<p>We study the large-time behavior of global energy class (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>) solutions of the one-dimensional nonlinear Schrödinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction Morawetz estimate localized to an exterior region, we prove that these solutions decompose into a free wave and a weakly localized part, which is asymptotically orthogonal to any fixed free wave. We further show that the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm of this weakly localized part is concentrated in the region <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|x| \le t^{1/2+}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <msup> <mi>t</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>+</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and that the energy (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\dot{H}^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>) norm is concentrated in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$|x| \le t^{1/3+}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <msup> <mi>t</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>3</mn> <mo>+</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. Our results hold for solutions with arbitrarily large initial data.</p>

Scattering and Localized States for Defocusing Nonlinear Schrödinger Equations with Potential

期刊论文
发表日期：

2025年12月16日

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Avy Soffer,
Gavin Stewart

摘要

We study the large-time behavior of global energy class ( $H^1$ $H^{1}$ ) solutions of the one-dimensional nonlinear Schrödinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction Morawetz estimate localized to an exterior region, we prove that these solutions decompose into a free wave and a weakly localized part, which is asymptotically orthogonal to any fixed free wave. We further show that the $L^2$ $L^{2}$ norm of this weakly localized part is concentrated in the region $|x| \le t^{1/2+}$ $| x | \leq t^{1 / 2 +}$ and that the energy ( $\dot{H}^1$ ${\dot{H}}^{1}$ ) norm is concentrated in $|x| \le t^{1/3+}$ $| x | \leq t^{1 / 3 +}$ . Our results hold for solutions with arbitrarily large initial data.