相关论文: On the soliton dynamics under a slowly varying med…
We review the theory of modulation equations or Whitham equations for the travelling wave solution of KdV. We then apply the Whitham modulation equations to describe the long-time asymptotics and small dispersion asymptotics of the KdV…
We analyze the gKdV equation, a generalized version of Korteweg-de Vries with an arbitrary function $f(u)$. In general, for a function $f(u)$ the Lie algebra of symmetries of gKdV is the $2$-dimensional Lie algebra of translations of the…
We establish the nonlinear stability of solitary waves (solitons) and periodic traveling wave solutions (cnoidal waves) for a Korteweg-de Vries (KdV) equation which includes a fifth order dispersive term. The traveling wave solutions which…
Uniform estimates for the decay structure of the $n$-soliton solution of the Korteweg-deVries equation are obtained. The KdV equation, linearized at the $n$-soliton solution is investigated in a class $\WW$ consisting of sums of travelling…
We study the problem of 2-soliton collision for the generalized Korteweg-de Vries equations, completing some recent works of Y. Martel and F. Merle. We classify the nonlinearities for which collisions are elastic or inelastic. Our main…
We study the dynamics of bright and dark matter-wave solitons in the presence of a spatially varying nonlinearity. When the spatial variation does not involve zero crossings, a transformation is used to bring the problem to a standard…
We study the behavior of nonlinear waves in a two-dimensional medium with density and stress relation that vary periodically in space. Efficient approximate Riemann solvers are developed for the corresponding variable-coefficient…
We prove a full asymptotic stability result for solitary wave solutions of the mKdV equation. We consider small perturbations of solitary waves with polynomial decay at infinity and prove that solutions of the Cauchy problem evolving from…
We present a detailed numerical study of solutions to the Zakharov-Kuznetsov equation in three spatial dimensions. The equation is a three-dimensional generalization of the Korteweg-de Vries equation, though, not completely integrable. This…
We use a class of trial wave functions which are generalizations of gaussians to study single soliton approximate analytic solutions to the KdV equations. The variational parameters obey a Hamiltonian dynamics obtained from the Principle of…
We propose an alternative method for one-dimensional continuum diffusion models with spatially variable (heterogeneous) diffusivity. Our method, which extends recent work on stochastic diffusion, assumes the constant-coefficient homogenized…
We study the behavior of the soliton which, while moving in non-dissipative medium encounters a barrier with dissipation. The modelling included the case of a finite dissipative layer as well as a wave passing from a dissipative layer into…
We consider a quasilinear KdV equation that admits compactly supported traveling wave solutions (compactons). This model is one of the most straightforward instances of degenerate dispersion, a phenomenon that appears in a variety of…
We consider the effects of varying dispersion and nonlinearity on the stability of periodic traveling wave solutions of nonlinear PDE of KdV-type, including generalized KdV and Benjamin-Ono equations. In this investigation, we consider the…
In this short note, we review recent results concerning the long time dynamics of large data solutions to several dispersive models. Starting with the KdV case and ending with the KP models, we review the literature and present new results…
We show how to apply ideas from the theory of rough paths to the analysis of low-regularity solutions to non-linear dispersive equations. Our basic example will be the one dimensional Korteweg--de Vries (KdV) equation on a periodic domain…
We present a spatially dependent framework for the existence and propagation of nonlinear kinetic Alfv\'en (KA) structures in the interstellar medium (ISM). Using a multi-component analytical model that incorporates the diffuse warm ionized…
We analyze how the interaction between local and nonlocal dispersions, combined with different types of nonlinearities, influences the smoothing effects of solutions. To achieve this goal, we consider a model that generalizes the KdV and…
We investigate whether the recently proposed PT-symmetric extensions of generalized Korteweg-de Vries equations admit genuine soliton solutions besides compacton solitary waves. For models which admit stable compactons having a width which…
We study existence of helical solitons in the vector modified Korteweg-de Vries (mKdV) equations, one of which is integrable, whereas another one is non-integrable. The latter one describes nonlinear waves in various physical systems,…