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We introduce matrix coupled (local and nonlocal) dispersionless equations, construct wide classes of explicit multipole solutions, give explicit expressions for the corresponding Darboux and wave matrix valued functions and consider their…
We study the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with a step-like initial data: $q(x,0)=q_0(x)$, where $q_0(x)=o(1)$ as $x\to-\infty$…
For nonlinear dispersive systems, the nonlinear Schr\"odinger (NLS) equation can usually be derived as a formal approximation equation describing slow spatial and temporal modulations of the envelope of a spatially and temporally…
We investigate non-autonomous solitons in a general coherently coupled nonlinear Schr\"odinger (CCNLS) system with temporally modulated nonlinearities and with an external harmonic oscillator potential. This general CCNLS system encompasses…
We consider PT-symmetric, nonlocal nonlinear Schrodinger equation on metric graphs. Vertex boundary conditions are derived from the conservation laws. Soliton solutions are obtained for simplest graph topologies, such as star and tree…
A new nonlinear model is proposed for tuneable lasers. Using the generalized nonlinear Schr\"odinger equation as a starting point, expressions for the transformations undergone by the pulse are derived for each of the five components (gain,…
In this paper we study radial solutions of certain two-dimensional nonlinear Schr\"odinger equation with harmonic potential, which is supercritical with respect to the initial data. By combining the nonlinear smoothing effect of Schr…
A new integrable system of two symmetrically coupled derivative nonlinear Schroedinger equations is detected by means of the singularity analysis. A nonlinear transformation is proposed which uncouples the equations of the new system.
We present some physically interesting, in general non-stationary, one-dimensional solutions to the nonlinear phase modification of the Schr\"{o}dinger equation proposed recently. The solutions include a coherent state, a phase-modified…
We prove finite time supercritical blowup in a parity-time-symmetric system of the two coupled nonlinear Schr\"odinger (NLS) equations. One of the equations contains gain and the other one contains dissipation such that strengths of the…
The large phase shift of strongly nonlocal spatial optical soliton(SNSOS) in the (1+1)-dimensional [(1+1)D] lead glass is investigated using the perturbation method. The fundamental soliton solution of the nonlocal nonlinear Schodinger…
We construct symmetry preserving and symmetry broken N-bright, dark and antidark soliton solutions of a nonlocal nonlinear Schr\"{o}dinger equation. To obtain these solutions, we use appropriate eigenfunctions in Darboux transformation (DT)…
Darboux transformation plays a key role in constructing explicit closed-form solutions of completely integrable systems. This paper provides an algebraic construction of generalized Darboux matrices with the same poles for the $2\times2$…
We consider vector Non-linear Schrodinger Equation(NLSE) with balanced loss-gain(BLG), linear coupling(LC) and a general form of cubic nonlinearity. We use a non-unitary transformation to show that the system can be exactly mapped to the…
In this paper, the inverse scattering transform for the integrable discrete nonlocal PT symmetric nonlinear Schr\"odinger equation with nonzero boundary conditions is presented. According to the two different signs of symmetry reduction and…
Using the generalized perturbation reduction method the scalar nonlinear Schr\"odinger equation is transformed to the coupled nonlinear Schr\"odinger equations for auxiliary functions. A solution in the form of a two-component vector…
The mixed nonlinear Schr\"odinger (MNLS) equation is a model for the propagation of the Alfv\'en wave in plasmas and the ultrashort light pulse in optical fibers with two nonlinear effects of self-steepening and self phase-modulation(SPM),…
We prove modified Strichartz estimates on the one-dimensional torus, that are adapted to a fourth-order dispersion relation, and use them to show global well-posedness of nonlinear fourth-order Schr\"odinger equations. This extends the (low…
This article provides a naturel sequel of previous works [6, 4] regarding the stability of travelling waves for a general one-dimensional Schr\"odinger equation (N LS) with non-zero condition at infinity. The aim of this article is twofold.…
We propose an integrable system of coupled nonlinear Schrodinger equations with cubic-quintic terms describing the effects of quintic nonlinearity on the ultra-short optical soliton pulse propagation in non-Kerr media. Lax pair, conserved…