The paper addresses a new “inverse” problem for reconstructing the amplitudes of 2D surface waves from observation of the wave patterns (formed by wave crests). These patterns will depend on the amplitudes because of nonlinear effects. We show that the inverse problem can be solved when the waves are modelled by an equation that supports soliton solutions. Specifically, the explicit solution to...

We consider two exit problems for the Korteweg-de Vries equation perturbed by an additive white in time and colored in space noise of amplitude ǫ. The initial datum gives rise to a soliton when ǫ = 0. It has been proved recently that the solution remains in a neighborhood of a randomly modulated soliton for times at least of the order of ǫ. We prove exponential upper and lower bounds for the sm...

In this paper, we present exact N-soliton solution by employing simple, straightforward Darboux transformation based on the Lax pair for Hirota equation, a higher-order nonlinear Schr€ odinger (HNLS) equation. As examples, oneand two-soliton solutions in explicit forms are given and their properties are also analyzed. A bound solution without interaction will be theoretically predicted if one c...

A new resonance version of NLS equation is found and embedded to the reactiondiffusion system, equivalent to the anti-de Sitter valued Heisenberg model, realizing a particular gauge fixing condition of the Jackiw-Teitelboim gravity. The space-time points where dispersion change the sign correspond to the event horizon, and the soliton solutions to the AdS black holes. The soliton with velocity ...

variational iteration method using he's polynomials can be used to construct solitary solution and  compacton-like solution for nonlinear dispersive equatioons. the chosen initial solution can be determined in compacton-like form or in solitary form with some compacton-like or solitary forms with some unknown parameters, which can be determined in the solution procedure. the compacton-like solu...

We consider two exit problems for the Korteweg-de Vries equation perturbed by an additive white in time and colored in space noise of amplitude ǫ. The initial datum gives rise to a soliton when ǫ = 0. It has been proved recently that the solution remains in a neighborhood of a randomly modulated soliton for times at least of the order of ǫ. We prove exponential upper and lower bounds for the sm...

The so-called sporadic sodium layers (SSLs or Na S ) are proposed to be strongly related wave fluctuations. solitary is a particular solution of the partial differential equation whose energy travels as localized packet. A soliton, on other hand, special type that exhibits particle-like behavior with strong stable form. For first time, theory has been used in this research study fine structure ...

In ultrafast optics, optical pulses are generated to be of shorter pulse duration, which has enormous significance to industrial applications and scientific research. The ultrashort pulse evolution in fiber lasers can be described by the higher-order Ginzburg-Landau (GL) equation. However, analytic soliton solutions for this equation have not been obtained by use of existing methods. In this pa...

We consider two exit problems for the Korteweg-de Vries equation perturbed by an additive white in time and colored in space noise of amplitude . The initial datum gives rise to a soliton when = 0. It has been proved recently that the solution remains in a neighborhood of a randomly modulated soliton for times at least of the order of −2. We prove exponential upper and lower bounds for the smal...

In this paper, a general bright-dark soliton solution in the form of Pfaffian is constructed for an integrable semi-discrete vector NLS equation via Hirota’s bilinear method. Oneand two-bright-dark soliton solutions are explicitly presented for two-component semidiscrete NLS equation; two-bright-one-dark, and one-bright-two-dark soliton solutions are also given explicitly for three-component se...