Some new travelling wave solutions with singular or nonsingular character for the higher order wave equation of KdV type (III)

Solitons and Other Solutions for Two Higher-Order Nonlinear Wave Equations of KdV Type Using the Unified Auxiliary Equation Method

“…Different types of these solutions via solitary wave solutions, shock wave solutions, singular solitary wave solutions and trigonometric wave solutions are found analytically and represented successfully via graphics. By comparing our new results in this article with the well known results obtained in [43][44][45][46][47], we conclude that our results are new and not found elsewhere. From these discussions, we conclude that the proposed method of Sect.…”

“…These two equations arise as models in the theory of water wave which were first proposed by Fokas [44]. Also these equations have been discussed in [45] when α = β = 1 using the integral bifurcation theory, where some traveling wave solutions with singular or non singular characters are ob-…”

The(G′/G)-Expansion Method and Its Application for Higher-Order Equations of KdV (III)

“…The physical meaning of other solutions for this equation is still unclear. For instance, the relationship between the solutions (21)- (25) and the movement of water waves and the dynamic behaviors that these solutions can demonstrate are both unknown to us. However, the rest of these questions is still worthy of further observation and investigation by the researchers in the territory of experimental physics, and we also hope that more researchers should pay attention to the investigation of this area.…”

“…For example, in [24], Li et al obtained some exact explicit parametric representations of solitary wave, kink and antikink wave solutions, and breaking wave solutions of (1) under special parametric conditions. Some peculiar exact traveling wave solutions including solitary wave, cusp wave, and loop solution with singular or nonsingular character of (1) were discussed under some particular parameters in [25]. By using the bifurcation theory of planar dynamical system and elliptic function integral method, the explicit and implicit solutions of periodic wave and solitary cusp wave of (1) were obtained in [26].…”

A Generalized KdV Equation of Neglecting the Highest-Order Infinitesimal Term and Its Exact Traveling Wave Solutions

“…In [8], Bi also obtained some results of numerical simulation of (10); it is a pity that the exact traveling wave solutions of (10) were not obtained yet. In [4,5], we obtained some exact traveling wave solutions of (4) under the parametric conditions 3 = ( + 1) 2 or 3 = 2 . In [15], under a new ansätze, Khuri studied (4); some exact solitary wave solutions and periodic wave solutions were obtained.…”