Invariance Analysis of the (2+1) Dimensional Long Dispersive Wave Equation

Transcription

1 Nonlinear Mathematical Physics 997 V.4 N Invariance Analysis of the + Dimensional Long Dispersive Wave Equation M. SENTHIL VELAN and M. LAKSHMANAN Center for Nonlinear Dynamics Department of Physics Bharathidasan University Tiruchirapalli India Abstract In this paper we bring out the Lie symmetries and associated similarity reductions of the recently proposed + dimensional long dispersive wave equation. We point out that the integrable system admits an infinite-dimensional symmetry algebra along with Kac-Moody-Virasoro-type subalgebras. We also bring out certain physically interesting solutions. Introduction Professor Wilhelm Fushchych has been stressing for several decades the importance of symmetry analysis of nonlinear evolution equations to understand the basic properties of the underlying physical systems. It is a great pleasure to contribute the present article on his sixtieth birthday. Soliton equations in + dimensions exhibit very interesting symmetry properties both Lie and Lie-Bäcklund type [-3]. These symmetries help one to understand the integrability properties of underlying nonlinear dynamical systems clearly. In + dimensions solutions have much richer structures [4 ]. As a consequence the identification and study of symmetries will play a crucial role here also. Concentrating on Lie symmetries for the present it has been realized that important classes of + dimensional extensions of soliton equations admit typically Lie symmetries involving infinite-dimensional symmetry algebras often of the Kac-Moody-Virasoro type. Typical systems are the following: i Kadomtsev-Petviashvili equation [6] ii Davey-Stewartson equation [7] iii Nizhnik-Novikov-Veselov equation [8] iv nonlinear Schrödinger equation introduced by Fokas and the sine-gordon equation [8]. However there are certain integrable evolution equations which though admit infinite-dimensional Lie algebras do not seem to possess a Virasoro- type subalgebra [9]. The typical examples being i the breaking soliton equation and ii nonlinear Schrödinger type equation were studied by Strachan recently. So the connection between integrability and Virasoro-type algebras deserves much study further. In this contribution we analyze the invariance properties of an important new evolution equation in + dimensions namely the + dimensional long dispersive wave Copyright c 997 by Mathematical Ukraina Publisher. All rights of reproduction in form reserved.

2 M. SENTHIL VELAN and M. LAKSHMANAN equation and bring out the existence of an infinite-dimensional Lie algebra of symmetries along with Kac-Moody-Virasoro-type subalgebras. We also deduce the possible similarity reductions and some particular solutions. The plan of the paper is as follows. In Sec. we discuss the Lie symmetries and Virasoro type subalgebras for the + dimensional long dispersive wave equation. In Sec.3 we construct the similarity variables and obtain the similarity reductions. In Sec.4 the nontrivial subcases are presented. In Sec. we present the invariance analysis of the reduced pde. In Sec.6 we present our conclusions. Lie symmetries and Kac-Moody-Virasoro algebras of the + dimensional long dispersive wave LDW equation Recently Chakravarty Kent and Newman[0] have introduced a new + dimensional long dispersive wave equation by symmetrically reducing the self-dual Yang-Mills equation. The resultant equation can be written in the form λq t + q xx q λr t r xx + r qr x dη = 0 qr x dη = 0 where η = x λ y and λ is a constant parameter. Eq. is the + dimensional generalization of the one-dimensional long dispersive wave equation [ ]. It is interesting to note that eq. reduces to the single nonlocal equation introduced recently by Fokas [3] iλq t + q xx q q xdη = 0 when r = q and t it. Eq. arises in plasma physics under appropriate circumstances [4] and it admits exponentially localized solutions and satisfies the Painlevé property []. Recently the integrability aspects of eq. have been studied by Radha and Lakshmanan [6]. Using the bilinear approach they have brought out the peculiar localization properties of solutions of eq. by generating dromions for the physical quantity rq composite field.. Lie symmetries By introducing the transformation qr x = v η where v is some arbitrary potential we can rewrite eq. as λq t + q xx qv = 0 λr t r xx + rv = 0 qr x v η = 0. 3

3 INVARIANCE ANALYSIS OF THE + DIMENSIONAL 3 Now one can apply the Lie algorithm to eq.3 and study the invariance properties. However for our present study we have considered the above equation in the form q t + q xx qv = 0 r t r xx + rv = 0 v y rq x qr x = 0 wherein we have introduced the notational change η y for convenience. The invariance of eq.4 under the infinitesimal point transformations x X = x + εξ t x y q r v y Y = y + εξ t x y q r v q Q = q + εφ t x y q r v r R = r + εφ t x y q r v t T = t + εξ 3 t x y q r v v V = v + εφ 3 t x y q r v ε leads to the following expressions for infinitesimals ξ = x ft + gt ξ = my ξ 3 = ft [ φ = xġt + 8 ftx m y ] ft Ny t q φ = [ xġt ] 8 ftx + Ny t r φ 3 = vft + 4 x gt + d 3 f 6 dt 3 x + ht where ft gt ht are arbitrary functions of t and Ny t is an arbitrary function of y t and dot and prime denote differentiations with respect ot t and y respectively. In the above the arbitrary functions ft ht and Ny t are constrained by the following equation ft + 4Ṅy t + 8ht = 0. 6 The infinitesimals given in eqs.-6 are actually obtained using the symbolic program LIE [7].. Lie algebras The presence of arbitrary functions ft gt my and Ny t necessarily leads to an infinite-dimensional Lie algebra of symmetries. We can write a general element of this Lie algebra as where V = V f + V g + V 3 m + V N V f = x ft x + ft t + 8 ftqx q ft q 8 ftx r r + d 3 f 6 dt 3 x vft v V g = gt x + ġtxq q ġtxr r + 4 gtx v V 3 m = my y m yq q V 4N = qny t q + rny t r. 4

4 4 M. SENTHIL VELAN and M. LAKSHMANAN The associated Lie algebra between these vector fields becomes [V f V f ] = V f f f f q q r r [V g V g ] = g ġ g ġ [V 3 m V 3 m ] = V 3 m m m m [V f V g] = V fġ g f [V f V 4 N] = V 4 fṅ [V 3 m V 4 N] = V 4 mn + g g g g 4 which is obviously an infinite-dimensional Lie algebra of symmetries. A Virasoro-Kac- Moody-type subalgebra is immediately obtained by restricting the arbitrary functions f and m to Laurent polynomials so that we have the commutators v [V t n V t m ] = m nv t n+m [V 3 y n V 3 y m ] = m nv 3 y n+m. It is interesting to note that a similar type of algebras also exist in other integrable systems mentioned in Introduction namely the Nizhnik-Novikov-Veselov equation + dimensional nonlinear Schrödinger equation and sine-gordon equation [8]. 3 Similarity variables and similarity reductions The similarity variables associated with the infinitesimal symmetries can be found by solving the characteristic equation dx x ft + gt = dy my = dt ft = dq xġt + 8 ftx m y ft Ny tq 7 dr = xġt + ftx 8 Ny tr = dv vft + 4 x gt + d 3 f 6 x dt + ht. 3 Integrating eq.7 with the condition that ft 0 we get the following similarity variables: τ = x t gt f t y dy t dt f 3 t dt τ = my ft F = qe w G = re w H = vft w 3 where F G and H are functions of τ and τ and w = gt t gt f / t f 3/ t dt + τ ft t + τ ft gt 8 4f 3/ t dt m y t g t f t dt + gtτ f / t + ft [ t gt dt ] 8 f 3/ t t ft dt t log ft Ny t ft dt w = gt t gt t g t f / t f 3/ t dt + f t dt gtτ f / t ft [ t gt dt 8 f 3/ t τ ft t τ ft gt t Ny t 8 4f 3/ t dt + ft dt ]

5 INVARIANCE ANALYSIS OF THE + DIMENSIONAL t gt w 3 = ġf / fg 4f 3/ t dt t gt f / 4f 3/ t dt + τ ġ 4 f / τ fgt 8f / t g 8f + τ f f 6 + τ f f t gt [ t gt + f f ] f 4f 3/ 4f 3/ t dt [ t gt ] 4f 3/ t dt τ f 3 τ f t gt t 6f 3/ t dt + ht dt. Under the above similarity transformations eq.4 gets reduced to a system of pde in two independent variables τ and τ : F τ F τ τ + F H + kf = 0 G τ + G τ τ GH kg = 0 H τ F G τ GF τ = 0 8 where k is an arbitrary constant obtained by integrating eq.6 t f + 4Ny t + 8 ht dt = 0. t Since the original + dimensional pde 4 satisfies the Painlevé property [7] for a general manifold the + dimensional similarity reduced pde 8 is also expected to satisfy the P-property. 4 Subcases In addition to the above general similarity reduction one can also investigate particular cases by assuming one or more of vector fields to be zero. We list below some of important nontrivial cases. Case: ft=0: The similarity variables are τ = t H = v g g 4 y dy τ = x gt y [ y my my F = qe w G = re w dy ] my dy τ g y dy 4 my ht y dy my where w = gġ [ y y dy ] my my dy + τ ġ y dy y my log my Ny tdy my w = gġ [ y y dy ] my my dy τ ġ y dy y my + Ny tdy my. The reduced pde takes the form F τ + F τ τ F H = 0 G τ G τ τ + GH = 0 F G τ + GF τ gτ H τ hτ τ /4g τ = 0.

6 6 M. SENTHIL VELAN and M. LAKSHMANAN In the above prime denotes differentiation with respect to the variable τ. Case: ft=gt=0: The similarity variables are y N/mdy τ = x τ = t F = qe w y dy G = re H = v ht my where y Ny tdy w = log my my. The reduced pde takes the form F τ + F τ τ F H = 0 G τ G τ τ + GH = 0 F G τ = hτ. Case:3 ft=my=0: The similarity variables are where τ = t τ = y F = qe w G = re w H = v g 8g x h g x w = ġ 4g x N g x. The reduced pde takes the form F τ + g τ gτ F + N τ τ g F F H = 0 τ G τ g τ gτ G N τ τ g G + GH = 0 τ H τ = 0. Eq.9 can be integrated as follows. Eq.9a and 9b admit an integral G = P τ F where P τ is an arbitrary function of τ. Integrating eq.9c we get H = Hτ 9a 9b 9c 0 where Hτ is an arbitrary function of τ. Substituting eq. in eq.9a and integrating it we get [ F = f τ exp Hτ g τ gτ N ] τ τ g dτ τ where f τ is arbitrary function of τ. Substituting the expression of F in eq.0 we get G = P τ [ f τ exp Hτ g τ gτ N ] τ τ g dτ. 3 τ

7 INVARIANCE ANALYSIS OF THE + DIMENSIONAL 7 Eqs. 3 form the solution to pde 9. From this one can also write down the solution to pde 4 as [ ġt Ny t q = f y exp 4gt x x + Ht ġt gt gt N ] t y g dt t r = P y [ f y exp ġt Ny t 4gt x + x Ht ġt gt gt N ] t y g dt t v = Ht + g 8g x + h g x. Lie symmetries and similarity reduction of eqs.8 Now the reduced pde 8 in two independent variables can itself be further analyzed for its symmetry properties by looking at its own invariance property under the classical Lie algorithm again. In this case we obtain the following five-parameter Lie symmetries ξ = c c F 3 τ c ξ = c 3 τ c 3 φ = c φ = 3 kτ + c 4 τ c 6 + c G φ 3 = 3 kτ + c 4 τ + 6 c + c c 3 H c 4 where c c c 3 c 4 and c are arbitrary constants. The associated vector fields are V = τ + τ k 3 τ 3 τ 3 τ F + k 6 F F + 3 τ 6 G G H 3 H V = τ F F + τ G G + H V 3 = F F + G G V 4 = τ V = τ. The nonzero commutation relations between the vector fields are [V V ] = 3 V [V V 4 ] = 3 V 4 [V V ] = 3 V 3 V [V V ] = V 3. Solving the characteristic equation associated with the similarity variables we obtain z = τ 3c τ 3c 3 c c w z = G τ 3c p 3/ [ 3 exp c where p = 3c 3 c k + 3c 4 c w z = F + 3c c + 4. k + 3c 4 c The associated similarity reduced ode turns out to be zw + z + 4 w + p 4 w + w w 3 = 0 zw z 4 w + p 6 4 w w w 3 = 0 zw 3 + w 3 + z / w w + w w = 0. τ 3c p [ 3 exp k + 3c 4 c c ] τ w 3 z = H 3c 4 c τ ] τ 3c 3 c

8 8 M. SENTHIL VELAN and M. LAKSHMANAN Even though it is very difficult to find a solution for the above equation one can obtain interesting solutions by assuming one or more of the constants c i i =... be zero. The following are some of nontrivial cases. Case: c = 0: The similarity variables are z = τ c τ c 3 [ c4 w = G exp τ + c ] τ c 3 c 3 w = F exp w 3 = H + c 4 c 3 τ. The reduced ode takes the form w + c w w w 3 k + c w = 0 c 3 c 3 w c w w w 3 k + c w = 0 c 3 c 3 w 3 + c 3 w w + c 4 = 0. c c This equation has not yet been fully analyzed. Case: c = c = 0: The similarity variables are z = τ w = G exp [ c4 τ + c ] τ c 3 c 3 [ w = F exp [ c4 τ + c ] τ w 3 = H + c 4 τ. c 3 c 3 c 3 The reduced ode takes the form w w w 3 k + c w = 0 c 3 w w w 3 k + c w = 0 c 3 w w + c 4 = 0. c 3 Integrating eq.4 we get the solution w = I 3 z I c 3 I c 3 c 4 c 4 w = c 4 z I c 3 + I c 3 c 4 c 3 I 3 c 4 w 3 = k + c c 3 8 I c 3 8c z I c 3 4 c 4 where I I and I 3 are integration constants. Case:3 c = c 3 = 0: The similarity variables are z = τ w = G exp w = F exp [ c4 τ τ + c ] τ w 3 = H + c 4 τ. c c c c4 τ + c ] τ c 3 c 3 [ c4 τ τ + c ] τ c c 4

9 INVARIANCE ANALYSIS OF THE + DIMENSIONAL 9 The reduced ode takes the form w + w w 3 + kw = 0 w w w 3 kw = 0 w 3 = 0. Eq. admits the following solution w = I exp[ I + kz] w = I 3 exp[i + kz] w 3 = I where I I and I 3 are integration constants. 6 Conclusions In this paper we have carried out an invariance analysis and similarity reductions of the + dimensional long dispersive wave LDW equation and obtained particular solutions. We have pointed out the fact that the LDW equation admits an infinitedimensional symmetry algebra and Kac-Moody-Virasoro type subalgebras which typically exist in many other integrable + dimensional systems. It is yet to be clearly understood as to what is the significance of the existence or nonexistence of Kac-Moody-Virasoro-type subalgebras to nonlinear evolution equations as far as integrability is concerned. Such an understanding can throw some light on the classification of integrable systems. Currently we are investigating the possible similarity reductions of the above said equation through the nonclassical method and direct method of Clarkson and Kruskal. Acknowledgements: The work forms a part of the research project of the Department of Science and Technology Government of India. References [] Olver P.J. Applications of Lie Groups to Differential Equations Springer New York 986. [] Bluman G.W and Kumei S. Symmetries and Differential Equations Springer New York 989. [3] Lakshmanan M and Kaliappan P. Lie transforms nonlinear evolution equations and Painlevé forms J. Math. Phys. 983 V [4] Ablowitz M.J and Clarkson P.A. Solitons Nonlinear Evolution Equations and Inverse Scattering Transform Cambridge University Press Cambridge 990. [] Konopelchenko B.G. Solitons in Multidimensions World Scientific Singapore 993. [6] David D. Kamran N. Levi D. and Winternitz P. Symmetry reductions for the Kadomtsev- Petviashvili equation using a loop algebra J. Math. Phys. 986 V.7. [7] Champagne B. and Winternitz P. On the infinite dimensional symmetry groups of the Davey- Stewartson equations J. Math. Phys. 988 V.9. [8] Lakshmanan M. and Senthil Velan M. Lie symmetries infinite dimensional Lie algebras and similarity reductions of certain + dimensional nonlinear evolution equations J. Nonlin. Math. Phys. 996 V.3 4. [9] Lakshmanan M. and Senthil Velan M. Lie Symmetries Kac-Moody-Virasoro algebras and integrability of certain higher dimensional nonlinear evolution equations in preparation

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