Radiationless Higher-Order Embedded Solitons

Abstract

The soliton solutions of a higher-order nonlinear Schrödinger equation including nonlinearities of orders \(2b+1\) and \(4b+1\), as well as second- and fouth-order dispersive terms, are obtained. It is shown that these solitons may be (or may not be) embedded, depending on the coefficients of the equation. The radiationless character of the embedded solitons is explained by analyzing the Fourier transform of the equation. The stability of these solitons is studied analytically and numerically. In the analytical approach, we combined the Vakhitov–Kolokolov stability criterion with the standard variational method. This combined technique predicts that the standard solitons are unstable and the embedded ones might be stable. The numerical results confirm these predictions, and show that the embedded and standard solitons respond very differently to perturbations.