Abstract
A combined numerical and analytic study is made of the nonlinear wave equationu tt , −u xx +u 3 = 0, with homogeneous Dirichlet boundary conditions atx = 0 andx = 1. Numerically, it is observed that solutions tend to have a recurrence property, and in particular that they do not decay to zero as time tends to infinity. Analytically, it is proved that if a solution decays to zero, it must do so very slowly. Moreover, the analysis shows that for large initial data, oscillations are much faster than for the corresponding linear equation. Finally, for positive initial positionu (t, 0) and zero initial speedu t (t, 0), a detailed analysis of the behavior of the solution at the moment it first becomes negative is carried out. Surprisingly, this behavior depends on the initial position in a more delicate way than expected.
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Cazenave, T., Haraux, A., Vazquez, L. et al. Nonlinear effects in the wave equation with a cubic restoring force. Computational Mechanics 5, 49–72 (1989). https://doi.org/10.1007/BF01046879
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DOI: https://doi.org/10.1007/BF01046879