# Modeling of Fluid Flows by Nonlinear Schrödinger Equation

## Abstract

Fluid flows exhibit diverse ways of their behavior, from ordered to chaotic and turbulent motion. The Navier-Stokes or Euler equations governing such motion are formidable as they are, and even the highest performance computers have difficulty in producing accurate and therefore useful solutions. Effort has constantly been made for mathematically modeling flow phenomena by simplified equations, deriving them from the Navier-Stokes equations and solving them. In this note, we illustrate how we model the nonlinear modulation of a traveling wave, as observed in water waves, by the nonlinear Schrödinger equation. The wave modulation is captured as the instability and bifurcation of a plane-wave solution. Behind this lies the Hamiltonian structure of the Euler equations, and Krein’s theory of the Hamiltonian spectra is applicable to it. We build on it a striking aspect of dissipation and diffusion that drives instability for an otherwise stable solution.

### Keywords

Nonlinear Schrödinger equation Ginzburg-Landau equation Stokes wave Benjamin-Feir instability Dissipation induced instability### References

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