On a class of initial-boundary value problems for equations of Korteweg-de Vries type

We consider a special class of initial-boundary value problems on the positive halfline x > 0 for the Korteweg-de Vries equation and its generalizations. For this class, we prove theorems on the nonexistence of global solutions for t > 0.     

On the nonexistence of global solutions of the Cauchy problem for the Korteweg-de Vries Equation

We establish conditions on the initial data under which the Cauchy problem for the Korteweg-de Vries equation does not admit a solution global in t > 0. The proof of the results is based on the nonlinear capacity method [7]. In closing, we provide an example.     

On the singular solutions of the Korteweg-de Vries equation

We consider two classes of singular solutions of the KdV equation: singular solutions of the Cauchy problem and singular traveling waves. In both cases, we establish sufficient conditions for their existence.     

On the Korteweg-de Vries equation

Existence, uniqueness, and continuous dependence on the initial data are proved for the local (in time) solution of the (generalized) Korteweg-de Vries equation on the real line, with the initial function in the Sobolev space of order s>3/2 and the solution u(t) staying in the same space, s= being included For the proper KdV equation, existence of global solutions follows if s2. The proof is based on the theory of abstract quasilinear evolution equations developed elsewhere.Dedicated to Hans Lewy and Charles B. Morrey Jr.Partially supported by NSF Grant MCS76-04655.     

Bounded solutions of the Korteweg-de Vries equation

New types of bounded nondecreasing solutions of the equation are found and it is proved that they are limits of N-soliton solutions.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 49, pp. 59–70, 1988.     

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

     

On the generalized Korteweg-de Vries equation

We prove that the local L² norm of the solution of the generalized Korteweg-de Vries equation $$u_t + (F(u) + \sum\limits_{s = 0}^m {( - 1)^s D_x^{2s} u)_x = 0,m \geqslant 2,} $$ with nice initial datum, where F satisfies certain general conditions, for example, P(u) = u^p, where p is an odd integer ≧3, decays t o zero as time goes to infinity.     

Exact solutions of the modified Korteweg-de Vries equation

We use the inverse scattering method to obtain a formula for certain exact solutions of the modified Korteweg-de Vries (mKdV) equation. Using matrix exponentials, we write the kernel of the relevant Marchenko integral equation as W( x + y;t ) = Ce ^{- ( x + y )A} e^{8A3 t} B\Omega \left( {x + y;t} \right) = Ce^{ - \left( {x + y} \right)A} e^{8A^3 t} BB, where the real matrix triplet (A,B,C) consists of a constant p×p matrix A with eigenvalues having positive real parts, a constant p×1 matrix B, and a constant 1× p matrix C for a positive integer p. Using separation of variables, we explicitly solve the Marchenko integral equation, yielding exact solutions of the mKdV equation. These solutions are constructed in terms of the unique solution P of the Sylvester equation AP + PA = BC or in terms of the unique solutions Q and N of the Lyapunov equations A^°Q + QA = C^°C and AN + NA^° = BB^°, where B^°denotes the conjugate transposed matrix. We consider two interesting examples.     

Exact soliton solutions for the general fifth Korteweg-de Vries equation

With the aid of computer symbolic computation system such as Maple, the extended hyperbolic function method and the Hirota’s bilinear formalism combined with the simplified Hereman form are applied to determine the soliton solutions for the general fifth-order KdV equation. Several new soliton solutions can be obtained if we taking parameters properly in these solutions. The employed methods are straightforward and concise, and they can also be applied to other nonlinear evolution equations in mathematical physics. The article is published in the original.     

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

     

On difference splitting schemes for the Korteweg-de Vries equation

The convergence of difference splitting schemes for the solution of the Korteweg-de Vries equation is considered. A method is developed for obtaining convergence bounds in C for the case when the scheme does not satisfy the maximum, principle. The proposed method is applied to prove convergence theorems for splitting schemes with sufficiently smooth initial values.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 56–61, 1985.     

Breather-type soliton and two-soliton solutions for modified Korteweg-de Vries equation

In this Letter, different kinds of solutions including breather-type soliton and two-soliton solutions, are obtained for the modified Korteweg-de Vries (M-KdV) equation by using bilinear form, the extended homoclinic test approach and dependent variable transformations. Moreover,we point out that the author did not obtain so-called periodic two-soliton solutions in W. Long (in press) [1].     

On the nonexistence of global solutions of the Hamilton-Jacobi equation

Although examples of nonexistence of global solutions of Hamilton-Jacobi equations have long been known, the general theory of nonexistence of global solutions (unlike the general theory of existence of global viscous solutions) does not yet exist.In the present paper, we use examples of model Hamilton-Jacobi equations to demonstrate how simple methods for the analysis of the problem of nonexistence of global solutions (in the corresponding function class) can be applied.     

On a family of initial-boundary value problems for the heat equation

We consider the heat problem with nonlocal boundary conditions containing a real parameter. For the zero value of the parameter, this problem is well known as the Samarskii-Ionkin problem and has been comprehensively studied. We analyze the spectral problem for the operator of second derivative subjected to the boundary conditions of the original problem. By separation of variables, we prove the existence and uniqueness of a classical solution for any nonzero value of the parameter. The obtained a priori estimates for a solution imply the stability of the problem with respect to the initial data.     

An efficient method for analyzing the solutions of the Korteweg-de Vries equation

A new iterative method is applied to study the solutions of the Korteweg-de Vries (KdV) equation. The method is a modified form of the well known Adomian decomposition method (ADM), where it avoids the difficulty of computing the Adomian polynomials. We prove the existence of a unique solution of the KdV equation. And then, we show that the new method generates an infinite series which converges uniformly to the exact solution of the problem. Soliton solutions of the KdV equation are obtained by the new method. Numerical calculations indicate the effectiveness of the new method where the obtained results are very accurate and better than the ones obtained by the ADM.     

Asymptotic behavior of solutions of the Korteweg-de Vries equation for large times

For the KdV equation a complete asymptotic expansion of the dispersive tail for large times is described, and generalized wave operators are introduced. The asymptotics for large times of the spectral Schrödinger equation with a potential of the type of a solution of the KdV equation is studied. It is shown that the KdV equation is connected in a specific manner with the structure of the asymptotics of solutions of the spectral equation. As a corollary, known explicit formulas for the leading terms of the asymptotics of solutions of the KdV equation in terms of spectral data corresponding to the initial conditions are obtained. A plan for justifying the results listed is outlined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 120, pp. 32–50, 1982.