Some new nonlinear wave solutions and their convergence for the (2+1)‐dimensional Broer–Kau–Kupershmidt equation

Abstract

We use the bifurcation method of dynamical systems to study the (2+1)-dimensional Broer-Kau-Kupershmidt equation. We obtain some new nonlinear wave solutions, which contain solitary wave solutions, blow-up wave solutions, periodic smooth wave solutions, periodic blow-up wave solutions, and kink wave solutions. When the initial value vary, we also show the convergence of certain solutions, such as the solitary wave solutions converge to the kink wave solutions and the periodic blow-up wave solutions converge to the solitary wave solutions.

Bounded Traveling Wave Solutions and Their Relations for the Generalized HD Type Equation

“…Comparing with [26], we obtained some new results of (2). For example, we found some new periodic loop soliton, loop soliton, and kink-like wave solutions as (23)- (25), (30) and (31), and (37)-(42), respectively. Obviously, the solutions which have been presented in this paper and [26] almost are exact implicit traveling wave solutions.…”

“…Example 8. Taking = 1.2, = 0.1, we get the approximations of , in formula (31), where ≐ −1.07143, ≐ 0.612372. The profile of (31) is shown in Figure 5.…”

“…where is given in (31). Substituting (79) into / = and integrating it along the open curves yield equation…”

“…In this paper, by employing the bifurcation method of dynamical systems [27][28][29][30][31][32][33][34], we will obtain some new bounded traveling wave solutions for (3). These solutions contain periodic loop soliton solutions, periodic cusp wave solutions, smooth periodic wave solutions, loop soliton solutions, cuspon solutions, smooth solitary wave solutions, and kink-like wave solutions.…”

Extension on peakons and periodic cusp waves for the generalization of the Camassa-Holm equation

“…In our previous work , we studied the peakons and periodic cusp wave solutions of Eq. when n = 1, m ≥2 and n ≥2, m = n + 1, by exploiting the bifurcation method and qualitative theory of dynamical systems , and setting the integral constant to be zero. The results of are summarized as follows: When n = 1, m = 2, Eq.…”