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 , critical point (CP) model , modified Lorentz (m-Lo) model , and quadratic complex rational function
(QCRF) model .
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
In , 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.
The aeroelastic equations of motion are formulated in the time-domain, through the Rational Function
Approximation and application of the Balanced Truncation method.
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: