New exact and numerical solutions for the KdV system arising in physical applications

Abstract

The Kortweg-de Vries (KdV) equation is more appropriate to simulate some natural phenomena and gives more accurate results for some physical systems such as the movement of water waves. In this work, novel analytical traveling wave solutions for a nonlinear KdV system are explored using the sech method. The exact solutions are presented in the form of hyperbolic functions. These solutions show the propagation of water waves on the surface. We also implement the numerical adaptive moving technique (MMPDEs) to construct the relevant system's numerical solutions. A detailed comparison between the numerical and analytical solutions is also presented to confirm the accuracy of the numerical technique. We present some new 2D and 3D figures to illustrate the behaviour of the exact and numerical solutions. The obtained numerical solutions verify the accuracy of the considered methods qualitatively and quantitatively. The stability of the obtained solutions is investigated using the Hamiltonian system. The achieved results can be applied for some new observations in the ocean, coastal water, macroscopic phenomena, and processes. The proposed techniques are potent tools to solve many other non-linear partial differential equations in applied mathematics.

New analytical solutions to the stochastic (2 + 1)-dimensional combined KdV-mKdV equation

“…In recent years, nonlinear partial differential equations (NPDEs) have gained significant relevance in the study of nonlinear phenomena due to their prevalence across various scientific and engineering fields, such as Geophysics [13], Quantum Mechanics [3], Nonlinear Optics [19], Condensed matter Physics [9] A variety of powerful methods have been used to study nonlinear evolution equations, for analytic and numerical solutions. Examples of these methods are The sech method [2,10], Tanh-Sech method [10,35,36], Sin-Cosine method [33,34], F-expansion method [40,42,43], Generalized Kudryashov technique [12,39], Exp-function method [12,15,28,41], (G'/G) expansion method [12,27,32], Generalized (G'/G) expansion method [17,27,29], mapping method [24], darboux transformation method [14,22], Hirota bilinear method [21,39], Painlevé analysis [21], Rational (G'/G)-expansion method [8,16,26], among other methods.…”