Stability of normalized solitary waves for three coupled nonlinear Schrodinger equations

In this paper we establish existence and stability results concerning fully nontrivial solitary-wave solutions to 3-coupled nonlinear Schr"odinger system $$i{\mathrm{\partial }}_t{u}_j+{\mathrm{\partial }}_{xx}{u}_j+\left(\sum _{k=1}^3{a}_{kj}\mathrm{\mid }{u}_k{\mathrm{\mid }}^p\right)\mathrm{\mid }{u}_j{\mathrm{\mid }}^{p-2}{u}_j=0,j=1,2,3,$$ where $u_j$ are complex-valued functions of $(x,t)\in {\mathbb{R}}^2$ and $a_{kj}$ are positive constants satisfying $a_{kj}=a_{jk}$ (symmetric attractive case). Our approach improves many of the previous known results. In all methods used previously to study solitary waves, which we are aware of, the variational problem has consisted of finding the extremum of an energy functional subject to the constraints that were not independently chosen. Here we study a problem of minimizing the energy functional subject to three independent $L^2$ mass constraints and establish existence and stability results for a true three-parameter family of solitary waves.