Related papers: The wave breaking for Whitham-type equations revis…
An effective field theory framework is used to investigate some Lorentz-violating effects on the generation of electromagnetic and gravitational waves, complementing previous work on propagation. Specifically we find solutions to a…
Dynamic response of the one-dimensional viscoelastic rod of finite length, that has one end fixed and the other subject to prescribed either displacement or stress, is analyzed by the analytical means of Laplace transform, yielding the…
We consider two types of the time-dependent Ginzburg-Landau equation in 2D bounded domains: the heat-flow equation and the Schroedinger equation. The system of ordinary differential equations is obtained that describes the evolution of the…
For $0<\alpha<\frac{1}{3}$ we construct unique solutions to the fractal Burgers equation $\partial_t u + u\partial_xu + (-\Delta)^\alpha u = 0$ which develop a first shock in finite time, starting from smooth generic initial data. This…
In this paper, a fractional generalization of the wave equation that describes propagation of damped waves is considered. In contrast to the fractional diffusion-wave equation, the fractional wave equation contains fractional derivatives of…
The decay of Burgers turbulence with compactly supported Gaussian "white noise" initial conditions is studied in the limit of vanishing viscosity and large time. Probability distribution functions and moments for both velocities and…
We present a method to compute dispersive shock wave solutions of the Korteweg-de Vries equation that emerge from initial data with step-like boundary conditions at infinity. We derive two different Riemann-Hilbert problems associated with…
In order to improve the frequency dispersion effects of irrotational shallow water models in coastal oceanography, several full dispersion versions of classical models were formally derived in the literature. The idea, coming from G.…
Wave-breaking is studied analytically first and the results are compared with accurate numerical simulations of 3D wave-breaking. We focus on the time dependence of various quantities becoming singular at the onset of breaking. The power…
The goal of this note is to construct a class of traveling solitary wave solutions for the compound Burgers-Korteweg-de Vries equation by means of a hyperbolic ansatz.
We consider the initial value problem for the viscous Fornberg-Whitham equation which is one of the nonlinear and nonlocal dispersive-dissipative equations. In this paper, we establish the global existence of the solutions and study its…
We propose and demonstrate, with the one-dimensional Korteweg-de Vries-Burgers model, the scenarios of transfer loop and \textit{all-scale} statistical equilibrium, the former being associated to shock formation and the latter to Gaussian…
In the present work we analyse the structure of the Hamiltonian field theory in the neighbourhood of the wave equation $q_{tt}=q_{xx}$. We show that, restricting to ``graded'' polynomial perturbations in $q_x$, $p$ and their space…
In this article we shall go over recent work in proving dispersive and Strichartz estimates for the Dirichlet-wave equation. We shall discuss applications to existence questions outside of obstacles and discuss open problems.
The Korteweg-de Vries equation is known to yield a valid description of surface waves for waves of small amplitude and large wavelength. The equation features a number of conserved integrals, but there is no consensus among scientists as to…
We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (e.g. the Whitham equation, the generalized KDV equation,…
We prove the existence of non-trivial standing wave solutions of the complex Ginzburg-Landau equation $\phi_t - e^{i\theta}(\rho I- \Delta) \varphi - e^{i\gamma} |\phi |^\alpha \varphi =0 $ in $\Rn$, where $(N-2)\alpha <4$, $\theta ,\gamma…
The Lamb-Bateman integral equation was introduced to study the solitary wave diffraction and its solution was written in terms of an integral transform. We prove that it is essentially the Abel integral equation and its solution can be…
The Fisher and Burgers' equations with finite memory transport, describing reaction-diffusion and convection-diffusion processes, respectively have recently attracted a lot of attention in the context of chemical kinetics, mathematical…
The numerical simulation of the inviscid Burgers' equation is often hindered by spurious oscillations near discontinuities. To mitigate this issue, a viscous term can be introduced, leading to the viscous Burgers' equation. In this work,…