Lump Solutions and Interaction Solutions for the Dimensionally Reduced Nonlinear Evolution Equation

Abstract

In this paper, by means of the Hirota bilinear method, a dimensionally reduced nonlinear evolution equation is investigated. Through its bilinear form, lump solutions are obtained. We construct interaction solutions between lump solutions and one soliton solution by choosing quadratic functions and exponential function. Interaction solutions with the combinations of exponential functions and sine function are also given. Meanwhile, the figures of these solutions are plotted. The dynamical characteristics and properties of obtained solutions are discussed, respectively. The results show that the corresponding physical quantities and properties of nonlinear waves are associated with the values of the parameters.

Multiple lump solutions and their interactions for an integrable nonlinear dispersionless PDE in vector fields

“…Many researchers have shown significant contributions in the interactions between lump, periodic, multi solitons and rogue wave [34]. Guo et al found lump solutions, lump with one-strip solution and observed their interactions with the different combinations of exponential functions of dimensionally reduced NLEEs [13]. In [17], Liu worked on lump soliton solutions and their interactions with the double exponential function and lump with two kink for Korteweg-De Vries equation (KdV).…”

Symmetry Groups, Similarity Reductions, and Conservation Laws of the Time-Fractional Fujimoto–Watanabe Equation Using Lie Symmetry Analysis Method

“…Nowadays, nonlinear partial differential equations (NPDEs) have become more and more significant in fluid mechanics, mathematical physics, oceanography, and so on [1][2][3][4]. As we know, NPDEs are usually of integer order and researchers have proposed abundant methods to obtain solutions of NPDEs, including inverse scattering transformation [5], Riemann-Hilbert method [6][7][8][9][10], Hirota direct method [11,12], Darboux transformation [13,14], Bäcklund transformation [15], Frobenius integrable decompositions [16,17], and so on [18][19][20][21][22][23][24]. As a generalization, the notions of fractional derivatives are put forward and the classical are Riemann-Liouville and Caputo fractional derivatives.…”

Computational Simulation and Modeling of Freak Waves Based on Longuet-Higgins Model and Its Electromagnetic Scattering Calculation

“…e microwaves are highly penetrative and can work under any weather conditions. For the safety of maritime navigation and offshore platforms, it is of great significance to explore the physical mechanism of the occurrence, evolution, and extinction of freak waves [3][4][5][6]. Moreover, the use of SAR to monitor and predict freak waves is a disaster reduction technology that should be valued [6].…”