An analytical solution for a coupled partial differential equation

• Published in: Applied mathematics and computation Vol. 212; no. 1; pp. 245 - 250
• Main Author: Kumar, Arun
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.06.2009; Elsevier
• Subjects: Calculus of variations and optimal control; Elliptic functions; Exact sciences and technology; Mathematical analysis; Mathematics; Non-linear Schrödinger Equation; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use; Trial function; Trial function; Elliptic functions; Non-linear Schrödinger Equation; Initial value problem; Optimization method; Periodic function; Partial differential equation; Non linear equation; Numerical analysis; Boundary value problem; Applied mathematics; Analytical solution; Mathematical programming
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2009.02.025

A variational method given by Ritz has been applied to the Coupled Partial Differential Equation to construct an analytical solution. The solution of Partial Differential Equation gives a good description of both linear and non-linear evolution of instabilities generated in waves due to modulation. The spatially periodic trial function is chosen in the form of combination of Jacobian Elliptic Functions with the dependence of its parameters subject to optimization. This equation reduced to stationary form of Non-linear Schrödinger Equation in the static limit.