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Solving the generalized nonlinear Schrödinger equation via quartic spline approximation. (English) Zbl 0979.65082
The authors study a new method to compute solutions of the generalized nonlinear Schrödinger equation $i{\partial u\over\partial t}+ {\partial^2u\over\partial x^2}+ f(|u|^2)u= 0,\quad -\infty< x<\infty,\quad t\geq t_0$ together with an initial condition for $$t= t_0$$. Here $$i$$ is the complex unit and $$f$$ is a sufficiently smooth function for which $$f(0)= 0$$.
The proposed method is based on a semidiscretization in space where quartic spline collocations are used to deal with the space singularities. Two methods are actually discussed, distinguished only by the approximation strategies for the Neumann boundary conditions. This method allows to compute long-time solitary solutions. The paper contains a careful analysis of the continuous and discrete conservation properties as well as the stability of the method.
Two numerical tests are included, a single soliton case and a collision of two solitons case. Both methods work well and there is no significant difference in the numerical results.

##### MSC:
 65M20 Method of lines 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 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q51 Soliton equations
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