Related papers: Dispersive and diffusive-dispersive shock waves fo…
Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian…
In this paper we study the general relationship between the evolutionary conditions for discontinuous solutions of hyperbolic conservation laws with a concave entropy function and the existence and uniqueness of steady dissipative shock…
We present a phenomenological approach to dispersion in nonlinear elasticity. A simple, thermomechanically sound, constitutive model is proposed to describe the (non-dissipative) properties of a hyperelastic dispersive solid, without…
We consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension, and we investigate the behavior of bounded L^2 perturbations of shock wave solutions to the Riemann problem using the relative entropy…
Diffraction is one of the universal phenomena of physics, and a way to overcome it has always represented a challenge for physicists. In order to control diffraction, the study of structured waves has become decisive. Here, we present a…
We introduce a dispersive regularization of the compressible Euler equations in Lagrangian coordinates, in the one-dimensional torus. We assume a Van der Waals pressure law, which presents both hyperbolic and elliptic zones. The dispersive…
We consider the so-called Naiver-Stokes-Korteweg(NSK) equations for the dynamics of compressible barotropic viscous fluids with internal capillarity. We handle the time-asymptotic stability in 1D of the viscous-dispersive shock wave that is…
Interactions of dispersive shock (DSWs) and rarefaction waves (RWs) associated with the Korteweg-de Vries equation are shown to exhibit multiphase dynamics and isolated solitons. There are six canonical cases: one is the interaction of two…
We present a way to deal with dispersion-dominated ``shock-type'' transition in the absence of completely integrable structure for the systems that one may characterize as strictly hyperbolic regularized by a small amount of dispersion. The…
The one-dimensional piston shock problem is a classical result of shock wave theory. In this work, the analogous dispersive shock wave (DSW) problem for a dispersive fluid described by the nonlinear Schr\"odinger equation is analyzed.…
We propose a hierarchy of nonlinearly dispersive generalized Korteweg--de Vries (KdV) evolution equations based on a modification of the Lagrangian density whose induced action functional the KdV equation extremizes. It is shown that two…
The Korteweg-de Vries (KdV) equation is a fundamental partial differential equation that models wave propagation in shallow water and other dispersive media. Accurately solving the KdV equation is essential for understanding wave dynamics…
We consider a general reaction--nonlinear-diffusion equation with a region of negative diffusivity, and show how a nonlinear regularisation selects a shock position. Negative diffusivity can model population aggregation, but leads to…
The truncated Korteweg-De Vries (TKdV) system, a shallow-water wave model with Hamiltonian structure that exhibits weakly turbulent dynamics, has been found to accurately predict the anomalous wave statistics observed in recent laboratory…
Debris flows often exhibit coherent wave structures-shock-like roll waves on steeper slopes and weaker, more sinusoidal dispersive pulses on gentler slopes. Coarse-rich heads raise basal resistance, whereas fines-rich tails lower it; in…
For a class of nonconservative hyperbolic systems of partial differential equations endowed with a strictly convex mathematical entropy, we formulate the initial value problem by supplementing the equations with a kinetic relation…
Collisionless regime kinetic models for coherent nonlinear Alfven wave dynamics are studied using fluid moment equations with an approximate closure anzatz. Resonant particle effects are modelled by incorporating an additional term…
In this paper the stability of the Korteweg-de Vries (KdV) equation is investigated. It is shown analytically and numerically that small perturbations of solutions of the KdV-equation introduce effects of dispersion, hence the perturbation…
The structure of a signed fundamental solution of a conservation law is studied without the convexity assumption. The types of shocks and rarefaction waves are classified together with their interactions. A comprehensive picture of a global…
This work deals with exact solutions to the wave equations. We start by introducing the Non-Diffracting Waves (NDW), and by a definition of NDWs. Afterwards we recall -besides ordinary waves (gaussian beams, gaussian pulses)- the simplest…