On the orbital stability of Gaussian solitary waves in the log-KdV equation
@article{Carles2014OnTO, title={On the orbital stability of Gaussian solitary waves in the log-KdV equation}, author={R{\'e}mi Carles and Dmitry E. Pelinovsky}, journal={Nonlinearity}, year={2014}, volume={27}, pages={3185 - 3202} }
We consider the logarithmic Korteweg–de Vries (log-KdV) equation, which models solitary waves in anharmonic chains with Hertzian interaction forces. By using an approximating sequence of global solutions of the regularized generalized KdV equation in with conserved L2 norm and energy, we construct a weak global solution of the log-KdV equation in a subset of . This construction yields conditional orbital stability of Gaussian solitary waves of the log-KdV equation, provided that uniqueness and…
16 Citations
On the linearized log-KdV equation
- Physics, Mathematics
- 2016
The logarithmic KdV (log-KdV) equation admits global solutions in an energy space and exhibits Gaussian solitary waves. Orbital stability of Gaussian solitary waves is known to be an open problem. We…
Orbital Stability of Periodic Waves for the Log-KdV Equation
- Mathematics
- 2014
In this paper we establish the orbital stability of periodic waves related to the logarithmic Korteweg-de Vries equation. Our motivation is inspired in the recent work \cite{carles}, in which the…
Justification of the log-KdV Equation in Granular Chains: The Case of Precompression
- MathematicsSIAM J. Math. Anal.
- 2014
For travelling waves with nonzero boundary conditions, we justify the logarithmic Korteweg-de Vries equation as the leading approximation of the Fermi-Pasta-Ulam lattice with Hertzian nonlinear…
Existence and stability of a two-parameter family of solitary waves for a logarithmic NLS–KdV system
- MathematicsNonlinear Analysis
- 2019
Asymptotic stability of viscous shocks in the modular Burgers equation
- MathematicsNonlinearity
- 2021
Dynamics of viscous shocks is considered in the modular Burgers equation, where the time evolution becomes complicated due to singularities produced by the modular nonlinearity. We prove that the…
The Gaussian soliton in the Fermi–Pasta–Ulam chain
- MathematicsNonlinear Dynamics
- 2021
We consider a Fermi–Pasta–Ulam (FPU) chain with the homogeneous fully nonlinear interaction potential which describes the propagation of acoustic wave in chains of touching beads without…
On a class of solutions to the generalized KdV type equation
- MathematicsCommunications in Contemporary Mathematics
- 2019
We consider the IVP associated to the generalized KdV equation with low degree of nonlinearity [Formula: see text] By using an argument similar to that introduced by Cazenave and Naumkin [Local…
Orbital stability of periodic standing waves for the logarithmic Klein-Gordon equation
- Mathematics
- 2019
Well-Posedness and Orbital Stability of Periodic Traveling Waves for Schamel's Equation
- Mathematics
- 2018
In this paper we will prove results of global well-posedness and orbital stability of periodic traveling-wave solutions related to the Schamel equation. The global solvability will be established by…
References
SHOWING 1-10 OF 21 REFERENCES
Gaussian solitary waves and compactons in Fermi–Pasta–Ulam lattices with Hertzian potentials
- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2014
A class of fully nonlinear Fermi–Pasta–Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order α>1, is considered, describing acoustic wave propagation in chains of touching beads in the absence of precompression.
Spectral stability of nonlinear waves in KdV-type evolution equations
- Mathematics
- 2012
This paper concerns spectral stability of nonlinear waves in KdV-type evolution equations. The relevant eigenvalue problem is defined by the composition of an unbounded self-adjoint operator with a…
A Hamiltonian–Krein (Instability) Index Theory for Solitary Waves to KdV‐Like Eigenvalue Problems
- Mathematics
- 2014
The Hamiltonian–Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem JLu=λu , where J is skew‐symmetric and…
A Hamiltonian-Krein (instability) index theory for KdV-like eigenvalue problems
- Mathematics
- 2012
The Hamiltonian-Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem $ J L u=\lambda u$, where $J$ is…
Asymptotic solution for solitary waves in a chain of elastic spheres.
- MathematicsPhysical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 1999
In this work, n is treated as "slightly" greater than 1, and an asymptotic solution is developed in terms of the associated small parameter, which is substantially more accurate than the presently available approximate solution given by Nesterenko.
Finite Time Extinction by Nonlinear Damping for the Schrödinger Equation
- Mathematics
- 2010
We consider the Schrödinger equation on a compact manifold, in the presence of a nonlinear damping term, which is homogeneous and sublinear. For initial data in the energy space, we construct a weak…
Finite Time Extinction for Nonlinear Schrödinger Equation in 1D and 2D
- Mathematics
- 2014
We consider a nonlinear Schrödinger equation with power nonlinearity, either on a compact manifold without boundary, or on the whole space in the presence of harmonic confinement, in space dimension…
On the Korteweg-de Vries equation
- Mathematics
- 1979
Existence, uniqueness, and continuous dependence on the initial data are proved for the local (in time) solution of the (generalized) Korteweg-de Vries equation on the real line, with the initial…
Semilinear Schrodinger Equations
- Mathematics
- 2003
Preliminaries The linear Schrodinger equation The Cauchy problem in a general domain The local Cauchy problem Regularity and the smoothing effect Global existence and finite-time blowup Asymptotic…
Applied asymptotic analysis
- Mathematics
- 2006
Fundamentals: Themes of asymptotic analysis The nature of asymptotic approximations Asymptotic analysis of exponential integrals: Fundamental techniques for integrals Laplace's method for asymptotic…