Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 83, pp. 1-7.
Title: Solutions to over-determined systems of partial differential equations
related to Hamiltonian stationary Lagrangian surfaces
Author: Bang-Yen Chen (Michigan State Univ., East Lansing, MI, USA)
Abstract:
This article concerns the over-determined system of partial
differential equations
$$
\Big(\frac{k}{f}\Big)_x+\Big(\frac{f}{k}\Big)_y=0, \quad
\frac{f_{y}}{k}=\frac{k_x}{f},\quad
\Big(\frac{f_y}{k}\Big)_y+\Big(\frac{k_x}{f}\Big)_x=-\varepsilon fk\,.
$$
It was shown in [6, Theorem 8.1] that this system
with $\varepsilon=0$ admits traveling wave solutions as well as
non-traveling wave solutions.
In this article we solve completely this system when $\varepsilon\ne 0$.
Our main result states that this system admits only traveling wave
solutions, whenever $\varepsilon \ne 0$.
Submitted December 1, 2011. Published May 23, 2012.
Math Subject Classifications: 35N05, 35C07, 35C99.
Key Words: Over-determined PDE system; traveling wave solution;
exact solution; Hamiltonian stationary Lagrangian surfaces.