8,892 research outputs found
The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations
In the present study we prove rigorously that in the long-wave limit, the
unidirectional solutions of a class of nonlocal wave equations to which the
improved Boussinesq equation belongs are well approximated by the solutions of
the Camassa-Holm equation over a long time scale. This general class of
nonlocal wave equations model bidirectional wave propagation in a nonlocally
and nonlinearly elastic medium whose constitutive equation is given by a
convolution integral. To justify the Camassa-Holm approximation we show that
approximation errors remain small over a long time interval. To be more
precise, we obtain error estimates in terms of two independent, small, positive
parameters and measuring the effect of nonlinearity and
dispersion, respectively. We further show that similar conclusions are also
valid for the lower order approximations: the Benjamin-Bona-Mahony
approximation and the Korteweg-de Vries approximation.Comment: 24 pages, to appear in Discrete and Continuous Dynamical System
Models for instability in inviscid fluid flows, due to a resonance between two waves
In inviscid fluid flows instability arises generically due to a resonance between two wave modes. Here, it is shown that the structure of the weakly nonlinear regime depends crucially on whether the modal structure coincides, or remains distinct, at the resonance point where the wave phase speeds coincide. Then in the weakly nonlinear, long-wave limit the generic model consists either of a Boussinesq equation, or of two coupled Korteweg-de Vries equations, respectively. For short waves, the generic model is correspondingly either a nonlinear Klein-Gordon equation for the wave envelope, or a pair of coupled first-order envelope equations
The Camassa-Holm equation as the long-wave limit of the improved Boussinsq equation and of a class of nonlocal wave equations
In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters ϵ and δ measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximatio
Stability of periodic waves in Hamiltonian PDEs of either long wavelength or small amplitude
Stability criteria have been derived and investigated in the last decades for
many kinds of periodic traveling wave solutions to Hamiltonian PDEs. They
turned out to depend in a crucial way on the negative signature of the Hessian
matrix of action integrals associated with those waves. In a previous paper
(Nonlinearity 2016), the authors addressed the characterization of stability of
periodic waves for a rather large class of Hamiltonian partial differential
equations that includes quasilinear generalizations of the Korteweg--de Vries
equation and dispersive perturbations of the Euler equations for compressible
fluids, either in Lagrangian or in Eulerian coordinates. They derived a
sufficient condition for orbital stability with respect to co-periodic
perturbations, and a necessary condition for spectral stability, both in terms
of the negative signature - or Morse index - of the Hessian matrix of the
action integral. Here the asymptotic behavior of this matrix is investigated in
two asymptotic regimes, namely for small amplitude waves and for waves
approaching a solitary wave as their wavelength goes to infinity. The special
structure of the matrices involved in the expansions makes possible to actually
compute the negative signature of the action Hessian both in the harmonic limit
and in the soliton limit. As a consequence, it is found that nondegenerate
small amplitude waves are orbitally stable with respect to co-periodic
perturbations in this framework. For waves of long wavelength, the negative
signature of the action Hessian is found to be exactly governed by the second
derivative with respect to the wave speed of the Boussinesq momentum associated
with the limiting solitary wave
The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations
In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters is an element of and delta measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation
Comparison of solutions of some pairs of nonlinear wave equations
In this thesis, we compare solutions of the Camassa-Holm equation with solutions of the Double Dispersion equation and the Hunter-Saxton equation. In the rst part of this thesis work, we determine a class of Boussinesq-type equations from which can be asymptotically derived. We use an expansion determined by two small positive parameters measuring nonlinear and dispersive e ects. We then rigorously show that solutions of the Camassa-Holm equation are well approximated by corresponding solutions of a certain class of the Double Dispersion equation over a long time scale. Finally we show that any solution of the Double Dispersion equation can be written as the sum of solutions of the two decoupled Camassa-Holm equations moving in opposite directions up to a small error. We observe that the approximation error for the decoupled problem is greater than the approximation error characterized by single Camassa-Holm approximation. We also obtain similar results for Benjamin-Bona-Mahony approximation to the Double Dispersion equation in the long wave limit. In the literature, Hunter- Saxton equation arises as high frequency limit of the Camassa-Holm equation. In the second part of this thesis work, we establish convergence results between the solutions of the Hunter-Saxton equations and the solutions of the Camassa-Holm equation in periodic setting providing a precise estimate for the approximation error
Boussinesq Solitary-Wave as a Multiple-Time Solution of the Korteweg-de Vries Hierarchy
We study the Boussinesq equation from the point of view of a multiple-time
reductive perturbation method. As a consequence of the elimination of the
secular producing terms through the use of the Korteweg--de Vries hierarchy, we
show that the solitary--wave of the Boussinesq equation is a solitary--wave
satisfying simultaneously all equations of the Korteweg--de Vries hierarchy,
each one in an appropriate slow time variable.Comment: 12 pages, RevTex (to appear in J. Math Phys.
Visco-potential free-surface flows and long wave modelling
In a recent study [DutykhDias2007] we presented a novel visco-potential free
surface flows formulation. The governing equations contain local and nonlocal
dissipative terms. From physical point of view, local dissipation terms come
from molecular viscosity but in practical computations, rather eddy viscosity
should be used. On the other hand, nonlocal dissipative term represents a
correction due to the presence of a bottom boundary layer. Using the standard
procedure of Boussinesq equations derivation, we come to nonlocal long wave
equations. In this article we analyse dispersion relation properties of
proposed models. The effect of nonlocal term on solitary and linear progressive
waves attenuation is investigated. Finally, we present some computations with
viscous Boussinesq equations solved by a Fourier type spectral method.Comment: 29 pages, 13 figures. Some figures were updated. Revised version for
European Journal of Mechanics B/Fluids. Other author's papers can be
downloaded from http://www.lama.univ-savoie.fr/~dutyk
- …