Search

We study the gap solitons and their stability properties in a Bose-Einstein condensation (BEC) under three-body interaction loaded in a Jacobian elliptic sine potential, which can be described by a cubic-quintic Gross-Pitaevskii equation (GPE) in the mean-field approximation. Firstly, the GPE is transformed into a stationary cubic-quintic nonlinear Schrödinger equation (NLSE) by the multi-scale method. A class of analytical solution of the NLSE is presented to describe the gap solitons. It is shown analytically that the amplitude of the gap soliton decreases as the two-body or three-body interaction strength increases. Secondly, many kinds of gap solitons, including the fundamental soliton and the sub-fundamental soliton, are obtained numerically by the Newton-Conjugate-Gradient (NCG) method. There are two families of fundamental solitons: one is the on-site soliton and the other is the off-site soliton. All of them are bifurcated from the Bloch band. Both in-phase and out-phase dipole solitons for off-site solitons do exist in such a nonlinear system. The numerical results also indicate that the amplitude of the gap soliton decreases as the nonlinear interaction strength increases, which accords well with the analytical prediction. Finally, long-time dynamical evolution for the GPE is performed by the time-splitting Fourier spectrum method to investigate the dynamical stability of gap solitons. It is shown that the on-site solitons are always dynamically stable, while the off-site solitons are always unstable. However, both stable and unstable in-phase or out-phase dipole solitons, which are not bifurcated from the Bloch band, indeed exist. For a type of out-phase soliton, there is a critical value \begin{document}$ q_c$\end{document} when the chemical potential μ is fixed. The solitons are linearly stable as \begin{document}$ q>q_c$\end{document}, while they are linearly unstable for \begin{document}$ q. Therefore, the modulus q plays an important role in the stability of gap solitons. One can change the dynamical behavior of gap solitons by adjusting the modulus of external potential in experiment. We also find that there exists a kind of gap soliton, in which the soliton is dynamically unstable if only the two-body interaction is considered, but it becomes stable when the three-body interaction is taken into account. This indicates that the three-body interaction has influence on the stability of gap solitons.

Both the homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions of the two-dimensional and time-independent Gross-Pitaevskii equation, a nonlinear Schrodinger equation used in describing the system of Bose-Einstein condensates trapped in a harmonic potential. The approximate analytical solutions are obtained successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are agreement very well with each other when the atomic interaction is not too strong.

In order to investigate the effect of an arbitrary dust size distribution for vortex-like ion distribution dusty plasma, we use a reasonable polynomial-expressed function to represent an arbitrary dust size distribution. The numerical results of linear dispersion relation, nonlinear solitary wave amplitude, width and velocity for polynomial expressed dust size distribution dusty plasma with vortex-like ion distribution have been studied.

Linear dispersion relation for linear wave and a Kadomtsev–Petviashvili (KP) equation for nonlinear wave are given for the unmagnetized two-ion-temperature cold dusty plasma with many different dust grain species. The numerical results of variations of linear dispersion with respect to the different dust size distribution are given. Moreover, how the amplitude, width, and propagation velocity of solitary wave vary vs different dust size distribution is also studied numerically in this paper.

We discuss the possible nonlinear waves of atomic matter waves in a cigar-shaped Bose-Einstein condensate with dissipation. The waves can be described by a KdV-type equation. The KdV-type equation has a solitary wave solution. The amplitude, speed, and width of the wave vary exponentially with time t. The dissipative term of γ plays an important role for the wave amplitude, speed, and width. Comparisons have been given between the analytical solutions and the numerical results. It is shown that both are in good agreement.

We discuss the possible nonlinear waves of atomic matter wave in a Bose–Einstein condensate. One and two of two-dimensional (2D) dark solitons in the Bose–Einstein condensed system are investigated. A rich dynamics is studied for the interactions between two solitons. The interaction profiles of two solitons are greatly different if the angle between them are different. If the angle is small enough, the maximum amplitude during the interaction between two solitons is even less than that of a single soliton. However, if the angle is large enough, the maximum amplitude of two solitons can gradually attend to the sum of two soliton amplitudes.

In this paper, the Fisher equation is analysed. One of its travelling wave solution is obtained by comparing it with KdV–Burgers (KdVB) equation. Its amplitude, width and speed are investigated. The instability for the higher order disturbances to the solution of the Fisher equation is also studied.

With the aid of computerized symbolic computation, the new modified Jacobi elliptic function expansion method for constructing exact periodic solutions of nonlinear mathematical physics equation is presented by a new general ansatz. The proposed method is more powerful than most of the existing methods. By use of the method, we not only can successfully recover the previously known formal solutions but also can construct new and more general formal solutions for some nonlinear evolution equations. We choose the (3+1)-dimensional Kadomtsev–Petviashvili equation to illustrate our method. As a result, twenty families of periodic solutions are obtained. Of course, more solitary wave solutions, shock wave solutions or triangular function formal solutions can be obtained at their limit condition.

By using the equivalent particle theory, the adiabatic approximation solutions of the Korteweg–de Vries type equation (including KdV equation, cylindrical KdV equation and spherical KdV equation) in dust ion-acoustic solitary waves were obtained. The method can be extended to other nonlinear evolution equations.

In this work Janssen ratio is measured in a dense granular pack. The pressure on the side walls as a function of the depth of the pack with top load under gravity is measured by photoelastic method. The samples are prepared by point source method with and without tapping. A non-monotonic distribution of the side pressure along the depth is found. Numerical simulation is performed and shows qualitative consistency with the experimental finding. The apparent weight of the sample is measured for different filling heights and for different top loads. Comparing with the normal stresses on the silo wall for different heights, we obtain the Janssen ratio J =xx/zz as a function of height. We find that although uJ = 0.11 is a constant as is expected, the Janssen Ratio is height dependent. It becomes height independent only when the top load is large enough.

We study the nonlinear solitary wave solution under the transverse perturbations for a system of coupled nonlinear electrical transmission lines. In the continuum limit and suitably scaled coordinates, the voltage on the system is described by a modified Zakharov-Kuznetsov equation. The cut-off frequency of the growth rate for the solitary waves under transverse perturbations has been analytically obtained. It is in agreement with the cases p = 1/2 and p = 1 which have been studied previously.