Abstract

Non-linear spectra for the blow-up profiles of the nonlinear Schrodinger equation C. Budd The cubic nonlinear Schrodinger equation arises applications in mathematical physics. When posed in two or three dimensions it can have solutions which become infinite in a finite time, with a singularity forming at a point. For dimensions greater than two the behaviour close to this point is self-similar and can be described in terms of a nonlinear ordinary differential equation with a nonlinear eigenvalue and an integral constraint on the solution. The nonlinear eigenvalue expresses the dgree of coupling between the amplitude and the phase of the solution. It has long been thought that this equation has a unique solution, howver I shall give numerical and asymptotic evidence for the existence of an infinite set of eigenvalues corresponding to multi-bump solutions which correspond to blow-up solutions with many maxima and minima. By considering radially symmetric solutions and letting the dimension approach 2 continously, we can identify at least two families of solution branches which have a non-analytic bifurcation point at dimension 2. At this point the self-similar solutions cease to exist and are replaced by an approximately self-similar solution profile. In this talk I will describe both how the solutions of the ordinary differential equation are computed and also how ther fully time dependent profiles are determined by using an adaptive method. Careful comparisons will be made with the results of a formal asymptotic calculation and excellent agreement is found.

General Enquiries Dr. Malcolm Brown. Computer Science Cardiff University PO Box 916 Cardiff CF24 3XF, U.K. Malcom@cs.cf.ac.uk

N.B. All Mathematics appears in TeX/LaTeX source.

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