Stability of solitary waves in dispersive media described by a fifth- order evolution equation

DOI10.1007/BF00417931zbMath0755.76022MaRDI QIDQ1194846

Andrej Il'ichev, Andrei Yu. Semenov

Publication date: 4 October 1992

Published in: Theoretical and Computational Fluid Dynamics (Search for Journal in Brave)

zbMATH Keywords

existenceHamiltonian systemsFréchet spacesKorteweg-de Vries equationorbital stabilityPicard methodpositive operator theory

Mathematics Subject Classification ID

Stability in context of PDEs (35B35) KdV equations (Korteweg-de Vries equations) (35Q53) Capillarity (surface tension) for incompressible inviscid fluids (76B45) Solitary waves for incompressible inviscid fluids (76B25)

Related Items

Orbital stability of internal waves, On the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping, Unnamed Item, Pontryagin's theorem and spectral stability analysis of solitons, Global existence of solutions for the Kawahara equation in Sobolev spaces of negative indices, Solitary waves in a cold plasma, Conditional stability of solitary waves propagating in elastic rods, High-speed transmission in long-haul electrical systems, Generalized beam model for the analysis of wave propagation with a symmetric pattern of deformation in planar pantographic sheets, On the normalized ground states for the Kawahara equation and a fourth order NLS, Conservation laws, symmetries, and line solitons of a Kawahara-KP equation, Higher-order derivative nonlinear Schrödinger equation in the critical case, On the nonlinear stability of solitary wave solutions of the fifth-order Korteweg-de Vries equation, Orbital stability of periodic traveling wave solutions for the Kawahara equation, Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework, Strichartz estimates for dispersive equations and solvability of the Kawahara equation, The unfolding of equivariant bifurcation problems with two types of state variables in the presence of parameter symmetry, Global existence of solutions for the Cauchy problem of the Kawahara equation with \(L^2\) initial data, Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics, Soliton-like structures on a liquid surface under an ice cover, Modified KdV equation with higher order dispersion terms, Time decay estimates for solutions of the Cauchy problem for the modified Kawahara equation, Well-posedness issues on the periodic modified Kawahara equation, Effective wavelength of envelope waves on the water surface beneath an ice sheet: small amplitudes and moderate depths, On the propagation of regularities in solutions of the fifth order Kadomtsev-Petviashvili II equation, Three-dimensional solitary waves in the presence of additional surface effects, The stability of exact solitary wave solutions for simplified modified Camassa-Holm equation, Solitary waves in media with dispersion and dissipation. (A review), Local and global existence of solutions to initial value problems of modified nonlinear Kawahara equations

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Weakly non-local solitons for capillary-gravity waves: Fifth-degree Korteweg-de Vries equation
• Existence of perturbed solitary wave solutions to a model equation for water waves
• Structural stability of the Korteweg-de Vries solitons under a singular perturbation
• Nonlocal models for nonlinear, dispersive waves
• Sufficient conditions for stability of solitary-wave solutions of model equations for long waves
• Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation
• Stability theory of solitary waves in the presence of symmetry. I
• A theory of solitary water-waves in the presence of surface tension
• Sur quelques généralisations de l'équation de Korteweg-de Vries
• On solitary water-waves of finite amplitude
• Finite-amplitude steady waves in stratified fluids
• Remarks on the Korteweg-de Vries equation
• Small internal waves in two-fluid systems
• A regularity theorem for parabolic equations
• An extension of Tietze's theorem
• The existence of periodic waves which degenerate into a solitary wave
• Solitary-wave solutions of nonlinear problems
• The existence of internal solitary waves in a two-fluid system near the KdV limit
• The stability of internal solitary waves
• Lyapunov stability of ground states of nonlinear dispersive evolution equations
• A Global Theory of Internal Solitary Waves in Two-Fluid Systems
• Symmetry breaking in periodic and solitary gravity-capillary waves on water of finite depth
• Stability and instability of solitary waves of Korteweg-de Vries type
• Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation
• On periodic water-waves and their convergence to solitary waves in the long-wave limit
• On the stability theory of solitary waves
• Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces
• The existence of solitary water waves
• On the existence theory for irrotational water waves
• Total Positivity and the Stability of Internal Waves in Stratified Fluids of Finite Depth
• Internal waves of finite amplitude and permanent form
• On the theory of steady-state waves of finite amplitude
• The Leray-Schauder index and the fixed point theory for arbitrary ANRs
• Degree of Mapping in Convex Linear Topological Spaces
• The existence of solitary waves