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Classification of solutions of the forced periodic nonlinear Schrödinger equation

2010; IOP Publishing; Volume: 23; Issue: 9 Linguagem: Inglês

10.1088/0951-7715/23/9/008

1361-6544

Eli Shlizerman, Vered Rom‐Kedar,

Nonlinear Waves and Solitons

The integrable structure of the periodic one-dimensional nonlinear Schr?dinger equation is utilized to gain insights regarding the perturbed near-integrable dynamics. After recalling the known results regarding the structure and stability of the unperturbed standing and travelling waves solutions, two new stability results are presented: (1) it is shown numerically that the stability of the 'outer' (cnoidal) unperturbed solutions depends on their power (the L2 norm): they undergo a finite sequence of Hamiltonian?Hopf bifurcations as their power is increased. (2) another proof that the 'inner'(dnoidal) unperturbed solutions with multiplicity ?2 are linearly unstable is presented. Then, to study the global phase-space structure, an energy?momentum bifurcation diagram (PDE-EMBD) that consists of projections of the unperturbed standing and travelling waves solutions to the energy?power plane and includes information regarding their linear stability is constructed. The PDE-EMBD helps us to classify the behaviour near the plane wave solutions: the diagram demonstrates that below some known threshold amplitude, precisely three distinct observable chaotic mechanisms arise: homoclinic chaos, homoclinic resonance and, for some parameter values, parabolic-resonance. Moreover, it appears that the dynamics of the PDE chaotic solutions that exhibit the parabolic-resonance instability may be qualitatively predicted: these exhibit the same dynamics as a recently derived parabolic-resonance low-dimensional normal form. In particular, these solutions undergo adiabatic chaos: they follow the level lines of an adiabatic invariant till they reach the separatrix set at which the adiabatic invariant undergoes essentially random jumps.