Two model equations with a second degree logarithmic nonlinearity and their Gaussian solutions

Abstract: We study the optical chirped envelope patterns of sub-pico-second pulse propagation through an optical fiber, which is modeled by the Kaup–Newell equation with the form of derivative nonlinear Schrodinger equation. The complete discrimination system for polynomial method is applied to give all chirped envelope solutions. A family of propagation patterns including the bright, dark, singular and elliptical type solutions are obtained.

Abstract: We consider the optical chirped envelope solutions of propagation of short pulse in highly nonlinear optical fibers, which is modeled by a kind of derivative nonlinear Schrodinger equation. We use the complete discrimination system for polynomial method to give a complete analysis of all chirped envelope solutions, and obtain a series of typical exact chirped patterns which include solitons, periodic solutions and singular solutions in terms of explicit and implicit functions.

Abstract: What the paper will show is how we attempt to get analytical solutions of the perturbed nonlinear Schrodinger equation in the presence of time- and space-dependent dissipation (or gain) and nonlinear dispersion with Kerr law by complete discriminant system for polynomial method. The analytical solutions which include solitary wave solutions, Jacobian elliptic function solutions, triangular function solutions and rational solutions, act on the propagations of optical pulse in dissipation (or gain) optical fibers with GOD, IMD, self-steepening and nonlinear dispersion.

Abstract: In this paper, the classification of all envelope traveling wave solutions of Gerdjikov–Ivanov model with full nonlinearity is obtained by using complete discriminant system for polynomial method and trial equation method. All the results show that the Gerdjikov–Ivanov model has abundant traveling wave solutions, including isolated solutions, periodic solutions, rational singular solutions, double periodic continuous solutions and discontinuous solutions. Among them, some new solutions are obtained.

Abstract: In this paper, according to the complete discriminant system for polynomial method, a series of analytical solutions of the perturbed nonlinear Schrodinger equation with variable coefficient and Kerr law are obtained. These solutions show the abundant propagation patterns of optical waves with Kerr law.

Abstract: Properties of the Schrodinger equation with the logarithmic nonlinearity are briefly described. This equation possesses soliton-like solutions in any number of dimensions, called gaussons for their Gaussian shape. Excited, stationary states of gaussons of various symmetries, in two and three dimensions are found numerically. The motion of gaussons in uniform electric and magnetic fields is studied and explicit solutions describing linear and rotational internal oscillations are found and analyzed.

Abstract: On considere l'equation de Schrodinger suivante: (u t +Δu+uLog|u| 2 =0 dans R×R N . On donne des resultats sur le comportement asymptotique des solutions

Abstract: A new method, that is, trial equation method, was given to obtain the exact trav eling wave solutions for nonlinear evolution equations. As an example, a class o f fifth-order nonlinear evolution equations was discussed. Its exact traveling w ave solutions, which included rational form solutions, solitary wave solutions, triangle function periodic solutions, polynomial type Jacobian elliptic function periodic solutions and fractional type Jacobian elliptic function periodic solu tions, were given.

Abstract: Recently a nonlinear Schr\"odinger equation (NLSE) with an inhomogeneous term proportional to $b\mathrm{ln}({|\ensuremath{\psi}|}^{2}|{a}^{3})\ensuremath{\psi}$ has been put forward. It has been proposed to apply it to atomic physics. Subsequent neutron interferometer experiments designed to test the physical reality of such a nonlinearity were not conclusive, thus rejecting it as unphysical. In the present paper it is pointed out that the different length scales $a$ associated with atomic and nuclear physics, for example, lead to different typical energies $b$ for these systems. Guided by the experience with phenomenological NLSE's, the constant $b$ is for the following applications to nuclear physics identified with the compressibility of finite nuclear matter, $C=\frac{K}{9}$, i.e., $b\ensuremath{\equiv}C$. Thus we obtain consistent qualitative and quantitative answers related to the concepts of microworlds and mesoworlds as well as, e.g., the prediction $130l~Kl~250$ MeV. However, this necessitates the interpretation of the respective NLSE as an equation for extended objects.

Abstract: A trial equation method to nonlinear evolution equation with rank inhomogeneous is given. As applications, the exact traveling wave solutions to some higher-order nonlinear equations such as generalized Boussinesq equation, generalized Pochhammer?Chree equation, KdV-Burgers equation, and KS equation and so on, are obtained. Among these, some results are new. The proposed method is based on the idea of reduction of the order of ODE. Some mathematical details of the proposed method are discussed.