A refinement of the Strichartz inequality on the saddle and applications
The paper is dealing with the problem of existence of singular solutions to the two-dimensional nonlinear Schrödinger (NLS) equation with the cubic nonlinearity and spatial two-dimensional linear operator of a hyperbolic (i.e., non-elliptic) type: \[ iu_t+u_{xy} +\gamma | u| ^2u=0, \] where \(\gamma\) is a constant. It is commonly known that the NLS equation with the elliptic two-dimensional spatial operator and the sign of \(\gamma\) corresponding to self-focusing gives rise to collapse, i.e., formation of a singularity in the solution in a finite time, if the properly defined norm of the initial state exceeds a certain critical value. It is not known whether collapsing solutions may be generated by the NLS equation with the hyperbolic spatial operator. To find an approach to the solution of this challenging problem, the present paper considers a phenomenon related to the collapse, i.e., mass concentration: in the case of collapsing solutions in the equation with an elliptic operator, a known fact is that, approaching the collapse moment, the solution tends to concentrate a finite part of the initial quadratic norm, \(\int\int| u(x,y)| ^2\,dx\,dy\), in an infinitely small region surrounding the collapse point. The paper produces a prove of the fact that, \text{ if} collapsing solutions do exist in the NLS equation with the hyperbolic operator, then the mass-concentration effect takes place too. The proof is based on the use of the Fourier transform, division of the integration domain into a set of rectangles, and application of a specially refined form of the so-called Strichartz inequality, which is a specific estimate for the Fourier transform.
scientific article; zbMATH DE number 5083467
scientific article