Abstract
Abstract A large family of explicit exact solutions to both Korteweg- de Vries and modified Korteweg- de Vries equations are determined by the implementation of the new extended direct algebraic method. The procedure starts by reducing both equations to related ODEs by compatible travelling wave transforms. The balance between the highest degree nonlinear and highest order derivative terms gives the degree of the finite series. Substitution of the assumed solution and some algebra results in a system of equations are found. The relation between the parameters is determined by solving this system. The solutions of travelling wave forms determined by the application of the approach are represented in explicit functions of some generalized trigonometric and hyperbolic functions and exponential function. Some more solutions with different characteristics are also found.
Highlights
The Korteweg-de Vries (KdV) family of nonlinear PDEs attracts many researchers due to having solitontype solutions that preserve their shapes and heights after interacting with each other
We are concerned with implementing a new extended direct algebraic approach to derive a large family of exact solutions to both the KdV equation represented in Eq (1) and the mKdV in Eq (2)
One can deduce that the new extended direct algebraic approach should derive more exact solutions
Summary
The Korteweg-de Vries (KdV) family of nonlinear PDEs attracts many researchers due to having solitontype solutions that preserve their shapes and heights after interacting with each other. The KdVE has multiple soliton-type solutions having particle-like behaviours that maintain their velocity, shape, and amplitudes after collision [2, 3]. A plenty of efficient methods have been implemented to derive solutions in various forms to the KdVE- and mKdv-type equations. There are many different techniques used to derive exact solutions to nonlinear PDEs of both integer and fractional order due to their importance in modelling physical phenomenon. We are concerned with implementing a new extended direct algebraic approach to derive a large family of exact solutions to both the KdV equation represented in Eq (1) and the mKdV in Eq (2).
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