Multi‐speed solitary wave solutions for nonlinear Schrödinger systems

We prove the existence of a new type of solutions to a nonlinear Schrodinger system. These solutions, which we call "multi-speeds solitary waves", are behaving at large time as a couple of scalar solitary waves traveling at different speeds. The proof relies on the construction of approximations of the multi-speeds solitary waves by solving the system backwards in time and using energy methods to obtain uniform estimates.

[1]  Michael I. Weinstein,et al.  Modulational Stability of Ground States of Nonlinear Schrödinger Equations , 1985 .

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

[3]  S. Terracini,et al.  Multipulse Phases in k-Mixtures of Bose–Einstein Condensates , 2008, 0807.1979.

[4]  Frank Merle,et al.  Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations , 2009, 0905.0470.

[5]  P. Schuur.  Asymptotic analysis of soliton problems , 1986 .

[6]  Thierry Cazenave,et al.  The Cauchy problem for the critical nonlinear Schro¨dinger equation in H s , 1990 .

[7]  B. Pellacci,et al.  Orbital Stability Property for Coupled Nonlinear Schrödinger Equations , 2008, 0809.3320.

[8]  B. Sirakov.  Least Energy Solitary Waves for a System of Nonlinear Schrödinger Equations in $${\mathbb{R}^n}$$ , 2007 .

[9]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, I existence of a ground state , 1983 .

[10]  J. Shatah,et al.  Stability theory of solitary waves in the presence of symmetry, II☆ , 1990 .

[11]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, II existence of infinitely many solutions , 1983 .

[12]  T. Cazenave.  Semilinear Schrodinger Equations , 2003 .

[13]  小澤 徹,et al.  Nonlinear dispersive equations , 2006 .

[14]  Govind P. Agrawal,et al.  Nonlinear Fiber Optics , 1989 .

[15]  P. Lions,et al.  Orbital stability of standing waves for some nonlinear Schrödinger equations , 1982 .

[16]  B. Pellacci,et al.  Positive solutions for a weakly coupled nonlinear Schrödinger system , 2006 .

[17]  Stability of solitary waves for coupled nonlinear Schro¨dinger equations , 1996 .

[18]  G. Lamb.  Elements of soliton theory , 1980 .

[19]  S. V. Manakov.  On the theory of two-dimensional stationary self-focusing of electromagnetic waves , 1973 .

[20]  Antonio Ambrosetti,et al.  Standing waves of some coupled nonlinear Schrödinger equations , 2007 .

[21]  Frank Merle,et al.  Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity , 1990 .

[22]  Catherine Sulem,et al.  The nonlinear Schrödinger equation , 2012 .

[23]  M. García-Huidobro,et al.  ON THE UNIQUENESS OF SIGN CHANGING BOUND STATE SOLUTIONS OF A SEMILINEAR EQUATION , 2010, 1809.07711.

[24]  M. Squassina,et al.  Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schr\ , 2009, 0905.4618.

[25]  Tobias Weth,et al.  Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations , 2008 .

[26]  M. Weinstein.  Lyapunov stability of ground states of nonlinear dispersive evolution equations , 1986 .

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

[28]  M. García-Huidobro,et al.  On the uniqueness of the second bound state solution of a semilinear equation , 2009 .

[29]  Chris H. Greene,et al.  Hartree-Fock Theory for Double Condensates , 1997 .

[30]  W. Rother,et al.  Nonlinear scalar field equations , 1992, Differential and Integral Equations.

[31]  Stefan Le Coz,et al.  High speed excited multi-solitons in nonlinear Schr\ , 2010, 1007.3034.

[32]  Frank Merle,et al.  Multi solitary waves for nonlinear Schrödinger equations , 2006 .

[33]  Masahito Ohta,et al.  Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction , 2009 .