## THERMAL SCIENCE

International Scientific Journal

### BILINEARIZATION AND FRACTIONAL SOLITON DYNAMICS OF FRACTIONAL KADOMTSEV-PETVIASHVILI EQUATION

**ABSTRACT**

Kadomtsev-Petviashvili equation is a mathematical model with many important applications in fluids. In this paper, a local fractional Kadomtsev-Petviashvili equation with Lax integrability is derived and solved by extending Hirota’s bilinear method. More specifically, the local fractional Kadomtsev-Petviashvili equation is derived from a local fractional Lax equation. With the help of a suitable transformation, the local fractional Kadomtsev-Petviashvili equation is then bilinearized. Based on the bilinearized form, n-soliton solution with Mittag-Leffler functions is obtained. In order to gain more insights into the fractional n-soliton solution, the velocity of the fractional one-soliton solution is simulated. It is shown that the velocity of the fractional one-soliton changes with the fractional order.

**KEYWORDS**

PAPER SUBMITTED: 2018-08-15

PAPER REVISED: 2018-11-21

PAPER ACCEPTED: 2019-01-24

PUBLISHED ONLINE: 2019-05-26

**THERMAL SCIENCE** YEAR

**2019**, VOLUME

**23**, ISSUE

**Issue 3**, PAGES [1425 - 1431]

- Fujioka, J., et al., Fractional Optical Solitons, Physics Letters A, 374 (2010), 9, pp. 1126-1134
- Yang, X. J., et al., Modelling Fractal Waves on Shallow Water Surfaces via Local Fractional Korteweg-de Vries Equation, Abstract and Applied Analysis, 2014 (2014), ID 278672
- Hirota, R., Exact Solution of the sine-Gordon Equation for Multiple Collisions of Solitons, Journal of P hysical Society of Japan, 33 (1972), 5, pp. 1459-1463
- Hirota, R., Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons, Physics Review Letters, 27 (1971), 18, pp. 1192-1194
- Ablowitz, M. J., Clarkson, P. A., Solitons, Non-Linear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, Mass., USA, 1991
- Mcarthur, I. N., Yung, C. M., Hirota Bilinear Form for the Super-KdV Hierarchy, Modern Physics Letters A, 8 (1993), 18, pp. 1739-1745.
- Ma, W. X., You, Y. C., Solving the Korteweg-de Vries Equation by Its Bilinear Form: Wronskian Solutions, Transactions of the American Mathematical Society, 357 (2005), 5, pp. 1753-1778
- Chen, D. Y., Introduction to Soliton (in Chinese), Science Press, Beijing, China, 2006
- Wazwaz, A.M.: The Hirota's Bilinear Method and the Tanh-Coth Method for Multiple-Soliton Solutions of the Sawada-Kotera-Kadomtsev-Petviashvili Equation, Applied Mathematics and Computation, 200 (2008), 1, pp. 160-166
- Zhang, S., Cai, B., Multi-Soliton Solutions of a Variable-Coefficient KdV Hierarchy, Nonlinear Dynamics, 78 (2014), 3, pp. 1593-1600
- Zuo, D. W, et al., Multi-Soliton Solutions for the Three-Coupled KdV Equations Engendered by the Neumann System, Nonlinear Dynamics, 75 (2014), 4, pp. 701-708
- Zhang, S. Wang, Z. Y., Bilinearization and New Soliton Solutions of Whitham-Broer-Kaup Equations with Time-Dependent Coefficients, Journal of Nonlinear Sciences and Applications, 10 (2017), 5, pp. 2324-2339
- Zhang, S., Gao, X. D., Analytical Treatment on a New Generalized Ablowitz-Kaup-Newell-Segur Hierarchy of Thermal and Fluid Equations, Thermal Science, 21 (2017), 4, pp. 1607-1612
- Zhang, S., et al., New Multi-Soliton Solutions of Whitham-Broer-Kaup Shallow-Water-Wave Equations, Thermal Science, 21 (2017), Suppl.1, pp. S137-S144
- Yang, X. J., et al., Local Fractional Integral Transforms and their Applications, Elsevier, London, UK, 2015
- Yang, X. J.. Srivastava, H. M., An Asymptotic Perturbation Solution for a Linear Oscillator of Free Damped Vibrations in Fractal Medium Described by Local Fractional Derivatives, Communications in Nonlinear Science and Numerical Simulation, 29 (2015), 1-3, pp. 499-504
- Yang, X. J., et al., On Exact Traveling-Wave Solutions for Local Fractional Korteweg-de Vries Equation, Chaos, 26 (2016), 8, ID 084312
- Yang, X. J., et al., Exact Travelling Wave Solutions for the Local Fractional Two-Dimensional Burgers-Type Equations, Computers and Mathematics with Applications, 73 (2017), 2, pp. 203-210
- Yang, X. J., et al., On a Fractal LC-Electric Circuit Modeled by Local Fractional Calculus, Communications in Nonlinear Science and Numerical Simulation, 47 (2017), 6, pp. 200-206
- Hu, Y., He, J. H, On Fractal Space-Time and Fractional Calculus, Thermal Science, 20, (2016), 3, pp. 773-777
- He, J. H., Fractal Calculus and its Geometrical Explanation, Results in Physics, 10 (2018), 1, pp. 272-276
- Zhang, S., Zhang, H. Q., Fractional Sub-Equation Method and its Applications to Non-linear Fractional PDEs, Physics Letters A, 375 (2011), 7, pp. 1069-1073
- Zhang, S., et al., Exact Solutions of Time Fractional Heat-Like and Wave-Like Equations with Variable Coefficients, Thermal Science, 20 (2016), Suppl.3, pp. S689-S693
- Zhang, S., Hong, S. Y., Variable Separation Method for a Non-Linear Time Fractional Partial Differential Equation with Forcing Term, Journal of Computational and Applied Mathematics, 339 (2018), 1, pp. 297-305
- Garder, C. S., et al., Method for Solving the Korteweg-de Vries Equation, Physical Review Letters, 19 (1967), 19, pp. 1095-1097
- Wang, M. L., Exact Solutions for a Compound KdV-Burgers Equation, Physics Letters A, 213 (1996), 5-6, pp. 279-287
- Fan, E. G., Travelling Wave Solutions in Terms of Special Functions for Non-linear Coupled Evolution Systems, Physics Letters A, 300 (2002), 2-3, pp. 243-249
- He, J. H., Wu, X. H., Exp-Function Method for Non-Linear Wave Equations, Chaos, Solitons and Fractals, 30 (2006), 3, pp. 700-708
- Zhang, S., Xia, T. C., A Generalized F-expansion Method and New Exact Solutions of Konopelchenko-Dubrovsky Equations, Applied Mathematics and Computation, 183 (2006), 3, pp. 1190-1200