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Starting from the governing equations for a quantum magnetoplasma including the electron spin −1/2 effects and quantum Bohm potential, we derive Korteweg-de Vries (KdV) equation of the system of quantum magnetohydrodynamics (QMHD). The amplitude and width of magnetosonic soliton with different parameters in the system are studied. It is found that the normalized Zeeman energy E plays a crucial role, for E ≥ 1 the amplitude rmξ and the width wξ of solitary wave all decrease as E increases. That is, the introduction of spin quantum force modifies the shape of solitary magnetosonic waves and makes them more narrower and shallower.

Driven dynamics of a two-dimensional Frenkel-Kontorova model is studied in the paper. In our numerical simulations, it is found that the movement direction of the center of mass is not consistent with that of the external driving force except for some special symmetric directions at the lower driving force. Our results also indicate that the movement direction of the center of mass strongly depends on both the magnitude and the direction of the external driving force as well as the misfit angle between two layers.

The dry friction force between two contacting surface layers is studied. The upper layer is arranged on a two-dimensional square lattice and driven by an external driving force. The lower layer is approximated by a two-dimensional periodic substrate potential. This model, usually called two-dimensional Frenkel–Kontorova model, is applied to study the friction forces in this paper. The behaviors of different substrate potential strongly affect the static friction force. It is found that the system has strong anisotropic character. The possibility to obtain superlubricity is suggested.

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.

A (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation for nonlinearly interacting intense laser pulses with an electron-positron (e-p) plasma is derived. Taking into account the combined action of the relativistic particle mass increase and the relativistic light ponderomotive force, using the perturbation method, and allowing different types solution, we discuss the analytical solution of (3+1)-dimensional KP-I equation, and give the approximate solutions of vector potential of the intense laser pulse in e-p plasma. Our results may be significantly useful in understanding the nonlinear wave propagation and interaction of intense laser beams in an e-p plasma.

Based on active control theory, anti-synchronization between two different hyperchaotic systems is investigated. The sufficient conditions for achieving anti-synchronization of two different hyperchaotic systems are derived. Moreover, numerical simulations are presented for hyperchaotic Lorenz–Chen system, hyperchaotic Lorenz–Lu system, and hyperchaotic Chen–Lu system to verify the effectiveness and feasibility of the proposed anti-synchronization scheme.

For two-component disk-shaped Bose–Einstein condensates with repulsive atom-atom interaction, the small amplitude, finite and long wavelength nonlinear waves can be described by a Kadomtsev–Petviashvili-I equation at the lowest order from the original coupled Gross–Pitaevskii equations. One- and two-soliton solutions of the Kadomtsev–Petviashvili-I equation are given, therefore, the wave functions of both atomic gases are obtained as well. The instability of a soliton under higher-order long wavelength disturbance has been investigated. It is found that the instability depends on the angle between two directions of both soliton and disturbance.

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.

Based on the multi-linear variable separation approach, a class of exact, doubly periodic wave solutions for the (3+1)-dimensional Jimbo–Miwa equation is analytically obtained by choosing the Jacobi elliptic functions and their combinations. Limit cases are considered and some new solitary structures (new dromions) are derived. The interaction properties of periodic waves are numerically studied and found to be inelastic. Under long wave limit, two sets of new solution structures (dromions) are given. The interaction properties of these solutions reveal that some of them are completely elastic and some are inelastic.

In this paper, (2+1)-dimensional electron acoustic waves (EAW) in an unmagnetized collisionless plasma have been studied by the linearized method and the reductive perturbation technique, respectively. The dispersion relation and a modified Kadomtsev–Petviashvili (KP) equation have been obtained for the EAW in the plasma considering a cold electron fluid and a vortex-like hot electrons. It is found from some numerical results that the parameter β (the ratio of the free hot electron temperature to the hot trapped electron temperature) effects on the amplitude and the width of the electron acoustic solitary waves (EASW). It can be indicated that the free hot electron temperature and the hot trapped electron temperature have very important effect on the characters of the propagation for the EASW.

A modified Zakharov–Kuznetsov equation for small but finite amplitude dust acoustic waves in a magnetized votex-like ion distribution dusty plasma is obtained in this paper. It seems that there are instability for a soliton under higher-order transverse perturbations in this system. There is a certain critical value 4λ0. If the ratio of the wave length of the higher-order perturbations to the width of the soliton is larger than this critical value, the solitary wave is unstable, otherwise it is stable.

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.

The governing equation of the dust fluid with non-thermal ions and variable dust charge on dust particles in hot dust plasmas is obtained. Both the compressive and rarefactive waves in this system are investigated. They can be determined by plasma parameters including the temperatures of dust fluid, ions and electrons, as well as the non-thermal parameter of ions, and the number densities of the dust particles, the ions and the electrons, etc.

The dispersion relation is derived for three-dimensional dust-acoustic waves in a current-driven dusty plasmas with both positively and negatively charged dust particles. The dependencies of the frequency and the growth rate on the wave number K, the intensity of magnetic field B, and the inclination angle θ have been numerically shown in this paper. The growth rate is negative for the laboratory dusty plasma, but it is positive for the cosmic dusty plasma. It is found that when the inclination angle θ = π/2, there is no instability. The effect of the electrostatic field E0 has also been studied in this paper.

By using the potential method and the perturbation method under the condition of small amplitude and shallow water waves, we analytically get the -type equation for a viscous shallow water. It indicates that for one soliton-like solution, its amplitude will decrease as it propagates away due to the viscous effects of water.

In a thin-walled, homogeneous, straight, long, circular, and incompressible fluid filled elastic tube, small but finite long wavelength nonlinear waves can be describe by a KdV (Korteweg de Vries) equation, while the carrier wave modulations are described by a nonlinear Schrodinger equation (NLSE). However if the elastic tube is slowly inhomogeneous, then it is found, in this paper, that the carrier wave modulations are described by an NLSE-like equation. There are soliton-like solutions for them, but the stability and instability regions for this soliton-like waves will change, depending on what kind of inhomogeneity the tube has.

By using the potential method to a fluid filled elastic tube, we obtained a solitary wave solution. Compared with reductive perturbation method, this method can be used for larger amplitude solitary waves. The result is in agreement with that of small amplitude approximation from reductive perturbation method when the amplitude is small enough.

The analytical solution of a bubble collapse close to a solid boundary in a compressible water is investigated by means of a perturbation method to first order in the bubble wall Mach number. It is shown, in this paper, that it is the Rayleigh–Plesset equation for incompressible liquid to zero order solution or similar to the Gilmore equation for compressible water to first order solution when the effect of solid boundary is negligibly small enough, i.e., sufficiently far away from the bubble center.

A new approach to nonlinear wave in one-dimensional discontinuous fluid-filled elastic tube is presented. As-a model, an elastic tube which has a discontinuity of radius, thickness and Young's modulus is considered. The incident, reflected and transmitted waves are described by KdV equations. The reflected and transmitted waves are constructed from incident wave analytically. Fission and reflection of a soliton due to the discontinuity are explicitly shown in the lowest order

By deriving a Korteweg-de Vries (KdV) equation for particular dusty plasma, we find the existing regions of solitary waves. The regions of both rarefactive and compressive solitary waves have also been given in this paper for a two-temperature ions and a hot adiabatic dusty fluid, where the dust charges can vary.

By employing the reductive perturbation technique we derived a Kadomtsev–Petviashvili equation for unmagnetized dusty plasmas. It suggests that the nonlinear dust acoustic solitary waves with adiabatic variation of dust charge are stable even there are some higher order transverse perturbatoins. There are only rarefactive solitary waves for this system which has been verified analytically in this paper.

Korteweg, de Vries–Burges equation is obtained for an incompressible and viscous fluid which is flowing in one direction for the shallow water. We assume that the wave amplitude is small but finite, the viscosity of the fluid is also small enough.