Pseudo-relativistic nonlinear Schroedinger equations

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Pseudo-relativistic nonlinear Schroedinger EquationsAbstract of Proposed ResearchGigliola Staffilani and Enno LenzmannThis proposal sets forth work on a novel class of dispersive partial differential equations (PDEs) which are called pseudo-relativistic nonlinear Schroedinger equations. These equations, whose mathematical study is still in its infancy, have recently found a significant application as effective descriptions for the dynamical evolution of self-gravitating, relativistic matter. Based on this physical motivation, the PI's proposed research aims at understanding the qualitative behavior of solutions to these model equations. In particular, great emphasis is put on the pseudo-relativistic Hartree equation whose focusing nonlinearity is of critical strength in the sense that large initial data can lead to finite-time blow-up of the solution. Here it is of paramount interest to gain analytical insight into blow-up rates and dynamics, as well as to prove existence of non-radial blow-up solutions, thereby extending the PI's previous results on radial blow-up for the pseudo-relativistic Hartree equation (in collaboration with J. Froehlich). Apart from the issue of finite-time blow-up, the PI proposes a detailed study of global-in-time solutions and their asymptotic behavior as time tends to infinity. Dispersive partial differential equations provide meeting grounds for theoretical and empirical branches of the natural sciences. The mathematical study of pseudo-relativistic Schroedinger equations substantiates intuitive and numerical insights into physical models of theoretical astrophysics. One of the most outstanding problems that astrophysics is facing today is the so-called dark matter problem. Various theoretical models set forth the existence of boson stars as possible candidates for explaining this empirical puzzle. The pseudo-relativistic Hartree equation considered in this proposal serves as an effective description for the dynamical evolution of boson stars. Therefore, this nonlinear dispersive PDE provides analytical testing grounds for these theoretical objects