Tappert transformation in nonlinear wave theory

Abstract

We review the Tappert transformation for the nonlinear Schrödinger equation and generalize it for the higher- order nonlinear Schrödinger equation (hoNLSE) model of femtosecond nonlinear optics, in which the para- meterized gravitational-like potential e¤ectively simulates the Raman self-scattering e¤ect (the soliton Raman self-frequency shift). We present the explicit gauge transformation of this model into the "free" hoNSLSE with varying in time dispersion and nonlinearity, but without external potentials. With a certain choice of the dispersion and nonlinearity parameters, hoNLSE is turning into the completely integrable physical models, for example, the Hirota equation with gravitational-like potential, for which the same reversible gauge transformation and the change of variables are fulfilled. The transformations between the proposed integrable models allowed one to obtain directly the soliton solutions of the forced nonisospectral equations - accelerating in the gravitational- like potential nonautonomous solitons - without resolving the nonisospectral IST problem with varying in time spectral parameter.