Concentration Compactness and the Stability of Solitary-Wave Solutions to Nonlocal Equations

In their proof of the stability of standing-wave solutions of nonlinear Schrödinger equations, Cazenave and Lions used the principle of concentration compactness to characterize the standing waves as solutions of a certain variational problem. In this article we first review the techniques introduced by Cazenave and Lions, and then discuss their application to solitary-wave solutions of nonlocal nonlinear wave equations. As an example of such an application, we include a new result on the stability of solitary-wave solutions of the Kubota-Ko-Dobbs equation for internal waves in a stratified fluid.

[1]  J. Boussinesq,et al.  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. , 1872 .

[2]  T. Benjamin.  Internal waves of permanent form in fluids of great depth , 1967, Journal of Fluid Mechanics.

[3]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[4]  Ronald Smith,et al.  Nonlinear Kelvin and continental-shelf waves , 1972, Journal of Fluid Mechanics.

[5]  J. Bona.  On the stability theory of solitary waves , 1975, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[6]  R. Joseph.  Solitary waves in a finite depth fluid , 1977 .

[7]  Ronald R. Coifman,et al.  Au delà des opérateurs pseudo-différentiels , 1978 .

[8]  T. Kubota,et al.  Weakly-Nonlinear, Long Internal Gravity Waves in Stratified Fluids of Finite Depth , 1978 .

[9]  R. Joseph,et al.  Multi-soliton solutions in a finite depth fluid , 1978 .

[10]  T. Kubota,et al.  Resonant Transfer of Energy between Nonlinear Waves in Neighboring Pycnoclines , 1980 .

[11]  M. Ablowitz,et al.  Nonlinear Intermediate Long-Wave Equation: Analysis and Method of Solution , 1981 .

[12]  P. Lions,et al.  Orbital stability of standing waves for some nonlinear Schrödinger equations , 1982 .

[13]  Elliott H. Lieb,et al.  On the lowest eigenvalue of the Laplacian for the intersection of two domains , 1983 .

[14]  D. Lebedev,et al.  Generalized internal long waves equations: Construction, Hamiltonian structure, and conservation laws , 1983 .

[15]  P. Lions.  The concentration-compactness principle in the calculus of variations. The locally compact case, part 1 , 1984 .

[16]  Alexander M. Rubenchik,et al.  Soliton stability in plasmas and hydrodynamics , 1986 .

[17]  Equations of the Korteweg-de Vries type with non-trivial conserved quantities , 1989 .

[18]  J. Bona,et al.  Nonlocal models for nonlinear, dispersive waves , 1990 .

[19]  J. Shatah,et al.  Stability theory of solitary waves in the presence of symmetry, II☆ , 1990 .

[20]  J. F. Toland,et al.  Uniqueness of benjamin's solitary-wave solution of the benjamin-ono equation , 1991 .

[21]  J. Bona,et al.  Total Positivity and the Stability of Internal Waves in Stratified Fluids of Finite Depth , 1991 .

[22]  J. Albert.  Positivity Properties and Stability of Solitary–Wave Solutions of Model Equations For Long Waves , 1992 .

[23]  R. Pego,et al.  On asymptotic stability of solitary waves , 1992 .

[24]  T. Benjamin.  A new kind of solitary wave , 1992, Journal of Fluid Mechanics.

[25]  J. F. Toland,et al.  Homoclinic orbits in the dynamic phase-space analogy of an elastic strut , 1992, European Journal of Applied Mathematics.

[26]  J. Maddocks,et al.  On the stability of KdV multi‐solitons , 1993 .

[27]  J. Bona,et al.  On the stability of solitary-wave solutions of model equations for long waves , 1994 .

[28]  J. Albert,et al.  On the exact solutions of the intermediate long-wave equation , 1994, Differential and Integral Equations.

[29]  J. Bona,et al.  Model equations for waves in stratified fluids , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.