期刊
NONLINEARITY
卷 23, 期 9, 页码 2183-2218出版社
IOP PUBLISHING LTD
DOI: 10.1088/0951-7715/23/9/008
关键词
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资金
- Israel Science Foundation [273/07]
- Minerva foundation
- Russian-Israeli joint grant [MNTI-RFBR No 06-01-72023]
The integrable structure of the periodic one-dimensional nonlinear Schrodinger 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 L-2 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.
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