Related papers: Singularity for Solutions of Linearized KdV Equati…
In this paper we study weak continuity of the dynamical systems for the KdV equation in H^{-3/4}(R) and the modified KdV equation in H^{1/4}(R). This topic should have significant applications in the study of other properties of these…
We show how the Laplace transform can be used to give a solution of the time-dependent Schr\"odinger equation for an arbitrary initial wave packet if the solution of the stationary equation is known. The solution is obtained without summing…
We consider Schr\"odinger equations with variable coefficients, and it is supposed to be a long-range type perturbation of the flat Laplacian on $R^n$. We characterize the wave front set of solutions to Schr\"odinger equations in terms of…
In this article, we prove that small localized data yield solutions to Kawahara type equation which have linear dispersive decay on a finite time. We use the similar method used to derive the dispersive decay bound of the solutions to the…
In this paper we establish a new uniqueness result of weak solutions for the 3D Navier-Stokes equations. Under assumption that there is not uniqueness of weak solution in singular time, we prove that if two weak solutions $u$ and $v$ of 3D…
In this paper, we consider forward and inverse problems for subdiffusion equations with time-dependent coefficients. The fractional derivative is taken in the sense of Riemann-Liouville. Using the classical Fourier method, the theorem of…
The general space-time evolution of the scattering of an incident acoustic plane wave pulse by an arbitrary configuration of targets is treated by employing a recently developed non-singular boundary integral method to solve the Helmholtz…
We consider an inverse boundary value problem for diffusion equations with multiple fractional time derivatives. We prove the uniqueness in determining a number of fractional time-derivative terms, the orders of the derivatives and…
We consider a tsunami wave equation with singular coefficients and prove that it has a very weak solution. Moreover, we show the uniqueness results and consistency theorem of the very weak solution with the classical one in some appropriate…
In this paper, we consider a wave equation on a bounded domain with a Sturm-Liouville operator with a singular intermediate coefficient and a singular potential. To obtain and evaluate the solution, the method of separation of variables is…
We consider the Korteweg-de Vries (KdV) equation, and prove that small localized data yields solutions which have dispersive decay on a quartic time-scale. This result is optimal, in view of the emergence of solitons at quartic time, as…
In this article, we explore convolutions of distributions with distributions given by (weighted) line integration. We also explore the scattering of singularities of such convolutions.
A representation of solutions of the wave equation with two spatial coordinates in terms of localized elementary ones is presented. Elementary solutions are constructed from four solutions with the help of transformations of the affine…
In the paper, we experimentally study the inverse problem with the resonant scattering determinant. We analyze the structure of characteristics of perturbed linear waves. Assuming there is the common part of potential perturbation…
Asymptotic stability is with no doubts an essential property to be studied for any system. This analysis often becomes very difficult for coupled systems and even harder when different timescales appear. The singular perturbation method…
Our aim is to study the effect of a continuous prescribed density variation on the propagation of ocean waves. More precisely, we derive KdV-type shallow water model equations for unidirectional flows along the Equator from the full…
In this paper, we treat the Fisher-KPP equation with a Caputo-type time fractional derivative and discuss the propagation speed of the solution. The equation is a mathematical model that describes the processes of sub-diffusion,…
The evolution of a solitary wave with very weak nonlinearity which was originally investigated by Miles [4] is revisited. The solution for a one-dimensional gravity wave in a water of uniform depth is considered. This leads to finding the…
In this paper we consider a singular wave equation with distributional and more singular non-distributional coefficients and develop tools and techniques for the phase-space analysis of such problems. In particular we provide a detailed…
We consider the problem of the soliton propagation, in a slowly varying medium, for a generalized Korteweg - de Vries equations (gKdV). We study the effects of inhomogeneities on the dynamics of a standard soliton. We prove that slowly…