Abstract
In this study, the generalized formula of the Hirota–Satsuma coupled KdV equation derived by Hirota and Satsuma in 1981 [Hirota and Satsuma, Phys. Lett. A 85, 407−408 (1981)] is analytically and semi-analytically investigated. This model is formulated to describe the interaction of two long undulations with diverse dispersion relations; that is why it is also known with a generalized model of the well-known KdV equation. The generalized Kudryashov and Adomian decomposition methods construct novel soliton wave and semi-analytical solutions. These solutions are represented in some distinct graphs to show the waves’ interactions. In addition, the accuracy of solutions is verified by comparing the obtained analytical and semi-analytical solutions that show the matching between them. All solutions are checked by putting them back into the original model through Mathematica 12.
Highlights
The generalized formula of the Hirota–Satsuma coupled KdV equation derived by Hirota and Satsuma in 1981 [Hirota and Satsuma, Phys
A 85, 407−408 (1981)] is analytically and semi-analytically investigated. This model is formulated to describe the interaction of two long undulations with diverse dispersion relations; that is why it is known with a generalized model of the well-known KdV equation
The generalized formula of the Hirota–Satsuma coupled KdV equation[1] is handled by applying the generalized Kudryashov (GKud) and Adomian decomposition (AD) methods to construct some novel closed solutions that represent the interactions of two long waves with different dispersion relations.[2–4]
Summary
The generalized formula of the Hirota–Satsuma coupled KdV (gHS-cKdV) equation[1] is handled by applying the generalized Kudryashov (GKud) and Adomian decomposition (AD) methods to construct some novel closed solutions that represent the interactions of two long waves with different dispersion relations.[2–4] The HS-cKdV equation is given by the following system:[5,6]. The generalized formula of the Hirota–Satsuma coupled KdV (gHS-cKdV) equation[1] is handled by applying the generalized Kudryashov (GKud) and Adomian decomposition (AD) methods to construct some novel closed solutions that represent the interactions of two long waves with different dispersion relations.[2–4]. The HS-cKdV equation is given by the following system:[5,6]. Where U = U(x, t), V = V(x, t) represent the field of horizontal velocity and the height deviating from the equilibrium position of liquid, respectively.[7,8]. (V Vx) represents the force part on the KdV wave system with the linear dispersion relation (k = √2 ω).[9]. For V = 0, the system (1) converts into the ordinary KdV equation that has the following soliton solutions:[10,11]. The gHS-cKdV system that describes the dispersive long wave in shallow water is given by[12,13]
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