A supraconvergent scheme for the Korteweg-de Vries equation

Summary A finite-difference method for the integration of the Korteweg-de Vries equation on irregular grids is analyzed. Under periodic boundary conditions, the method is shown to be supraconvergent in the sense that, though being inconsistent, it is second order convergent. However, such a convergence only takes place on grids with an odd number of points per period. When a grid with an even number of points is used, the inconsistency of the method leads to divergence. Numerical results backing the analysis are presented.     

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.     

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 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.     

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.     

Neumann problem for the Korteweg-de Vries equation

We consider Neumann initial-boundary value problem for the Korteweg-de Vries equation on a half-line

(0.1)     

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 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.     

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 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.     

A new efficient energy-preserving finite volume element scheme for the improved Boussinesq equation

In this paper, a new linearized energy-preserving Crank-Nicolson finite volume element scheme is derived for the improved Boussinesq equation. The fully discrete scheme can be shown to conserve both mass and energy in the discrete setting. It is proved that the scheme is uniquely solvable and convergent with the rate of order two in a discrete L₂ norm. At last, a series of numerical experiments on typical improved Boussinesq and Sine–Gordon equations are provided to verify our theoretical results and to show the efficiency and accuracy of the proposed scheme.     

Scattering for the quartic generalised Korteweg-de Vries equation

We show that the quartic generalised KdV equation

ut+uxxx+(u⁴x₎=0     

Soliton interaction for the extended Korteweg-de Vries equation

Soliton interactions for the extended Korteweg-de Vries (KdV)equation are examined. It is shown that the extended KdV equationcan be transformed (to its order of approximation) to a higher-ordermember of the KdV hierarchy of integrable equations. This transformationis used to derive the higher-order, two-soliton solution forthe extended KdV equation. Hence it follows that the higher-ordersolitary-wave collisions are elastic, to the order of approximationof the extended KdV equation. In addition, the higher-ordercorrections to the phase shifts are found. To examine the exactnature of higher-order, solitary-wave collisions, numericalresults for various special cases (including surface waves onshallow water) of the extended KdV equation are presented. Thenumerical results show evidence of inelastic behaviour wellbeyond the order of approximation of the extended KdV equation;after collision, a dispersive wavetrain of extremely small amplitudeis found behind the smaller, higher-order solitary wave.     

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.     

Numerical solution of complex modified Korteweg-de Vries equation by collocation method

The collocation method using quintic B-spline is derived for solving the complex modified Korteweg-de Vries (CMKdV). The method is based on Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The von Neumann stability is used to prove that the scheme is unconditionally stable. Newton’s method is used to solve the nonlinear block pentadiagonal system obtained. Numerical tests for single, two, and three solitons are used to assess the performance of the proposed scheme.     

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.