Solutions to over-determined systems of partial differential equations related to Hamiltonian stationary Lagrangian surfaces
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 $v...
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| Format: | Article |
| Language: | English |
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Texas State University
2012-05-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/Volumes/2012/83/abstr.html |
| Summary: | 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 $varepsilon e 0$. |
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| ISSN: | 1072-6691 |