The large-time development of the solution to an initial-value problem for the Korteweg-de Vries equation: III. Pure soliton solutions

In this paper, we consider an initial-value problem for the Korteweg-de Vries equation. The normalized Korteweg-de Vries equation considered is given by

     

Existence and asymptotic behavior of traveling wave solution for Korteweg-de Vries-Burgers equation with distributed delay

In this paper, we investigate the existence and asymptotic behavior of traveling wave solution for delayed Korteweg-de Vries-Burgers (KdV-Burgers) equation. Using geometric singular perturbation theory and Fredholm alternative, we establish the existence of traveling wave solution for this equation. Employing the standard asymptotic theory, we obtain asymptotic behavior of traveling wave solution of the equation.     

Stability of a traveling-wave solution to the Cauchy problem for the Korteweg-de Vries-Burgers equation

The asymptotic behavior of the solution to the Cauchy problem for the Korteweg-de Vries-Burgers equation u _t + (f(u)) _x + au _xxx − bu _xx = 0 as t → ∞ is analyzed. Sufficient conditions for the existence and local stability of a traveling-wave solution known in the case of f(u) = u ² are extended to the case of an arbitrary sufficiently smooth convex function f(u).     

The large-time development of the solution to an initial-boundary value problem for the Korteweg-de Vries equation on the negative quarter-plane

In this paper, we consider an initial-boundary value problem for the Korteweg-de Vries equation on the negative quarter-plane. The normalized Korteweg-de Vries equation considered is given by

     

Convergence rate of solutions toward traveling waves for the Cauchy problem of generalized Korteweg-de Vries-Burgers equations

In this paper we investigate the exponential time decay rate of solutions toward traveling waves for the Cauchy problem of generalized Korteweg-de Vries-Burgers equations

(E)     

The large-time development of the solution to an initial-boundary value problem for the Korteweg-de Vries equation. I. Steady state solutions

In this paper, we consider an initial-boundary value problem for the Korteweg-de Vries equation on the positive quarter-plane. The normalized Korteweg-de Vries equation considered is given by

     

Initial boundary value problem and asymptotic stabilization of the Camassa-Holm equation on an interval

We investigate the nonhomogeneous initial boundary value problem for the Camassa-Holm equation on an interval. We provide a local in time existence theorem and a weak-strong uniqueness result. Next we establish a result on the global asymptotic stabilization problem by means of a boundary feedback law.     

A non-homogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain II

Studied here is an initial- and boundary-value problem for the Korteweg-de Vries equation

     

A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane

The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem

studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0.1) by making use of modern methods for the study of nonlinear dispersive wave equations. Speaking technically, local well-posedness is obtained for initial data in the class for \frac34$"> and boundary data in , whereas global well-posedness is shown to hold for when , and for when . In addition, it is shown that the correspondence that associates to initial data and boundary data the unique solution of (0.1) is analytic. This implies, for example, that solutions may be approximated arbitrarily well by solving a finite number of linear problems.

     

Global existence of solution of Cauchy problem for nonlinear pseudo-parabolic equation

In this paper, we prove that the Cauchy problem for the nonlinear pseudo-parabolic equation

vt−αvxxt−βvxx+γvx+fx(v)=φx(vx)+g(v)−αg(v)_xx     

Study of classical solution of a one-dimensional mixed problem for one class of fifth-order semilinear equations of the Korteweg-de Vries-Burgers type

As is well known, many problems of mathematical physics are reduced to one- and multidimensional initial and initial-boundary value problems for, generally speaking, strongly nonlinear pseudoparabolic equations. The existence (local and global) and uniqueness of a classical solution to a one-dimensional mixed problem with homogeneous Riquier-type boundary conditions are analyzed for a class of fifth-order semilinear pseudoparabolic equations of the Korteweg-de Vries-Burgers type. For the classical solution of the mixed problem, a uniqueness theorem is proved using the Gronwall-Bellman inequality, a local existence theorem is proved by combining the generalized contraction mapping principle with the Schauder fixed point principle, and a global existence theorem is proved by applying the method of a priori estimates.     

Asymptotic expansion of solutions to the drift-diffusion equation with large initial data

We consider the large-time behavior of the solution to the initial value problem for the Nernst-Planck type drift-diffusion equation in whole spaces. In the Lp-framework, the global existence and the decay of the solution were shown. Moreover, the second-order asymptotic expansion of the solution as t→∞ was derived. We also deduce the higher-order asymptotic expansion of the solution. Especially, we discuss the contrast between the odd-dimensional case and the even-dimensional case.     

A fundamental solution to the Cauchy problem for a fourth-order pseudoparabolic equation

The Cauchy problem for a fourth-order pseudoparabolic equation describing liquid filtration problems in fissured media, moisture transfer in soil, etc., is studied. Under certain summability and boundedness conditions imposed on the coefficients, the operator of this problem and its adjoint operator are proved to be homeomorphism between certain pairs of Banach spaces. Introduced under the same conditions, the concept of a θ-fundamental solution is introduced, which naturally generalizes the concept of the Riemann function to the equations with discontinuous coefficients; the new concept makes it possible to find an integral form of the solution to a nonhomogeneous problem.     

On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation

We prove the global existence and uniqueness of admissible weak solutions to an asymptotic equation of a nonlinear hyperbolic variational wave equation with nonnegative L ²(ℝ) initial data. The work of Ping Zhang is supported by the Chinese postdoctor’s foundation, and that of Yuxi Zheng is supported in part by NSF DMS-9703711 and the Alfred P. Sloan Research Fellows award.     

Multiple positive solutions to a singular boundary value problem for a superlinear Emden-Fowler equation

A multiplicity result for the singular ordinary differential equation y^″+λx⁻²yσ=0, posed in the interval (0,1), with the boundary conditions y(0)=0 and y(1)=γ, where σ>1, λ>0 and γ?0 are real parameters, is presented. Using a logarithmic transformation and an integral equation method, we show that there exists Σ^?∈(0,σ/2] such that a solution to the above problem is possible if and only if λγσ−1?Σ^?. For 0<λγσ−1<Σ^?, there are multiple positive solutions, while if γ=(λ⁻¹Σ^?)^1/(σ−1) the problem has a unique positive solution which is monotonic increasing. The asymptotic behavior of y(x) as x→⁰⁺ is also given, which allows us to establish the absence of positive solution to the singular Dirichlet elliptic problem −Δu=d⁻²(x)uσ in Ω, where Ω⊂RN, N?2, is a smooth bounded domain and d(x)=dist(x,∂Ω).     

The global attractivity and asymptotic stability of solution of a nonlinear integral equation

In this paper, we employ the fixed point theorem to study the existence of an integral equation and obtain the global attractivity and asymptotic stability of solutions of the equation. Some new results are given.     

Central symmetric solution to the Neumann problem for a time-fractional diffusion-wave equation in a sphere

In this paper, a time-fractional central symmetric diffusion-wave equation is investigated in a sphere. Two types of Neumann boundary condition are considered: the mathematical condition with the prescribed boundary value of the normal derivative and the physical condition with the prescribed boundary value of the matter flux. Several examples of problems are solved using the Laplace integral transform with respect to time and the finite sin-Fourier transform of the special type with respect to the spatial coordinate. Numerical results are illustrated graphically.     

Formulation and well-posedness of the Cauchy problem for a diffusion equation with discontinuous degenerating coefficients

The choice of a differential diffusion operator with discontinuous coefficients that corresponds to a finite flow velocity and a finite concentration is substantiated. For the equation with a uniformly elliptic operator and a nonzero diffusion coefficient, conditions are established for the existence and uniqueness of a solution to the corresponding Cauchy problem. For the diffusion equation with degeneration on a half-line, it is proved that the Cauchy problem with an arbitrary initial condition has a unique solution if and only if there is no flux from the degeneration domain to the ellipticity domain of the operator. Under this condition, a sequence of solutions to regularized problems is proved to converge uniformly to the solution of the degenerate problem in L ₁(R) on each interval.