A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation

Abstract

A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the expfunction method, the mapping method, and the F -expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3 + 1 dimensional Jimbo-Miwa equation is treated, together with a Bäcklund transformation. MSC: 35Q58, 37K10, 35Q53

The Modified (G'/G)-Expansion Method for Nonlinear Evolution Equations

“…To keep the solution process as simple as possible, the function Q should not be a total ξ -derivative of another function. Otherwise, taking integration with respect to ξ further reduces the transformed equation [21].…”

“…Among of them, (21), (23), and (24) are equivalent to (27), (28), and (29), respectively. However, as demonstrated above, (20) and (22) with an explicit linear function in ξ are different from (27) - (29) and cannot be obtained by the (G /G)-expansion method [25, 26, 28 -31] and its improvements [27,36] if we don't transform (13) into (14) but directly solving (13).…”

Pulse Propagation Models with Bands of Forbidden Frequencies or Forbidden Wavenumbers: A Consequence of Abandoning the Slowly Varying Envelope Approximation and Taking into Account Higher-Order Dispersion

“…It is worth mentioning that the new methods to obtain exact solutions for nonlinear and evolution equations described in Refs. [47][48][49], and references therein, might be helpful to investigate if Equations (A12) and (A13) can be indeed related to modular forms, or to Weierstrass ℘ functions, which are closely related to elliptic curves [39].…”

“…It would be interesting to investigate if the new methods described in Refs. [47][48][49] and the references therein, might be successful in finding new solutions for Equation (6).…”

Construction of exact solutions for fractional-type difference-differential equations via symbolic computation

“…The essential part of our method, which can be thought as a special case of the transformed rational function method [23], is based on the proposal that the special functions that one takes to expand the exact solution are the general solution of simpler ODE with a higher order than the original differential equation with eminent solution. The main point of the transformed rational function method, which successfully unifies some already known methods, is to look for rational solutions to variable-coefficient ODEs deduced from given PDEs.…”