Rogue waves in fluid dynamics and optical waveguides are unexpectedly large displacements from a background state, and occur in the nonlinear Schrödinger equation with positive linear dispersion in the regime of positive cubic nonlinearity. Rogue waves of a derivative nonlinear Schrödinger equation are calculated in this work as a long-wave limit of a breather (a pulsating mode), and can occur in the regime of negative cubic nonlinearity if a sufficiently strong self-steepening nonlinearity is also present. This critical magnitude is shown to be precisely the threshold for the onset of modulation instabilities of the background plane wave, providing a strong piece of evidence regarding the connection between a rogue wave and modulation instability. The maximum amplitude of the rogue wave is three times that of the background plane wave, a result identical to that of the Peregrine breather in the classical nonlinear Schrödinger equation model. This amplification ratio and the resulting spectral broadening arising from modulation instability correlate with recent experimental results of water waves. Numerical simulations in the regime of marginal stability are described.

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The extended Korteweg de Vries model governs the evolution of weakly dispersive waves under the combined influence of quadratic and cubic nonlinearities, and is relevant to finite-amplitude wave motions in the atmosphere and the ocean. Analytic expressions for a multi-soliton are obtained by the Hirota bilinear method, and are shown to agree with those for isolated solitary waves or breathers obtained earlier in the literature. In particular, the interaction of a breather and a soliton can now be studied. Both the soliton and the breather retain their identities after interaction except for some phase shifts. Detailed examination of the interaction process shows that the profile of the breather will depend critically on the polarity of the colliding soliton.3

The phenomenon of dissipative soliton resonance (DSR) predicts that an increase of pulse energy by orders of magnitude can be obtained in laser oscillators. Here, we prove that DSR is achievable in a realistic ring laser cavity using nonlinear polarization evolution as the mode-locking mechanism, whose nonlinear transmission function is adjusted through a set of waveplates and a passive polarizer. The governing model accounts explicitly for the arbitrary orientations of the waveplates and the polarizer, as well as the gain saturation in the amplifying medium. It is shown that DSR is achievable with realistic laser settings. Our findings provide an excellent design tool for optimizing the mode-locking performance and the enhancement of energy delivered per pulse by orders of magnitude.