Stability analysis of multipulses in nonlinearly-coupled Schrödinger equations

Indiana University Mathematics Journal

Volumn 49, Issue 3, 2000, Pages 1079-1124

  Yew, A C ^a  

^a OHIO STATE UNIVERSITY   (United States)

Author keywords

Multipulses; Nonlinear Schr dinger equations; Second harmonic generation; Solitary waves; Stability

Indexed keywords

EID: 0004158480     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2000.49.1826     Document Type: Article

References (36)

1. A. C. YEW, Multipulses of nonlinearly-coupled Schrödinger equations, Jour. Diff. Equations (to appear).
2. A. C. YEW, A. R. CHAMPNEYS, & P. J. MCKENNA, Multiple solitary-waves due to second harmonic generation in quadratic media, J. Nonlinear Sci. 9 (1999), 33-52.
3. H. HE, M. J. WERNER & P. D. DRUMMOND, Simultaneous solitary-wave solutions in a nonlinear parametric waveguide, Phys. Rev. E 54 (1996), 896-911.
4. D. MIHALACHE, F. LEDERER, D. MAZILU & L.-C. CRASOVAN, Multiple-humped bright solitary waves in second-order nonlinear media, Opt. Eng. 35 (1996), 1616-1623.
5. M. HAELTERMAN, S. TRILLO & P. FERRO, Multiple soliton bound states and symmetry breaking in quadratic media, Opt. Lett. 22 (1997), 84-86.
6. M. GRILLAKIS, J. SHATAH & W. STRAUSS, Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal. 74 (1987), 160-197.
7. _ ,Stability theory of solitary waves in the presence of symmetry, II, J. Funct. Anal. 94 (1990), 308-348.
8. W. A. STRAUSS, Nonlinear Wave Equations, In: CBMS Regional Conference Series in Mathematics, American Mathematical Society, 1989.
9. A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences Volume 44, Springer-Verlag, 1983.
10. T. KATO, Quasilinear equations of evolution, with applications to partial differential equations, In: Spectral Theory and Differential Equations, Lecture Notes in Mathematics, Volume 448 (A. Dold & B. Eckmann, eds.) Springer-Verlag, (1975), pp 25-70.
11. R. L. PECO & M. I. WEINSTEIN, Asymptotic stability of solitary waves, Commun. Math. Phys. 164 (1994), 305-349.
12. M. GRILLAKIS, Linearized instability for nonlinear Schrödinger and Klein-Gordon equations, Comm. Pure Appl. Math. 41 (1988), 747-774.
13. J. ALEXANDER, R. GARDNER & C. JONES, A topological invariant arising in the stability analysis of travelling waves, J. reine angew. Math. 410 (1990), 167-212.
14. A. C. YEW, An Analytical Study of Solitary-Waves in Quadratic Media, Ph. D. Thesis, Brown University, 1999.
15. D. HENRY, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Volume 804 Springer-Verlag, 1981.
16. A. A. KANASHOV & A. M. RUBENCHIK, On diffraction and dispersion effect on three wave interaction, Physica D 4 (1981), 122-134.
17. L. BERGÉ, V. K. MEZENTSEV, J. J. RASMUSSEN & J. WYLLER, Formation of stable solitons in quadratic nonlinear media, Phys. Rev. A 52 (1995), R28-R31.
18. D. E. PELINOVSKY, A. V. BURYAK & Y. S. KIVSHAR, Instability of solitons governed by quadratic nonlinearities, Phys. Rev. Lett. 75 (1995), 591-595.
19. A. V. BURYAK, Solitons due to quadratic nonlinearities, PhD thesis, Australian National University, 1996.
20. J. SHATAH & W. STRAUSS, Instability of nonlinear bound states, Commun. Math. Phys. 100 (1985), 173-190.
21. S. NII, A topological proof of stability of N-front solutions of the Fitzhugh-Nagumo equations, J. Dynam. Diff. Eq. 11 (1999), 515-555.
22. B. SANDSTEDE, Stability of multiple-pulse solutions, Trans. Am. Math. Soc. 350 (1998), 429-472.
23. _ , Verzweigungstheorie homokliner Verdopplungen, PhD thesis, University of Stuttgart, 1993.
24. B. SANDSTEDE, C. K. R. T. JONES ® J. C. ALEXANDER, Existence and stability of N-pulses on optical fibers with phase-sensitive amplifiers, Physica D 106 (1997), 167-206.
25. C. K. R. T. JONES, Instability of standing waves for nonlinear Schrödinger type equations, Ergod. Th. & Dynam. Sys. 8* (1988), 119-138.
26. M. GRILLAKIS, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math. 43 (1990), 299-333.
27. M. HAELTERMAN & A. SHEPPARD, Bifurcation phenomena and multiple solitonbound states in Isotropic Kerr media, Phys. Rev. E 49 (1994), 3376-3381.
28. N. AKHMEDIEV & J. M. SOTO-CRESPO, Dynamics of solitonlike pulse propagation in birefringent optical fibers, Phys. Rev. E 49 (1994), 5742-5754.
29. L. BERGÉ, O. BANG, J. J. RASMUSSEN & V. K. MEZENTSEV, Self-focusing and solitonlike structures in materials with competing quadratic and cubic nonlinearities, Phys. Rev. E 55 (1997), 3555-3570.
30. A. V. BURYAK, Y. S. KIVSHAR & S. TRILLO, Stability of three-wave parametric solitons in diffractive quadratic media. Phys. Rev. Lett. 77 (1996), 5210-5213.
31. U. PESCHEL, C. ETRICH, F. LEDERER, & B. A. MALOMED, Vectorial solitary waves in optical media with a quadratic nonlinearity, Phys. Rev. E 55 (1997), 7704-7711.
32. Y. S. CHOI, P. J. MCKENNA & A. C. YEW, Uniqueness of positive localized solutions for a system of coupled nonlinear Schrödinger equations, In preparation, (1999).
33. X.-B. LIN, Using Melnikov's method to solve Silnikov'sproblems, Proc. Roy. Soc. Edinburgh 116A (1990), 295-325.
34. T. KAPITULA & B. SANDSTEDE, Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations, Physica D 124 (1998), 58-103.
35. S. TRILLO, M. HAELTERMAN & A. SHEPPARD, Stable topological spatial solitons in optical parametric oscillators, Opt. Lett. 22 (1997), 970-972.
36. B. SANDSTEDE, Weak interaction of pulses in dissipative systems, In preparation (2000).