Construction of ( n + 1 ) $(n+1)$ -dimensional dual-mode nonlinear equations: multiple shock wave solutions for ( 3 + 1 ) $(3+1)$ -dimensional dual-mode Gardner-type and KdV-type

Under investigations into this paper is a higher-dimensional model, namely the time fractional [Formula: see text]-dimensional Korteweg–de Vries (KdV)-type equation, which can be usually used to express shallow water wave phenomena. At the beginning, the symmetry of the time fractional [Formula: see text]-dimensional KdV-type equation via the group analysis scheme is obtained. The definition of the fractional derivative in the sense of the Riemann–Liouville is considered. Then, the one-parameter Lie group and invariant solutions of this considered equation are constructed. Subsequently, we applied a direct method to construct the optimal system of one-dimensional of this considered equation. Next, this considered higher-dimensional model can be reduced into the lower-dimensional fractional differential equations (FDEs) with the help of the three-parameter and two-parameter Erdélyi–Kober fractional differential operators (FDOs). Lastly, conservation laws of this discussed equation by using a new conservation theorem are also found. A series of results of the above obtained can provide strong support for us to reveal the mysterious veil of this viewed equation.

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

Singh

Ray

2023

EPL

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Nonlinear evolution equations exhibit a variety of physical behaviours, which are clearly illustrated by their exact solutions. In this view, this article is concerned with the study of dual-mode (2 + 1)-dimensional Kadomtsev-Petviashvili (KP) and Zakharov-Kuznetsov (ZK) equations. These models describe the propagation of two-wave solitons traveling simultaneously in the same direction and with mutual interaction dependent on an embedded phase-velocity parameter. The considered nonlinear evolution equations have been solved analytically for the ﬁrst time using the Paul-Painlevé approach method. As a result, new abundant analytic solutions have been derived successfully for both the considered equations. The 3D dynamics of each of the solution has been plotted by opting suitable constant values. These graphs show the dark-soliton, bright-soliton, complex dual-mode bright-soliton and complex dual-mode dark-soliton solutions.

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Construction of ( n + 1 ) $(n+1)$ -dimensional dual-mode nonlinear equations: multiple shock wave solutions for ( 3 + 1 ) $(3+1)$ -dimensional dual-mode Gardner-type and KdV-type

Cited by 3 publications

References 44 publications

Hyers–Ulam Stability for Cayley Quantum Equations and Its Application to h-Difference Equations

Hyers–Ulam Stability for Cayley Quantum Equations and Its Application to h-Difference Equations

Group Analysis of the Time Fractional (3 + 1)-Dimensional KDV-Type Equation

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

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