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Particle trajectories in nonlinear Schrödinger models. (English) Zbl 1452.76033

Summary: The nonlinear Schrödinger equation is well known as a universal equation in the study of wave motion. In the context of wave motion at the free surface of an incompressible fluid, the equation accurately predicts the evolution of modulated wave trains with low to moderate wave steepness. While there is an abundance of studies investigating the reconstruction of the surface profile \(\eta\), and the fidelity of such profiles provided by the nonlinear Schrödinger equation as predictions of real surface water waves, very few works have focused on the associated flow field in the fluid. In the current work, it is shown that the velocity potential \(\phi\) can be reconstructed in a similar way as the free surface profile. This observation opens up a range of potential applications since the nonlinear Schrödinger equation features fairly simple closed-form solutions and can be solved numerically with comparatively little effort. In particular, it is shown that particle trajectories in the fluid can be described with relative ease not only in the context of the nonlinear Schrödinger equation, but also in higher-order models such as the Dysthe equation, and in models incorporating certain types of viscous effects.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76D33 Waves for incompressible viscous fluids
35Q55 NLS equations (nonlinear Schrödinger equations)
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References:

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