Traveling wave solutions of a gradient system: solutions with a prescribed winding number. I

Abstract:

Consideration is given to a system of equations of the form ${u_t} = {u_{xx}} + \nabla F(u)$, $u \in {{\mathbf {R}}^2}$. In a previous paper [6], conditions of $F$ were given which guarantee that the system possesses infinitely many traveling wave solutions. The solutions are now characterized by how many times they wind around in phase space. A winding number for solutions is defined. It is demonstrated that for each positive integer $K$, there exists at least two traveling wave solutions, each with winding number $K$ or $K + 1$.