Multiple-soliton solutions and a generalized double Wronskian determinant to the \((2+1)\)-dimensional nonlinear Schrödinger equations
© The Author(s) 2017
Received: 2 March 2017
Accepted: 7 June 2017
Published: 19 June 2017
A \((2+1)\)-dimensional nonlinear Schrödinger equation is mainly discussed. Based on the Hirota direct method and the Wronskian technique, multiple-soliton solutions and a generalized double Wronskian determinant are obtained, respectively.
One of vital aspects of soliton theory is searching exact solutions to soliton equations. Generally speaking, the Hirota bilinear method and the Wronskian technique are efficient and direct methods to construct exact solutions [11–14]. Recently, Chen et al.  extended the traditional condition equation to the arbitrary matrix equation and established the rational solutions and complexitons in terms of double Wronskian forms for the AKNS system. At present, general rational soliton solutions to various soliton equations are also discussed within the Casoratian structure and the Grammian or Pfaffian formulation [16–18].
In this paper, using the Hirota bilinear method and Chen’s method, we discuss multiple-soliton solutions and a generalized double Wronskian determinant solution to system (2), respectively. The paper is organized as follows. In Section 2, by a dependent variable transformation, system (2) is transformed into a bilinear equation. Utilizing the perturbation method, we derive soliton solutions of system (2) based on the Hirota bilinear form. In Section 3, we present a double Wronskian form of system (2) whose entries satisfy a general matrix equation. A conclusion and remarks are given in Section 4.
2 Multi-soliton solutions
2.1 One-soliton solution
2.2 Two-soliton solution
Before collision (limit \(t \to-\infty\)):
2.3 N-soliton solution
3 Generalized double Wronskian solution
For convenience of proof, we first give the following lemma.
Proof of Theorem 3.1
4 Conclusion and remarks
In summary, by using the Hirota method and the Wronskian technique, the multiple-soliton solutions and the double Wronskian form satisfying a matrix equation to system (2) have been presented, respectively. As is well known, the Wronskian technique can also be applied to construct rational solutions, positons, negatons, complexitons, and interaction solutions of the nonlinear equations. We also point out that the N-soliton solutions of system (2) may be expressed by the Grammian determinants. Therefore, there is a lot of work to be done in these directions, and it should be interesting yet difficult to construct more solution formulas for system (2).
The authors express their sincere thanks to the Referees and Editor for their valuable comments.
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- Ablowitz, MJ, Clarkson, PA: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991) View ArticleMATHGoogle Scholar
- Zhang, HQ, Tian, B, Li, LL, Xue, YS: Darboux transformation and soliton solutions for the \((2+1)\)-dimensional nonlinear Schrödinger hierarchy with symbolic computation. Physica A 388(1), 9-20 (2009) MathSciNetView ArticleGoogle Scholar
- Maruno, KI, Ohta, Y: Localized solitons of a \((2 +1)\)-dimensional nonlocal nonlinear Schrödinger equation. Phys. Lett. A 372(24), 4446-4450 (2008) View ArticleMATHGoogle Scholar
- Zhu, HJ, Chen, YM, Song, SH, Hu, HY: Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations. Appl. Numer. Math. 61(3), 308-321 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Malomed, B, Torner, L, Wise, F, Mihalache, D: On multidimensional solitons and their legacy in contemporary atomic, molecular and optical physics. J. Phys. B, At. Mol. Opt. Phys. 49, 170502 (2016) View ArticleGoogle Scholar
- Malomed, BA: Multidimensional solitons: well-established results and novel findings. Eur. Phys. J. Spec. Top. 225(13), 2507-2532 (2016) View ArticleGoogle Scholar
- Mihalache, D: Localized structures in nonlinear optical media: a selection of recent studies. Rom. Rep. Phys. 67(4), 1383-1400 (2015) Google Scholar
- Bagnato, VS, Frantzeskakis, DJ, Kevrekidis, PG, Malomed, BA, Mihalache, D: Bose-Einstein condensation: twenty years after. Rom. Rep. Phys. 67(1), 5-50 (2015) Google Scholar
- Ruan, HY, Chen, YX: Symmetries and dromion solution of a \((2+1)\)-dimensional nonlinear Schrödinger equation. Acta Phys. Sin. 8(4), 241-251 (1999) Google Scholar
- Zhang, JF: Generalized dromions of the \((2+1)\)-dimensional nonlinear Schrödinger equations. Commun. Nonlinear Sci. Numer. Simul. 6(1), 50-53 (2001) MathSciNetView ArticleMATHGoogle Scholar
- Ma, WX, You, Y: Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions. Trans. Am. Math. Soc. 357(5), 1753-1778 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Ma, WX, Li, CX, He, JS: A second Wronskian formulation of the Boussinesq equation. Nonlinear Anal. TMA 70(12), 4245-4258 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Ma, WX: Complexiton solutions to integrable equations. Nonlinear Anal. 63, e2461 (2005) View ArticleMATHGoogle Scholar
- Ma, WX: Complexiton solutions to the Korteweg-de Vries equation. Phys. Lett. A 301(1-2), 35-44 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Chen, DY, Zhang, DJ, Bi, JB: New double Wronskian solutions of the AKNS equation. Sci. China Math. 51(1), 55-69 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Chen, S, Grelu, P, Mihalache, D, Baronio, F: Families of rational soliton solutions of the Kadomtsev-Petviashvili I equation. Rom. Rep. Phys. 68(4), 1407-1424 (2016) Google Scholar
- Ma, WX, You, Y: Rational solutions of the Toda lattice equation in Casoratian form. Chaos Solitons Fractals 22, 395-406 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Ohta, Y, Yang, JK: Dynamics of rogue waves in the Davey-Stewartson II equation. J. Phys. A, Math. Theor. 46, 105202 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Hirota, R: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004) View ArticleMATHGoogle Scholar
- Freeman, NC, Nimmo, JJC: Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: the Wronskian technique. Phys. Lett. A 95(1), 1-3 (1983) MathSciNetView ArticleMATHGoogle Scholar
- Nimmo, JJC, Freeman, NC: A method of obtaining the N-soliton solution of the Boussinesq equation in terms of a Wronskian. Phys. Lett. A 95(1), 4-6 (1983) MathSciNetView ArticleGoogle Scholar