A fundamental question in wave turbulence theory is to understand how the "wave kinetic equation" (WKE) describes long-time dynamics of its associated nonlinear dispersive equation. Formal derivations physics literature date back work Pieirls 1928. For cubic Schr\"odinger equation, it expected that such a description should hold, limiting regime where size $L$ domain goes infinity, and strength...

We study the well posedness of nonlinear Schrödinger (NLS) equation with a point interaction and power nonlinearity in dimension two three. Behind autonomous interest problem, this is model evolution so called singular solutions that are known analysis semilinear elliptic equations. show Cauchy problem for NLS considered enjoys local existence uniqueness strong (operator domain) solutions, depe...

The cubic non-linear Schrödinger equation 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)−1, where a and b are constants. This is explained by transforming the time-dependent system into the constant-coefficient NLS by means of a time-dependent non-linear transformation, related to th...

We propose a general method to derive kinetic equations for dense soliton gases in physical systems described by integrable nonlinear wave equations. The kinetic equation describes evolution of the spectral distribution function of solitons due to soliton-soliton collisions. Owing to complete integrability of the soliton equations, only pairwise soliton interactions contribute to the solution, ...

In this paper we develop a bilinearisation-reduction approach to derive solutions the classical and nonlocal nonlinear Schr\"{o}dinger (NLS) equations with nonzero backgrounds. We start from second order Ablowitz-Kaup-Newell-Segur coupled as an unreduced system. With pair of $(q_0,r_0)$ bilinearize system obtain in terms quasi double Wronskians. Then implement reductions by introducing constrai...

The nonlinear Schrodinger (NLS) equation in a self-defocusing Kerr medium supports dark solitons. Moreover the mean field description of a dilute Bose-Einstein condensate (BEC) is described by the Gross-Pitaevskii equation, which for a highly anisotropic (cigar-shaped) magnetic trap reduces to a one-dimensional (1D) cubic NLS in an external potential. A quantum lattice algorithm is developed fo...

In this paper we study the cubic fractional nonlinear Schrödinger equation (NLS) on the torus and on the real line. Combining the normal form and the restricted norm methods we prove that the nonlinear part of the solution is smoother than the initial data. Our method applies to both focusing and defocusing nonlinearities. In the case of full dispersion (NLS) and on the torus, the gain is a ful...

In this paper we propose a microscopic model to study the polymerization of microtubules (MTs). Starting from fundamental reactions during MT’s assembly and disassembly processes, we systematically derive a nonlinear system of equations that determines the dynamics of microtubules in 3D. We found that the dynamics of a MT is mathematically expressed via a cubic-quintic nonlinear Schrödinger (NL...

Abstract In this paper, for the first time, a three-dimensional treatment of microtubules’ polymerization is presented. Starting from fundamental biochemical reactions during microtubule’s assembly and disassembly processes, we systematically derive a nonlinear system of equations that determines the dynamics of microtubules in three dimensions. We found that the dynamics of a microtubule is ma...