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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Traveling wave solutions of a gradient system: solutions with a prescribed winding number. I


Author: David Terman
Journal: Trans. Amer. Math. Soc. 308 (1988), 369-389
MSC: Primary 35K57; Secondary 20E05, 35B99
Part II: Trans. Amer. Math. Soc. (1) (1988), 391-412
MathSciNet review: 946449
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Abstract: Consideration is given to a system of equations of the form $ {u_t} = {u_{xx}} + \nabla F(u)$, $ u \in {{\mathbf{R}}^2}$. In a previous paper [6], conditions of $ F$ were given which guarantee that the system possesses infinitely many traveling wave solutions. The solutions are now characterized by how many times they wind around in phase space. A winding number for solutions is defined. It is demonstrated that for each positive integer $ K$, there exists at least two traveling wave solutions, each with winding number $ K$ or $ K + 1$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-99924-9
PII: S 0002-9947(1988)99924-9
Article copyright: © Copyright 1988 American Mathematical Society