<p>This paper establishes the existence of solitary wave solutions for a singularly perturbed Camassa-Holm equation with Atangana’s conformable derivative, where the singular perturbation is given by a Kuramoto-Sivashinsky (KS) term <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\tau (D_{xx}v + D_{xxxx}v)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow> <mi mathvariant="italic">xx</mi> </mrow> </msub> <mi>v</mi> <mo>+</mo> <msub> <mi>D</mi> <mrow> <mi mathvariant="italic">xxxx</mi> </mrow> </msub> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Physically, the KS perturbation encodes small-scale dispersive effects due to viscosity and surface tension, while the conformable time derivative <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$D_t^{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>D</mi> <mi>t</mi> <mi>α</mi> </msubsup> </math></EquationSource> </InlineEquation> captures memory effects. The parameter <i>k</i> shifts the linear wave speed near the critical shallow-water velocity, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$m=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> corresponds to quadratic nonlinear advection. Using traveling wave coordinates, solitary waves are characterized as homoclinic orbits; by combining geometric singular perturbation, invariant manifold theory, and Fredholm orthogonality, this paper proves the existence of solitary waves under the KS perturbation. In this way, this result extends earlier existence theories by simultaneously accommodating the linear term <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$2kD_x v$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>k</mi> <msub> <mi>D</mi> <mi>x</mi> </msub> <mi>v</mi> </mrow> </math></EquationSource> </InlineEquation>, quadratic nonlinearity <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$m=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and the KS perturbation.</p>

Solitary Waves in a Singularly Perturbed Camassa-Holm Equation Involving Atangana’s Conformable Derivative

期刊论文
发表日期：

2026年2月19日

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Zhenglong Zhang,
Xiaowan Li

摘要

This paper establishes the existence of solitary wave solutions for a singularly perturbed Camassa-Holm equation with Atangana’s conformable derivative, where the singular perturbation is given by a Kuramoto-Sivashinsky (KS) term $\tau (D_{xx}v + D_{xxxx}v)$ $τ (D_{xx} v + D_{xxxx} v)$ . Physically, the KS perturbation encodes small-scale dispersive effects due to viscosity and surface tension, while the conformable time derivative $D_t^{\alpha }$ $D_{t}^{α}$ captures memory effects. The parameter k shifts the linear wave speed near the critical shallow-water velocity, and $m=2$ $m = 2$ corresponds to quadratic nonlinear advection. Using traveling wave coordinates, solitary waves are characterized as homoclinic orbits; by combining geometric singular perturbation, invariant manifold theory, and Fredholm orthogonality, this paper proves the existence of solitary waves under the KS perturbation. In this way, this result extends earlier existence theories by simultaneously accommodating the linear term $2kD_x v$ $2 k D_{x} v$ , quadratic nonlinearity $m=2$ $m = 2$ , and the KS perturbation.