The homotopy analysis method for approximating the solution of the modified Korteweg-de Vries equation

In this paper, the homotopy analysis method for solving the nonlinear modified Korteweg-de Vries equation is implemented with approximate initial conditions. We discuss the case when the problem has solitons or breathers. Some numerical examples are presented.     

Stability of periodic traveling waves for complex modified Korteweg-de Vries equation

We study the existence and stability of periodic traveling-wave solutions for complex modified Korteweg-de Vries equation. We also discuss the problem of uniform continuity of the data-solution mapping.     

On the numerical solution of Korteweg-de Vries equation by the iterative splitting method

In this paper, we apply the method of iterative operator splitting on the Korteweg-de Vries (KdV) equation. The method is based on first, splitting the complex problem into simpler sub-problems. Then each sub-equation is combined with iterative schemes and solved with suitable integrators. Von Neumann analysis is performed to achieve stability criteria for the proposed method applied to the KdV equation. The numerical results obtained by iterative splitting method for various initial conditions are compared with the exact solutions. It is seen that they are in a good agreement with each other.     

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.     

Self-similar planar curves related to modified Korteweg-de Vries equation

We exhibit a time reversible geometric flow of planar curves which can develop singularities in finite time within the uniform topology. The example is based on the construction of selfsimilar solutions of modified Korteweg-de Vries equation of a given (small) mean.     

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.     

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].     

The discrete Korteweg-de Vries equation

We review the different aspects of integrable discretizations in space and time of the Korteweg-de Vries equation, including Miura transformations to related integrable difference equations, connections to integrable mappings, similarity reductions and discrete versions of Painlevé equations as well as connections to Volterra systems.     

Numerical solution of second-order one-dimensional hyperbolic equation by exponential B-spline collocation method

In this paper, we propose a method based on collocation of exponential B-splines to obtain numerical solution of a nonlinear second-order one-dimensional hyperbolic equation subject to appropriate initial and Dirichlet boundary conditions. The method is a combination of B-spline collocation method in space and two-stage, second-order strong-stability-preserving Runge–Kutta method in time. The proposed method is shown to be unconditionally stable. The efficiency and accuracy of the method are successfully described by applying the method to a few test problems.     

A new high-order energy-preserving scheme for the modified Korteweg-de Vries equation

In this paper, a new high-order energy-preserving scheme is proposed for the modified Korteweg-de Vries equation. The proposed scheme is constructed by using the Hamiltonian boundary value methods in time, and Fourier pseudospectral method in space. Exploiting this method, we get second-order and fourth-order energy-preserving integrators. The proposed schemes not only have high accuracy, but also exactly conserve the total mass and energy in the discrete level. Finally, single solitary wave and the interaction of two solitary waves are presented to illustrate the effectiveness of the proposed schemes.     

Remarks on the Korteweg-de Vries equation

We show for the Korteweg-de Vries equation an existence uniqueness theorem in Sobolev spaces of arbitrary fractional orders≧2, provided the initial data is given in the same space.     

The Korteweg-de Vries equation and beyond

We review a new method for linearizing the initial-boundary value problem of the KdV on the semi-infinite line for decaying initial and boundary data. We also present a novel class of physically important integrable equations. These equations, which include generalizations of the KdV, of the modified KdV, of the nonlinear Schrödinger and of theN-wave interactions, are as generic as their celebrated counterparts and, furthermore it appears that they describe certain physical situations more accurately.     

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.     

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.     

Complex solutions of general Korteweg-de Vries equation: Inverse problem method

The inverse method of scattering problem has been applied to find complex solutions of the general Korteweg-de Vries equation. The direct and inverse problem have been considered for nonself-adjoint one-dimensional Schrödinger operator (with complex potential) in L₂(). The used technique of inverse problems for nonself-adjoint operators has been developed by V. É. Lyantse and his disciples.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 223–230, February, 1990.