Numerical Computation of Solutions of the Critical Nonlinear Schrödinger Equation after the Singularity

We present numerical results for the solution of the one-dimensional critical nonlinear Schrodinger with periodic boundary conditions and initial data that give rise to a finite time singularity. We construct, through the Mori–Zwanzig formalism, a reduced model which allows us to follow the solution after the formation of the singularity. The computed postsingularity solution exhibits the same characteristics as the postsingularity solutions constructed recently by Terence Tao.

[1]  Panagiotis Stinis.  A PHASE TRANSITION APPROACH TO DETECTING SINGULARITIES OF PARTIAL DIFFERENTIAL EQUATIONS , 2009 .

[2]  T. Tao.  Nonlinear dispersive equations : local and global analysis , 2006 .

[3]  Alexandre J. Chorin,et al.  Optimal prediction with memory , 2002 .

[4]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[5]  G. Perelman.  On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D , 2000 .

[6]  Alexandre J. Chorin,et al.  Problem reduction, renormalization, and memory , 2005 .

[7]  Gadi Fibich,et al.  Proof of a Spectral Property related to the singularity formation for the L2 critical nonlinear Schrödinger equation , 2006 .

[8]  A. Stuart,et al.  Extracting macroscopic dynamics: model problems and algorithms , 2004 .

[9]  Jean Bourgain,et al.  Construction of blowup solutions for the nonlinear Schr ? odinger equation with critical nonlineari , 1997 .

[10]  Panagiotis Stinis,et al.  Optimal prediction and the rate of decay for solutions of the Euler equations in two and three dimensions , 2007, Proceedings of the National Academy of Sciences.

[11]  Takayoshi Ogawa,et al.  Blow-up of solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary condition , 1990 .

[12]  Frank Merle,et al.  Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schrödinger Equation , 2005 .

[13]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[14]  Frank Merle,et al.  On universality of blow-up profile for L2 critical nonlinear Schrödinger equation , 2004 .

[15]  T. Tao.  Global existence and uniqueness results for weak solutions of the focusing mass-critical non-linear Schr\"odinger equation , 2008, 0807.2676.

[16]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .

[17]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[18]  A J Chorin,et al.  Optimal prediction and the Mori-Zwanzig representation of irreversible processes. , 2000, Proceedings of the National Academy of Sciences of the United States of America.