Explicit Lump and New Closed-Form Analytical Outcomes for the Nonlinear Hyperbolic Schrödinger Equation

The hyperbolic Schrödinger equation (SE) is a variant of nonlinear SE and is used in the study of ultrashort optical pulse prorogation in nonlinear media when the paraxial envelop approximation is not valid. In this work, we investigate a nonlinear hyperbolic Schrödinger equation (SE) via two distinct analytical approaches: the first one is the Hirota method (HM), and the other is the \(\frac{\texttt{G}^{\prime }\left( \xi _{0}\right) }{\texttt{G}^{\prime }\left( \xi _{0}\right) +\texttt{G}\left( \xi _{0}\right) +c_{2}}\) -expansion procedure (GEP), both of which highlight the importance of wave behaviour. The proposed strategies are reliable and effective. The nonlinear hyperbolic SE admits a wide variety of innovative solutions. These solutions include multi-order lump, lump-wave, and trigonometric forms, which are obtained via HM and the \(\frac{\texttt{G}^{\prime }\left( \xi _{0}\right) }{\texttt{G}^{\prime }\left( \xi _{0}\right) +\texttt{G}\left( \xi _{0}\right) +c_{2}}\) -expansion procedure. Through simulation graphs, the dynamic behaviour of the outcomes is demonstrated. The derived solutions are visualized via 3D surface plots and 2D contour graphs.