<p>A fractional space–time Cahn–Hilliard (FSTCH) model using Jumarie’s modified Riemann-Liouville derivative with order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\delta _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\delta _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> is examined in this work. This model has extensive applications in various domains, including optics and plasma physics. The <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Phi ^{6}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Φ</mi> <mn>6</mn> </msup> </math></EquationSource> </InlineEquation> approach is used to obtain an advanced analytical solution for it. The model framework can yield accurate and efficient solutions for various wave structures based on this methodology. By utilising the method, we acquired different behaviours of solutions such as hyperbolic, trigonometric and rational waves. The accuracy and adaptability of the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Phi ^{6}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Φ</mi> <mn>6</mn> </msup> </math></EquationSource> </InlineEquation> approach in fractional model modelling are also validated. As a result, this study demonstrates that the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Phi ^6$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Φ</mi> <mn>6</mn> </msup> </math></EquationSource> </InlineEquation> approach is an effective method for analysing wave propagation in fractional models. The <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Phi ^6$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Φ</mi> <mn>6</mn> </msup> </math></EquationSource> </InlineEquation> approach, which is used in this study to construct solutions, is important in mathematical and physical fields.</p>

Analysis of soliton behaviour to fractional space–time Cahn–Hilliard model with stability analysis

期刊论文
发表日期：

2026年5月22日

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Marriam Anayat,
Asghar Ali,
Jamshad Ahmad

摘要

A fractional space–time Cahn–Hilliard (FSTCH) model using Jumarie’s modified Riemann-Liouville derivative with order $\delta _1$ $δ_{1}$ and $\delta _2$ $δ_{2}$ is examined in this work. This model has extensive applications in various domains, including optics and plasma physics. The $\Phi ^{6}$ $Φ^{6}$ approach is used to obtain an advanced analytical solution for it. The model framework can yield accurate and efficient solutions for various wave structures based on this methodology. By utilising the method, we acquired different behaviours of solutions such as hyperbolic, trigonometric and rational waves. The accuracy and adaptability of the $\Phi ^{6}$ $Φ^{6}$ approach in fractional model modelling are also validated. As a result, this study demonstrates that the $\Phi ^6$ $Φ^{6}$ approach is an effective method for analysing wave propagation in fractional models. The $\Phi ^6$ $Φ^{6}$ approach, which is used in this study to construct solutions, is important in mathematical and physical fields.