New periodic wave solutions of (3+1)-dimensional soliton equation

Abstract

In this paper, associating with the Hirota bilinear form, the three-wave method, which is applied to construct some periodic wave solutions of (3+1)-dimensional soliton equation, is a powerful approach to obtain periodic solutions for many non-linear evolution equations in the integrable systems theory.

Lump periodic wave, soliton periodic wave, and breather periodic wave solutions for third-order (2+1)-dimensional equation

“…In this paper, we have constructed a new family of periodic wave solutions of the third-order (2 + 1)dimensional equation, using the bilinear method of Hirota and three-wave method [31,32], with the help of Maple. Using a choice of the appropriate parameters, we have plotted the graphs that illustrate these solutions.…”

Localized waves and interaction solutions to the fractional generalized CBS-BK equation arising in fluid mechanics

“…The multiple rogue waves method used by some of powerful authors for the various nonlinear equations, including constructing rogue waves with a controllable center in the nonlinear systems [41], a (3 + 1)-dimensional Hirota bilinear equation [42], the generalized (3 + 1)-dimensional KP equation [43], and the Boussinesq equation [44]. There are many new papers about this fields, such as lump solutions (constructing the lump-soliton and mixed lump strip solutions of (3 + 1)-dimensional soliton equation [45]; utilizing the linear superposition principle to discuss the (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation [46]; obtaining periodic solutions for many non-linear evolution equations in the integrable systems theory [47]), rogue wave solutions (utilizing the Hirota bilinear form of the extended (3 + 1)-dimensional JM equation to find 30 classes of rogue wave type solutions [48]; resonant multiple wave solutions to some integrable soliton equations [49]). Some important work related with recent development in fractional calculus and its applications can be pointed out referring to the valuable papers containing studies of general fractional derivatives: theory, methods and applications by Yang [50]; anomalous diffusion equations with the decay exponential kernel by the Laplace transform [51]; new fractal nonlinear Burgers' equation arising in the acoustic signals propagation by Yang and Machado [52]; time fractional nonlinear diffusion equation from diffusion process by fractional Lie group approach [53]; the generalized time fractional diffusion equation by symmetry analysis [54]; investigating a time fractional nonlinear heat conduction equation with applications in mathematics physics, integrable system, fluid mechanics and nonlinear areas, by means of applying the fractional symmetry group method [55]; and determining the time fractional extended (2 + 1)-dimensional Zakharov-Kuznetsov equation in quantum magneto-plasmas by using a group analysis approach [56].…”

Soliton solutions, travelling wave solutions and conserved quantities for a three-dimensional soliton equation in plasma physics

“…He found out that the lump solutions are rationally localized in every direction within the space. In [52], the authors secured some periodic wave solutions of the equation using the Hirota bilinear technique together with three-wave approaches. Furthermore, in [53],…”