Stability and symmetry of solitary-wave solutions to systems modeling interactions of long waves

Journal des Mathematiques Pures et Appliquees

Volumn 79, Issue 3, 2000, Pages 195-226

  Albert, John ^a   Linares, Felipe ^b  

^a UNIVERSITY OF OKLAHOMA   (United States)

^b IMPA   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034148625     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-7824(00)00157-4     Document Type: Article

References (33)

1. M. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, PA, 1981.
2. J. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, In: Applied Analysis (Baton Rouge, LA, 1996), Contemporary Mathematics, Vol. 221, Amer. Math. Soc., Providence, RI, 1999, pp. 1-29.
3. J. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations, In: Applied Analysis (Baton Rouge, LA, 1996), Contemporary Mathematics, Vol. 221, Amer. Math. Soc., Providence, RI, 1999, pp. 1-29.
4. J. Albert, J. Bona and J.-C. Saut, Model equations for waves in stratified fluids, Proc. Roy. Soc. London Ser. A 453 (1997) 1233-1260.
5. J. Albert and F. Linares, Stability of solitary-wave solutions to long-wave equations with general dispersion, Mat. Contemp. 15 (1998) 1-19.
6. J. Albert and J. Toland, On the exact solutions of the intermediate long-wave equation, Differential Integral Equations 7 (1994) 601-612.
7. J. Angulo Pava, Existence and stability of solitary wave solutions of the Benjamin equation, J. Differential Equations 152 (1999) 136-159.
8. D. Benney, Long non-linear waves in fluid flows, J. Math. Phys. 45 (1966) 52-63.
9. J. Bergh and J. Löfstrom, Interpolation Spaces: An Introduction, Springer-Verlag, New York, 1976.
10. J. Bona, On solitary waves and their role in the evolution of long waves, in: Applications of Nonlinear Analysis in the Physical Sciences, H. Amann, N. Bazley, and K. Kirchgässner (Eds), Pitman, New York, 1981, pp. 183-205.
11. J. Bona and H. Chen, Solitary Waves in Stratified Fluid Systems, to appear.
12. J. Bona, G. Ponce, J.-C. Saut and M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Phys. 143 (1992) 287-313.
13. J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl. 17 (2) (1872) 55-108.
14. T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982) 549-561.
15. H. Chen and J. Bona, Existence and asymptotic properties of solitary-wave solutions of Benjamin-type equations, Adv. Differential Equations 3 (1998) 51-84.
16. R. Coifman and Y. Meyer, Au Delà des Opérateurs Pseudo-différentiels, Asterisque, No. 57, Société Mathématique de France, Paris, 1978.
17. W. Craig and P. Sternberg, Symmetry of solitary waves, Comm. Partial Differential Equations 13 (1988) 603-633.
18. W. Craig and P. Sternberg, Symmetry of free-surface flows, Arch. Rational Mech. Anal. 118 (1992) 1-36.
19. A. De Bouard and J.-C. Saut, Remarks on the stability of generalized KP solitary waves, in: Mathematical Problems in The Theory of Water Waves (Luminy, 1995), Contemporary Mathematics, Vol. 200, Amer. Math. Soc., Providence, RI, 1996, pp. 75-84.
20. A. De Bouard and J.-C. Saut, Remarks on the stability of generalized KP solitary waves, in: Mathematical Problems in The Theory of Water Waves (Luminy, 1995), Contemporary Mathematics, Vol. 200, Amer. Math. Soc., Providence, RI, 1996, pp. 75-84.
21. A. De Bouard and J.-C. Saut, Solitary waves of generalized Kadomtsev-Petviashvili equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 211-236.
22. J. Gear and R. Grimshaw, Weak and strong interactions between internal solitary waves, Stud. Appl. Math. 70 (1984) 235-258.
23. B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243.
24. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
25. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1976.
26. T. Kubota, D. Ko and L. Dobbs, Weakly nonlinear interval gravity waves in stratified fluids of finite depth, J. Hydrodynamics 12 (1978) 157-165.
27. C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. Partial Differential Equations 16 (1991) 585-615.
28. P.-L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case, Part 1, Ann. Inst. H. Poincaré 1 (1984) 109-145.
29. A. Liu, T. Kubota and D. Ko, Resonant transfer of energy between nonlinear waves in neighboring pycnoclines, Stud. Appl. Math. 63 (1980) 25-45.
30. D. Luenberger, Optimization by Vector Space Methods, Wiley, New York, 1969.
31. L. Maia, Symmetry of internal waves, Nonlinear Anal. 28 (1997) 87-102.
32. R. Pego and M. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys. 164 (1994) 305-349.
33. M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations 12 (1987) 1133-1173.