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An Extension to Direct Method of Clarkson and Kruskal and Painlev\(\acute{e}\) Analysis for the System of Variable Coefficient Nonlinear Partial Differential Equations

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Abstract

In this work, direct method of Clarkson and Kruskal has been extended for the system of variable coefficient nonlinear partial differential equations. This extension can be applied to various higher order systems with variable coefficients to obtain novel exact solutions. An example of coupled KdV-Burgers system with variable coefficients has been considered to obtain the new exact solutions by utilizing proposed direct method. The coupled KdV-Burgers system with variable coefficients is especially relevant for modeling shallow water waves in channels with variable width or depth. Moreover, it plays a crucial role in studying the interactions between long-wave and short-wave phenomena in fluid flows with varying viscosity or density. The previously known exact solutions of considered system with constant coefficients have been exploited to derive new solutions of considered system. In this manuscript, the direct method is applied in a generalized manner for the first time to a system of partial differential equations with variable coefficients. The novel exact solutions are in the form of arbitrary function from which the different types of solutions of governed equation can be obtained. The obtained exact solutions have been displayed graphically by taking particular values of arbitrary constants and function. The comprehensive graphical analysis of the wave solutions has been conducted by extracting various standard wave configurations, including kink, bright-dark soliton, dark-bright soliton and periodic waves. The Painlev\(\acute{e}\) analysis of governing system has been also performed by utilizing WTC-Kruskal algorithm which describes non-integrability of system.

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Acknowledgements

Manjeet Sharma is gratefully thanks to Council of Scientific and Industrial Research, New Delhi, India (CSIR file no. 09/1152(0018)/2019-EMR-1) for the financial support.

Funding

This research has received funding from the Council of Scientific and Industrial Research, New Delhi, India (CSIR file no. 09/1152(0018)/2019-EMR-1). The support of the CSIR is greatly acknowledged.

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  1. Department of Mathematics, School of Basic Sciences, Central University of Haryana, Mahendergarh, Haryana, 123031, India

    Rajesh Kumar Gupta & Manjeet Sharma

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RKG—Conceptualization, Writing—review & editing, Visualization, Supervision. MS—Investigation, Writing—original draft, Methodology, Software.

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Correspondence to Manjeet Sharma.

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Gupta, R.K., Sharma, M. An Extension to Direct Method of Clarkson and Kruskal and Painlev\(\acute{e}\) Analysis for the System of Variable Coefficient Nonlinear Partial Differential Equations. Qual. Theory Dyn. Syst. 23, 115 (2024). https://doi.org/10.1007/s12346-024-00994-1

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