Electron. J. Diff. Equ., Vol. 2012 (2012), No. 83, pp. 1-7.

Solutions to over-determined systems of partial differential equations related to Hamiltonian stationary Lagrangian surfaces

Bang-Yen Chen

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 $\varepsilon=0$.

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.

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Bang-Yen Chen
Department of Mathematics, Michigan State University
619 Red Cedar Road
East Lansing, Michigan 48824-1027, USA
email: bychen@math.msu.edu

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