<p>This paper studies the long-time asymptotic behavior of the Cauchy problem for the <i>n</i>-component coupled higher-order nonlinear Schrödinger equation with initial data in the Schwartz space. By spectral analysis of the associated <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(n+1) \times (n+1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> Lax pair, the solution is formulated as a matrix Riemann–Hilbert (RH) problem. Applying the Deift–Zhou nonlinear steepest descent method, we analyze the RH problem in two different regimes of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\zeta =\frac{x}{t}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ζ</mi> <mo>=</mo> <mfrac> <mi>x</mi> <mi>t</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\zeta < \frac{1}{3\varepsilon }-\varpi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ζ</mi> <mo><</mo> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mi>ε</mi> </mrow> </mfrac> <mo>-</mo> <mi>ϖ</mi> </mrow> </math></EquationSource> </InlineEquation> (with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varpi >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϖ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> a constant), the asymptotic behavior is characterized by explicit expressions involving parabolic cylinder functions in an oscillatory manner. For <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\zeta > \frac{1}{3\varepsilon }+\varpi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ζ</mi> <mo>></mo> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mi>ε</mi> </mrow> </mfrac> <mo>+</mo> <mi>ϖ</mi> </mrow> </math></EquationSource> </InlineEquation>, the solution decays rapidly. Rigorous asymptotic formulas are obtained with controlled error estimates.</p>

Long-Time Asymptotics for the n-Component Coupled Higher-Order Nonlinear Schrödinger Equation

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发表日期：

2026年7月22日

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Bing Chen,
Wenxia Chen,
Chaosheng Zhang

摘要

This paper studies the long-time asymptotic behavior of the Cauchy problem for the n-component coupled higher-order nonlinear Schrödinger equation with initial data in the Schwartz space. By spectral analysis of the associated $(n+1) \times (n+1)$ $(n + 1) \times (n + 1)$ Lax pair, the solution is formulated as a matrix Riemann–Hilbert (RH) problem. Applying the Deift–Zhou nonlinear steepest descent method, we analyze the RH problem in two different regimes of $\zeta =\frac{x}{t}$ $ζ = \frac{x}{t}$ . For $\zeta < \frac{1}{3\varepsilon }-\varpi $ $ζ < \frac{1}{3 ε} - ϖ$ (with $\varpi >0$ $ϖ > 0$ a constant), the asymptotic behavior is characterized by explicit expressions involving parabolic cylinder functions in an oscillatory manner. For $\zeta > \frac{1}{3\varepsilon }+\varpi $ $ζ > \frac{1}{3 ε} + ϖ$ , the solution decays rapidly. Rigorous asymptotic formulas are obtained with controlled error estimates.