We investigate the Cauchy problem for the spin-1 Gross–Pitaevskii (GP) equation, which is a model instrumental in characterizing the soliton dynamics within spinor Bose-Einstein condensates. Recently, Geng et al. (Commun Math Phys 382:585–611, 2021) reported the long-time asymptotic result with error <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {O}(\frac{\ln t}{t})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mfrac> <mrow> <mo>ln</mo> <mi>t</mi> </mrow> <mi>t</mi> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the spin-1 GP equation for the case of the purely continuous spectrum. Based on the previous work, we conduct in-depth research on the soliton resolution conjecture and asymptotic analysis of the spin-1 GP equation. Compared with the previous work, we improve the asymptotic error accuracy from <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {O}(\frac{\ln t}{t})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mfrac> <mrow> <mo>ln</mo> <mi>t</mi> </mrow> <mi>t</mi> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {O}(t^{-3/4})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow> <mo>-</mo> <mn>3</mn> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. More importantly, through the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\bar{\partial }$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mrow> <mi>∂</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </math></EquationSource> </InlineEquation>-nonlinear steepest descent method and the Deift–Zhou’s nonlinear steepest descent method, we obtain effective asymptotic errors and successfully carry out a full asymptotic analysis of the spin-1 GP equation based on the characteristics of the spectral problem, including three cases: (i) coexistence of discrete and continuous spectrum; (ii) the purely continuous spectrum, as considered in the work of Geng et al. with error <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {O}(\frac{\ln t}{t})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mfrac> <mrow> <mo>ln</mo> <mi>t</mi> </mrow> <mi>t</mi> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>; (iii) the purely discrete spectrum. For the case (i), the corresponding asymptotic approximations can be characterized by an N-solitons as well as an interaction term between soliton solutions and the dispersion term with diverse residual error order <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {O}(t^{-3/4})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow> <mo>-</mo> <mn>3</mn> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In the case (ii), we strictly prove that the solution of the spin-1 GP equation can be characterized by the soliton solution and an error term with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {O}(t^{-3/4})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow> <mo>-</mo> <mn>3</mn> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. For the case (iii), we rigorously prove the localization of multiple degenerate soliton groups (DSGs), which is comprised of inseparable solitons with identical velocities, and calculate the long-time asymptotics for an arbitrary N-soliton solutions of the spin-1 GP equation. Finally, our results confirm the soliton resolution conjecture of the spin-1 GP equation and show that the soliton solutions of the spin-1 GP equation become a linear combination of multiple DSGs with different sizes.

On Cauchy Problem for the Spin-1 Gross–Pitaevskii Equation: Soliton Resolution Conjecture and Asymptotic Analysis

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发表日期：

2026年2月5日

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Shou-Fu Tian,
Jia-Fu Tong

摘要

We investigate the Cauchy problem for the spin-1 Gross–Pitaevskii (GP) equation, which is a model instrumental in characterizing the soliton dynamics within spinor Bose-Einstein condensates. Recently, Geng et al. (Commun Math Phys 382:585–611, 2021) reported the long-time asymptotic result with error $\mathcal {O}(\frac{\ln t}{t})$ $O (\frac{ln t}{t})$ for the spin-1 GP equation for the case of the purely continuous spectrum. Based on the previous work, we conduct in-depth research on the soliton resolution conjecture and asymptotic analysis of the spin-1 GP equation. Compared with the previous work, we improve the asymptotic error accuracy from $\mathcal {O}(\frac{\ln t}{t})$ $O (\frac{ln t}{t})$ to $\mathcal {O}(t^{-3/4})$ $O (t^{- 3 / 4})$ . More importantly, through the $\bar{\partial }$ $\bar{\partial}$ -nonlinear steepest descent method and the Deift–Zhou’s nonlinear steepest descent method, we obtain effective asymptotic errors and successfully carry out a full asymptotic analysis of the spin-1 GP equation based on the characteristics of the spectral problem, including three cases: (i) coexistence of discrete and continuous spectrum; (ii) the purely continuous spectrum, as considered in the work of Geng et al. with error $\mathcal {O}(\frac{\ln t}{t})$ $O (\frac{ln t}{t})$ ; (iii) the purely discrete spectrum. For the case (i), the corresponding asymptotic approximations can be characterized by an N-solitons as well as an interaction term between soliton solutions and the dispersion term with diverse residual error order $\mathcal {O}(t^{-3/4})$ $O (t^{- 3 / 4})$ . In the case (ii), we strictly prove that the solution of the spin-1 GP equation can be characterized by the soliton solution and an error term with $\mathcal {O}(t^{-3/4})$ $O (t^{- 3 / 4})$ . For the case (iii), we rigorously prove the localization of multiple degenerate soliton groups (DSGs), which is comprised of inseparable solitons with identical velocities, and calculate the long-time asymptotics for an arbitrary N-soliton solutions of the spin-1 GP equation. Finally, our results confirm the soliton resolution conjecture of the spin-1 GP equation and show that the soliton solutions of the spin-1 GP equation become a linear combination of multiple DSGs with different sizes.