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Newly constructed closed-form soliton solutions, conservation laws and modulation instability for a (2+1)-dimensional cubic nonlinear Schrödinger’s equation using optimal system of Lie subalgebra

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Abstract

The primary goal of this work is to derive newly constructed invariant solutions, conservation laws, and modulation instability in the context of the (2+1)-dimensional cubic nonlinear Schrödinger’s equation (cNLSE), which explains the phenomenon of soliton propagation along optical fibers. The nonlinear Schrödinger equations and their various formulations hold significant importance across a wide spectrum of scientific disciplines, especially in nonlinear optics, optical fiber, quantum electronics, and plasma physics. In this context, we utilize Lie symmetry analysis to determine the vector fields and assess the optimality of the governing equation. Upon establishing the optimal system, we derive similarity reduction equations, thereby converting the system of partial differential equations into a set of ordinary differential equations. Through the simplification of these ordinary differential equations, we are able to construct several optically invariant solutions for the governing equation. Furthermore, through the utilization of the generalized exponential rational function (GERF) approach, we have derived additional intriguing closed-form solutions. Conservation laws are derived for the governing equation by utilizing the resulting symmetries introduced by the Ibragimov scheme. Furthermore, modulation instability and gain spectrum are derived for this equation to understand the correlation between nonlinearity and dispersive effects. To provide a comprehensive and insightful portrayal of our findings, we have created three-dimensional (3D) visualizations of these solutions, which reveal the periodic waves and the solitary wave structures. The form of cubic nonlinear Schrödinger’s equation discussed in this article and the optical solutions obtained have never been studied before. Also, these attained solutions can be beneficial to study analytically the identical models arising in fluid dynamics, birefringent fibers, plasma physics, and other optical areas.

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Acknowledgements

The authors are pleased to acknowledge the editor and the referees for their informative and helpful suggestions. The third author, Sachin Kumar, would like to recognize the Institution of Eminence, University of Delhi, India, for funding this research under the Faculty Research Programme Grant (IoE) with Ref. No./IoE/2023-24/12/FRP.

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  1. Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, 110007, India

    Setu Rani, Shubham Kumar Dhiman & Sachin Kumar

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  1. Setu Rani
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  2. Shubham Kumar Dhiman
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  3. Sachin Kumar
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Each author contributed equally to the study conception and design of this manuscript. All authors would have read and approved their approval to the final manuscript.

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Correspondence to Sachin Kumar.

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Rani, S., Dhiman, S.K. & Kumar, S. Newly constructed closed-form soliton solutions, conservation laws and modulation instability for a (2+1)-dimensional cubic nonlinear Schrödinger’s equation using optimal system of Lie subalgebra. Opt Quant Electron 56, 532 (2024). https://doi.org/10.1007/s11082-023-06085-w

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