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The low dimensionality of time-periodic standing waves in water of finite and infinite depth. (English) Zbl 1263.37084

Summary: Time-periodic standing waves are demonstrated to be low-dimensional by use of the proper orthogonal decomposition (POD). Moreover, the nonlinear dynamics of the system restricted to this low-dimensional linear subspace are shown to accurately recover the spatio-temporal full PDE dynamics. A global set of modes, generated with sequential POD, are then used to produce time-periodic standing wave branches as a function of the period. This representation quantitatively reproduces the entire branch, including both large- and small amplitude solutions, using only a few POD modes. This technique offers a new direction of exploration for this challenging problem, including an efficient way to characterize the bifurcation structure and stability of these solutions.

MSC:

37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
37C27 Periodic orbits of vector fields and flows
62H25 Factor analysis and principal components; correspondence analysis
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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