## THERMAL SCIENCE

International Scientific Journal

### LUMP SOLUTIONS TO THE (2+1)-DIMENSIONAL SHALLOW WATER WAVE EQUATION

**ABSTRACT**

Through symbolic computation with MAPLE, a class of lump solutions to the (2+1)-D shallow water wave equation is presented, making use of its Hirota bi-linear form. The resulting lump solutions contain six free parameters, two of which are due to the translation invariance of the (2+1)-D shallow water wave equation and the other four of which satisfy a non-zero determinant condition guaranteeing analyticity and rational localization of the solutions.

**KEYWORDS**

PAPER SUBMITTED: 2016-08-16

PAPER REVISED: 2016-08-29

PAPER ACCEPTED: 2016-09-18

PUBLISHED ONLINE: 2017-09-09

**THERMAL SCIENCE** YEAR

**2017**, VOLUME

**21**, ISSUE

**4**, PAGES [1765 - 1769]

- Ma, H. C., et al., Rational Solution to a Shallow Water Wave-Like Equation, Thermal Science, 20 (2016), 2, pp. 875-880
- Ma, H. C., et al., Rational Solutions to an Caudrey-Dodd-Gibbon-Sawada-Kotera-Like Equation, Ther-mal Science, 20 (2016), 2, pp. 871-874
- Ma, H. C., Deng, A. P., Lump Solution of (2+1)-Dimensional Boussinesq Equation, Communications in Theoretical Physics, 65 (2016), 2, pp. 546-552
- Ma, W. X., You, Y., Solving the Korteweg-de Vries Equation by its Bilinear Form: Wronskian Solu-tions, Transactions of the American Mathematical Society, 357 (2005), 2, pp. 1753-1778
- Ma, W. X., et al., A Second Wronskian Formulation of the Boussinesq Equation, Nonlinear Analysis, 70 (2009), 2, pp. 4245-4258
- Tamizhmani, T., et al., Wronskian and Rational Solutions of the Differential-Difference KP Equation, Journal of Physics A General Physics, 31 (1998), 2, pp. 7627-7633
- Hu, X. B., Tam, H. W., Some Recent Results on Integrable Bilinear Equations, Journal of Nonlinear Mathematical Physics, 8 (2001), 2, pp. 149-155
- Hu, X. B., et al., Hirota Bilinear Approach to a New Integrable Differential-Difference System, Journal of Mathematical Physics 40 (1999), 2, pp. 2001-2010
- Liu, P., Lou, S., A (2+1)-Dimensional Displacement Shallow Water Wave System, Chinese Physics Let-ters, 25 (2008), 2, pp. 3311-3314
- Wei, H., et al., The Explicit Solutions of a Shallow Wave Equation, Journal of Zhoukou Normal Univer-sity, 29 (2012), 2, pp. 23-31
- Ma, W. X., You, Y., Rational Solutions of the Toda Lattice Equation in Casoratian Form, Chaos Solitons & Fractals, 22 (2004), 2, pp. 395-406
- Zhang, Y., Ma, W. X., A Study on Rational Solutions to a KP-like Equation, Zeitschrift Naturforschung, Teil A, 70 (2015), 2, pp. 263-268
- Wadati, M., Introduction to Solitons, Pramana, 57 (2001), 2, pp. 841-847
- Chakravarty, S., Kodama, Y., Soliton Solutions of the KP Equation and Application to Shallow Water Waves, Studies in Applied Mathematics, 123 (2009), 2, pp. 83-151
- Hirota, R., The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, UK, 2004
- Ye, C., Zhang, W., New Explicit Solutions for (2+1)-Dimensional Soliton Equation, Chaos Solitons & Fractals, 44 (2011), 2, pp. 1063-1069
- Zhang, Y., et al., Wronskian and Grammian Solutions for (2+1)-Dimensional Soliton Equation, Com-munications in Theoretical Physics, 55 (2011), 2, pp. 20-24
- Xie, Z., Zhang, H. Q., New Soliton-Like Solutions for (2+1)-Dimensional Breaking Soliton Equation, Communications in Theoretical Physics ,43 (2005), 2, pp. 401-406
- Ma, W. X., Bilinear Equations, Bell Polynomials and Linear Superposition Principle, Journal of Physics Conference Series, 411 (2013), ID 012021
- Thacker, W.C., Some Exact Solutions to the Nonlinear Shallow-Water Wave Equations, Journal of Flu-id Mechanics, 107 (1981), 2, pp. 499-508