Nonstationary probabilities and time correlation functions for an asymmetric exclusion process

We give the complete solution of the master equation for a system of interacting particles with finite density. We obtain the solution using a new form of the Bethe ansatz for an asymmetric simple exclusion process on the ring. We first find the one-point time correlation function for the discrete version of the process. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 499–508, March, 2006.     

Exact traveling wave solutions of some nonlinear physical models. II. Coupled KdV type equations

New exact traveling wave solutions are derived for two coupled nonlinear water wave equations by using a delicate way of rank analysis two-step ansatz method.     

Dark solitons for a combined potential KdV and Schwarzian KdV equations with t-dependent coefficients and forcing term

In this work we formally derive the dark soliton solutions for the combined potential KdV and Schwarzian KdV equations. The combined KdV and Schwarzian KdV equations with time-dependent coefficients and forcing term are then investigated to obtain dark soliton solutions. The solitary wave ansatz is used to carry out the analysis for both models.     

Scalar products of the open XYZ chain with non-diagonal boundary terms

With the help of the F-basis provided by the Drinfeld twist or factorizing F-matrix of the eight-vertex solid-on-solid (SOS) model, we obtain the determinant representations of the scalar products of Bethe states for the open XYZ chain with non-diagonal boundary terms. By taking the on shell limit, we obtain the determinant representations (or Gaudin formula) of the norms of the Bethe states.     

Bright and dark soliton solutions for some nonlinear fractional differential equations

In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space–time fractional modified Benjamin–Bona–Mahoney(m BBM) equation, the time fractional m Kd V equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional derivatives are described in the modified Riemann–Liouville sense.     

Exact travelling wave solutions to the space-time fractional Calogero-Degasperis equation using different methods

In this paper, we employed the ansatz method, the exp-function method and the $\left( \frac{G^{\prime }}{G}\right) $-expansion method for the first time to obtain the exact and traveling wave solutions of the space time fractional Calogero Degasperis equation. As a result, we obtained some soliton and traveling wave solutions for this equation by means of proposed three analytical methods and the aid of commercial software Maple. The results show that these methods are effective and powerful mathematical tool for solving nonlinear FDEs arising in mathematical physics.     

Dipole and Combo Optical Solitons in Birefringent Fibers in the Presence of Four-Wave Mixing

The ultimate theme of this study is to develop dipole and combo optical solitons in birefringent fibers (BF) along with the effect of four-wave mixing (4WM). Two types of rough mediums are used which are Kerr law and parabolic law. Ansatz method of Choudhuri is applied to obtain dark in the bright (dipole) soliton solutions providing bright background for the propagation of optical dark pulse. Ansatz method of Li is applied to obtain combo soliton solutions providing bright solitary wave and dark solitary wave solutions. These results also exist to hold in non-Kerr media with higher order dispersion.     

Stationary wave packets in the Kerr nonlinear media with imaginary harmonic potential and linear gain

The existence of stationary wave packets in the nonlinear Kerr media with an imaginary harmonic potential and a linear gain is investigated. By employing a variational approach the existence of stable bright solitons is shown for the case of a defocusing nonlinearity. In focusing nonlinear media, the bright solitons have been shown to be unstable. The predictions of variational approach are confirmed by numerical simulations of the full modified NLS equation. The predicted stationary localized wave packets can be observed in a quasi-one-dimensional BEC with an imaginary optical potential and atoms feeding.