Electronic Journal of Differential Equations (May 2012)
Solutions to over-determined systems of partial differential equations related to Hamiltonian stationary Lagrangian surfaces
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 $varepsilone 0$. Our main result states that this system admits only traveling wave solutions, whenever $varepsilon e 0$.
