Abstract
Traveling wave solutions, including localized and periodic structures (e.g., solitary waves, cnoidal waves, and periodic waves), to a symmetry Korteweg–de Vries equation (KdV) with integer and rational power law nonlinearity are reported using several approaches. In the case of the localized wave solutions, i.e., solitary waves, to the evolution equation, two different methods are devoted for this purpose. In the first one, new hypotheses with Cole–Hopf transformation are employed to find general solitary wave solutions. In the second one, the ansatz method with hyperbolic sech algorithm are utilized to obtain a general solitary wave solution. The obtained solutions recover the solitary wave solutions to all one-dimensional KdV equations with a power law nonlinearity, such as the KdV equation with quadratic nonlinearity, the modified KdV (mKdV) equation with cubic nonlinearity, the super KdV equation with quartic nonlinearity, and so on. Furthermore, two different approaches with two different formulas for the Weierstrass elliptic functions (WSEFs) are adopted for deriving some general periodic wave solutions to the evolution equation. Additionally, in the form of Jacobi elliptic functions (JEFs), the cnoidal wave solutions to the KdV-, mKdV-, and SKdV equations are obtained. These results help many authors to understand the mystery of several nonlinear phenomena in different branches of sciences, such as plasma physics, fluid mechanics, nonlinear optics, Bose Einstein condensates, and so on.
Highlights
Mathematical modeling of most real-life problems usually yields functional equations, such as ordinary differential equations (ODEs), partial differential equations (PDEs), fractional equations, integral equations, and so on
These equations arise in several fields of science, such as fluid dynamics, physics of plasmas, biological models, nonlinear optics, chemical kinetics, quantum mechanics, ecological systems, electricity, ocean, and sea, and many others
The objectives of this paper are divided into two parts: in the first part, the general solitary wave solutions to the evolution equation using two different schemes have been obtained
Summary
Mathematical modeling of most real-life problems usually yields functional equations, such as ordinary differential equations (ODEs), partial differential equations (PDEs), fractional equations, integral equations, and so on. Studying the propagation of ion acoustic waves (IAWs), such as solitary waves in different plasma models with negative ions has been extensively made both experimentally [22,23,24] and theoretically [24,25,26,27,28,29,30] by using the family of KdV equation, mostly the. G 0 /G method [41], tanh method [2,3,42], exp function method [43], variational iteration methods [44], Bäcklund and Darboux transforms [45], sn–ns expansion method [46], the Hirota’s bilinear method [47], elliptic functions expansion method [48], and many others [2] Motivated by these studies, in this work, we consider the generalized KdV equation. The solutions for some particular cases related to Equation (5) such as the KdV Equation (1), mKdV Equation (2), super mKdV Equation (3), SKdV Equation (4), and so on, are obtained
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