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<p>In this paper, the wave solutions of the unstable and modified unstable nonlinear Schrödinger equations with a truncated M-fractional derivative are discovered via the modified <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(({G'}/{G^2})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">/</mo> <msup> <mi>G</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-expansion procedure. This procedure is based on the transformation of the given partial differential equation to an ordinary differential equation via wave transforms. Results are given in different forms: rational, trigonometric, and hyperbolic functions. Graphical representations of some obtained results are illustrated by 3D plots, contour plots, and 2D plots. Graphical representations help us to decide the changes in the movements of the solutions that are obtained. Finally, bifurcation and sensitivity analysis are given for two equations. Results are useful for physical mathematics, physics, and engineering.</p>

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Bifurcation and sensitivity analysis, wave solutions of the two types of M-truncated unstable Schrödinger equations

  • 期刊论文

  • 发表日期：

    2026年4月7日

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  • Waseem Razzaq,
  • Asim Zafar,
  • Arzu Akbulut

摘要

In this paper, the wave solutions of the unstable and modified unstable nonlinear Schrödinger equations with a truncated M-fractional derivative are discovered via the modified \(({G'}/{G^2})\) ( G ′ / G 2 ) -expansion procedure. This procedure is based on the transformation of the given partial differential equation to an ordinary differential equation via wave transforms. Results are given in different forms: rational, trigonometric, and hyperbolic functions. Graphical representations of some obtained results are illustrated by 3D plots, contour plots, and 2D plots. Graphical representations help us to decide the changes in the movements of the solutions that are obtained. Finally, bifurcation and sensitivity analysis are given for two equations. Results are useful for physical mathematics, physics, and engineering.