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

DOI:
https://doi.org/10.1090/S0002-9947-1988-99924-9

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 , . In a previous paper [**6**], conditions of 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 , there exists at least two traveling wave solutions, each with winding number or .

**[1]**C. Conley,*Isolated invariant sets and the Morse index*, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R. I., 1978. MR**511133 (80c:58009)****[2]**R. Franzosa,*Index filtrations and connection matrices for partially ordered Morse decompositions*, Ph.D. dissertation, Univ. of Wisconsin, Madison, 1984.**[3]**K. Mischaikow,*Classification of traveling wave solutions of reaction-diffusion systems*, Brown Univ., LCDS #86-5, 1985.**[4]**J. Reineck,*Connecting orbits in one parameter families of flow*, preprint. MR**967644 (89i:58128)****[5]**D. Terman,*Directed graphs and traveling waves*, Trans. Amer. Math. Soc.**289**(1985), 809-847. MR**784015 (86e:35072)****[6]**-,*Infinitely many traveling wave solutions of a gradient system*, Trans. Amer. Math. Soc.**301**(1987), 537-556. MR**882703 (88i:35082)****[7]**-,*Traveling wave solutions of a gradient system*:*Solutions with a prescribed winding number*. II, Trans. Amer. Math. Soc. (to appear). MR**940216 (89m:35116)****[8]**-,*Infinitely many radial solutions of an elliptic system*, Ann. Inst. H. Poincaré, Analyse Non Lineaire**4**(1987), 549-604. MR**929475 (89k:35084)****[9]**-,*Radial solutions of an elliptic system*:*Solutions with a prescribed winding number*, Houston J. Math. (to appear). MR**1032401 (91k:35078)**

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DOI:
https://doi.org/10.1090/S0002-9947-1988-99924-9

Article copyright:
© Copyright 1988
American Mathematical Society