Abstract In this paper, we establish the optimal convergence for a second-order exponential-type integrator from Hofmanová & Schratz (2017, An KdV equation. Numer. Math., 136, 1117–1137) solving Korteweg–de Vries equation with rough initial data. The scheme is explicit and efficient to implement. By rigorous error analysis, show that provides accuracy in $H^\gamma $ data $H^{\gamma +4}$ any...

Studied here is the Kawahara equation, a fifth-order Korteweg-de Vries type with time-delayed internal feedback. Under suitable assumptions on time delay coefficients, we prove that solutions of this system are exponentially stable. First, considering damping and delayed system, some restriction spatial length domain, energy goes to $ 0 as t \rightarrow \infty $. After that, by introducing more...

We investigate the limit behavior of the solutions to the Kawahara equation ut + u3x + εu5x + uux = 0 , ε > 0 as ε → 0. In this equation, the terms u3x and εu5x do compete together and do cancel each other at frequencies of order 1/ √ ε. This prohibits the use of a standard dispersive approach for this problem. Nervertheless, by combining different dispersive approaches according to the range o...

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...

We present an elementary method to obtain the equations of the shallow-water solitary waves in different orders of approximation. The first two of these equations are solved to get the shapes and propagation velocities of the corresponding solitary waves. The first order equation is shown to be equivalent to the Korteweg−de Vries (KdV) equation, while the second order equation is solved numeric...

In this paper, using the Lie group analysis method, we study the invariance properties of the general time fractional fifth-order Korteweg-de Vries (KdV) equation. A systematic research to derive Lie point symmetries of the equation is performed. In the sense of point symmetry, all of the geometric vector fields and the symmetry reductions of the equation are obtained, the exact power series so...

Some of the well-known convergence acceleration algorithms, when viewed as two-variable difference equations, are equivalent to discrete soliton equations. It is shown that the η−algorithm is nothing but the discrete KdV equation. In addition, one generalized version of the ρ−algorithm is considered to be integrable discretization of the cylindrical KdV equation. ‡ E-mail: [email protected]...

In this paper we establish the nonlinear stability of solitary travelling-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 travelling-waves will be the theory developed by Albert in [1].

In this Letter, we consider a coupled Schrödinger–Korteweg–de Vries equation (or Sch–KdV) equation with appropriate initial values using the Adomian’s decomposition method (or ADM). In this method, the solution is calculated in the form of a convergent power series with easily computable components. The method does not need linearization, weak nonlinearity assumptions or perturbation theory. Th...