Blow-up profile for the complex Ginzburg–Landau equation

AbstractWe construct a solution to the complex Ginzburg–Landau equation, which blows up in finite time T only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite-dimensional one, and the use of index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint and it has a second neutral mode. In the last section, the interpretation of the parameters of the finite-dimensional problem in terms of the blow-up time and the blow-up point gives the stability of the constructed solution with respect to perturbations in the initial data

An Adaptive Method for Calculating Blow-Up Solutions

Reactive-diffusive systems modeling physical phenomena in certain situations develop a singularity at a finite value of the independent variable referred to as blow-up. The attempt to find the blow-up time analytically is most often impossible, thus requiring a numerical determination of the value. The numerical methods often use a priori knowledge of the blow-up solution such as monotonicity or self-similarity. For equations where such a priori knowledge is unavailable, ad hoc methods were constructed. The object of this research is to develop a simple and consistent approach to find numerically the blow-up solution without having a priori knowledge or resorting to other ad hoc methods. The proposed method allows the investigator the ability to distinguish whether a singular solution or a non-singular solution exists on a given interval. Step size in the vicinity of a singular solution is automatically adjusted. The programming of the proposed method is simple and uses well-developed software for most of the auxiliary routines. The proposed numerical method is mainly concerned with the integration of nonlinear integral equations with Abel-type kernels developed from combustion problems, but may be used on similar equations from other fields. To demonstrate the flexibility of the proposed method, it is applied to ordinary differential equations with blow-up solutions or to ordinary differential equations which exhibit extremely stiff structure