Painlevé analysis and inelastic interactions of the lumps for a generalized (2+1)-dimensional Korteweg-de Vries system for the shallow-water waves

Abstract

Water waves, one of the common natural phenomena, are recognized as complex
and often turbulent. A generalized (2+1)-dimensional Korteweg-de Vries system for
the shallow-water waves is conducted in this paper. We perform the Painlev´e analysis
and find that the system is Painlev´e integrable. We study the inelastic interactions of
the lumps for the system. We find that two lumps, which propagate along the curves
with the equal amplitude, are symmetric about the x axis before the interaction, where
x is a scaled spatial variable. After the interaction, amplitudes of the two lumps are
different, but in the process of moving, the lower lump gradually increases, while the
higher lump gradually decreases, and the velocities of two lumps at the infinity are
equal. We observe two different inelastic interactions of the three lumps: (1) the
three lumps are symmetric in time and space, and they slowly contract (before the
interaction) and swell (after the interaction); (2) the three lumps slowly fuse and after
the interaction they form a straight line forward, and their amplitudes are gradually
equal when t → ∞.

Analytical study of soliton dynamics in the realm of fractional extended shallow water wave equations

“…It has been determined that our work is a generalization of works found in the literature. For example, Faisal and Kumar solved problems (1) and (2) using an improved ( ) ¢ G G -expansion approach [28]. The solutions in [28] are special instances of the solutions acquired in the current research effort; that is, for α = β = γ = 1 in the current solutions, many equations are the same.…”

“…For example, Faisal and Kumar solved problems (1) and (2) using an improved ( ) ¢ G G -expansion approach [28]. The solutions in [28] are special instances of the solutions acquired in the current research effort; that is, for α = β = γ = 1 in the current solutions, many equations are the same. If we examine our answers in fractional order, the findings acquired thus far are novel.…”

“…These findings have important significance for the discipline, offering vital insights into the intricate dynamics of FESWWEs. The effectiveness of KM is demonstrated by its capacity to produce varied solutions and contribute to a thorough knowledge of such complex phenomena.¢ G G -expansion method [21][22][23][24][25], KM [26,27], and many others [28][29][30].Within the arena of analytical approaches, several researchers have used the KM to construct travelling wave solutions for nonlinear FPDEs. The KM does a strategic variable transformation to turn the FPDE into an integer-ordered nonlinear ODE (NODE).…”