From Nonlinear PDEs to Singular ODEs

E. Weinmüller, Vienna University of Technology, C. Budd, University of Bath,and O. Koch, Vienna University of Technology

We discuss the quasi-linear parabolic partial differential equation

$$\frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left( u^\sigma \frac{\partial u}{\partial x}\right)+u^\beta,$$

which is a model for the temperature profile of a fusion reactor plasma with one source term [6]. We also consider the classical nonlinear Schrödinger equation in three space dimensions,

$$&& \ensuremath{\mathrm{i}}\frac{\partial u}{\partial t} +\Delta u+\vert u\vert^2u=0,\quad t>0,$$

which occurs in various important applications for example in nonlinear optics [4] or plasma physics [5]. Both problems have solutions that become unbounded in finite time. This occurs at a single point at which there is a growing and increasingly narrow peak. Moreover, the solutions blow up in a self-similar way, which means that the above differential equations are invariant under certain Lie group transformations.

The problem of the computation of this self-similar solution profile reduces to a nonlinear, ordinary differential equation on an unbounded domain, where the boundary conditions at infinity are carefully chosen to avoid unsmooth solutions. We show that a transformation of the independent variable to the interval [0,1] yields a boundary value problem with an essential singularity. We use our MATLAB solver sbvp [1] based on polynomial collocation to treat the resulting system of ordinary differential equations. In case of the Schrödinger equation we can show the stability of the collocation routine, cf. [2]. Also, our code provides a reliable estimate of the global error of the collocation solution. This is possible because the boundary conditions for the transformed problem yield a boundary value problem with a singularity of the first kind, see [3].