can be expressed in terms of elementary functions if and only if there exists some

rational function h such that it is a solution to the differential equation:

In recent years, some novel dispersive models have been introduced, for example, the complex-conjugate pole-residue (CCPR) model [3], critical point (CP) model [4], modified Lorentz (m-Lo) model [5], and quadratic complex

rational function (QCRF) model [6].

Lump solutions are a type of

rational function solutions, localized in all directions in the space.

Let R be a

rational function (or a polynomial) that belongs asymptotically to [kr.sup.[alpha]] as r [right arrow] [infinity], where k [not equal to] 0, a are constants.

Furthermore, for the nonlinear problem, the multiple exp-function method [18, 19], the transformed

rational function method [20-22], and invariant subspace method [23, 24] are three systematical approaches to handle the nonlinear terms.

One of them is that, given a non-constant

rational function f, with rational coefficients, if [xi] is a Liouville number, then so is f([xi]).

Several types of approximations are available in the literature, for example, by use of Functional approach, Sampling approach, Geometric approach, Weight function approach, Adomain approach, Composition approach and

Rational function approach.

In [31], Ma and Lee have obtained rational solutions of (1) including travelling wave solutions, variable separated solutions, and polynomial solutions by using

rational function transformation and Backlund transformation.

This method uses the

rational function of the PDN impedance in the time domain based on measurements.

and for f, g [member of] K[[X.sub.1],...,[X.sub.n]], all the degrees above may be extended for the

rational function f / g as the maximum between the corresponding degrees of f and g: