On the singularity and numerical instability of flow fields associated with the nonlinear Schrödinger equation

The nonlinear Schrödinger equation (NLS) is a well-known equation in the study of wave motion. In the context of nonlinear water waves, the NLS has been proven to accurately model deep-water waves with a narrow spectral bandwidth and moderate wave steepness. While many studies have focused on reconstructing free surface profiles using the NLS, the associated flow field within the fluid has received less attention. Recently, a paper was published in which the flow field associated with solutions of the NLS was constructed. Here, it has been noted that the associated modeled flow field can contain singularities below the water surface. This work analyzes the occurrence and location of such artificial singularities for different analytical NLS solutions. It is found that the flow field can generally become singular at any depth below the surface and may even appear periodically in water depth. Additionally, it is shown that the numerical computation of the flow field poses considerable challenges due to the instability of numerical schemes. These findings are crucial for the accurate analysis of flow fields corresponding to the NLS.

Subjects

Deep-water waves

Flow field

Nonlinear Schrödinger equation

Numerical instability

Singularity

Velocity potential

DDC Class

530: Physics

515: Analysis

519: Applied Mathematics, Probabilities

Funding(s)

Effiziente Simulation zur Analyse und Optimierung eines Wellenenergiekonverters in nichtlinearen, zufälligen Seegängen

Lizenz

https://creativecommons.org/licenses/by/4.0/

Publication version

publishedVersion

Name

1-s2.0-S0029801826008991-main.pdf

Type

Main Article

Size

6.57 MB

Format

Adobe PDF

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On the singularity and numerical instability of flow fields associated with the nonlinear Schrödinger equation