Numerical simulations show that higher order KdV equation under certain conditions has a self-focusing singularity, which means that the solution of the equation blows up in finite time. In this paper, two numerical schemes: the split-step Fourier transform and the pseudospectral methods are...

The variable-coefficient Korteweg-de Vries equation that governs the dynamics of weakly nonlinear long waves in a periodically variable dispersion management media is considered. For general bit patterns, an analytic expression describing the evolution of the timing shift produced by nonlinear...

In this paper, a fractional Korteweg-de Vries equation (KdV for short) with initial condition is introduced by replacing the first order time and space derivatives by fractional derivatives of order α and β with 0α,β≤1, respectively. The fractional derivatives are described in the Caputo...

It is shown that if the dispersion of the KdV equation is replaced by a higher order dispersion ∂xm, where m≥3 is an odd integer, then the critical Sobolev exponent for local well-posedness on the circle does not change. That is, the resulting equation is locally well-posed in Hs(T), s≥−1/2.

Solutions of a boundary value problem for the Korteweg–de Vries equation are approximated numerically using a finite-difference method, and a collocation method based on Chebyshev polynomials. The performance of the two methods is compared using exact solutions that are exponentially small at...

In this paper, the weak form of the mathematical model of the traffic flow is adopted. Theoretical analyses of the discontinuous solutions of the red-and-green light models are discussed. The discontinuous Galerkin finite element method is used to numerically simulate the problems. The numerical...