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<p>This paper explores the cubic-quintic Schrödinger equation in the entire Euclidean space. Our objectives are twofold: first, to advance the understanding of unresolved issues related to this equation, which are well known in the extensively studied Gross-Pitaevskii equation. Second, to consolidate existing results on the cubic-quintic equation, providing partial contributions. Specifically, we determine the explicit constant for the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1109_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> a priori bound and establish a partial existence result for finite energy traveling waves in suitable approximate domains of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1109_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>.</p>

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On the cubic-quintic Schrödinger equation

  • 期刊论文

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  • 发表日期：

    2025年7月30日

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  • Laura Baldelli

摘要

This paper explores the cubic-quintic Schrödinger equation in the entire Euclidean space. Our objectives are twofold: first, to advance the understanding of unresolved issues related to this equation, which are well known in the extensively studied Gross-Pitaevskii equation. Second, to consolidate existing results on the cubic-quintic equation, providing partial contributions. Specifically, we determine the explicit constant for the \(L^\infty \) L ∞ a priori bound and establish a partial existence result for finite energy traveling waves in suitable approximate domains of \(\mathbb {R}^d\) R d .