingly. Higher-order derivatives in these terms must be introduced according to these expressions. 4. From the steps above, we obtain an overdetermined system of differential equations for ξ 1,ξ,ξ 3,η u, η ω,η ψ by setting equal to zero terms in different orders of the derivatives of the fields, if we pursue Lie point symmetries. Steps 1 5 are nontrivial, leading to cumbersome equations which have been manipulated with a symbolic calculus package. In our case, we make use of MAPLE c. We extend transformation 5) to first-, second- and higher-order derivatives appearing in Eq. 4), u x u x ɛη u x] Oɛ ), u y u y ɛη u y] Oɛ ), ω x ω x ɛη ω x] Oɛ ), ω y ω y ɛη ω y] Oɛ ), u xx u xx ɛη u xx] Oɛ ), u xy u xy ɛη u xy] Oɛ ), u xt u xt ɛη u xt] Oɛ ), ω yy ω yy ɛη ω yy] Oɛ ), u xxxy u xxxy ɛη u xxxy] Oɛ ). 8) And derivatives appearing in the Lax pair ) ψ x ψ x ɛη x] ψ Oɛ ), ψ y ψ y ɛη y] ψ Oɛ ), ψ t ψ t ɛη t] ψ Oɛ ), ψ xx ψ xx ɛη xx] ψ Oɛ ), ψ yy ψ yy ɛη yy] ψ Oɛ ). 9) η xy] u u u u, η u xt], and for the Lax The prolongations needed for Eq. 4)areη u xxxy], η xx], η ω yy], η x], η y], η ω x], η ω y] pair ), the prolongations are η yy] ψ, ηxx] ψ, ηt] ψ, ηy] η x]. These prolongations can be calculated according ψ ψ, to the Lie method explained in textbooks 31]. From the original equations, we make use of ω yy = u xt u xxxy 8u x u xy 4u xx u y, u yy = ω xy, 10) and the original the Lax pair ψ xx = iψ y u x ψ, ψ t = iψ yy 4u y ψ x u xy iω y )ψ. 11) Also, their further derivatives must be computed using the given expressions. Introducing such relations we arrive at the classical Lie symmetries ξ 1 = Ȧ3t) x A 1 t), 4 ξ = Ȧ3t) y A t), ξ 3 = A 3 t),

4 16 P. G. Estévez et al. Table 1 Reductions for 1-BKP Case I: A 3 t) = 0 Case II: A 3 t) = 0 1. A 1 t) = 0 A t) = 0 1. A 1 t) = 0 A t) = 0. A 1 t) = 0 A t) = 0. A 1 t) = 0 A t) = 0 3. A 1 t) = 0 A t) = 0 3. A 1 t) = 0 A t) = 0 η u = Ȧ3t) u Ȧt) 4 8 A 1 t)y B 1 t), 4 η ω = Ȧ3t) w Ȧ1t) 4 A B 3 t)y t) 16 y,... A3t) η ψ = A x 3 t) 16 xy A x 3 t) 3 x 96 y3 B t), λ Ȧ3t) Ȧ t) i y 8 4 Ä3t) 16 y B 3 t)dt )] ψ. 1) These symmetries depend on a constant λ and six arbitrary functions of time, A 1 t), A t), A 3 t) and B 1 t), B t), B 3 t), which shall serve us as a way to classify the possible reductions. Indeed, we have the possible reductions attending to Table 1). 3 Reduction to 1 1 dimensions To reduce the problem, we have to solve the Lagrange- Charpit system by the method of characteristics, that is integration of the characteristic system 8,31] dx ξ 1 = dy ξ = dt ξ 3 = du η u = dω η ω = dψ η ψ. 13) We shall use the next notation for the reduced variables x, y, t x 1, x 14) and for the reduced fields ωx, y, t) x 1, x ), ux, y, t) Ux 1, x ), ψx, y, t) x 1, x ). 15) As a matter of simplification, we shall drop the dependency on x 1, x ) of certain fields in the forthcoming expressions. Case I.1. A 3 t) = 0, A 1 = 0, A = 0 x x 1 = A 3 t) 1/4 y x = A 3 t) 1/ A 1 t) dt, A 3 t) 5/4 A t) dt. A 3 t) 3/ ux, y, t) = Ux 1, x ) F 1 t)dt A 3 t) 1/4 Ȧ3t)xy A t)x 4A 1 t)y, 16A 3 t) ωx, y, t) = x 1, x ) F t)dt A 3 t) 1/ A 3t)Ä 3 t) Ȧ 3 t) 19A 3 t) y 3 A 3t)Ȧ t) A t)ȧ 3 t) 3A 3 t) y B3 t)dt A t) ) A 3 t) 16A 3 t) y Ȧ3t) 3A 3 t) x A 1t) 4A 3 t) x, where we have used the definitions F 1 t) = B 1t) A 3 t) 3/4 3A 1t)A t) 8A 3 t) 7/4, F t) = B t) A 3 t) 1/ A t) B3 t)dt A 3 t) 3/ A t) 3 16A 3 t) 5/ A 1t) 4A 3 t) 3/. U x x x1 x = 0, x x = U x1 x 1 x 1 x 8U x1 x U x1 4U x1 x 1 U x. 16) These two equations can be summarized as ) U x x 1 x 1 x 1 x 1 4 U x1 U x ) x1 U x1 U x1 x U x x x x = 0, 1 17)

5 Symmetry computation and reduction of a wave model 17 which appears in 3,33] and has multiple soliton solutions 39]. Reduced eigenfunction ψx, y, t) = x 1, x ) A 3 t) 1/8 Ȧ3 t)y ) 4A t)y exp i 16A 3 t) ] i F 3 t)dt. where F 3 t) = B 3 t)dt A 3 t) iλ A 3 t) A t) 4A 3 t). x1 x 1 i x U x1 = 0, i x x U x x1 U x1 x i x λ ) = 0. 18) Notice that λ plays the role of the spectral parameter in 1 1 dimensions. Cases I.. and I.3. will be omitted for being equivalent to I.1. Case II.1. A 3 t) = 0, A 1 t) = 0, A t) = 0 x 1 = A t)x A 1 t)y A t) 3/ A 1 t) x = dt A t) 5/ F 1 t)dt ) Ux 1, x ) B3 t)dt A 1t) 8A t) x 1 ux, y, t) = A t) B1 t) A t) Ȧt) ) 8A t) x y Ȧ1t) A t) A ) 1t)Ȧ t) y A t) 16 wx, y, t) = A 1 t) x 1, x ) A t) 3/ Ät) 48A t) y3 Ȧt) 16 x 1 with 4 B 3t) A t) A ) 1t)Ȧ 1 t) y A t) 8 B t) A t) y Ȧ1t) 4A t) xy F 1 t) = 5 A 1 t) 3 A t) 7/ 4 B 1t) A t) 3/ 1 A 1t) A t) 5/ B 3 t)dt U x1 x 1 x1 x 1 = 0 x1 = U x1 Gx ) U x1 x 1 x 1 x 1 1U x1 x 1 U x1 U x1 x = 0. where Gx ) is an arbitary function of x.the above system can be equivalently rewritten as U x1 x 1 x 1 x 1 x 1 1U x 1 x 1 1U x1 U x1 x 1 x 1 U x1 x 1 x = 0. 19) This reduced equation corresponds with the potential KdV equation in 1 1 dimensions 11]. Therefore, we can conclude that the 1)-BKP equation is a generalization of the potential KdV equation to 1 dimensions. Reduced Eigenfunction x, y, t) = x 1, x ) A t) 1/4 )] A1 t) exp i A t) 1/ x 1 F t)dt Ȧ t) B3 t)dt iλ ) )] exp i 8A t) y y A t) where F t) = B t) A t) A 1t) 4 A t) 4 A 1t) A t) 3 Gx t)) 4iλ) Ȧ1t) B3 t)dt iλ A t) 1/ F 1 t)dt 8 A t) ) x1 x 1 iλ U x1 ) = 0 0) x 4iλ U x1 ) x1 U x1 x 1 = 0 1)

6 18 P. G. Estévez et al. which is the Lax pair corresponding with the KdV equation. Again, λ plays the role of the spectral parameter. Case II.. A 3 t) = 0, A 1 t) = 0, A t) = 0 x 1 = y, x = t. ux, y, t) = ia 1t)Ux 1, x ) 8 B 3 t)dt iλ ) B 3t) A 1 t) x 1 B1 t) A 1 t) Ȧ1t) ) 4A 1 t) y x wx, y, t) = i x 1, x ) Ȧ1t) B t) A 1 t) B 3t) A 1 t) y 8A 1 t) x ) x Ȧ1t) A 1 t)ä 1 t) 4A 1 t) x 3 1 B 1t)Ȧ 1 t) A 1 t)ḃ 1 t) A 1 t) x 1 These reduced fields lead to a trivial reduction of the equations. U x1 x 1 = 0, x1 x 1 = 0 Reduced eigenfunction x, y, t) = x 1, x ) A1 t) exp i exp i B3 t)dt iλ ) A 1 t) Ȧ1 t) 4A 1 t) x 1 x 8 )] F 1 t)dt A 1t)B 1 t) B 3 t)dt iλ ) )] A 1 t) x 1 F 1 t) = B ) ] 1t) B3 A 1 t) t)dt iλ A 1 t) x1 = 0, x ) U x1 x1 = 0 ) Case II.3. A 3 t) = 0, A 1 t) = 0, A t) = 0 x 1 = x A t) 4 B 1 t) A t) 3/ dt x = t, ux, y, t) = Ux 1, x ) x 1 B3 t)dt A t) B1 t) A t) Ȧt) ) 8A t) x y wx, y, t) = x 1, x ) A t) 16 x 1 Ät) 48A t) y3 B 3t) A t) y B t) A t) y These reduced fields lead to trivial a reduction of the equation. U x1 x = 0 3) Reduced eigenfunction x, y, t) = x 1, x ) A t) 1/4 Ȧ t) B3 t)dt iλ exp i 8A t) y y A t) )] F 3 t)dt F 3 t) = B t) A t) 8 B3 t)dt iλ A t) x = 0 x1 x 1 U x1 iλ ) = 0 ) 3.1 Reduction of a Lax pair in 1 1 dimensions Let us now study the nontrivial reduction I.1. obtained in the past section. We consider this reduction of inter-

7 Symmetry computation and reduction of a wave model 19 est from a possible physical viewpoint. In this way, we aim to perform another symmetry search on the equation and its corresponding Lax pair in 11 dimensions. We reconsider the equation given in 17) U x x = x1 x, U x1 x 1 x 1 x 8U x1 x U x1 4U x1 x 1 U x x x = 0, 4) which is integrable in the Painlevé sense and possesses an associated linear spectral problem or Lax pair 18), which takes the form x1 x 1 i x U x1 = 0, i x x U x x1 U x1 x i x λ ) = 0. 5) whose compatibility condition x1 x 1 x x = x x x x 1 ) recovers 4). This Lax pair presents a constant parameter λ that plays the role of the spectral parameter. We aim at studying its classical Lie point symmetries and further reduction under the action of the symmetries. We propose a general transformation in which the infinitesimal generator depends on any independent and dependent variables as x 1 x 1 ɛξ 1 x 1, x, U,) Oɛ ), x x ɛξ x 1, x, U,) Oɛ ), λ λ ɛη λ x 1, x,λ,) Oɛ ), U U ɛη U x 1, x, U,) Oɛ ), ɛη x 1, x, U,) Oɛ ). 6) Here, we can see that we have considered λ as an independent variable in order to make the reductions properly. If we want to achieve symmetries of equation 4) and those of the corresponding Lax pair 5)at the same time, we must include the transformed eigenfunctions ɛη x 1, x, U,,λ,) Oɛ ), where we have only specified the transformation for one of the eigenfunctions, since the complex conjugate version was not considered in the former reductions. Similar results can be obtained for the complex conjugate by extension of previous results. By definition of symmetry, transformation 6) must leave invariant equation 4) and its associated Lax pair 5). The associated symmetry vector field has the expression X = ξ 1 ξ η λ x 1 x λ η U U η η. 7) From the terms in ɛ = 0 we retrieve the original equations, U x x = x1 x, U x1 x 1 x 1 x = x x 8U x1 x U x1 4U x1 x 1 U x, x1 x 1 = i x U x1, i x x = U x x1 U x1 x i x λ ) 8) that shall be used in the forthcoming steps. First introduce transformation 6) into the system of differential equations in 4), 5) and set the linear term in ɛ equal to zero. Introduce the needed prolongations for Eq. 4), that are η x 1x 1 x 1 x ] U, η x 1x ] U, η x 1x 1 ] U, η x x ] U, η x 1] U, ηx ] U, ηx x ], η x 1x ], η x ], and the prolongations neeeded for the Lax pair 5), that are η x 1x 1 ], η x x ], η x 1], ηx ], calculated following Lie s formula 31] and U x x, U x1 x 1 x 1 x, x1 x 1, x x from 8). We come up with the classical Lie point symmetries ξ 1 x 1, x, U,)= 1 k 1x 1 k, ξ x 1, x, U,)= k 1 x, η λ = k 1 λ, η u x 1, x, U,)= 1 k 1U k 5, η x 1, x, U,)= k 1 i x k 6 x 1 ), η x 1, x,λ,u,,,λ)= Bλ). 9) These symmetries depend on 5 arbitrary constants of integration k 1, k,,, k 5 and two arbitrary functions k 6 x 1 ) and Bλ) Table ). We introduce the following notation for the reduced variables. In this case, λ is an independent variable. x 1, x,λ z, 30)

8 0 P. G. Estévez et al. Table Reductions for 1 1-BKP equation Case I: k 1 = 0 Case II: k 1 = 0 1. k = 0 = 0 1. k = 0 = 0. k = 0 = 0 3. k = 0 = 0 and the reduced fields and eigenfunctions Ux 1, x ) V z), x 1, x ) W z), x 1, x ) ϕz,). 31) We find the following reductions Case I.1. k 1 = 0 z = k 1k 1 x ) k 1 x 1 k ), = k1 5 k 1λ )k 1 x 1 k ) 4. V z) Ux 1, x ) = k 1 k 1 x 1 k ), x 1, x ) = i k1 3 k 1 x 1 k ) z k 1 Reduced Eigenfunction W z) k 1 x 1 k ) z x 1, x,λ)= ϕz,)e Bλ) k 1 λ dλ z ϕ zz 8iz V z ϕ zϕ z ) W zw z 6izV z ) ] iz 4zV zz ) ϕ = 0, 16 ϕ 16zϕ z 43 8iz V z )ϕ i 6z 16iz 3 V z )ϕ z W zw z V z1 3iz)V z ] iz 4zV zz ) ϕ = 0.. In this case, a constant appears and plays the role of a spectral parameter. s W zz = V zz, 16z 6 V zzzz 144z 5 V zzz 8z V )V z zw z W 1 300z 3z V 96z 3 V z )z V zz 176z 4 V z = 0. These equations can be integrated as W = a 1 z a 16z 5 V zzz 80z 4 V zz 48z 4 V z z 3 3V 60)V z zv z V a a 3 z = 0. where a 1, a, a 3 are constants. Case II.1. k 1 = 0, k = 0, = 0, = 0 z = k k = k x x 1 k λ x 1 ) ). Ux 1, x ) = k V z) k 5 k x 1, x 1, x ) = k k3 W z) i k3 k 3 1 k Reduced Eigenfunction k 6 x 1 )dx 1. x 1, x,λ)= ϕz,)e 1 Bλ)dλ. z k ) x 1 x 1 ϕ ϕ z ϕ zz iϕ z C 1 V z ) ϕ = 0, ic W z iv zz )ϕ i ϕ ϕ z ) V z ϕ zz = 0. 3)

9 Symmetry computation and reduction of a wave model 1 In this reduced spectral problem, the constant plays the role of the spectral parameter. W zz = V zz ic, V zzzz 4 C 1 3V z ) V zz W zz = 0. 33) where C 1 = k 5 and C k 3 = k5 3 k 6 The above equations can be integrated as W = V a 1 z a, P zz 6Pz 8C 1 1)P z ic z a3 = 0, P = V z. 34) This last equation is nothing but the celebrated Painlevé II equation whose solutions has been extensively studied 8,11]. Case II.. k 1 = 0, k = 0, = 0, = 0, k 5 = 0 z = k 5 x, k ) 1/ ) k5 k = λ x 1 k ) 1/ k5 Ux 1, x ) = V z) k 5 x 1, k k x 1, x ) = k 5 W z) i x x 1 k k 1 k 6 x 1 )dx 1. k Reduced Eigenfunction ψx 1, x ) = ϕz,)e 1 Bλ)dλ. ϕ iϕ z ϕ = 0, ϕ zz iv z ϕ W z V zz ) ϕ = 0. 35) k 5/ 5 V zz = i k3/, W zz = 0. 36) Case II.3. k 1 = 0, k = 0, = 0, = 0, k 5 = 0 z = k5 ) 1/3 x 1, = k1/3 3 k /3 5 λ k ) 4 x i k5 ) 1/3 k5 Ux 1, x ) = V z) k 5 x, x 1, x ) = W z) i x k B 0 x. 3 where B 0 is a constant. Reduced Eigenfunction ψx 1, x ) = ϕz,)e 1 Bλ)dλ. ϕ iϕ z V zz ϕ = 0, ϕ zz iϕ V z ϕ = 0. ) /3 B 0 Here, the constant plays the role of the spectral paramenter. V zz = i k5, W zz = 0. 37) 4 Conclusions and brief comments We have calculated the classical point) symmetries of a nonlinear PDE and its corresponding two-component, nonisospectral Lax pair in 1 dimensions. The spectral parameter has been conveniently introduced as

10 P. G. Estévez et al. dependent variable obeying the nonisospectral condition. We have reduced both the equation and its Lax pair attending to different choices of the arbitrary functions and constants of integration present in the calculated symmetries. Of a total of 6 reductions, of them happen to be knowledgeable equations in the literature. One is the KdV equation. From this result, we can say that 1)-BKP equation is a generalization of the KdV equation. The nonclassical version when ξ 3 = 0 has also been obtained although it is not included in this manuscript), leading to the same set of symmetries. The other interesting reduction is an equation showing multisoliton solutions. In this equation and its associated Lax pair, we have performed a second classical Lie symmetry analysis. In this case, the spectral parameter has been conveniently introduced in the reduction and it needs to be considered as an independent variable. Another 4 possible reductions are contemplated. After the computation of the classical Lie symmetry results, the nonclassical symmetries for the nontrivial reduction I.1. studied in Sect Equation 4) and those of its corresponding Lax pair 5) have also been obtained, provided that ξ = 0, leading us to equivalent results not listed in this paper). Therefore, the classical and nonclassical symmetries of Eqs. 4) and 5) are equivalent. Nonetheless, this is not always the rule. Indeed, nonclassical symmetries are usually more general and contain the classical ones as a particular case. As a future perspective, it would be desirable to be able to find ways of solving nonisospectral Lax pairs in 1 1 dimensions, since the methods applicable are only applicable in the cases of isospectral ones. Acknowledgements P. G. Estévez, J. D. Lejarreta and C. Sardón acknowledge partial financial support by research projects MAT C-1-R MINECO) and SA6013 JCYL). References 1. Ablowitz, M.J., Clarkson, P.: Solitons, Non-linear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge 1991). Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia 1981) 3. Arrigo, D., Beckham, J.R.: Nonclassical symmetries of evolutionary partial differential equations and compatibility conditions. J. Math. Anal. Appl. 89, ) 4. Bluman, G.W., Cole, J.D.: The general similarity solution of the heat equation. J. Math. Mech. 18, ) 5. Bluman, G.W., Yan, Z.: Nonclassical potential solutions of partial differential equations. Eur. J. Appl. Math. 16, ) 6. Bogoyanlevskii, O.I.: Overturning solitons in new twodimensional integrable equations. Lett. Nuovo Cimento Math. URSS. Izv. 34, ) 7. Calogero, F.: A method to generate solvable non-linear evolution equations. Lett. Nuovo Cimento 14, ) 8. Clarkson, P.A.: NIST Handbook of Mathematical Functions, pp Cambridge University Press, Cambridge 010) 9. Clarkson, P.A., Winternitz, P.: Nonclassical symmetry reductions for the Kadomtsev Petviashvili equation. Phys. D 49, ) 10. Drazin, P.G.: Solitons. London Mathematical Society Lecture Note Series, vol. 85. Cambridge University Press, Cambrigde 1983) 11. Drazin, P.G., Johnson, R.S.: Solitons: An Introduction. Cambridge University Press, Cambridge 1990) 1. Druyma, V.S.: On the analytical solution of the two dimensional Korteweg de Vries equation. Sov. Phys. JETP Lett. 19, ) 13. Estévez, P.G.: Construction of lumps with non-trivial interaction. In: Proceedings of 6th Workshop Group Analysis of Differential Equations & Integrable Systems Protaras, Cyprus, 01), pp University of Cyprus, Nicosia 013) 14. Estévez, P.G., Hernáez, G.A.: Non-isospectral problem in 1 dimensions. J. Phys. A Math. Gen. 33, ) 15. Estévez, P.G., Lejarreta, J.D., Sardón, C.: Non isospectral 11 hierarchies arising from a Camassa-Holm hierarchy in 1 dimensions. J. Nonlinear Math. Phys. 8, ) 16. Estévez, P.G., Prada, J.: Singular manifold method for an equation in 1 dimensions. J. Nonlinear Math. Phys. 1, ) 17. Estévez, P.G., Prada, J.: Lump solutions for PDE s: algorithmic construction and classification. J. Nonlinear Math. Phys. 15, ) 18. Kadomtsev, B.B., Petviashvili, V.I.: On stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15, ) 19. Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal and on a new type of stationary waves. Philos. Mag. 39, ) 0. Lie, S.: Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen. Arch. Math. 6, ) 1. Lie, S.: Theorie der Transformationgruppen, vol.. Teubner, Leipzig 1890). Lie, S.: Theorie der Transformationgruppen, vol. 3. Teubner, Leipzig 1893) 3. Liu, J., Mu, G., Dai, Z., Lao, H.: Spaciotemporal deformation of multi-soliton to 1)-dimensional KdV equation. Nonlinear Dyn. 83, ) 4. Minzoni, A.A., Smyth, N.F.: Evolution of lump solutions for the KP equation. Wave Motion 4, ) 5. Nucci, M.C.: The role of symmetries in solving differential equations. Math. Comput. Model. 5, )

11 Symmetry computation and reduction of a wave model 3 6. Nucci, M.C.: Nonclassical symmetries as special solutions of heir-equations. Math. Anal. Appl. 79, ) 7. Nucci, M.C., Ames, W.F.: Classical and nonclassical symmetries of the Helmholtz equation. J. Math. Anal. Appl. 178, ) 8. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, Berlin 1993) 9. Pucci, E., Saccomandi, G.: On the reduction methods for ordinary differential equations. J. Phys. A 35, ) 30. Sardón, C., Estévez, P.G.: Miura-Reciprocal transformations for two Integrable hierarchies in 1 1 dimensions. In: Proceedings of 6th Workshop Group Analysis of Differential Equations & Integrable Systems Protaras, Cyprus, 01), pp University of Cyprus, Nicosia 013) 31. Stephani, H.: Differential Equations: Their Solution Using Symmetries. Cambridge University Press, Cambridge 1990) 3. Wazwaz, A.M.: New higher dimensional fifth-order nonlinear equations with multiple soliton solutions. Phys. Scr. 84, ) 33. Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with algorithmic nonlinearities. Nonlinear Dyn. 83, ) 34. Weiss, J.: The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs and the Schwartzian derivative. J. Math. Phys. 4, ) 35. Weiss, J., Tabor, M., Carnevale, G.: The Painlevé property for partial differential equations. J. Math. Phys. 4, ) 36. Yan, Z., Konotop, V.V., Akhmediev, N.: Three-dimensional rogue waves in nonstationary parabolic potentials. Phys. Rev. E 8, ) 37. Yan, Z.: Two-dimensional vector rogue wave excitations and controlling parameters in the two-component Gross- Pitaevskii equations with varying potentials. Nonlinear Dyn. 79, ) 38. Yu, S.J., Toda, K., Fukuyama, T.: N-soliton solutions to a 1 dimensional integrable equation. J. Phys. A Math. Gen. 31, ) 39. Zhang, S., Gao, X.: Exact N-soliton solution and dynamics of a new AKNS equation with time-dependent coefficients. Nonlinear Dyn. 83, )