<p>In this article we develop the direct and inverse scattering theory of the AKNS system <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_944_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="156" /> </InlineMediaObject> <EquationSource Format="TEX">${\varvec{V}}_x=(ik\Sigma +{\mathcal {Q}}(x)){\varvec{V}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">V</mi> </mrow> <mi>x</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>k</mi> <mi mathvariant="normal">Σ</mi> <mo>+</mo> <mi mathvariant="script">Q</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="bold-italic">V</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_944_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="129" /> </InlineMediaObject> <EquationSource Format="TEX">$\Sigma =\text {diag}(1,0,-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Σ</mi> <mo>=</mo> <mtext>diag</mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_944_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {Q}}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is an off-diagonal <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_944_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$3\times 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_944_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\varvec{\sigma }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">σ</mi> </mrow> </math></EquationSource> </InlineEquation>-focusing potential with entries in <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11587_2025_944_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$L^1({\mathbb {R}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We derive the time evolution of the scattering data which, through the inverse scattering transform, lead to the solution of the initial-value problem for a generalized long-wave-short-wave equation. We determine the essential spectrum of the AKNS system rigorously.</p>

Three-way focusing Zakharov-Shabat systems with zero diagonal entry

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2025年2月25日

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Francesco Demontis,
Cornelis van der Mee

摘要

In this article we develop the direct and inverse scattering theory of the AKNS system ${\varvec{V}}_x=(ik\Sigma +{\mathcal {Q}}(x)){\varvec{V}}$ $V_{x} = (i k Σ + Q (x)) V$ , where $\Sigma =\text {diag}(1,0,-1)$ $Σ = diag (1, 0, - 1)$ and ${\mathcal {Q}}(x)$ $Q (x)$ is an off-diagonal $3\times 3$ $3 \times 3$ $\varvec{\sigma }$ $σ$ -focusing potential with entries in $L^1({\mathbb {R}})$ $L^{1} (R)$ . We derive the time evolution of the scattering data which, through the inverse scattering transform, lead to the solution of the initial-value problem for a generalized long-wave-short-wave equation. We determine the essential spectrum of the AKNS system rigorously.