Exp-function Method and Reduction Transformations for Rogue ‎Wave Solutions of the Davey-Stewartson Equations‎

Document Type : Research Paper

Authors

1 School of Mathematical‎ Sciences, Bohai University, Jinzhou, 121013, China‎

2 School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, China

3 School of Educational Sciences, Bohai University, Jinzhou, 121013, China

Abstract

A pair of rogue wave solutions of the Davey-Stewartson (DS) equations are obtained by using the exp-function method and reduction transformations. Firstly, the Davey-Stewartson equations are transformed into two easy-to-solve equations, one of which is the deformed nonlinear Schrödinger (NLS) equation and the other is a polynomial equation. Secondly, based on the existing known solutions of the deformed NLS equation constructed by the exp-function method, rogue wave solutions of the DS equations are obtained. Finally, some spatial and spatiotemporal structures and dynamical evolutionary plots of the obtained rogue wave solutions are shown.

Keywords

Main Subjects

‎[1] Solli, D., Ropers, C., Koonath, P., Jalali B., Optical Rogue Waves, Nature, 450, 2007, 1054-1057.‎
‎[2] Gao, L., Liang, J. Y., Li, C. Y., Wang, L. H., Single-shot Compressed Ultrafast Photography at One Hundred Billion Frames per Second, ‎Nature, 516, 2014, 74-77. ‎
‎[3] Patsyk, A., Sivan, U., Segev, M., Bandres, M. A., Observation of Branched Flow of Light, Nature, 583, 2020, 60-65. ‎
‎[4] Yan, Z. Y., Financial Rogue Waves, Communications in Theoretical Physics, 54(5), 2010, 947-949.‎
‎[5] Guo, B.L., Tian, L.X., Yan Z.Y., et al. Rouge Wave and its Mathematical Theory, Zhejiang Science and Technology Press, Hangzhou, 2015. (in ‎Chinese)‎
‎[6] Zhang, S. Zhang, L.J., Xu, B., Rational Waves and Complex Dynamics: Analytical Insights into a Generalized Nonlinear Schrödinger ‎Equation with Distributed Coefficients, Complexity, 2019, 2019, Article ID 3206503.‎
‎[7] He, J.H., Wu, X.H., Exp-function Method for Nonlinear Wave Equations, Chaos, Solitons & Fractals, 30(3), 2006, 700-708. ‎
‎[8] Wu, X.H., He, J.H. Solitary Solutions, Periodic Solutions and Compacton-like Solutions using the Exp-function Method, Computers & Mathematics with Applications, 54(7-8), 2007, 966-986.‎
‎[9] Zhang, S. Application of Exp-function Method to a KdV Equation with Variable Coefficients, Physics Letters A, 365(5-6), 2007, 448-453.‎
‎[10] Zhu, S.D. Exp-function Method for the Hybrid-lattice System, International Journal of Nonlinear Sciences and Numerical Simulation, 8(3), ‎‎2007, 461-464. ‎
‎[11] Zhang, S. You, C.H., Xu, B. Simplest Exp-function Method for Exact Solutions of Mikhauilov-Novikov-Wang Equations, Thermal Science, ‎‎23(4), 2019, 2381-2388.‎
‎[12] Zhang, S. Li, J.H., Zhang, L.Y. A Direct Algorithm of Exp-function Method for Non-linear Evolution Equations in Fluids, Thermal Science, ‎‎20(3), 2016, 881-884.‎
‎[13] Zhang, S. Gao, Q., Zong, Q.A., et al. Multi-wave Solutions for a Non-isospectral KdV-type Equation with Variable Coefficients, Thermal Science, 16(5), 2012, 1576-1579.‎
‎[14] Zhang, S, Zhang, H.Q. Exp-function Method for N-soliton Solutions of Nonlinear Differential-difference Equations, Zeitschrift für Naturforschung A, 65(11), 2010, 924-934.‎
‎[15] Ji, F. Y., He, C. H., Zhang, J. J., He, J. H., A Fractal Boussinesq Equation for Nonlinear Transverse Vibration of a Nanofiber-reinforced ‎Concrete Pillar, Applied Mathematical Modelling, 82, 2020, 437-448. ‎
‎[16] He, J. H., Ji, F.Y., Mohammad-Sedighi, H., Difference Equation vs Differential Equation on Different Scales, International Journal of Numerical Methods for Heat and Fluid Flow, 2020, DOI: 10.1108/HFF-03-2020-0178.‎
‎[17] Zhang, S. You, C.H., Inverse Scattering Transform for New Mixed Spectral Ablowitz-Kaup-Newell-Segur Equations, Thermal Science, 24(4), ‎‎2020, 2437-2444.‎
‎[18] Zhang, S., Liu, D.D., Darboux Transform and Conservation Laws of New Differential-difference Equations, Thermal Science, 24(4), 2020, ‎‎2519-2527. ‎
‎[19] Zhang, S., You, C.H, Xu, Bo., Simplest Exp-function Method for Exact Solutions of Mikhauilov-Novikov-Wang Equations, Thermal Science, ‎‎23(4), 2019, 2381-2388. ‎
‎[20] Xu, B., Zhang, S., Exact Solutions with Arbitrary Functions of the (4+1)-dimensional Fokas Equation, Thermal Science, 23(4), 2019, 2403-‎‎2411. ‎
‎[21] Li, Y.S., Soliton and integrable system, Shanghai Scientific and Technological Education Publishing House, Shanghai, 1999. (in Chinese)‎
‎[22] Zhang, J.F., Dai C.Q., Wang Y.Y., Rogue Wave Theory Based on Nonlinear Schrödinger Equation and its Application, Science Press, Beijing, 2016. ‎‎(in Chinese)‎