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Supersymmetric Ito Equation: Bosonization and Exact Solutions Bo Ren, Ji Lin, and Jun Yu

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Supersymmetric Ito Equation: Bosonization and Exact Solutions Bo Ren, Ji Lin, and Jun Yu Supersymmetric Ito equation: Bosonization and exact solutions Bo Ren, Ji Lin, and Jun Yu Citation: AIP Advances 3, 042129 (2013); doi: 10.1063/1.4802969 View online: http://dx.doi.org/10.1063/1.4802969 View Table of Contents: http://aipadvances.aip.org/resource/1/AAIDBI/v3/i4 Published by the American Institute of Physics. Additional information on AIP Advances Journal Homepage: http://aipadvances.aip.org Journal Information: http://aipadvances.aip.org/about/journal Top downloads: http://aipadvances.aip.org/most_downloaded Information for Authors: http://aipadvances.aip.org/authors Downloaded 20 Apr 2013 to 117.147.163.195. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ AIP ADVANCES 3, 042129 (2013) Supersymmetric Ito equation: Bosonization and exact solutions Bo Ren,1 Ji Lin,2 and Jun Yu1,a 1Institute of Nonlinear Science, Shaoxing University, Shaoxing 312000, China 2Institute of Nonlinear Physics, ZheJiang Normal University, Jinhua, 321004, China (Received 13 September 2012; accepted 10 April 2013; published online 19 April 2013) Based on the bosonization approach, the N = 1 supersymmetric Ito (sIto) system is changed to a system of coupled bosonic equations. The approach can effectively avoid difficulties caused by intractable fermionic fields which are anticommuting. By solving the coupled bosonic equations, the traveling wave solutions of the sIto system are obtained with the mapping and deformation method. Some novel types of exact solutions for the supersymmetric system are constructed with the solu- tions and symmetries of the usual Ito equation. In the meanwhile, the similarity reduction solutions of the model are also studied with the Lie point symmetry theory. Copyright 2013 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4802969] I. INTRODUCTION The study of supersymmetric integrable systems has assumed great importance in the past few years. It has played a more and more important role in physics and mathematics, especially, in modern particle physics and quantum mechanics.1, 2 Over the last three decades, a number of important soliton systems, such as Korteweg-de Vries, Sine-Gorden, KP hierarchy and nonlinear Schordinger¨ equation have been embedded into their supersymmetric counterparts.3–8 Though the supersymmetric integrable systems have been studied by many authors in the both quantum and classical levels, many important problems are still open.9 How to find a proper bosonization procedure is an important problem for quantum and classical supersymmetric integrable models.10 Recently, a simple bosonization approach to treat the super integrable systems has been proposed.11, 12 One essential advantage of the method is that it can effectively avoid difficulties caused by intractable fermionic fields. In this letter, we shall use the bosonization approach on the N = 1 supersymmetric Ito (sIto) system. And then many kinds of solutions for the sIto system are obtained through the coupled bosonic equations. The well known Ito equation is utt + 6uxxut + 6ux uxt + uxxxt = 0, (1) which was first proposed by Ito, and its bilinear Backlund¨ transformation, Lax pair and multi-soliton solutions were obtained.13 Since Ito equation possesses typical properties of a soliton equation, a great deal of research works on the Ito equation have been carried out. The integrable properties of this equation such as the nonlinear superposition formula, Kac-Moody algebra, bi-Hamiltonian structure have been further found.14–16 Recently, the supersymmetric version of the Ito equation and its one, two and three soliton solutions were obtained.17 The N = 1 sIto system reads as17 Dt t + 6(x (Dt ))x + Dt xxx = 0, (2) aElectronic mail: [email protected]. 2158-3226/2013/3(4)/042129/12 3, 042129-1 C Author(s) 2013 Downloaded 20 Apr 2013 to 117.147.163.195. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ 042129-2 Ren, Lin, and Yu AIP Advances 3, 042129 (2013) where Dt = ∂θ + θ∂t is the covariant derivative. It is established with usual spatial variable x and super temporal variables (t, θ), where θ is a Grassmann variable, and the field u leads to a fermionic superfield (θ, x, t) = ξ(x, t) + θu(x, t). The component version of (2) reads as utt + 6uxxut + 6ux uxt + uxxxt + 6(ξx ξ)x = 0, (3a) ξt + 6ξx ux + 6uxxξ + ξxxx = 0, (3b) where u and ξ are bosonic and fermionic component fields, respectively. When the fermionic sectors vanish, the supersymmetric system degenerates to the known classical one. The structure of this paper is organized as follows. In section II, we shall use the bosonization approach on the N = 1 sIto system by expanding the superfields with introducing two fermionic parameters. And then the exact solutions of the model are found using the mapping and deformation method and Lie point symmetry. In Sections III and IV, we extend the bosonization approach for the sIto system to the case of three fermionic parameters and N fermionic parameters, respectively. The exact solutions are obtained with the same methods. The last section is a simple summary and discussion. II. TWO FERMIONIC PARAMETERS BOSIONIZATION AND ITS SOLUTIONS A. Bosonization approach with two fermionic parameters The supersymmetric extension of the Ito equation is a system of coupled equations for a commut- ing and an anticommuting field. In order to avoid the difficulties in dealing with the anticommutative fermionic field of the supersymmetric equations, we expand the component fields ξ and u with the two fermionic parameters11, 12 ξ(x, t) = p1ζ1 + p2ζ2, (4a) u(x, t) = u0 + u12ζ1ζ2, (4b) where ζ 1 and ζ 2 are two Grassmann parameters, while the coefficients p1 = p1(x, t), p2 = p2(x, t), u0 = u0(x, t) and u12 = u12(x, t) are four usual real or complex functions with respect to the spacetime variables x and t. Then, substituting (4) into the Ito system (3), we get the equations u0,tt + u0,xxxt + 6u0,xxu0,t + 6u0,x u0,xt = 0, (5a) p1,t + p1,xxx + 6p1,x u0,x + 6p1u0,xx = 0, (5b) p2,t + p2,xxx + 6p2,x u0,x + 6p2u0,xx = 0, (5c) u12,tt + u12,xxxt + 6u12,xxu0,t + 6u0,xxu12,t + 6(u0,x u12,x )t = 6(p1 p2,xx − p2 p1,xx). (5d) The above way is just the bosonic procedure for the sIto system (3) with two fermionic parameters. Equation (5a) is exactly the usual Ito equation which has been widely studied.14–16 Equations (5b) and (5c) are linear homogeneous in p1 and p2, respectively. Equation (5d) is linear nonhomogeneous in u12. These equations which are usual pure bosonic systems can be easily solved in principle. This is just one of the advantages of the bosonization approach. B. Traveling wave solutions with mapping and deformation method Now let us consider the traveling wave solutions of the bosonic (5). Introducing the traveling wave variable X = kx + ωt + c0 with constants k, ω and c0, (5) is transformed to the ordinary Downloaded 20 Apr 2013 to 117.147.163.195. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ 042129-3 Ren, Lin, and Yu AIP Advances 3, 042129 (2013) differential equations (ODEs) 3 2 k u0,XXXX + (ω + 12k u0,X )u0,XX = 0, (6a) 3 2 2 k p1,XXX + ωp1,X + 6k p1,X u0,X + 6k p1u0,XX = 0, (6b) 3 2 2 k p2,XXX + ωp2,X + 6k p2,X u0,X + 6k p2u0,XX = 0, (6c) 3 2 2 2 k ωu12,XXXX + ω u12,XX + 12k ω(u0,X u12,X )X = 6k p1 p2,XX − p2 p1,XX . (6d) As the well know exact solutions of (6a), we try to build the mapping and deformation relation- ship between the traveling wave solutions of the usual Ito equation and the sIto equation, then to get the exact solutions of the sIto equation with the known solutions of Ito equation. At first, we solve out u0, X from (6a) 2 ω u0 u , =− − . (7) 0 X k 4k2 To simplify (6d), we integrate the linear inhomogeneous equation once 3 2 2 2 k ωu12,XXX + (ω + 12k ωu0,X )u12,X = 6k (p1 p2,X − p2 p1,X ) + c1, (8) where c1 is the integral constant. In order to get the mapping relationship between p1, p2, u12 and u0, we introduce the variable transformations as follows p1(X) = P1(u0(X)), p2(X) = P2(u0(X)), u12(X) = U12(u0(X)). (9) Using the transformation (9) and vanishing u0, X via (7), the linear ODEs (6b-6c) as well as (8) become d3 P d2 P 2 4 + ω 2 + ω2 1 + 2 3 + ω 1 − 2 = , 16k u0 8k u0 3 96k u0 24k u0 2 192k u0 P1 0 (10a) du0 du0 d3 P d2 P 2 4 + ω 2 + ω2 2 + 2 3 + ω 2 − 2 = , 16k u0 8k u0 3 96k u0 24k u0 2 192k u0 P2 0 (10b) du0 du0 d3U d2U dU 2 + ω 12 + 12 − 12 = , (4ku0 ) 3 24ku0 2 24k F(u0) (10c) du0 du0 du0 where 96k3 dP dP 64k3c F(u ) = 2 P − 1 P + 1 . 0 2 2 1 2 2 4ku ω + ω du0 du0 ω 2 + ω 0 4ku0 The mapping and deformation relations are constructed via (10) √ u0 c2 2 2 1 2 P = + 64c k ω kωu , dy + 4c k u , , (11a) 1 2 3 0 X 2 3 4 0 X 4k u0,X (4ky + ω) √ u0 c5 2 2 1 2 P = + 64c k ω kωu , dy + 4c k u , , (11b) 2 2 6 0 X 2 3 7 0 X 4k u0,X (4ky + ω) √ u0 1 120ku3 + 26ωu U = c 6 kω 20kωu2 + ω dy + 0 0 + c 20ku2 + ω , (11c) 12 8 0 2 + ω 2 + ω 9 0 4ky 4ku0 where ci (i = 2,3,...,9)arearbitraryconstants.
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