Consistent Riccati expansion solvability, symmetries, and analytic solutions of a forced variable-coefficient extended Korteveg–de Vries equation in fluid dynamics of internal solitary waves*

Abstract

We study a forced variable-coefficient extended Korteweg–de Vries (KdV) equation in fluid dynamics with respect to internal solitary wave. Bäcklund transformations of the forced variable-coefficient extended KdV equation are demonstrated with the help of truncated Painlevé expansion. When the variable coefficients are time-periodic, the wave function evolves periodically over time. Symmetry calculation shows that the forced variable-coefficient extended KdV equation is invariant under the Galilean transformations and the scaling transformations. One-parameter group transformations and one-parameter subgroup invariant solutions are presented. Cnoidal wave solutions and solitary wave solutions of the forced variable-coefficient extended KdV equation are obtained by means of function expansion method. The consistent Riccati expansion (CRE) solvability of the forced variable-coefficient extended KdV equation is proved by means of CRE. Interaction phenomenon between cnoidal waves and solitary waves can be observed. Besides, the interaction waveform changes with the parameters. When the variable parameters are functions of time, the interaction waveform will be not regular and smooth.

Semi-analytic solution of time-fractional Korteweg-de Vries equation using fractional residual power series method

“…Historically the KdV equation was rst developed in the study of shallow-water waves in canals [3]; since then it has been discovered to be involved in a variety of physical processes, especially those exhibiting shock waves, travelling waves, and solitons. The KdV model is used to explain many theoretical physical phenomena in the solitons [4], aerodynamics [5], turbulence [6], uid dynamics [7] etc. Thus the KdV equation has been studied and applied for many decades.…”

Propagation of two-wave solitons depending on phase-velocity parameters of two higher-dimensional dual-mode models in nonlinear physics

“…In addition, these NLPDEs describe a variety of natural and real-life scenarios, including shallow-water waves, plasma physics, ocean physics, fluid mechanics, heat conduction, etc. They also have numerous applications in the fields of science, biomathematics, engineering, biomedical, and other fields [5][6][7][8]. For finding the analytical solutions to these NLPDEs, various methods have been developed such as Painlevé analysis [9,10], auto-Bäcklund transformation [9,10], exp-function method [11], tanh-function method [12], simplified Hirota's method [13,14], Lie symmetry analysis [15], the (G /G)-expansion method [16], Paul-Painlevé approach (PPA) method [17][18][19][20][21] and so on.…”