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 | References | Similar Articles | Additional Information

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]**Charles Conley,*Isolated invariant sets and the Morse index*, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR**511133****[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]**James F. Reineck,*Connecting orbits in one-parameter families of flows*, Ergodic Theory Dynam. Systems**8***(1988), no. Charles Conley Memorial Issue, 359–374. MR**967644**, 10.1017/S0143385700009482**[5]**David Terman,*Directed graphs and traveling waves*, Trans. Amer. Math. Soc.**289**(1985), no. 2, 809–847. MR**784015**, 10.1090/S0002-9947-1985-0784015-6**[6]**David Terman,*Infinitely many traveling wave solutions of a gradient system*, Trans. Amer. Math. Soc.**301**(1987), no. 2, 537–556. MR**882703**, 10.1090/S0002-9947-1987-0882703-6**[7]**James F. Reineck,*Travelling wave solutions to a gradient system*, Trans. Amer. Math. Soc.**307**(1988), no. 2, 535–544. MR**940216**, 10.1090/S0002-9947-1988-0940216-8**[8]**D. Terman,*Infinitely many radial solutions of an elliptic system*, Ann. Inst. H. Poincaré Anal. Non Linéaire**4**(1987), no. 6, 549–604 (English, with French summary). MR**929475****[9]**David Terman,*Radial solutions of an elliptic system: solutions with a prescribed winding number*, Houston J. Math.**15**(1989), no. 3, 425–458. MR**1032401**

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

Article copyright:
© Copyright 1988
American Mathematical Society