In this paper we establish the nonlinear stability of solitary traveling-wave solutions for the Kawahara-KdV equation ut + uux + uxxx − γ1uxxxxx = 0, and the modified Kawahara-KdV equation ut + 3u 2ux + uxxx − γ2uxxxxx = 0, where γi ∈ R is a positive number when i = 1, 2. The main approach used to determine the stability of solitary traveling-waves will be the theory developed by Albert in [1].

We provide an elementary approach to integrable systems associated with hyperelliptic curves of infinite genus. In particular, we explore the extent to which the classical Burchnall-Chaundy theory generalizes in the infinite genus limit, and systematically study the effect of Darboux transformations for the KdV hierarchy on such infinite genus curves. Our approach applies to complex-valued peri...

In the present article, a numerical method is proposed for the numerical solution of the KdV equation by using a new approach by combining cubic B-spline functions. In this paper we convert the KdV equation to system of two equations. The method is shown to be unconditionally stable using von-Neumann technique. To test accuracy the error norms L2, L∞ are computed. Three invariants of motion are...

1. As was shown in the remarkable communication [4] the Cauchy problem for the Korteweg–de Vries (KdV) equation ut = 6uux−uxxx, familiar in theory of nonlinear waves, is closely linked with a study of the spectral properties of the Sturm–Liouville operator Lψ = Eψ, where L = −d/dx + u(x). For rapidly decreasing initial conditions u(x, 0), where ∫∞ −∞ u(x, 0)(1 + |x|)dx < ∞, the KdV equation can...

The second part of the notes are written jointly with my collaborator from University of Illinois, M. B. Erdogan. We developed the material with two goals in mind. First to prove existence and uniqueness results in the case of dispersive PDE evolving from initial data that are periodic in the space variable. Secondly we develop new tools to address the problem of wellposedness of solutions, in ...

Delay-difference and delay-differential analogues of the KdV Boussinesq (BSQ) equations are presented. Each them has N-soliton solution reduces to an already known soliton equation as delay parameter approaches 0. In addition, a analogue KP is proposed. We discuss its limit Finally, relationship between KdV, BSQ, clarified. Namely, reductions yield BSQ equations.

The Korteweg-de Vries (KdV) equation is widely recognized as a simple model for unidirectional weakly nonlinear dispersive waves on the surface of a shallow body of fluid. While solutions of the KdV equation describe the shape of the free surface, information about the underlying fluid flow is encoded into the derivation of the equation, and the present article focuses on the formulation of mas...

The object of this article is the comparison of numerical solutions of the so-called Whitham equation describing wave motion at the surface of a perfect fluid to numerical approximations of solutions of the full Euler free-surface water-wave problem. The Whitham equation ηt + 3 2 c0 h0 ηηx +Kh0∗ ηx = 0 was proposed by Whitham [33] as an alternative to the KdV equation for the description of sur...

We revisit existence and stability of two-pulse solutions in the fifth-order Korteweg–de Vries (KdV) equation with two new results. First, we modify the Petviashvili method of successive iterations for numerical (spectral) approximations of pulses and prove convergence of iterations in a neighborhood of two-pulse solutions. Second, we prove structural stability of embedded eigenvalues of negati...

The initial boundary-value problem for the Korteweg–de Vries (KdV) equation on the negative quarter-plane, x < 0 and t > 0, is considered. The formulation of this problem is different to the usual initial boundary-value problem on the positive quarter-plane, for which x > 0 and t > 0. Two boundary conditions are required at x = 0 for the negative quarter-plane problem, in contrast to the one bo...