A minimal energy tracking method for non-radially symmetric solutions of coupled nonlinear Schrödinger equations

We aim at developing methods to track minimal energy solutions of time-independent m-component coupled discrete nonlinear Schrodinger (DNLS) equations. We first propose a method to find energy minimizers of the 1-component DNLS equation and use it as the initial point of the m-component DNLS equations in a continuation scheme. We then show that the change of local optimality occurs only at the bifurcation points. The fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve. By combining all these techniques with a parameter-switching scheme, we successfully compute a non-radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly.

[1]  Andrea Sacchetti,et al.  Exponential times in the One-dimensional Gross–petaevskii Equation with Multiple Well Potential , 2007 .

[2]  Shu-Ming Chang,et al.  A continuation BSOR-Lanczos-Galerkin method for positive bound states of a multi-component Bose-Einstein condensate , 2005 .

[3]  Chen,et al.  Self-Trapping of Partially Spatially Incoherent Light. , 1996, Physical review letters.

[4]  Qiang Du,et al.  Computing the Ground State Solution of Bose-Einstein Condensates by a Normalized Gradient Flow , 2003, SIAM J. Sci. Comput..

[5]  D. Christodoulides,et al.  Discrete self-focusing in nonlinear arrays of coupled waveguides. , 1988, Optics letters.

[6]  A R Bishop,et al.  Discrete nonlinear Schrödinger equation with defects. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Yueh-Cheng Kuo,et al.  A hyperplane-constrained continuation method for near singularity in coupled nonlinear Schrödinger equations , 2010 .

[8]  Adrian Ankiewicz,et al.  PARTIALLY COHERENT SOLITONS ON A FINITE BACKGROUND , 1999 .

[9]  A. Davydov,et al.  The theory of contraction of proteins under their excitation. , 1973, Journal of theoretical biology.

[10]  Foek T Hioe,et al.  Special set and solutions of coupled nonlinear schrödinger equations , 2004 .

[11]  E. Allgower,et al.  Numerical path following , 1997 .

[12]  S.-L. Chang,et al.  Liapunov-Schmidt Reduction and Continuation for Nonlinear Schrödinger Equations , 2007, SIAM J. Sci. Comput..

[13]  Kwok Wing Chow.  Periodic solutions for a system of four coupled nonlinear Schrödinger equations , 2001 .

[14]  Yaron Silberberg,et al.  Discretizing light behaviour in linear and nonlinear waveguide lattices , 2003, Nature.

[15]  F Minardi,et al.  Josephson Junction Arrays with Bose-Einstein Condensates , 2001, Science.

[16]  Weizhu Bao.  Ground States and Dynamics of Multicomponent Bose-Einstein Condensates , 2004, Multiscale Model. Simul..

[17]  H. Keller.  Lectures on Numerical Methods in Bifurcation Problems , 1988 .

[18]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[19]  Foek T Hioe,et al.  Solitary Waves for N Coupled Nonlinear Schrodinger Equations , 1999 .

[20]  Shu-Ming Chang,et al.  Gauss-Seidel-type methods for energy states of a multi-component Bose-Einstein condensate , 2005 .

[21]  M. Kwong.  Uniqueness of positive solutions of Δu−u+up=0 in Rn , 1989 .

[22]  Tai-Chia Lin,et al.  Ground State of N Coupled Nonlinear Schrödinger Equations in Rn,n≤3 , 2005 .

[23]  M Lakshmanan,et al.  Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations. , 2001, Physical review letters.

[24]  A Smerzi,et al.  Discrete solitons and breathers with dilute Bose-Einstein condensates. , 2001, Physical review letters.