Construction of chirped propagation with Jacobi elliptic functions for the nonlinear Schrödinger equations with quadratic nonlinearity with inter-modal and spatio-temporal dispersions

At the end of the past century, a completely new sort of soliton was discovered: embedded solitons. They were discovered in optical systems first, and then in liquid crystal theory, discrete systems and hydrodynamic models. These unique solitary waves are intriguing because they exist in settings where solitons were previously assumed to be impossible to propagate. Initially, these nonlinear waves were thought to be inherently isolated and unstable, but it was later shown that they can be stable and may exist in families. In this article, we find embedded solitons in terms of chirped periodic wave (CPW) soliton solutions for nonlinear Schrödinger equations with quadratic nonlinearity (NLSE-QN). These solutions further degenerate to chirp-free solitons such as singular, hyperbolic, kink, anti-kink, bright–dark combo, bright, dark, and solitons are recovered with the help of Jacobi elliptic functions (JEFs) and show our result graphically in 3D and 2D form.