Physically relevant soliton solutions of the resonant nonlinear Schrodinger (RNLS) equation with nontrivial boundary conditions, recently proposed for description of uniaxial waves in a cold collisionless plasma, are considered in the Hirota bilinear approach. By the Madelung representation, the model transformed to the reaction-diffusion analog of the NLS equation for which the bilinear repres...

In this paper, we construct the discrete higher-order rogue wave (RW) solutions for a generalized integrable nonlinear Schrödinger (NLS) equation. First, based on modified Lax pair, version of Darboux transformation is constructed. Second, dynamical behaviors first-, second- and third-order RW are investigated in corresponding to unique spectral parameter, term coefficient, free constants. The ...

The aim of this work is to describe a geometric definition of localized solutions of NLS. In the linear case we have the RAGE Theorem, which relates localized solutions to the pure point spectrum of the Hamiltonian: Localized solutions of the linear Schroedinger equation are linear combinations of L eigenfunctions of the Hamiltonian. In particular, they are almost periodic functions of time. Fo...

1 An introduction to semi-classical analysis for a class of nonlinear Schrödinger equations We consider the asymptotic behavior of the solution to the following semi-classical nonlinear Schrödinger equation:    iε∂ t u ε + 1 2 ε 2 ∆u ε = ε α F (u ε), (t, x) ∈ R 1+N u ε |t=0 = u ε 0 (NLS ε) as positive parameter ε → 0. The parameter ε corresponds to the Planck constant and the limit ε → 0 is ...

We consider the problem of identifying sharp criteria under which radial H (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i∂tu + ∆u + |u|2u = 0 scatter, i.e. approach the solution to a linear Schrödinger equation as t → ±∞. The criteria is expressed in terms of the scale-invariant quantities ‖u0‖L2‖∇u0‖L2 and M [u]E[u], where u0 denotes the initial data,...

In this paper, the non-local reverse space−time fifth-order non-linear Schrödinger(NLS) equation has been investigated, which is proposed by reduction of Ablowitz–Kaup–Newell–Segur (AKNS) scattering problems. The determinant representation Nth Darboux transformation for NLS obtained. Some interesting wave solutions, including soliton, complexiton, and rogue are derived transformation. Moreover,...

By the recursion operator of the Kaup–Newell hierarchy we construct the relativistic derivative NLS (RDNLS) equation and the corresponding Lax pair. In the nonrelativistic limit c→∞ it reduces to DNLS equation and preserves integrability at any order of relativistic corrections. The compact explicit representation of the linear problem for this equation becomes possible due to notions of the q-...

In this paper, the generalized sub-equation method is extended to investigate localized nonlinear waves of the one-dimensional nonlinear Schrödinger equation (NLSE) with potentials and nonlinearities depending on time and on spatial coordinates. With the help of symbolic computation, three families of analytical solutions of this NLS-type equation are presented. Based on these solutions, period...

We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on twodimensional torus. The admits a special family of elliptic invariant quasiperiodic tori called finite gap solutions. These are inherited from integrable 1D model (cubic NLS circle) by considering solutions that depend only one variable. study long-time stability such for 2D and show that, under certain assumptions over ...

The cubic non-linear Schrödinger equation (NLS), where the coefficient of the non-linear term can be a function F (t, x), is shown to pass the Painlevé test of Weiss, Tabor, and Carnevale only for F = (a+ bt), where a and b constants. This is explained by transforming the time-dependent system into the ordinary NLS (with F = const.) by means of a time-dependent non-linear transformation, relate...