On conservation laws, their applications in stability analysis and chirped solitary wave solutions for the generalized Schrödinger–Hirota equation and its reductions

Abstract

Conserved quantities for a generalized version of the Schrödinger–Hirota (SH) equation and its reductions are deduced using an algorithmic approach which is easier to implement compared to more sophisticated mathematical methods. The conserved mass and the Hamiltonian, whenever it exists, are then used to analyse the stability of exact stationary wave solutions, by invoking the Vakhitov-Kolokolov criterion and variational methods. In addition we have focussed on two classes of chirped solitary wave solutions for reductions of the generalised SH equation, namely the algebraic and the hyperbolic classes. It is found that along with the regular dependance of the chirp on the intensity there may even be an additional dependence varying inversely on the intensity of the solitary wave. The chirped hyperbolic solutions are found to correspond to gray solitary waves for suitable parameter values which may also flip to anti-dark solitary waves depending on the ratio of the parameters involved in the solution.