×

Solitary wave interactions for the modified equal width equation. (English) Zbl 0951.65098

The numerical solution of the modified equal width equation in one space dimension, \[ u_t-\mu u_{xxt}+ 3u^2 u_x= 0,\quad \mu> 0, \] is considered. The exact solutions in an infinite domain with \(u\to 0\) as \(x\to\pm\infty\) are solitary waves having the same width. Moreover, they fulfill some invariant principles.
The spatial discretization is a collocation method using quintic B-splines and equidistant nodal points. For the temporal discretization, the Crank-Nicolson scheme is applied. The linearized method is shown to be unconditionally stable in the sense of von Neumann.
The author studies three different test cases: the motion of a single solitary wave that fulfills the invariance principles, the inelastic interaction of two solitary waves, and the birth of a soliton from the initial condition.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
76B25 Solitary waves for incompressible inviscid fluids
35Q51 Soliton equations
PDFBibTeX XMLCite
Full Text: DOI