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A class of nonlinear Schrödinger-type equations, including the Rangwala-Rao equation, the Gerdjikov-Ivanov equation, the Chen-Lee-Lin equation and the Ablowitz-Ramani-Segur equation are investigated, and the exact solutions are derived with the aid of the homogeneous balance principle, and a set of subsidiary higher order ordinary differential equations (sub-ODEs for short).
The auxiliary equation method is very useful for finding the exact solutions of the nonlinear evolution equations. In this paper, a new idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the auxiliary elliptic-like equation are derived using exp-function method, and then the exact solutions of the nonlinear evolution equations...
An irrational trial equation method was proposed to solve nonlinear differential equations. By this method, a number of exact travelling wave solutions to the Burgers-KdV equation and the dissipative double sine-Gordon equation were obtained. A more general irrational trial equation method was discussed, and many exact solutions to the Fujimoto-Watanabe equation were given.
We discuss recent findings about properties of quantum nonequilibrium steady states. In particular we focus on transport properties. It is shown that the time-dependent density matrix renormalization method can be used successfully to find a stationary solution of Lindblad master equation. Furthermore, for a specific model an exact solution is presented.
A new approach is taken to calculate the speed of front propagation at which the interface moves from a superconducting to a normal region in a superconducting sample. Using time-dependent Ginzburg–Landau (TDGL) equations we have calculated the speed by constructing a new exact solution. This approach is based on a method given by Di Bartolo and Dorsey. Our result for the speed agrees with the result...
By the complete discrimination system for the polynomial, we invest the classifications of single travelling wave solutions to the generalized Pochhammer–Chree (PC) equation with p = 1/2 and p = 3/2.
In this paper, a hierarchy of nonisospectral equations with variable coefficients is derived from the compatibility condition of Toda spectral problem and its time evolution. In order to solve the derived Toda lattice hierarchy, the inverse scattering transformation is utilized. As a result, new and more general exact solutions are obtained. It is shown that the inverse scattering transformation can...
In this paper, two integration schemes are employed to obtain solitons, singular periodic waves and other types of solutions of the Drinfel’d–Sokolov–Wilson equation. The two schemes studied in this paper are the Bäcklund transformation of Riccati equation and the trial function approach. The corresponding constraint conditions of the solutions are also given.
In this work, we present G ′ / G , 1 / G $\left (G^{\prime }/G, 1/G\right )$ -expansion method for solving fractional differential equations based on a fractional complex transform. We apply this method for solving space–time fractional Cahn–Allen equation and space–time fractional Klein–Gordon equation. The fractional derivatives are described in the sense of modified Riemann–Lioville...
The discrimination system for the polynomial method is applied to variant Boussinesq equations to classify single travelling wave solutions. In particular, we construct corresponding solutions to the concrete parameters to show that each solution in the classification can be realized.
In this paper, the famous Klein–Gordon–Zakharov (KGZ) equations are first generalized, and the new special types of KGZ equations with the positive fractional power terms (gKGZE) are presented. In order to derive exact solutions of the new special gKGZE, subsidiary higher-order ordinary differential equations (sub-ODEs) with the positive fractional power terms are introduced, and with the aid of the...
We study some thermodynamics quantities for the Klein–Gordon equation with a linear plus inverse-linear, scalar potential. We obtain the energy eigenvalues with the help of the quantization rule from the biconfluent Heun’s equation. We use a method based on the Euler–MacLaurin formula to analytically compute the thermal functions by considering only the contribution of positive part of the spectrum...
A new exact solution of embedding class I is presented for a relativistic anisotropic massive fluid sphere. The new exact solution satisfies Karmarkar condition, is well-behaved in all respects, and therefore is suitable for the modelling of superdense stars. Consequently, using this solution, we have studied in detail two compact stars, namely, XTE J1739-289 (strange star $$1.51M_{\odot }$$ ...
In this paper, by using the complete discrimination system of the polynomial method, the classification of the envelope travelling wave solutions to the Gerdjikov–Ivanov model is obtained. The complete result shows that there exist rich patterns of travelling wave solutions to the Gerdjikov–Ivanov model, including solitary solutions, periodic solutions, rational singular solutions and double periodic...
In this work, we have considered the Riccati–Bernoulli sub-ODE method for obtaining the exact solutions of nonlinear fractional-order differential equations. The fractional derivatives are described in Jumarie’s modified Riemann–Liouville sense. The space–time fractional modified equal width (mEW) equation and time-fractional generalised Hirota–Satsuma coupled Korteweg–de Vries (KdV) equations are...
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