Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBifurcations and Spectral Stability of Solitary Waves in Coupled Nonlinear Schrodinger Equations
283
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study bifurcations and spectral stability of solitary waves in coupled nonlinear Schrodinger equations (CNLS) on the line. We assume that the coupled equations possess a solution of which one component is identically zero, and call it a $textit{fundamental solitary wave}$. By using a result of one of the authors and his collaborator, the bifurcations of the fundamental solitary wave are detected. We utilize the Hamiltonian-Krein index theory and Evans function technique to determine the spectral or orbital stability of the bifurcated solitary waves as well as as that of the fundamental one under some nondegenerate conditions which are easy to verify, compared with those of the previous results. We apply our theory to CNLS with a cubic nonlinearity and give numerical evidences for the theoretical results.
rate research
Read More
We numerically study solitary waves in the coupled nonlinear Schrodinger equations. We detect pitchfork bifurcations of the fundamental solitary wave and compute eigenvalues and eigenfunctions of the corresponding eigenvalue problems to determine the spectral stability of solitary waves born at the pitchfork bifurcations. Our numerical results demonstrate the theoretical ones which the authors obtained recently. We also compute generalized eigenfunctions associated with the zero eigenvalue for the bifurcated solitary wave exhibiting a saddle-node bifurcation, and show that it does not change its stability type at the saddle-node bifurcation point.
We consider nonlinear Schrodinger equations with either power-type or Hartree nonlinearity in the presence of an external potential. We show that for long-range nonlinearities, solutions cannot exhibit scattering to solitary waves or more general localized waves. This extends the well-known results concerning non-existence of non-trivial scattering states for long-range nonlinearities.
We consider the stability problem for standing waves of nonlinear Dirac models. Under a suitable definition of linear stability, and under some restriction on the spectrum, we prove at the same time orbital and asymptotic stability. We are not able to get the full result proved by Cuccagna for the nonlinear Schrodinger equation, because of the strong indefiniteness of the energy.
In this paper, we characterize a family of solitary waves for NLS with derivative (DNLS) by the structue analysis and the variational argument. Since (DNLS) doesnt enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters $4omega>c^2$ and the critical parameters $4omega=c^2, c>0$, we show the existence and uniqueness of the solitary waves for (DNLS), up to the phase rotation and spatial translation symmetries. Secondly, for the critical parameters $4omega=c^2, cleq 0$ and the supercritical parameters $4omega<c^2$, there is no nontrivial solitary wave for (DNLS). At last, we make use of the invariant sets, which is related to the variational characterization of the solitary wave, to obtain the global existence of solution for (DNLS) with initial data in the invariant set $mathcal{K}^+_{omega,c}subseteq H^1(R)$, with $4omega=c^2, c>0$ or $4omega>c^2$. On one hand, different with the scattering result for the $L^2$-critical NLS in cite{Dod:NLS_sct}, the scattering result of (DNLS) doesnt hold for initial data in $mathcal{K}^+_{omega,c}$ because of the existence of infinity many small solitary/traveling waves in $mathcal{K}^+_{omega,c},$ with $4omega=c^2, c>0$ or $4omega>c^2$. On the other hand, our global result improves the global result in cite{Wu-DNLS, Wu-DNLS2} (see Corollary ref{cor:gwp}).
We consider nonlinear half-wave equations with focusing power-type nonlinearity $$ i pt_t u = sqrt{-Delta} , u - |u|^{p-1} u, quad mbox{with $(t,x) in R times R^d$} $$ with exponents $1 < p < infty$ for $d=1$ and $1 < p < (d+1)/(d-1)$ for $d geq 2$. We study traveling solitary waves of the form $$ u(t,x) = e^{iomega t} Q_v(x-vt) $$ with frequency $omega in R$, velocity $v in R^d$, and some finite-energy profile $Q_v in H^{1/2}(R^d)$, $Q_v ot equiv 0$. We prove that traveling solitary waves for speeds $|v| geq 1$ do not exist. Furthermore, we generalize the non-existence result to the square root Klein--Gordon operator $sqrt{-DD+m^2}$ and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds $|v| < 1$. Finally, we discuss the energy-critical case when $p=(d+1)/(d-1)$ in dimensions $d geq 2$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
