Bifurcation of nonlinear Bloch waves from the spectrum in the Gross-Pitaevskii equation. (English) Zbl 1339.37072
Summary: We rigorously analyze the bifurcation of stationary so-called nonlinear Bloch waves (NLBs) from the spectrum in the Gross-Pitaevskii (GP) equation with a periodic potential, in arbitrary space dimensions. These are solutions which can be expressed as finite sums of quasiperiodic functions and which in a formal asymptotic expansion are obtained from solutions of the so-called algebraic coupled mode equations. Here we justify this expansion by proving the existence of NLBs and estimating the error of the formal asymptotics. The analysis is illustrated by numerical bifurcation diagrams, mostly in 2D. In addition, we illustrate some relations of NLBs to other classes of solutions of the GP equation, in particular to so-called out-of-gap solitons and truncated NLBs, and present some numerical experiments concerning the stability of these solutions.
MSC:
| 37K50 | Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 37M05 | Simulation of dynamical systems |
| 65P30 | Numerical bifurcation problems |
Keywords:
periodic nonlinear Schrödinger equation; nonlinear Bloch wave; Lyapunov-Schmidt decomposition; asymptotic expansion; bifurcation; delocalizationSoftware:
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