Data assimilation and determining forms for weakly damped, dispersive systems
Abstract
In this work, we show that the global attractor of the 1D damped, driven, nonlinear Schrodinger equations (NLS) is embedded in the long-time dynamics of a determining form. The determining form for the NLS is an ordinary differential equation in a space of trajectories X = Cb 1(R,PmH2) where Pm is the L2-projector onto the span of the ?rst m Fourier modes. Similarly, we also find a determining form for the damped, driven Korteweg de-Vries equations (KdV). This time, the determining form for the KdV is an ordinary differential equation in a space of trajectories X = Cb 1(R,PmH2). In both cases, there is a one-to-one identi?cation with the trajectories in the global attractor of the underlying equations and the steady states of the determining form for the that equation. The determining form for both of these equations is dv(s, t)/ dt= - sup{s∈R} |v( s, t) - PmW (v( s, t))|2(v(s, t) - Pmu* (s, t)), where v( s) ∈ X, u* is a steady state of the underlying equation and W is a special map from X to a different Banach space which contains the relation between the underlying partial differential equation and the determining form. Additionally, we prove that the determining modes property holds for both of these equations. We give an improved estimate for the number of the determining modes for the NLS and we give an estimate for the number of determining modes for the KdV. Moreover, we give a continuous data assimilation algorithm via feedback control approach for the NLS and the KdV using only definitely many modes. The NLS and the KdV equations are ius + uxx + |u|2u + gammau = f, (NLS) us + uux + uxxx + gamma u = f, (KdV) respectively. We prove the following theorem: Theorem. Let u be a solution of the following equation us = F( u), with an initial data u(s 0), where the above equation is either (NLS) or (KdV), and let w be the solution of the corresponding data assimilation equation ws = F(w) - micro Pm(w - u), with an arbitrary initial data w(s0). For micro large enough (depending on gamma and f) and m large enough (depending on micro), we have |w( s) - u(s)|L2 → 0 at an exponential rate, as s → infinity.
- Publication:
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Ph.D. Thesis
- Pub Date:
- 2015
- Bibcode:
- 2015PhDT.......132S
- Keywords:
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- Applied mathematics;Mathematics;Theoretical mathematics