Periodic traveling wave solutions and their limit forms for the (n + 1)-dimensional double sinh-Gordon equation

In this paper, the (n + 1)-dimensional double sinh-Gordon equation∑j=1nuxjxj−utt−αsinh(u)−βsinh(2u)=0is studied by using the bifurcation method of dynamical systems. The periodic traveling wave solutions and their limit forms are investigated. When the first integral varies, we show the convergence of the smooth periodic wave solutions and the periodic blow-up wave solutions, such as the smooth periodic wave solutions converge to the solitary wave solution, the smooth periodic wave solutions converge to the smooth periodic wave solution, the periodic blow-up wave solutions converge to the blow-up wave solution and the periodic blow-up wave solutions converge to the periodic blow-up wave solution. All possible explicit exact parametric representations of various nonlinear traveling waves also are given.