Applications of a new function introduced by Kudryashov [Optik. 2020;206:163550] to obtain solitary wave solutions of nonlinear PDEs through their travelling wave reductions are considered. The Kudryashov function, R, satisfying a first-order second degree ODE has several features which significantly assist symbolic calculations, especially for highly dispersive nonlinear equations. A remarkable feature of the Kudryashov function R, is that its even order derivatives are polynomials in R only while its odd order derivatives turn out to be polynomials in R and Rz . The procedure has been illustrated by means of the Schrödinger–Hirota equation, a quartic NLS equation and the fifth-order Kawahara equation as examples. A comparison with the Rayleigh–Ritz variational approach has also been considered for the purposes of illustration. The results obtained here are novel and span the family of solutions for such kind of equations.

中文翻译：

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使用 Kudryashov 的 R 函数方法求解非线性偏微分方程的孤立波解

Kudryashov [Optik. 2020;206:163550] 考虑通过行波减少来获得非线性偏微分方程的孤立波解。满足一阶二阶常微分方程的 Kudryashov 函数 R 具有几个显着辅助符号计算的特征，尤其是对于高度色散非线性方程。Kudryashov 函数 R 的一个显着特征是它的偶数阶导数是 R 中的多项式，而它的奇数阶导数结果是 R 和 Rz 中的多项式。该过程已通过薛定谔-广田方程、四次 NLS 方程和五阶 Kawahara 方程作为例子加以说明。出于说明的目的，还考虑了与 Rayleigh-Ritz 变分方法的比较。