<p>We conduct a comprehensive analysis of asymmetric integrable turbulence and rogue waves emerging from the modulation instability (MI) of plane waves for the derivative nonlinear Schrödinger equation (DNLSE). Long-term turbulence is characterized via <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-th moments, energies, wave-action spectrum, intensity PDF, and autocorrelation. The <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-th moments and ensemble-averaged kinetic and potential energy exhibit oscillatory convergence toward their steady-state values. Specifically, the oscillation amplitude and the nonlinear phase shift approach the asymptotic limit algebraically with time. These oscillations are categorized by their phase relationship with the potential energy modulus <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ |\langle H_4\rangle | $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">⟨</mo> <msub> <mi>H</mi> <mn>4</mn> </msub> <mo stretchy="false">⟩</mo> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> as either synchronized or in anti-phase. Unlike the NLS equation, the wave-action spectrum in the DNLSE setting is asymmetric, primarily due to the asymmetry between the wavenumber of the plane wave from the MI and the perturbation wavenumber. The intensity PDF and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ n $</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-th-order moments converge to exponential and Gaussian limits <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ \Gamma (n/2+1) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in an oscillatory manner. While these one-point statistics superficially suggest a Gaussian random field, the intensity autocorrelation <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ g^{(2)}(x,t) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> deviates from standard Gaussian metrics, indicating that the asymptotic state is not strictly Gaussian. Furthermore, investigating the carrier wavenumber <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ k_0 $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> reveals a translational invariance in the spectral domain; the wave-action spectrum shifts by <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ k_0 $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> while retaining its functional form. Crucially, the intensity PDFs and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ n $</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-th-order moments remain strictly invariant, collapsing onto universal profiles independent of the carrier momentum.</p>

Asymmetric Integrable Turbulence and Rogue Wave Statistics for the Derivative Nonlinear Schrödinger Equation

期刊论文
发表日期：

2026年4月13日

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Ming Zhong,
Weifang Weng,
Zhenya Yan

摘要

We conduct a comprehensive analysis of asymmetric integrable turbulence and rogue waves emerging from the modulation instability (MI) of plane waves for the derivative nonlinear Schrödinger equation (DNLSE). Long-term turbulence is characterized via $n$ $n$ -th moments, energies, wave-action spectrum, intensity PDF, and autocorrelation. The $n$ $n$ -th moments and ensemble-averaged kinetic and potential energy exhibit oscillatory convergence toward their steady-state values. Specifically, the oscillation amplitude and the nonlinear phase shift approach the asymptotic limit algebraically with time. These oscillations are categorized by their phase relationship with the potential energy modulus $ |\langle H_4\rangle | $ $| ⟨ H_{4} ⟩ |$ as either synchronized or in anti-phase. Unlike the NLS equation, the wave-action spectrum in the DNLSE setting is asymmetric, primarily due to the asymmetry between the wavenumber of the plane wave from the MI and the perturbation wavenumber. The intensity PDF and $ n $ $n$ -th-order moments converge to exponential and Gaussian limits $ \Gamma (n/2+1) $ $Γ (n / 2 + 1)$ in an oscillatory manner. While these one-point statistics superficially suggest a Gaussian random field, the intensity autocorrelation $ g^{(2)}(x,t) $ $g^{(2)} (x, t)$ deviates from standard Gaussian metrics, indicating that the asymptotic state is not strictly Gaussian. Furthermore, investigating the carrier wavenumber $ k_0 $ $k_{0}$ reveals a translational invariance in the spectral domain; the wave-action spectrum shifts by $ k_0 $ $k_{0}$ while retaining its functional form. Crucially, the intensity PDFs and $ n $ $n$ -th-order moments remain strictly invariant, collapsing onto universal profiles independent of the carrier momentum.