Nonlinear Alfvén waves, propagating along a homogeneous magnetic field, are studied using relativistic isotropic hydrodynamics. Alfvén solitons of the moving-wave and wave packet types are considered for modified Korteweg–de Vries (mKdV) equation and the nonlinear Schrödinger (NLS) equation, respectively. The method of characteristics is used and the Bäcklund transformations (BTs) are employed to generate new solutions from the old ones. Thus, families of new solutions for the mKdV and the NLS equations are obtained. The question arises which solitons exist in the pulsar atmosphere.

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The motion of non-Newtonian fluid with heat and mass transfer through porous medium past a shrinking plate is discussed. The fluid obeys Casion model, heat generation, viscous dissipation, thermal diffusion and chemical reaction are taken in our considered. The motion is modulated mathematically by a system of non liner partial differential equations which describe the continuity, momentum, heat and mass equations. These system of non linear equations are transformed into ordinary differential equations by using a suitable transformations. These equations are solved numerically by using Mathematica package. The numerical distributions of the velocity, temperature and concentration are obtained as a functions of the physical parameters of the problem. Moreover the effects of these parameters on these solutions are discussed numerically and illustrated graphically through some figures. It is clear that these parameters play an important role to control the velocity, temperature and concentration of the fluid motion. It's found that the fluid velocity deceases with the increasing of electric parameter while it increases as the magnetic hartman parameter increases, these results is good agreament with the physical sitution. Also, the fluid temperature decreases and increases as the Prandtl number and Eckert number increases respeictively. At least the fluid concentration decreases with both of soret and schimdt numbers.

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Dusty plasma with inertial dust fluid and two-temperature ions admits both compressive and rarefactive solitary waves. The Korteweg-de Vries equations (KdV-type equations) with cubic nonlinearity at the critical density of low-temperature isothermal ions are considered to discuss properties of dust-acoustic solitary waves. In the vicinity of the critical density of low-temperature ions, a KdV-type equation with mixed nonlinearity is discussed. The method of characteristics is used and the Bäcklund transformations (BTs) are employed to generate new solutions from the old ones. Another new solution of the KdV–mKdV equation is obtained using a simple transformation between the sine-Gordon equation and a linear equation combined with an extension of the tanh method of Malfliet.

The nonlinear propagation of modified ion acoustic shock waves and double layers in a relativistic degenerate plasma is considered. This plasma system is proposed for containing inertial viscous positive and negative ion fluids, relativistic electron fluids, and negatively charged immobile heavy ions. The basic set of fluid equations is reduced to modified Burgers (MB) and further modified Burgers (FMB) or (Gardner) or Mamun and Zobaer (M-Z) equations by using the reductive perturbation method. The basic features of these shocks obtained from this analysis are observed to be significantly different from those obtained from the standard Burgers equation. By introducing two special functions and He's semi-inverse method, a variational principle and conservation laws for the Gardner (FmB) equation are obtained. A set of new exact solutions for the Gardner (FmB) equation are obtained by the auto-Bäcklund transformations. Finally, we will study the physical meanings of solutions.

A small-amplitude slow ion acoustic monotonic double layer in an unmagnetized plasma consisting of relativistic drifting cold electrons and nonrelativistic drifting thermal ions is investigated. By using the reductive perturbation method, Schamel–Korteweg–de Vries (SKdV) and Schamel equations are derived. We used the linearization transformation to obtained the solutions of the SKdV and Schamel equations. The method is based upon a linearization principle that can be applied on nonlinearities which have a polynomial form. We illustrate the potential of the method by finding solutions of the SKdV and Schamel equations. Furthermore, we show that the monotonic double-layer solution is a nonlinear extension of the slow ion acoustic solitary hole having a negative trapping parameter in a semi relativistic plasma.