<p>We study the Cauchy problem for the defocusing modified Korteweg–de Vries (mKdV) equation with step-like initial data approaching nonzero constants <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$c_l$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mi>l</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$c_r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$x \rightarrow -\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mo>-</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$x\rightarrow +\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, respectively. Assuming <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$c_l>c_r>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mi>l</mi> </msub> <mo>></mo> <msub> <mi>c</mi> <mi>r</mi> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the solution exhibits a rarefaction wave structure. We first develop the inverse scattering transform for the solution satisfying these step-like boundary conditions. Using the associated scattering data, we prove that there exists a unique global solution of the Cauchy problem and characterize it in terms of a Riemann–Hilbert (RH) problem. By applying the nonlinear steepest descent method to this RH problem, we rigorously obtain large-time asymptotics of rarefaction wave solution in three distinct space-time regions, each characterized by a different leading order behavior. They are: (I) a left-field region where the solution approaches the left background constant, modulo a small oscillatory correction, (II) a central region where the solution exhibits a slowly varying profile that transitions between the two constants, and (III) a right-field region where the solution tends to the right background constant, up to an algebraically small correction. Rigorous derivations of the leading terms, sub-leading terms as well as the error bounds are presented.</p>

On the Large-Time Asymptotics of the Defocusing mKdV Equation with Step-Like Initial Data

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发表日期：

2026年7月28日

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Taiyang Xu,
Yidan Zhang

摘要

We study the Cauchy problem for the defocusing modified Korteweg–de Vries (mKdV) equation with step-like initial data approaching nonzero constants $c_l$ $c_{l}$ and $c_r$ $c_{r}$ as $x \rightarrow -\infty $ $x \to - \infty$ and $x\rightarrow +\infty $ $x \to + \infty$ , respectively. Assuming $c_l>c_r>0$ $c_{l} > c_{r} > 0$ , the solution exhibits a rarefaction wave structure. We first develop the inverse scattering transform for the solution satisfying these step-like boundary conditions. Using the associated scattering data, we prove that there exists a unique global solution of the Cauchy problem and characterize it in terms of a Riemann–Hilbert (RH) problem. By applying the nonlinear steepest descent method to this RH problem, we rigorously obtain large-time asymptotics of rarefaction wave solution in three distinct space-time regions, each characterized by a different leading order behavior. They are: (I) a left-field region where the solution approaches the left background constant, modulo a small oscillatory correction, (II) a central region where the solution exhibits a slowly varying profile that transitions between the two constants, and (III) a right-field region where the solution tends to the right background constant, up to an algebraically small correction. Rigorous derivations of the leading terms, sub-leading terms as well as the error bounds are presented.