Multisoliton Bound States in the Fourth-Order Concatenation Model of the Nonlinear Schrödinger Equation Hierarchy

Abstract

We present a comparative analysis of exact soliton solutions that form multisoliton bound states arising in the hierarchy of the even-order nonlinear Schrödinger equations. Specifically, we consider two-soliton bound states in three models: the nonlinear Schrödinger equation, the fourth-order nonlinear equation, and the Lakshmanan-Porsezian-Daniel equation (LPDE). The LPDE may be viewed as one example of aninfinite concatenation hierarchy. The exact integrability of these equations is ensured by the Lax pairs constructed using the Ablowitz, Kaup, Newell, and Segur formalism of the Inverse Scattering Transform method. We confirm that the main property of solitons-to interact elastically and attract or repel each other in the collision region depending on the difference of their initial phases-is also preserved for LPDE solitons, and this property should take place in the entire hierarchy of concatenation models. We present the detailed dynamics of two-soliton bound states periodically breathing in space and time, the conditions of their formation, and analytical formulas for their oscillation periods in all the considered models.