Analysis of Multi-Dimensional Nonlinear Mathematical Physics Equations Traveling Wave Solutions and Initial Edge Value Problems

Exploring the traveling wave solutions of mathematical physics equations and their boundary problems is an important work in mathematical physics research. In this paper, three types of representative multidimensional nonlinear equations are selected, and the traveling wave solutions of the equations under different initial and side value conditions are studied. From the perspective of dynamical system theory, the treatment of parameter transformation is used to transform them into planar dynamical systems, and then the phase diagram of the planar system under the corresponding parameter conditions is obtained by using the dynamical system branching method, from which the existence of isolated wave solutions, isolated sharp waves and torsion waves of the perturbation equations and coupled field equations are obtained.

When
\[
\sigma f\!\left(\frac{c+\mu}{\sigma}\right) > 0,
\]
the Camassa–Holm system has 11 branches of traveling wave solutions.

When
\[
\sigma f\!\left(\frac{c+\mu}{\sigma}\right) < 0, \] the Camassa–Holm system has 6 branches of traveling wave solutions.

The Boussinesq equation has a unique non-extendable solution defined on the interval under the first boundary condition and a generalized solution under the second boundary condition, while the Klein–Gordon equation has three singular points in different branches of the traveling wave solution. The present study is important for further understanding the dynamical properties of nonlinear systems.