New application to Riccati equation

Abstract

To seek new infinite sequence of exact solutions to nonlinear evolution equations, this paper gives the formula of nonlinear superposition of the solutions and Bäcklund transformation of Riccati equation. Based on the tanh-function expansion method and homogenous balance method, new infinite sequence of exact solutions to Zakharov–Kuznetsov equation, Karamoto–Sivashinsky equation and the set of (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov equations are obtained with the aid of symbolic computation system Mathematica. The method is of significance to construct infinite sequence exact solutions to other nonlinear evolution equations.

A model of CT dose profiles in Banach space; with applications to CT dosimetry

“…Riccati's equation is a nonlinear ordinary differential equation (ODE) that has applications in various fields [17][18][19][20][21]. One type of scalar Riccati equation can be written in general form as…”

Constructing infinite sequence exact solutions of nonlinear evolution equations

“…Inserting infinite sequence solutions determined by the following solutions and Bäcklund transforma-tion (36) for the first kind of elliptic function into Eq. (54), we can search for three kinds of infinite sequence smooth soliton solutions of generalized pentavalent KdV equation…”

The New Infinite Sequence Complexion Solutions of a Kind of Nonlinear Evolutionary Equations

“…And the first integrals (21), (22) of the two ordinary differential equations are obtained. In Part 2.2, according to function transformation (12), a class of nonlinear evolutionary Equations (10) is changed into two kinds of ordinary differential Equations (25), (26). And then with the help of the function transformation, ordinary differential Equations (25), (26) can be changed to ordinary differential Equations (31), (32).…”

“…Step four, the new solutions, Bäcklund transformation and the nonlinear superposition formula of solutions of the ordinary differential equation that could be integrated [10]- [12] are applied to construct the new infinite sequence complexion solutions consisting of the Riemann θ function, the Jacobi elliptic function, the hyperbolic function, the trigonometric function and the rational function of a class of nonlinear evolutionary equations (10). These solutions are consisting of two-soliton solutions, two-period solutions and solutions composed of soliton solutions and period solutions.…”