Symmetry Reduction of (2+1)-Dimensional Lax Kadomtsev Petviashvili Equation

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1 Commun. Theor. Phys. 63 (2015) Vol. 63, No. 2, February 1, 2015 Symmetry Reduction of (2+1)-Dimensional Lax Kadomtsev Petviashvili Equation HU Heng-Chun ( ), 1, WANG Jing-Bo ( ), 1 and ZHU Hai-Dong ( ) 2 1 College of Science, University of Shanghai for Science and Technology, Shanghai , China 2 National Laboratory of High Power Laser and Physics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai , China (Received October 22, 2014; revised manuscript received December 18, 2014) Abstract The Lax Kadomtsev Petviashvili equation is derived from the Lax fifth order equation, which is an important mathematical model in fluid physics and quantum field theory. Symmetry reductions of the Lax Kadomtsev Petviashvili equation are studied by the means of the Clarkson Kruskal direct method and the corresponding reduction equations are solved directly with arbitrary constants and functions. PACS numbers: Ik, Jr, Yv Key words: Clarkson Kruskal direct method, Lax Kadomtsev Petviashvili equation, symmetry reduction, exact solution 1 Introduction Nonlinear partial differential equations have been studied extensively in the soliton theory because most of the nonlinear phenomena can be characterized by certain nonlinear evolution systems. In order to learn much more about the nonlinear phenomena in the nature world, mathematician and physicists have proposed many effective methods to obtain more exact solutions of the nonlinear partial differential equations, such as the inverse scattering transformation, the Painlevé analysis, the Hirota bilinear form, symmetry reduction, Darboux transformation, and Bäcklund transformation, [1 5] etc. On the other hand, it is well known that symmetry group techniques provide one method for obtaining exact and analytical solutions of nonlinear systems. The classical method for finding symmetry reductions of nonlinear systems is the Lie group method and its generalized forms. [6 10] The direct and algorithmic method to find symmetry reductions is called Clarkson Kruskal direct method (CK direct method), which can be used to obtain previously unknown reductions of nonlinear systems. [11 12] Many new symmetry reductions and exact solutions for a large number of physically significant nonlinear systems have been obtained by the means of the CK direct method. In this paper, the symmetry reductions for the (2+1)-dimensional Lax Kadomtsev Petviashvili (Lax- KP) equation are studied by the means of the CK direct method. The (2+1)-dimensional Lax Kadomtsev Petviashvili (Lax-KP) equation in the integral form is as follows p t +30p 2 p x +20p xx p x +10pp xxx +p xxxxx + 1 x p yy = 0, (1) which was derived by extending the Lax fifth order equation using the sense of the Kadomtsev Petviashvili equation in extending the KdV equation. [13 14] When p yy = 0, equation (1) is reduced to the Lax fifth-order nonlinear equation, which has been used widely in quantum mechanics and nonlinear optics. Setting p = u x, Eq. (1) becomes u xt + 30u 2 xu xx + 20u xx u xxx + 10u x u xxxx + u xxxxxx + u yy = 0. (2) In Ref. [13], the author has obtained the single soliton and multiple-soliton solutions of the Lax-KP equation (2) by using the tanh-coth method and Hirota bilinear method. The classical Lie point symmetry and group-invariant solutions of the Lax-KP equation (2) are studied in [15]. In this paper, we focus on the Lax-KP equation in the form of Eq. (2) and study the symmetry reductions of the Lax-KP equation (2) by the CK direct method. 2 Symmetry Reduction of the (2+1)- Dimensional Lax-KP Equation In the soliton theory, there is much interest in obtaining exact analytical solutions of nonlinear systems. Symmetry reduction is either by seeking a similarity solution in a special form or, more generally, by exploiting symmetries of the nonlinear equations. On the other hand, many high dimension nonlinear systems can be reduced to low dimension differential equations or ordinary differential equations. In order to find the similarity solutions of a nonlinear system, one can use the standard classical Lie approach, nonclassical Lie approach, the CK direct method and modified CK direct method. The CK direct Supported by National Natural Science Foundation of China under Grant Nos and , Shanghai Natural Science Foundation under Grant No. 10ZR , Shanghai Leading Academic Discipline Project under Grant No. XTKX2012 and the Hujiang Foundation of China under Grant No. B14005 Corresponding author, hhengchun@163.com c 2015 Chinese Physical Society and IOP Publishing Ltd

2 No. 2 Communications in Theoretical Physics 137 method is the most convenient one to obtain the similarity reduction of a nonlinear system. With the help of CK direct method, it is sufficient to seek a similarity reduction of the (2+1)-dimensional Lax- KP equation in the special form u(x, y, t) = α(x, y, t) + β(x, y, t)p (z(x, y, t)), (3) where α = α(x, y, t), β = β(x, y, t), and z = z(x, y, t) are functions to be determined. Substituting Eq. (3) into Eq. (2) yields βz 6 xp (6) + (15βz 4 xz xx + 6β x z 5 x)p (5) + (60β x z 3 xz xx + 45βz 2 xz 2 xx + 10α x βz 4 x + 20βz 3 xz xxx )P (4) + 10β 2 z 5 xp P (4) + 20β 2 z 5 xp P + (45β xx z 2 xx + 15β xxxx z 2 x + 10βz 2 xxx + βz 2 y + 60α x β xx z 2 x + 20βz 2 xα xxx + 15βz xx z xxxx + 30βα 2 xz 2 x + 40βα x z x z xxx + βz t z x + 60α xx βz x z xx + 120α x β x z x z xx + 30α x βz 2 xx + 60α xx β x z 2 x + 60β xxx z x z xx + 60β xx z x z xxx + 6βz x z xxxxx + 60β x z xx z xxx + 30β x z x z xxxx )P + (120β 2 xz 3 x + 60β 2 α x z 3 x + 120ββ xx z 3 x + 60β 2 z 2 xz xxx + 90β 2 z 2 xxz x + 360ββ x z 2 xz xx )P P + 30β 3 z 4 xp 2 P + + α xxxxxx + 30α 2 xα xx + 10α x α xxxx + 20α xx α xxx + α yy + α xt + (β xt + β xxxxxx + β yy + 10β x α xxxx + 10α x β xxxx + 20α xx β xxx + 20β xx α xxx + 30α 2 xβ xx + 60α x α xx β x )P = 0, where P (n) = d n P (z)/dz n. In order to require this equation be an ordinary differential equation for the function P (z), the ratios of the coefficients of different derivatives and powers of P (z) have to be functions of z only. We use the coefficient of P (6) as the normalizing coefficient and require that other coefficients be of the form βz 6 xγ(z), where Γ(z) is a function of z to be determined. We consider two cases z x 0 and z x = 0 in the following. Case 1 When z x 0, we have the following equations: 10β 2 zx 5 = βzxγ 6 1 (z), (4) 10ββ x zx 4 = βzxγ 6 2 (z), (5) 60β x zxz 3 xx + 45βzxz 2 xx α x βzx β xx zx βzxz 3 xxx = βzxγ 6 3 (z), (6) β xt + β xxxxxx + β yy + 10β x α xxxx + 10α x β xxxx + 20α xx β xxx + 20β xx α xxx + 30αxβ 2 xx + 60α x α xx β x = βzxγ 6 4 (z), (7) 45β xx zxx β xxxx zx βzxxx 2 + βzy α x β xx zx βzxα 2 xxx + 15βz xx z xxxx + 30βαxz 2 x βα x z x z xxx + βz t z x + 60α xx βz x z xx + 120α x β x z x z xx + 30α x βzxx α xx β x zx β xxx z x z xx + 60β xx z x z xxx + 6βz x z xxxxx + 60β x z xx z xxx + 30β x z x z xxxx = βzxγ 6 5 (z), (8) βz yy + βz xxxxxx + 6β x z xxxxx + 15β xxxx z xx + 15β xx z xxxx + 20β xxx z xxx + βz xt + β x z t + β t z x + 40β x z x α xxx + 60βα x α xx z x + 40α x β x z xxx + 10α x βz xxxx + 10βz x z xxxx + 60α xx β xx z x + 60α xx β x z xx + 20βα xx z xxx + 20βα xxx z xx + 40α x β xxx z x + 60α x β xx z xx + 60αxβ 2 x z x + 30αxβz 2 xx + 2β y z y = βzxγ 6 6 (z), (9) α xxxxxx + 30αxα 2 xx + 10α x α xxxx + 20α xx α xxx + α yy + α xt = βzxγ 6 7 (z), (10) 20β 2 zx 5 = βzxγ 6 8 (z), 30β 3 zx 4 = βzxγ 6 9 (z), (11) 120βxz 2 x β 2 α x zx ββ xx zx β 2 zxz 2 xxx + 90β 2 zxxz 2 x + 360ββ x zxz 2 xx = βzxγ 6 10 (z), (12) 40α x β x zx β xxx zx βz x z xx z xxx + 15βzxz 2 xxxx + 60βα x zxz 2 xx + 20βα xx zx β xx zxz 2 xx + 15βzxx β x z x zxx β x zxz 3 xxx = βzxγ 6 11 (z), (13) 80ββ x zx β 2 z xx zx 3 = βzxγ 6 12 (z), (14) 40βxz 2 x ββ xx zx ββ x zxz 2 xx = βzxγ 6 13 (z), (15) 60ββ x zx β 2 zxz 3 xx = βzxγ 6 14 (z), (16) 60α x ββ x zx ββ xxx zx β x β xx zx ββ xx z x z xx + 120ββxz 2 x z xx + 120βxz 2 x z xx + 30ββ x zxx ββ x z x z xxx = βzxγ 6 15 (z), (17) 30ββxz 2 x 2 = βzxγ 6 16 (z), 60β 2 β x zx 3 = βzxγ 6 17 (z), 30βxβ 2 xx = βzxγ 6 18 (z), (18)

3 138 Communications in Theoretical Physics Vol β 2 xα xx + 10β x β xxxx + 20β xx β xxx = βz 6 xγ 19 (z), (19) 60β 2 β x z 3 x + 30β 3 z 2 xz xx = βz 6 xγ 20 (z), 60β 2 β x z x z xx + 120ββ 2 xz 2 x = βz 6 xγ 21 (z), (20) 40βz 2 xβ xxx + 120ββ xx z x z xx + 60α x β 2 z x z xx + 120α x ββ x z 2 x + 10β 2 z x z xxxx + 30β 2 α xx z 2 x + 60ββ x z 2 xx + 20β 2 z xx z xxx + 80ββ x z x z xxx + 120β x β xx z 2 x + 120β 2 xz x z xx = βz 6 xγ 22 (z), 60z x β 3 x + 30β 2 xβz xx + 60ββ x β xx z x = βz 6 xγ 23 (z), (21) 40z xxx β 2 x + 20β xx βz xxx + 80ββ xxx z x + 20ββ xxx z xx + 60z x β 2 xx + 10ββ x z xxxx + 10ββ xxxx z x + 60α xx ββ x z x + 60α x ββ x z xx + 120β x β xx z xx + 120α x β 2 xz x = βz 6 xγ 24 (z), (22) 6β x z 5 x + 15βz 4 xz xx = βz 6 xγ 25 (z), (23) where Γ i (z) (i = 1, 2,..., 25) are functions of z to be determined later. There are three freedoms in the determination of functions α, β, z without loss of generality. Rule 1 If α(x, y, t) has the form α(x, y, t) = α 0 (x, y, t)+ β(x, y, t)q(z), then we can take Q(z) = 0 (by substituting P (z) P (z) Q(z)); Rule 2 If β(x, y, t) has the form β(x, y, t) = β 0 (x, y, t)q(z), then we can take Q(z) = 1 (by substituting Q(z) P (z)/q(z)); Rule 3 If z(x, y, t) is determined by an equation of the form Q(z) = z 0 (x, y, t), where Q(z) is any invertible function, then we can take Q(z) = z (by substituting z Q 1 (z)); Rule 4 We reserve uppercase Greek letters for undetermined functions of z so that after performing operations (differentiation, integration, exponentiation, rescaling, etc.) the result can be denoted by the same letter for simplicity. Now we shall determine the similarity reduction of the Lax-KP equation (2) using the direct method. Based on the Rule 2 and Eq. (4) β = z x, Γ 1 (z) = 10. (24) Substituting Eq. (24) into Eq. (5), we get 10z xx /z x = z x Γ 2 (z), which upon integration gives ln(z x ) + Γ 2 (z) = θ(y, t), where θ(y, t) is a function of integration. Exponentiating and integrating again gives Γ 2 (z) = xθ(y, t) + τ(y, t), with τ(y, t) is another function of integration. By Rule 3, we have z = xθ(y, t) + τ(y, t), Γ 2 (z) = 0, (25) with θ(y, t) and τ(y, t) to be determined. From Eqs. (24) and (25), we have β = θ(y, t). (26) Substituting Eqs. (25) and (26) into Eq. (6), we have 10α x = θ 2 Γ 3 (z). Integrating once and by Rule 1, we learn that α = f(y, t), Γ 3 (z) = 0, (27) with f(y, t) to be determined later. Substituting Eqs. (25), (26), and (27) into Eqs.(7) (23), we find Γ 8 (z) = 20, Γ 9 (z) = 30, Γ j (z) = 0, (j = 10,..., 25), and the undetermined functions {θ, τ, f} satisfy θ yy = θ 7 Γ 4 (z), (28) f yy = θ 7 Γ 7 (z), (29) (xθ y + τ y ) 2 + (xθ t + τ t )θ = θ 6 Γ 5 (z), (30) θθ yy x + θτ yy + 2θθ t + 2xθ 2 y + 2θ y τ y = θ 7 Γ 6 (z). (31) Since z = xθ(y, t) + τ(y, t) and f = f(y, t), the function Γ 7 (z) can only be constant in Eq. (29). Similarly, the right-hand side of Eq. (30) is a quadratic polynomial in x, consequently Γ 5 (z) = A 2 z 2 + Bz + D. (32) Substituting Eq. (32) and Γ 7 (z) = C into Eqs. (28) (31), we will obtain f yy = Cθ 7, θ y = Aθ 4, θ t + τ yy = (4A 2 τ + E B)θ 6, θ t + 2Aθ 3 τ y = (2A 2 τ + B)θ 6, θτ t + τ 2 y = (A 2 τ 2 + Bτ + D)θ 6, Γ 4 (z) = 4A 2, Γ 6 (z) = 6A 2 z + E, (33) where A, B, C, D, E are arbitrary constants. Substituting Eq. (3) with Eqs. (25) (27) into Eq. (2), the final corresponding symmetry reduction equation for P reads (A 2 z 2 + Bz + D)P + (6A 2 z + E)P + 4A 2 P + P (6) + 30P 2 P + 10P P (4) + 20P P + C = 0. When A = B = C = E = 0, from Eq. (33), we know θ = a, f = f 1 y + f 2, τ = by + Da6 b 2 t + c, a where f 1, f 2 are arbitrary functions of t and a, b, c are arbitrary constants. Thus the symmetry reduction of the Lax-KP equation has the form u = f 1 y + f 2 + ap (z), z = ax + by + Da6 b 2 t + c. a

4 No. 2 Communications in Theoretical Physics 139 Then the reduction equation becomes P (6) + 30P 2 P + 10P P (4) + 20P P (3) + DP = 0. (34) Integrating Eq. (34) once about z and supposing the integral constant to zero, we obtain P (5) + 10P P P (3) + 5P 2 + DP = 0. (35) Using the transformation U = P, Eq. (35) becomes U (4) + 10U UU + 5U 2 + DU = 0, (36) which is the same reduction equation as Ref. [15]. Using the tanh expansion method, one can obtain three types of traveling wave solutions of Eq. (36) which are listed as follows: U = 2 2 tanh 2 (z), D = 16, 5 U = tanh2 (z), D = , 5 U = tanh2 (z), D = Case 2 When z x = 0, the corresponding equation becomes (2β y z y + β x z t + βz yy )P + (β yy + β xt + 20α xxx β xx + 60α x α xx β x + 20α xx β xxx + 30α 2 xβ xx + 10α x β xxxx + 10α xxxx β x + β xxxxxx )P + 10α x α xxx + 30α 2 xα xx + 20α xx α xxx + α xxxxxx + α yy + α xt + βz 2 yp + (30β 2 xα xx + 10β x β xxxx + 20β xx β xxx + 60α x β x β xx )P β 2 xβ xx P 3 = 0. (37) In order that Eq. (37) to be an ordinary differential equation of P (z), there are two possibilities to be considered. Case 2a When z y 0, then we have the following formulae 2β y z y + β x z t + βz yy = βz 2 yr 1 (z), β yy + β xt + 20α xxx β xx + 60α x α xx β x + 20α xx β xxx + 30α 2 xβ xx + 10α x β xxxx + 10α xxxx β x + β xxxxxx = βz 2 yr 2 (z), 10α x α xxxx + 30α 2 xα xx + 20α xx α xxx + α xxxxxx + α yy + α xt = βz 2 yr 3 (z), 30α xx β 2 x + 10β x β xxxx + 20β xx β xxx + 60α x β x β xx = βz 2 yr 4 (z), 30β 2 xβ xx = βz 2 yr 5 (z), with r 1 (z), r 2 (z), r 3 (z), r 4 (z), r 5 (z) being functions of z to be determined. Since z y 0, we just consider z(y, t) = y. From 2β y = βr 1 (y) and Rule 2, we have β = β 0 (x, t). For the simplicity of reduction, we just consider β being constant and take β = 1. Then Eq. (38) becomes 10α x α xxxx + 30α 2 xα xx + 20α xx α xxx + α xxxxxx + α yy + α xt = r 3 (y), (39) and r 1 (z) = r 2 (z) = r 4 (z) = r 5 (z) = 0. So in this case, the symmetry reduction of the Lax-KP equation is u = α + P (y). Then the similarity reduction equation of the Lax-KP (2) is in the form of and the general solution of Eq. (40) is given by ( P (y) = (38) P + r 3 (y) = 0, (40) ) r 3 (y)dy dy + C 1 y + C 2, where the functions α and r 3 (y) satisfy Eq. (39) and C 1, C 2 are arbitrary constants. Case 2b When z y = 0, that is to say, z = z(t). The following equations become 30β 2 xβ xx = β x z t w 1 (z), β yy + β xt + 20α xxx β xx + 60α x α xx β x + 20α xx β xxx + 30α 2 xβ xx + 10α x β xxxx + 10α xxxx β x + β xxxxxx = β x z t w 2 (z), 10α x α xxxx + 30α 2 xα xx + 20α xx α xxx + α xxxxxx + α yy + α xt = β x z t w 3 (z), 30α xx β 2 x + 10β x β xxxx + 20β xx β xxx + 60α x β x β xx = β x z t w 4 (z), (41) with w 1 (z), w 2 (z), w 3 (z), w 4 (z) being functions of z to be determined. We take z(t) = t and α x = 0 for simplicity. Then Eq. (41) becomes α yy = β x w 3 (t), 30β 2 xβ xx = β x w 1 (t), β yy + β xt + β xxxxxx = β x w 2 (t), 10β x β xxxx + 20β xx β xxx = β x w 4 (t). (42)

5 140 Communications in Theoretical Physics Vol. 63 Solving Eq. (42), we have w 1 (t) = w 4 (t) = 0 and α = 1 6 y3 f 3 + y 2 f 4 w 3 + f 5 y + f 6, β = x(yf 3 + f 4 ) y3 (w 2 f 3 f 3t ) y2 (w 2 f 4 f 4t ) + yf 1 + f 2, where f 1, f 2, f 3, f 4, f 5, f 6, w 2 and w 3 are arbitrary functions of t. In this subcase, the similarity reduction of the Lax-KP equation will become u = 1 6 y3 f 3 + y 2 f 4 w 3 + f 5 y + f 6 + [x(yf 3 + f 4 ) y3 (w 2 f 3 f 3t ) + 1 ] 2 y2 (w 2 f 4 f 4t ) + yf 1 + f 2 P (t), (43) and the function P (t) satisfies the following equation P + w 2 P + w 3 = 0. (44) The general solution of Eq. (44) is given by ( P (t) = e w 2dt C 3 w 3 e w 2dt dt ), where C 3 is an arbitrary constant. Fig. 1 Solution of the Lax-KP equation for (43) at x = 0. In this section, we consider two special similarity reductions of the Lax-KP equation and obtain the solutions of reduction equation directly. Fixing the arbitrary constant and functions as C 3 = 0, w 2 = t, w 3 = sin t, f 1 = cos t, f 2 = 0, f 3 = sin 2t, f 4 = cos t, f 5 = 4t, f 6 = 0, and Fig. 1 is the detailed structure of the solutions of Eq. (43). 3 Conclusion In summary, we study the similarity reduction of the Lax-KP equation by the means of CK direct method. Several cases are considered for z x 0 or z x = 0 and the corresponding reduction ordinary differential equations are also obtained. A special case of the reduction equation of the Lax-KP equation is the same as the result in [15] which is obtained by the classical Lie group method. We also give the explicit expressions and the detailed structures of the general solutions for the reduced ordinary equations which include some arbitrary functions or constants. References [1] P.J. Olver, Application of Lie Groups to Differential Equation, Springer-Verlag, New York (1993). [2] H.W. Tam, W.X. Ma, X.B. Hu, and D.L. Wang, J. Phys. Soc. Jpn. 69 (2000) 45. [3] S.B. Leble and N.V. Ustinov, J. Phys. A: Math. Gen. 26 (1993) [4] R. Hirota, The Direct Method in Soliton Theory, Cambridge Unibersity Press, Cambridge (2004). [5] J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys. 24 (1983) 522. [6] G.W. Bluman and S.K. Kumei, Symmetries and Differential Equations, Springer, Berlin, Appl. Math. Sci. 81 (1989). [7] G.J. Reid, J. Phys. A: Math. Gen. 23 (1990) L853. [8] L.V. Ovsiannikov, Group Analysis of Differential Equations, (Russion Edition, NAUKA 1978), English traslation edited by W.F. Ames, Academic Press, New York (1982). [9] J.C. Chen, X.X. Peng, and Y. Chen, Commun. Theor. Phys. 62 (2014) 173. [10] Y. Jin, M. Jia, and S.Y. Lou, Commun. Theor. Phys. 58 (2012) 795. [11] S.F. Shen, Commun. Theor. Phys. 44 (2005) 964. [12] P.A. Clarkson and M.D. Kruskal, J. Math. Phys. 30 (1989) [13] A.M. Wazwaz, Appl. Math. Comput. 201 (2008) 168. [14] D. Kaya and S.M. El-Sayed, Phys. Lett. A 310 (2003) 44. [15] J.Q. Yu and T.T. Wang, Journal of Liaocheng University 22 (2009) 14.