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.