We present breather solutions of the quintic integrable equation of the Schrödinger hierarchy. This equation has terms describing fifth-order dispersion and matching nonlinear terms. Using a Darboux transformation, we derive first-order and second-order breather solutions. These include first- and second-order rogue-wave solutions. To some extent, these solutions are analogous with the correspo...

Adaptive wave model for financial option pricing is proposed, as a high-complexity alternative to the standard Black-Scholes model. The new option-pricing model, representing a controlled Brownian motion, includes two wave-type approaches: nonlinear and quantum, both based on (adaptive form of) the Schrödinger equation. The nonlinear approach comes in two flavors: for the case of constant volat...

Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based on the elimination of the unknown boundary values by proper restrictions of the functional space and of the spectral variable complex domain. Illustrative exa...

We give a proof of the existence of stationary bright soliton solutions of the cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearity. By using bifurcation theory, we prove that the norm of the positive solution goes to zero as the parameter λ, called chemical potential in the Bose-Einstein condensates’ literature, tends to zero. Moreover, we solve the time-dependent cubic...

A class of discrete nonlinear Schrödinger equations with arbitrarily high-order nonlinearities is introduced. These equations are derived from the same Hamiltonian using different Poisson brackets and include as particular cases the saturable discrete nonlinear Schrödinger equation and the Ablowitz-Ladik equation. As a common property, these equations possess three kinds of exact analytical sta...

It is shown that the collective nonlinear interactions between intense neutrino or anti-neutrino fluxes and a dense neutrino plasma are governed by a multidimensional coupled cubic Schrödinger equation in which the interaction potential is positive or negative depending on the neutrino type. The cubic Schrödinger equation describes the splitting and focusing of intense neutrino beams due to the...

Article history: Received 7 February 2012 Available online 13 August 2012 We derive and justify a normal form reduction of the nonlinear Schrödinger equation for a general pitchfork bifurcation of the symmetric bound state that occurs in a double-well symmetric potential. We prove persistence of normal form dynamics for both supercritical and subcritical pitchfork bifurcations in the timedepend...

Using the path-integral technique we examine the mutual information for the communication channel modelled by the nonlinear Schrödinger equation with additive Gaussian noise. The nonlinear Schrödinger equation is one of the fundamental models in nonlinear physics, and it has a broad range of applications, including fiber optical communications — the backbone of the Internet. At large signal-to-...

The existence of compactons in the discrete nonlinear Schrödinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. In the averaged discrete nonlinear Schrödinger equation, the resulting effective interwell tunneling depends on the modulation parameters and on the field amplitude. This introduces nonlinear dispersion in the system and can lead to a ...