Strongly nonlinear modal equations for nearly integrable PDEs. (English) Zbl 0797.35144
Summary: The purpose of this paper is the derivation of reduced, finite- dimensional dynamical systems that govern the near-integrable modulations of \(N\)-phase, spatially periodic, integrable wavetrains. The small parameter in this perturbation theory is the size of the nonintegrable perturbation in the equation, rather than the amplitude of the solution, which is arbitrary. Therefore, these reduced equations locally approximate strongly nonlinear behavior of the nearly integrable PDE. The derivation we present relies heavily on the integrability of the underlying PDE and applies, in general, to any \(N\)-phase periodic wavetrain. For specific applications, however, a numerical pretest is applied to fix the truncation order \(N\). We present one example of the reduction philosophy with the damped, driven sine-Gordon system and summarize our present progress toward application of the modulation equations to this numerical study.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
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