Emergence of polynomial external potentials in solitonic hierarchies: Applications to the nonisospectral LPDE model

Abstract

Inspired by the work of Akira Hasegawa, recently published in Optik 279 (2023) 170769, we set ourselves the task of explaining in detail how external nonstationary polynomial potentials appear in the nonisospectral NLSE hierarchy. We consider possible applications for observing solitonic regimes of propagation of ultrashort laser pulses in optical fibers. Our main goal is pedagogical, and our central point is the detailed derivation of new equations resulting from the application of the concept of nonautonomous solitons. It is revealed that complex potentials arise as polynomial series of the fourth degree of the spatial coordinate Uext(x,t)=α0(t)+α1(t)x+α2(t)x2+α3(t)x3+α4(t)x4, if the derivative of the spectral parameter is also expanded into a polynomial series of the spectral parameter up to the second order Λt(t)=λ0t+λ1(t)Λ+λ2(t)Λ2. It is predicted that new fourth-order nonlinear models arise if and only if the NLSE operator, the mKdV complex operator, and the fourth-order LPDE operators are connected by the extended Hirota constraint. We focus on exact analytical solutions and basic dynamic regimes of formation and interaction of new phase-modulated nonautonomous solitons of the nonisospectral LPDE model with fourth-order and time-dependent polynomial potentials.