## THERMAL SCIENCE

International Scientific Journal

### RATIONAL SOLUTION TO A SHALLOW WATER WAVE-LIKE EQUATION

**ABSTRACT**

Two classes of rational solutions to a shallow water wave-like non-linear differential equation are constructed. The basic object is a generalized bilinear differential equation based on a prime number, p = 3. Through this new transformation and with the help of symbolic computation with MAPLE, both the new equation and its rational solutions are obtained.

**KEYWORDS**

PAPER SUBMITTED: 2015-11-01

PAPER REVISED: 2015-12-10

PAPER ACCEPTED: 2016-02-01

PUBLISHED ONLINE: 2016-08-13

**THERMAL SCIENCE** YEAR

**2016**, VOLUME

**20**, ISSUE

**3**, PAGES [875 - 880]

- Zhang, Y., Ma, W. X., Rational Solutions to a KdV-like Equation, Applied Mathematics and Computation, 256 (2015), 1, pp. 252-256
- Ma, W. X., Trilinear Equations, Bell Polynomials, and Resonant Solutions, Frontiers of Mathematics in China, 8 (2013), 5, pp. 1139-1156
- Ma, W. X., Lump Solutions to the Kadomtsev-Petviashvili Equation, Physics Letters A, 379 (2015), 36, pp. 1975-1978
- Ma, W. X., You, Y., Solving the Korteweg-de Vries Equation by its Bilinear Form: Wronskian Solutions, Transactions of the American Mathematical Society, 357 (2005), 5, pp. 1753-1778
- He, J.-H., Li, Z. B., Converting Fractional Differential Equations into Partial Differential Equations, Thermal Science, 16 (2012), 2, pp. 331-334
- Ma, H. C., et al., Lie Symmetry and Exact Solution of (2+1)-Dimensional Generalized KP Equation with Variable Coefficients, Thermal Science, 17 (2013), 5, pp. 1490-1493
- Ma, H. C., et al., Exact Solutions of Non-Linear Fractional Partial Differential Equations by Fractional Sub-Equation Method, Thermal Science, 19 (2015), 4, pp. 1239-1244
- Hu, X. B., Wu, Y. T., Application of the Hirota Bilinear Formalism to a New Integrable Differential-Difference Equation, Physics Letters A, 246 (1998), 6, pp. 523-529
- Wang, X., et al., Generalized Darboux Transformation and Localized Waves in Coupled Hirota Equations, Wave Motion, 51 (2014), 7, pp. 1149-1160
- Elwakil, S. A., et al., Exact Travelling Wave Solutions for the Generalized Shallow Water Wave Equation, Chaos, Solitons & Fractals, 17 (2003), 1, pp. 121-126
- Ma, W. X., You, Y., Rational Solutions of the Toda Lattice Equation in Casoratian form, Chaos Solitons & Fractals, 22 (2004), 2, pp. 395-406
- Hietarinta, J., A Search for Bilinear Equations Passing Hirota's Three-Soliton Condition. IV. Complex Bilinear Equations, Journal of Mathematical Physics, 29 (1988), 3, pp. 628-635
- Wadati, M., Introduction to Solitons, Pramana, 57 (2001), 5-6, pp. 841-847
- Bagchi, B., et al., New Exact Solutions of a Generalized Shallow Water Wave Equation, Physica Scripta, 82 (2010), 2, pp. 1485-1502
- Hirota, R., The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, UK, 2004
- Thacker, W. C., Some Ex act Solutions to the Nonlinear Shallow-Water Wave Equations, Journal of Fluid Mechanics, 107 (1981), 6, pp. 499-508
- Free man, N. C., Nimmo, J. J. C., Soliton Solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili Equations: The Wronskian Technique, Physics Letters A, 95 (1983), 1, pp. 1-3
- Zhang, H., New Exact Travelling Wave Solutions for Some Nonlinear Evolution Equations, Chaos Solitons & Fractals, 26 (2005), 4, pp. 921-925
- Ma, W. X., Bilinear Equa tions, Bell Polynomials and Linear Superposition Principle, Journal of Physics Conference Series, 411 (2013), 1, pp. 594-597
- Ma, W. X., Generalized Bilinear Differential Equations, Studies in Nonlinear Sciences, 2 (2011), 4, pp. 140-144