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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.

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.

In order to better simulate a real model which can be realized in experiment, we study a two-dimensional(2D) Frenkel-Kontorova(FK) model driven by the disordered substrate potential which is charactered by the sum of identical Gaussian functions. Commensurate and incommensurate interfaces are discussed. In the paper, we mainly analyse the mechanism of friction, and the effects of static friction force on the disordered substrate potential.

The reductive perturbation method is employed to describe the behaviour of ion-acoustic waves for plasmas in the absence of magnetic field, leading to a type of modified Kadomtsev-Petviashvili equation. The stability of a special type of solitary wave solutions for the modified Kadomtsev-Petviashvili equation is investigated with a finite difference scheme. The numerical results show that this solitary wave is unstable under two particular initial perturbations.

Based on the ideas of the hyperbola function expansion method, we obtained some analytical solutions of the combined KdV-mKdV (cKdV) equations by introducing new expansion functions.One of the single soliton solutions has the kink-antikink structure, and this solution reduces to the kink-like solution and the bell-like solution under different limitations. Theoretical analysis shows that the cKdV equation has both propagated-type and non-propagated-type solitary wave solutions. We also investigated the stability of the single solitary wave solution with double kinks numerically. The results indicate that the solution may be stable or unstable, depending on different sets of parameters.

We obtained a class of approximations and analytical solutions for the KdV-Burgers equation by using the homotopy analysis method (HAM). The solutions we got agree very well with the exact solutions. The results indicate that the HAM is still valid for solving a class of dissipative systems.

Based on the idea of the hyperbola function expansion method, some analytical solutions of the modified Korteweg-de Vries (mKdV) equation are obtained by introducing new expansion functions. One of the single soliton solutions has a kink-antikink structure and it reduces to a kink-like solution and bell-like solution under different limitations. The stability of the single soliton solution with double kinks is investigated numerically. The results indicate that the soliton is stable under different disturbances.

We solved the Burgers equation by using the homotopy analysis method (HAM) and obtained its approximate kink solitary wave solutions. The solutions agree very well with the exact solutions. The results indicate that the HAM is valid for finding the solitary wave solutions of a class of nonlinear evolution equations. We also made some efforts to extend the HAM. As an example, we got the bell-like solitary wave solutions of KP equation by this approach. Our extended method can be used to find the approximate solitary wave solutions for more nonlinear evolution equations in an easy way.

We obtained a class of approximate periodic solutions for the (2+1)-dimensional modified Zakharov-Kuznetsov equation by using the homotopy analysis method (HAM). The solutions we got agree with the exact solutions. The results indicate that the HAM is still valid for solving a class of higher dimensional evolution equations. We also made some efforts to extend the HAM to find the analytical solutions for more nonlinear evolution equations in an easier way.

We generalized the Jacobi elliptic function expansion method and obtained some new exact periodic solutions, thus replenished the known results of a number of nonlinear wave equations by using this method.

With the aid of computer algebraic system “Mathematica” and by using the hyperbola function method,the exact and solitary wave solutions of the combined KdV equation are obtained,including the bell-like and the kink solitary wave solutions.The thought based on the hyperbola function method is extended,and more explicit and exact solutions are successfully derived,including shock wave type solutions with break point and exponential function type solutions.This method can be used also to solve other nonlinear evolution equation(s).