We study various properties of solutions of an extended nonlinear Schrödinger (ENLS) equation, which arises in the context of geometric evolution problems -including vortex filament dynamicsand governs propagation of short pulses in optical fibers and nonlinear metamaterials. For the periodic initial-boundary value problem, we derive conservation laws satisfied by local in time, weak H 2 (distributional) solutions, and establish global existence of such weak solutions. The derivation is obtained by a regularization scheme under a balance condition on the coefficients of the linear and nonlinear terms -namely, the Hirota limit of the considered ENLS model. Next, we investigate conditions for the existence of traveling wave solutions, focusing on the case of bright and dark solitons. The balance condition on the coefficients is found to be essential for the existence of exact analytical soliton solutions; furthermore, we obtain conditions which define parameter regimes for the existence of traveling solitons for various linear dispersion strengths. Finally, we study the modulational instability of plane waves of the ENLS equation, and identify important differences between the ENLS case and the corresponding NLS counterpart. The analytical results are corroborated by numerical simulations, which reveal notable differences between the bright and the dark soliton propagation dynamics, and are in excellent agreement with the analytical predictions of the modulation instability analysis. 1991 Mathematics Subject Classification. 35Q53, 35Q55, 35B45, 35B65, 37K40. 1 Sec. 3]. Briefly speaking, this derivation begins from a natural generalization of the localized induction equation (LIE) which governs the velocity of the vortex filament. The generalization takes account of the axial-flow effect up to the second-order (see [27, Eqs. (3.1)-(3.2)]). Then, the Hirota equation is derived by repeating the original Hasimoto's procedure [33], which proved the equivalence between LIE and the integrable NLS equation iφ t + φ xx + 1 2 |φ| 2 φ = 0 (see also [47,48]). This equivalence implies that LIE is completely integrable. Thus, since the generalized LIE of [27] preserves integrability, the equivalent evolution equation which is in this case ENLS equation (1.1), should be also integrable. This is the reason why condition (1.2) is essential for the association of Eq. (1.1) with the vortex filament dynamics.On the other hand, non-integrable versions of (1.1), i.e., when the balance condition (1.2) is violated, can be derived in the context of geometric evolution equations. In particular, it was shown in [53] that certain differential equations of one-dimensional dispersive flows into compact Riemann surfaces, may be reduced by a definition of a generalized Hasimoto transform, to ENLS type equations. See also [54,55], and references therein.Another important physical context, where certain versions of the ENLS equation have important applications, is that of nonlinear optics: such models have been used to describe femtosecond pulse pr...

The dynamical behavior of a higher-order cubic Ginzburg-Landau equation is found to include a wide range of scenarios due to the interplay of higher-order physically relevant terms. We find that the competition between the third-order dispersion and stimulated Raman scattering effects, gives rise to rich dynamics: this extends from Poincaré-Bendixson-type scenarios, in the sense that bounded solutions may converge either to distinct equilibria via orbital connections, or space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. One of our main results is that the third-order dispersion has a dominant role in the development of such complex dynamics, since it can be chiefly responsible (i.e., even in the absence of the other higher-order effects) for the existence of the periodic, quasi-periodic and chaotic spatiotemporal structures. Suitable lowdimensional phase space diagnostics are devised and used to illustrate the different possibilities and identify their respective parametric intervals over multiple parameters of the model.

We examine conditions for finite-time collapse of the solutions of the higher-order nonlinear Schrödinger (NLS) equation incorporating third-order dispersion, self-steepening, linear and nonlinear gain and loss, and Raman scattering; this is a system that appears in many physical contexts as a more realistic generalization of the integrable NLS. By using energy arguments, it is found that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. We identify a critical value of the linear gain, separating the possible decay of solutions to the trivial zero-state, from collapse. The numerical simulations, performed for a wide class of initial data, are found to be in very good agreement with the analytical results, and reveal long-time stability properties of localized solutions. The role of the higher-order effects to the transient dynamics is also revealed in these simulations.

We investigate, by direct numerical simulations and for certain parametric regimes, the dynamics of the damped and forced nonlinear Schrödinger (NLS) equation in the presence of a time-periodic forcing. It is thus revealed that the wave number of a plane-wave initial condition dictates the number of emerged Peregrine-type rogue waves at the early stages of modulation instability. The formation of these events gives rise to the same number of transient “triangular” spatiotemporal patterns, each of which is reminiscent of the one emerging in the dynamics of the integrable NLS in its semiclassical limit, when supplemented with vanishing initial conditions. We find that the L2-norm of the spatial derivative and the L4-norm detect the appearance of rogue waves as local extrema in their evolution. The impact of the various parameters and noisy perturbations of the initial condition in affecting the above behavior is also discussed. The long-time behavior, in the parametric regimes where the extreme wave events are observable, is explained in terms of the global attractor possessed by the system and the asymptotic orbital stability of spatially uniform continuous wave solutions.

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