Soliton decay of nonlinear Alfvén waves: Numerical studies

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Abstract
The derivative nonlinear Schrödinger equation is numerically solved for arbitrary initial conditions by an extension of the Ablowitz–Ladik scheme [Stud. Appl. Math 5 7, 1 (1977)]. The numerical nonlinear difference code, which takes advantage of the inverse scattering method, simulates the original differential equation reproducing common features, like solitons and an infinite set of constants of motion. The long‐time behavior is analyzed in terms of the sign of one of the constants of motion. The formation of a soliton train is seen whenever the constant has a negative value. This fact is the global expression of the Mj≂lhus local criterion to distinguish between modulationally stable and unstable cases.