Kinds Of Nonlinear Evolution Equations Research Articles

New solitons and periodic wave solutions for nonlinear physical models

In this paper, an extended tanh method with computerized symbolic computation is used for constructing the traveling wave solutions of coupled nonlinear equations arising in physics. The obtained solutions include solitons, kinks, and plane periodic solutions. The applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics.

A new method to construct Weierstrass elliptic function solutions for soliton equations

Utilizing the equation that has Weierstrass elliptic function solutions, we have obtained two groups of new solutions of projection Riccati equation. Because the projection Riccati equation can be used to solve nearly all kinds of non-linear evolution equations, thus we have given a method for constructing. Weierstrass elliptic function solutions for nearly all equations with soliton solutions.

Slow nonlinear waves in magnetic flux tubes

The theory of nonlinear slow body and surface waves is developed. The conventional long-wavelength approximation is lifted. A new kind of nonlinear evolution equations is obtained. Two complementary equations are obtained. One of them is of the highest accuracy. Another one has less accuracy but its analytical solutions are found. Two analytical solutions of a distinct kind are found. It is revealed that both dark and bright solitons are the solutions of the evolutionary equation for slow waves. One of these two solutions is not singular for arbitrary amplitudes. Another one is not singular only when the critical amplitude is exceeded. The canonic form of the new evolutionary equation is presented. As a by-product of the exploration, the equation of the highest accuracy for body waves in the long-wavelength approximation is obtained. The interplay of shocks and solitons in flux tubes is discussed.

New multi-order exact solutions to a kind of nonlinear evolution equations

Based on the Lamé equation and new Lamé functions, the perturbation method and Jacobi elliptic function expansion method are applied to get the multi-order ex act solutions of a kind of nonlinear evolution equations (such as mKdV equation, nonlinear Klein-Gordon equation Ⅱ etc.), where some more new multi-order exact solutions are found among different nonlinear evolution equations. These multi- order exact solutions correspond to different periodic solutions, which can dege nerate into different solitary wave solutions, such as band-soliton, bell-shaped solitary wave, etc.

A new application of Jacobi elliptic function expansion method

Nonlinear evolution equations are classified by introducing a concept of “rank”. A new application of the Jacobi elliptic function expansion method is given， and some exact periodic solutions of a new kind of nonlinear evolution equations are obtained.

Bifurcation and stability of spatially periodic solutions of nonlinear evolution equations with integral operators

A more general kind of nonlinear evolution equations with integral operators is discussed in order to study the spatially periodic static bifurcating solutions and their stability. At first, the necessary condition and the sufficient condition for the existence of bifurcation are studied respectively. The stability of the equilibrium solutions is analyzed by the method of semigroups of linear operators. We also obtain the principle of exchange of stability in this case. As an example of application, a concrete result for a special case with integral operators of exponential type is presented.

Classical Spin Systems, Nonlinear Evolution Equations and Nonlinear Excitations

Nonlinear excitations in classical generalized Heisenberg spin systems are studied with particular attention paid to domain-wall and solitary-wave excitations. Two kinds of nonlinear evolution equations obeyed either by the conventional two angles of rotation or the stereographic variables are derived in the form of nonlinear differential-difference equations for the isotropic and anisotropic Heisenberg models and the XY model, each of which is in an external magnetic field. It is shown by employing a continuum approximation that nonlinear excitations in classical spin systems described by various kinds of nonlinear differential equations are very rich in the sense that different spin model and also different spatial dimensionality yield different type of soliton-like excitations. The nonlinear differential equations written in terms of the stereographic variables have stronger nonlinearity than those appearing in other fields. A detailed study of static solutions of the equations is made in one-dimensional case for the anisotropic Heisenberg model and the XY model. It is shown that a variety of domain-wall solutions can exist, one is associated with symmetry-breaking states and the others are 2π -and π-domain-wall solutions. Brief discussions are also given to stationary solutions of the nonlinear differential equations in two- and three-dimensional cases to elucidate the existence of different nature of nonlinear excitations.