We study the dichotomy phenomena of solutions to the Kawahara equation with bounded variation initial data. The phenomena, called Talbot effect, are that at rational times the solution is quantized, while at irrational times it is a continuous nowhere differentiable function with fractal profile. Such is unknown for the Kawahara equation yet, which is a fifth-order KdV type equation. For the purpose, we obtain smoothing estimates for the nonlinear Duhamel solution, which, combined the known results on the linear solution, mathematically describes the Talbot effect. 2020 Mathematics Subject Classification. Primary: 35B45; Secondary: 35Q53.

We discuss the Lugiato-Lefever equation and its variant with third-order dispersion, which are mathematical models used to describe how a light beam forms patterns within an optical cavity. It is mathematically demonstrated that the solutions of these equations follow the Talbot effect, which is a phenomenon of periodic self-imaging of an object under certain conditions of diffraction. The Talbot effect is regarded as the underlying cause of pattern formation in optical cavities. 2020 Mathematics Subject Classification. Primary: 35B45; Secondary: 35B10.

We prove dispersive estimate for the elastic wave equation by which we extend the known Strichartz estimates for the classical wave equation to those for the elastic wave equation. In particular, the endpoint Strichartz estimates are deduced. For the purpose we diagonalize the symbols of the Lamé operator and its semigroup, which also gives an alternative and simpler proofs of the previous results on perturbed elastic wave equations. Furthermore, we obtain uniform Sobolev inequalities for the elastic wave operator.

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