<p>This study investigates the nonlinear dynamical behavior of dark soliton equations in Bose–Einstein condensates (BEC). By employing the perturbation method, we analytically solve the governing equations and examine soliton interactions and collision processes, revealing the complex dynamical properties of solitary waves in quantum fluids. Furthermore, we derive the corresponding Korteweg–de Vries (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{KdV}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>KdV</mtext> </math></EquationSource> </InlineEquation>) equation and explore its integrability using the prolongation structure method, successfully obtaining its Lax pair. This provides a novel framework for constructing soliton solutions to the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textrm{KdV}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>KdV</mtext> </math></EquationSource> </InlineEquation> equation. Additionally, we utilize <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathrm {B\ddot{a}cklund}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">B</mi> <mover accent="true"> <mi mathvariant="normal">a</mi> <mo>¨</mo> </mover> <mi mathvariant="normal">cklund</mi> </mrow> </math></EquationSource> </InlineEquation> transformation to derive exact solutions, validating the accuracy of our theoretical analysis. The key innovation of this work lies in the systematic application of prolongation structure theory as a core mathematical physics approach to study soliton dynamics in BEC. Notably, we advance the Lax pair derivation and exact solution acquisition for the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textrm{KdV}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>KdV</mtext> </math></EquationSource> </InlineEquation> equation, establishing a robust theoretical foundation for future experimental observations and technological applications. Our findings also suggest new research directions in nonlinear quantum systems.</p>

Description of dark soliton equations in Bose–Einstein condensates based on Bäcklund transformation

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发表日期：

2026年4月14日

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Lixiu Wang,
Zengangmao Ren,
Yangjie Jia

摘要

This study investigates the nonlinear dynamical behavior of dark soliton equations in Bose–Einstein condensates (BEC). By employing the perturbation method, we analytically solve the governing equations and examine soliton interactions and collision processes, revealing the complex dynamical properties of solitary waves in quantum fluids. Furthermore, we derive the corresponding Korteweg–de Vries ( $\textrm{KdV}$ $KdV$ ) equation and explore its integrability using the prolongation structure method, successfully obtaining its Lax pair. This provides a novel framework for constructing soliton solutions to the $\textrm{KdV}$ $KdV$ equation. Additionally, we utilize $\mathrm {B\ddot{a}cklund}$ $B \ddot{a} cklund$ transformation to derive exact solutions, validating the accuracy of our theoretical analysis. The key innovation of this work lies in the systematic application of prolongation structure theory as a core mathematical physics approach to study soliton dynamics in BEC. Notably, we advance the Lax pair derivation and exact solution acquisition for the $\textrm{KdV}$ $KdV$ equation, establishing a robust theoretical foundation for future experimental observations and technological applications. Our findings also suggest new research directions in nonlinear quantum systems.