Solitons, Bäcklund Transformation, Lax Pair, and Infinitely Many Conservation Law for a (2+1)-Dimensional Generalised Variable-Coefficient Shallow Water Wave Equation
-
Zhong-Zhou Lan
,
Jin-Wei Yang
Abstract
Under investigation in this article is a (2+1)-dimensional generalised variable-coefficient shallow water wave equation, which describes the interaction of the Riemann wave propagating along the y axis with a long-wave propagating along the x axis in a fluid, where x and y are the scaled space coordinates. Bilinear forms, Bäcklund transformation, Lax pair, and infinitely many conservation law are derived based on the binary Bell polynomials. Multi-soliton solutions are constructed via the Hirota method. Propagation and interaction of the solitons are illustrated graphically: (i) variable coefficients affect the shape of the multi-soliton interaction in the scaled space and time coordinates. (ii) Positions of the solitons depend on the sign of wave numbers after each interaction. (iii) Interaction of the solitons is elastic, i.e. the amplitude, velocity, and shape of each soliton remain invariant after each interaction except for a phase shift.
Acknowledgments
We express our sincere thanks to all the members of our discussion group for their valuable comments. This work has been supported by the National Natural Science Foundation of China under grant no. 11272023 and by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) under grant no. IPOC2013B008.
References
[1] H. L. Zhen, B. Tian, Y. F. Wang, and D. Y. Liu, Phys. Plasmas 22, 032307 (2015).10.1063/1.4913668Search in Google Scholar
[2] H. L. Zhen, B. Tian, Y. Sun, and J. Chai, Phys. Plasmas 22, 102304 (2015).10.1063/1.4932076Search in Google Scholar
[3] W. R. Sun, B. Tian, Y. Jiang, and H. L. Zhen, Phys. Rev. E 91, 023205 (2015).10.1103/PhysRevE.91.023205Search in Google Scholar
[4] W. R. Sun, B. Tian, H. L. Zhen, and Y. Sun, Nonlinear Dyn. 81, 725 (2015).10.1007/s11071-015-2022-4Search in Google Scholar
[5] X. Y. Xie, B. Tian, W. R. Sun, and Y. Sun, Nonlinear Dyn. 81, 1349 (2015).10.1007/s11071-015-2073-6Search in Google Scholar
[6] X. Y. Xie, B. Tian, W. R. Sun, Y. Sun, and D. Y. Liu, J. Mod. Opt. 62, 1374 (2015).10.1080/09500340.2015.1039944Search in Google Scholar
[7] A. R. Adem and C. M. Khalique, Comput. Fluids 81, 10 (2013).10.1016/j.compfluid.2013.04.005Search in Google Scholar
[8] A. Bekir, Chaos. Solitons. Fract. 32, 449 (2007).10.1016/j.chaos.2006.06.047Search in Google Scholar
[9] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York 1991.10.1017/CBO9780511623998Search in Google Scholar
[10] R. Hirota, Prog. Theor. Phys. 52, 1498 (1974).10.1143/PTP.52.1498Search in Google Scholar
[11] R. Hirota, The Direct Method in Soliton Theory, Springer, Berlin 1980.10.1007/978-3-642-81448-8_5Search in Google Scholar
[12] R. Hirota and Y. Ohta, I. J. Phys. Soc. Jpn. 60, 798 (1991).10.1143/JPSJ.60.798Search in Google Scholar
[13] R. Hirota, Phys. Rev. Lett. 27, 1192 (1971).10.1103/PhysRevLett.27.1192Search in Google Scholar
[14] R. Hirota, X. B. Hu, and X. Y. Tang, J. Math. Anal. Appl. 288, 326 (2003).Search in Google Scholar
[15] V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin 1991.10.1007/978-3-662-00922-2Search in Google Scholar
[16] M. Wadati, J. Phys. Soc. Jpn. 38, 673 (1975).10.1143/JPSJ.38.673Search in Google Scholar
[17] F. Caruello and M. Tabor, Phys. D 39 77 (1989).10.1016/0167-2789(89)90040-7Search in Google Scholar
[18] N. C. Freeman and J. J. Nimmo, Phys. Lett. A 95 1 (1983).10.1016/0375-9601(83)90764-8Search in Google Scholar
[19] E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its, and V. B. Matveev, Algebro-Geometrical Approach to Nonlinear Integrable Equations, Springer, Berlin, 1994.Search in Google Scholar
[20] B. Tian and Y. T. Gao, Eur. Phys. J. D 33, 243 (2005).10.1140/epjd/e2005-00037-5Search in Google Scholar
[21] Y. H. Wang and Y. Chen, Chin. Phys. B 22, 050509 (2013).10.1088/1674-1056/22/5/050509Search in Google Scholar
[22] L. Delisle and M. Mosaddeghi, J. Phys. A 46, 115203 (2013).10.1088/1751-8113/46/11/115203Search in Google Scholar
[23] X. Ceng and C. Cao, Chaos, Solitons Fract. 22, 683 (2004).10.1016/j.chaos.2004.02.025Search in Google Scholar
[24] Z. Y. Yan and H. Q. Zhang, Comput. Math. Appl. 44, 1439 (2002).Search in Google Scholar
[25] W. X. Ma, R. Zhou, and L. Gao, Mod. Phys. Lett. A 24, 1677 (2009).10.1142/S0217732309030096Search in Google Scholar
[26] B. Tian, K. Y. Zhao, and Y. T. Gao, Int. J. Eng. 35, 1081 (1997).10.1016/S0020-7225(97)00001-3Search in Google Scholar
[27] E. T. Bell, Ann. Math. 35, 258 (1934).10.2307/1968431Search in Google Scholar
[28] F. Lambert, I. Loris, J. Springael, and R. Willox, J. Phys. A 27, 5325 (1994).10.1088/0305-4470/27/15/028Search in Google Scholar
[29] F. Lambert and J. Springael, Chaos, Solitons Fract. 12, 2821 (2001).10.1016/S0960-0779(01)00096-0Search in Google Scholar
©2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Theoretical Investigations on the Elastic and Thermodynamic Properties of Rhenium Phosphide
- Lax Pair, Conservation Laws, Solitons, and Rogue Waves for a Generalised Nonlinear Schrödinger–Maxwell–Bloch System under the Nonlinear Tunneling Effect for an Inhomogeneous Erbium-Doped Silica Fibre
- Effect of Trace Fe3+ on Luminescent Properties of CaWO4: Pr3+ Phosphors
- Rogue-Wave Interaction of a Nonlinear Schrödinger Model for the Alpha Helical Protein
- Multi-Scale Long-Range Magnitude and Sign Correlations in Vertical Upward Oil–Gas–Water Three-Phase Flow
- Theoretical Study of Geometries, Stabilities, and Electronic Properties of Cationic (FeS)n+ (n = 1–5) Clusters
- Explanation of the Quantum-Mechanical Particle-Wave Duality through the Emission of Watt-Less Gravitational Waves by the Dirac Equation
- Closed Analytical Solutions of the D-Dimensional Schrödinger Equation with Deformed Woods–Saxon Potential Plus Double Ring-Shaped Potential
- Solitons, Bäcklund Transformation, Lax Pair, and Infinitely Many Conservation Law for a (2+1)-Dimensional Generalised Variable-Coefficient Shallow Water Wave Equation
- The Non-Alignment Stagnation-Point Flow Towards a Permeable Stretching/Shrinking Sheet in a Nanofluid Using Buongiorno’s Model: A Revised Model
- Rapid Communication
- Extrinsic and Intrinsic Contributions to Plasmon Peaks in Solids
Articles in the same Issue
- Frontmatter
- Theoretical Investigations on the Elastic and Thermodynamic Properties of Rhenium Phosphide
- Lax Pair, Conservation Laws, Solitons, and Rogue Waves for a Generalised Nonlinear Schrödinger–Maxwell–Bloch System under the Nonlinear Tunneling Effect for an Inhomogeneous Erbium-Doped Silica Fibre
- Effect of Trace Fe3+ on Luminescent Properties of CaWO4: Pr3+ Phosphors
- Rogue-Wave Interaction of a Nonlinear Schrödinger Model for the Alpha Helical Protein
- Multi-Scale Long-Range Magnitude and Sign Correlations in Vertical Upward Oil–Gas–Water Three-Phase Flow
- Theoretical Study of Geometries, Stabilities, and Electronic Properties of Cationic (FeS)n+ (n = 1–5) Clusters
- Explanation of the Quantum-Mechanical Particle-Wave Duality through the Emission of Watt-Less Gravitational Waves by the Dirac Equation
- Closed Analytical Solutions of the D-Dimensional Schrödinger Equation with Deformed Woods–Saxon Potential Plus Double Ring-Shaped Potential
- Solitons, Bäcklund Transformation, Lax Pair, and Infinitely Many Conservation Law for a (2+1)-Dimensional Generalised Variable-Coefficient Shallow Water Wave Equation
- The Non-Alignment Stagnation-Point Flow Towards a Permeable Stretching/Shrinking Sheet in a Nanofluid Using Buongiorno’s Model: A Revised Model
- Rapid Communication
- Extrinsic and Intrinsic Contributions to Plasmon Peaks in Solids
