Related papers: Weak asymptotics method
In this paper, we characterized resonant interaction of weakly nonlinear hyperbolic waves in gas dynamics with a real gas background. An asymptotic approach is used to study the interaction between waves, governed by the Euler equations of…
We study the weakly non-linear development of shear-driven gravity waves, and investigate the mixing properties of the finite amplitude solutions. Calculations to date have been restricted to the linear theory, which predicts that gravity…
The problem of propagating nonlinear acoustic waves is considered; the solution to which, both with and without damping, having been obtained to-date starting from the Navier-Stokes-Duhem equations together with the continuity and thermal…
We propose a theoretical approach to derive amplitude equations governing the weakly nonlinear evolution of nonnormal dynamical systems when they experience transient growth or respond to harmonic forcing. This approach reconciles the…
In this paper we consider an acoustic problem of wave propagation through a discontinuous medium. The problem is reduced to the dissipative wave equation with distributional dissipation. We show that this problem has a so-called very weak…
We consider a kinetic equation describing evolution of a particle distribution function in a system with nonlinear wave-particle interactions (trappings into a resonance and nonlinear scatterings). We study properties of its solutions and…
We present mathematical proofs on the existence and uniqueness of weak solutions for a special class of non linear parabolic and hyperbolic equations of mathematical physics subject to colored noise (structured turbulence) as random-…
We illustrate how to compute asymptotic interactions between discrete solitary waves of dispersive equations, using the approach proposed by Manton [Nucl. Phys. B 150, 397 (1979)]. We also discuss the complications arising due to…
A method for solving a quasilinear nonelliptical equation of the second order is developed and we give classification and parametrization of simple elements of the equation.We find exact solutions of an equation for potential stationary…
In this paper, the Lie symmetry method is performed for the nonlinear evolution equation for description nonlinear waves in a viscoelastic tube. we will find one and two-dimensional optimal system of Lie subalgebras. Furthermore,…
We establish a new criterion for exponential mixing of random dynamical systems. Our criterion is applicable to a wide range of systems, including in particular dispersive equations. Its verification is in nature related to several topics,…
We propose a new asymptotic expansion method for nonlinear filtering, based on a small parameter in the system noise. The conditional expectation is expanded as a power series in the noise level, with each coefficient computed by solving a…
We present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It applies to systems of coupled equations,…
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…
In this work, we derive reduced interface models for hydroelastic water waves coupled to a nonlinear viscoelastic plate. In a weakly nonlinear small-steepness regime we obtain bidirectional nonlocal evolution equations capturing the…
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Toda lattice with steplike initial data corresponding to a rarefaction wave.
Weakly-nonlinear waves in a layered waveguide with an imperfect interface (soft bonding between the layers) can be modelled using coupled Boussinesq equations. We assume that the materials of the layers have close mechanical properties, in…
A new nonlinear equation governing asymptotic dynamics of ripples is derived by using a short wave perturbative expansion on a generalized version of the Green-Naghdi system. It admits peakon solutions with amplitude, velocity and width in…
In this overview paper, we show existence of smooth solitary-wave solutions to the nonlinear, dispersive evolution equations of the form \begin{equation*} \partial_t u + \partial_x(\Lambda^s u + u\Lambda^r u^2) = 0, \end{equation*} where…
We introduce the notion of asymptotic integrability into the theory of nonlinear wave equations. It means that the Hamiltonian structure of equations describing propagation of high-frequency wave packets is preserved by hydrodynamic…