Nonlinear Density Waves in the Single-Wave Model

TL;DR

This paper derives amplitude equations for nonlinear density waves in the single-wave model using hydrodynamic closure and the renormalization group method, analyzing different root cases and validating results numerically.

Contribution

It introduces a novel analytical approach to derive amplitude equations for nonlinear density waves in the single-wave model, considering complex root cases.

Findings

Amplitude equations for three real roots case derived.

Amplitude equations for mixed real and complex roots obtained.

Analytical results validated against numerical simulations.

Abstract

The single-wave model equations are transformed to an exact hydrodynamic closure by using a class of solutions to the Vlasov equation corresponding to the waterbag model. The warm fluid dynamic equations are then manipulated by means of the renormalization group method. As a result, amplitude equations for the slowly varying wave amplitudes are derived. Since the characteristic equation for waves has in general three roots, two cases are examined. If all three roots of the characteristic equation are real, the amplitude equations for the eigenmodes represent a system of three coupled nonlinear equations. In the case, where the dispersion equation possesses one real and two complex conjugate roots, the amplitude equations take the form of two coupled equations with complex coefficients. The analytical results are then compared to the exact system dynamics obtained by solving the hydrodynamic equations numerically.