Nonlinear evolution of the modulational instability

• Published in: Physics of fluids. B, Plasma physics Vol. 2; no. 1; pp. 44 - 52
• Main Authors: Goldstein, P. P., Rozmus, W.
• Format: Journal Article
• Language: English
• Published: New York, NY American Institute of Physics 01.01.1990
• Subjects: 640410 - Fluid Physics- General Fluid Dynamics; CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; DIFFERENTIAL EQUATIONS; DISPERSION RELATIONS; EQUATIONS; Exact sciences and technology; FUNCTIONS; INSTABILITY; LAGRANGIAN FUNCTION; MODULATION; NONLINEAR PROBLEMS; OPTIMIZATION; P WAVES; PARTIAL DIFFERENTIAL EQUATIONS; PARTIAL WAVES; Physics; Physics of gases, plasmas and electric discharges; Physics of plasmas and electric discharges; PLASMA; PLASMA INSTABILITY; PLASMA WAVES; SCHROEDINGER EQUATION; VARIATIONAL METHODS; WAVE EQUATIONS; Waves, oscillations, and instabilities in plasmas and intense beams; PLASMA INSTABILITY; PLASMA; NONLINEAR PROBLEMS; SCHROEDINGER EQUATION; OPTIMIZATION; P WAVES; PLASMA WAVES; VARIATIONAL METHODS; MODULATION; LAGRANGIAN FUNCTION; DISPERSION RELATIONS; Non linear equation; Modulational instability; Plasma instability; Theoretical study; Elliptic function; Variational calculus; Schroedinger equation; Non linear Schrödinger equation; Ritz method; Electrostatic wave; Trial function
• ISSN: 0899-8221; 2163-503X
• DOI: 10.1063/1.859584

The Ritz variational method has been applied to the nonlinear Schrödinger equation to construct a model for the nonlinear evolution of the modulational instability. The spatially periodic trial function was chosen in the form of a combination of Jacobian elliptic functions, with the dependence of its parameters subject to optimization. The model predicts development of the instability through localization to a quasisoliton state and a periodic recurrence of the initial condition. Theoretical predictions compare well with numerical solutions to the nonlinear Schrödinger equation.