The Initial-Boundary Value Problem for the Korteweg–de Vries Equation

We prove local well-posedness of the initial-boundary value problem for the Korteweg–de Vries equation on right half-line, left half-line, and line segment, in the low regularity setting. This is accomplished by introducing an analytic family of boundary forcing operators.     

Initial Value Problem for a Generalized Korteweg－de Vries Equation with Singular Integral－Differential Terms

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Bifurcations in the Generalized Korteweg–de Vries Equation

We study the generalized Korteweg–de Vries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) equation with periodic in the spatial variable boundary conditions. For various values of parameters, in a sufficiently small neighborhood of the zero equilibrium state we construct asymptotics of periodic solutions and invariant tori. Separately we consider the case when the stability spectrum of the zero solution contains a countable number of roots of the characteristic equation. In this case we state a special nonlinear boundary-value problem which plays the role of a normal form and determines the dynamics of the initial problem.     

A Nonhomogeneous Boundary-Value Problem for the Korteweg–de Vries Equation Posed on a Finite Domain

Abstract

Studied here is an initial- and boundary-value problem for the Korteweg–de Vries equation posed on a bounded interval with nonhomogeneous boundary conditions. This particular problem arises naturally in certain circumstances when the equation is used as a model for waves and a numerical scheme is needed. It is shown here that this initial-boundary-value problem is globally well-posed in the L ₂-based Sobolev space H  ^s (0, 1) for any s ≥ 0. In addition, the mapping that associates to appropriate initial- and boundary-data the corresponding solution is shown to be analytic as a function between appropriate Banach spaces.     

An Initial-Boundary Value Problem for Parabolic Monge-Ampère Equation in Mathematical Finance

This paper deals with some parabolic Monge-Ampère equation raised from mathematical finance: V_sV_(yy)+ryV_yV_(yy)-θV_y~2= 0(V_(yy) 0). The existence and uniqueness of smooth solution to its initial-boundary value problem with some requirement is obtained.     

An Initial Boundary-Value Problem in a Half-Strip for the Korteweg–De Vries Equation in Fractional-Order Sobolev Spaces

Abstract

An initial boundary-value problem in a half-strip with one boundary condition for the Korteweg–de Vries equation is considered and results on global well-posedness of this problem are established in Sobolev spaces of various orders, including fractional. Initial and boundary data satisfy natural (or close to natural) conditions, originating from properties of solutions of a corresponding initial-value problem for a linearized KdV equation. An essential part of the study is the investigation of special solutions of a “boundary potential” type for this linearized KdV equation.     

Compacton Solutions of the Korteweg–de Vries Equation with Constrained Nonlinear Dispersion

Computational Mathematics and Mathematical Physics - The numerical solution of initial value problems is used to obtain compacton and kovaton solutions of K(f m, g n) equations...     

The Initial-Boundary Value Problem for the Boussinesq Equations

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On the Absence of Global Solutions of the Korteweg–de Vries Equation

This paper is devoted to the problem of existence of global solutions of the Korteweg–de Vries equation. For certain initial–boundary problems for the Korteweg–de Vries equation, we obtain necessary conditions of existence (in other words, sufficient conditions of nonexistence) of global solutions.     

On Complex-Valued Solutions of the General Loaded Korteweg–de Vries Equation with a Source

Differential Equations - Using the inverse scattering method, we derive the evolution of the scattering data of a nonself-adjoint Sturm–Liouville operator whose potential is a solution of the...     

Weak Solutions for the Vlasov-Poisson Initial-Boundary Value Problem with Bounded Electric Field

The aim of this work is to construct weak solutions for the three dimensional Vlasov-Poisson initial-boundary value problem with bounded electric field.The main in- gredient consists of estimating the change in momentum along characteristics of regular electric fields inside bounded spatial domains.As direct consequences,the propagation of the momentum moments and the existence of weak solution satisfying the balance of total energy are obtained.     

Integrating the Korteweg–de Vries Equation with a Self-Consistent Source and “Steplike” Initial Data

It is shown that steplike solutions of the Korteweg–de Vries equation with a self-consistent source can be found by the inverse scattering method for the Sturm–Liouville operator on the entire real line.     

Decay of Solutions to Damped Korteweg–de Vries Type Equation

In the present paper we establish results concerning the decay of the energy related to the damped Korteweg–de Vries equation posed on infinite domains. We prove the exponential decay rates of the energy when a initial value problem and a localized dissipative mechanism are in place. If this mechanism is effective in the whole line, we get a similar result in H ^k -level, k∈ℕ. In addition, the decay of the energy regarding a initial boundary value problem posed on the right half-line, is obtained considering convenient a smallness condition on the initial data but a more general dissipative effect.     

Complexiton solutions of the Korteweg–de Vries equation with self-consistent sources

Complexiton solutions to the Korteweg–de Vires equation with self-consistent sources are presented. The basic technique adopted is the Darboux transformation. The resulting solutions provide evidence that soliton equations with self-consistent sources can have complexiton solutions, in addition to soliton, positon and negaton solutions. This also implies that soliton equations with self-consistent sources possess some kind of analytical characteristics that linear differential equations possess and brings ideas toward classification of exact explicit solutions of nonlinear integrable differential equations.