To start with, having employed transformation wave, some nonlinear partial differential equations have been converted into an ODE. Then, using the infinite series method for equations with similar linear part, the researchers have earned the exact soliton solutions of the selected equations. It is required to state that the infinite series method is a well-organized method for obtaining exact s...

to start with, having employed transformation wave, some nonlinear partial differential equations have been converted into an ode. then, using the infinite series method for equations with similar linear part, the researchers have earned the exact soliton solutions of the selected equations. it is required to state that the infinite series method is a well-organized method for obtaining exact s...

An integrable 2-component Camassa-Holm 2-CH shallow water system is studied by using integral bifurcation method together with a translation-dilation transformation. Many traveling wave solutions of nonsingular type and singular type, such as solitary wave solutions, kink wave solutions, loop soliton solutions, compacton solutions, smooth periodic wave solutions, periodic kink wave solution, si...

We consider an extended nonlinear Schrödinger equation with higher-order odd (third order) and even (fourth order) terms with variable coefficients. The resulting equation has soliton solutions and approximate rogue wave solutions. We present these solutions up to second order. Moreover, specific constraints on the parameters of higher-order terms provide integrability of the resulting equation...

We consider a new type of integrable coupled nonlinear Schrodinger (CNLS)equations proposed by our self [submitted to Phys. Plasmas (2011)]. The explicitform of soliton solutions are derived using the Hirota's bilinear method.We show that the parameters in the CNLS equations only determine the regionsfor the existence of bright and dark soliton solutions. Finally, throughthe linear stability an...

Recently, many nonlinear coupled evolution equations, such as the coupled Korteweg–de Vries (KdV) equation, the coupled Boussinesq equation, and the coupled mKdV equation, appear in scientific applications [1 – 13]. The coupled evolution equations attracted a considerable research work in the literature. The aims of these works have been the determination of soliton solutions and the proof of c...

A class of dissipative complex Ginzburg–Landau (DCGL) equations that govern the wave propagation in dissipative nonlinear transmission lines is solved exactly by means of the Hirota bilinear method. Two-soliton solutions of the DCGL equations, from which the one-soliton solutions are deduced, are obtained in analytical form. The modified Hirota method imposes some restrictions on the coefficien...

Soliton equations are infinite-dimensional integrable systems described by nonlinear evolution equations. As one of the soliton equations, long wave equation takes on profound significance of theory and reality. By using the method of nonlinearization, the relation between long wave equation and second-order eigenvalue problem is generated. Based on the nonlinearized Lax pairs, Euler-Lagrange f...

In this paper, we investigate two types of nonlocal soliton equations with the parity-time (PT) symmetry, namely, a two dimensional nonlocal nonlinear Schrödinger (NLS) equation and a coupled nonlocal Klein-Gordon equation. Solitons and periodic line waves as exact solutions of these two nonlocal equations are derived by employing the Hirota's bilinear method. Like the nonlocal NLS equation, th...

Algebro-geometric solution is a kind of very important solution in the integrable system. The soliton solution, breather solution and rogue wave solution can be obtained from the algebrogeometric solution through special limit. Therefore, we agree that almost all the “new soliton solutions” of the vector NLS equation are contained in the general formula in [1] as particular cases. However, it i...