Parameterization method for unstable manifolds of standing waves on the line. (English) Zbl 1473.74075
Summary: We consider a linearly unstable standing wave solution of a parabolic partial differential equation (PDE) on the real line and develop a high order method for polynomial approximation of the local unstable manifold. The unstable manifold describes the breakdown of the nonlinear wave after the loss of stability. Our method is based on the parameterization method for invariant manifolds and studies an invariance equation describing a local chart map. This invariance equation is a PDE posed on the product of a disk and the line. The dimension of the disk is equal to the Morse index of the wave. We develop a formal series solution for the invariance equation, and show that the coefficients of the series solve certain boundary value problems (BVPs) on the line. We solve these BVPs numerically to any desired order. The result is a polynomial describing the dynamics of the PDE in a macroscopic neighborhood of the unstable standing wave. The method is implemented for a number of example problems. Truncation/numerical errors are quantified via a posteriori indicators.
MSC:
| 74J30 | Nonlinear waves in solid mechanics |
| 74H10 | Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics |
| 74H55 | Stability of dynamical problems in solid mechanics |
Keywords:
traveling wave; parabolic PDE; stability loss; invariance equation; formal series expansion; Morse indexReferences:
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