Stability of solitary solutions to the cubic-quintic nonlinear Schrodinger equation (CQNLSE)

Analytical investigation of the stability of solitary solutions to the Cubic-Quintic Nonlinear Schr ö dinger Equation (CQNLSE), which governs the propagation of electromagnetic fields in nonlinear fibers, is the main subject of this article. Assuming a perturbation of the form in the solutions and using the theory of operators, we show that for CQNLSE, is pure imaginary so that such perturbations do not destroy the stability of the solutions. We also introduce a Hamiltonian, from which CQNLSE is derivable, and calculate its first and second order variations under the aforementioned perturbation. It is shown that the first order variation vanishes and the second one is positive, which is a further indication of the stability of the solutions.