Traveling wave solutions of a gradient system: solutions with a prescribed winding number. I
HTML articles powered by AMS MathViewer
- by David Terman PDF
- Trans. Amer. Math. Soc. 308 (1988), 369-389 Request permission
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 $K + 1$.References
- 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 R. Franzosa, Index filtrations and connection matrices for partially ordered Morse decompositions, Ph.D. dissertation, Univ. of Wisconsin, Madison, 1984. K. Mischaikow, Classification of traveling wave solutions of reaction-diffusion systems, Brown Univ., LCDS #86-5, 1985.
- 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, DOI 10.1017/S0143385700009482
- David Terman, Directed graphs and traveling waves, Trans. Amer. Math. Soc. 289 (1985), no. 2, 809–847. MR 784015, DOI 10.1090/S0002-9947-1985-0784015-6
- David Terman, Infinitely many traveling wave solutions of a gradient system, Trans. Amer. Math. Soc. 301 (1987), no. 2, 537–556. MR 882703, DOI 10.1090/S0002-9947-1987-0882703-6
- James F. Reineck, Travelling wave solutions to a gradient system, Trans. Amer. Math. Soc. 307 (1988), no. 2, 535–544. MR 940216, DOI 10.1090/S0002-9947-1988-0940216-8
- 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
- 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
Additional Information
- © Copyright 1988 American Mathematical Society
- 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
- MathSciNet review: 946449