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<p>We deal with the construction of some special solutions for a two-component nonlinear Schrödinger system. More precisely, let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(R_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(R_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> be two traveling waves of a scalar Schrödinger equation; our goal is to construct a solution of the system which behave at infinity like the pair <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((R_1,R_2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Initially, we prove the existence of such solutions in dimensions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(1\le d\le 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>d</mi> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> with the assumption of relative high speeds between the solitary waves. We then finish our results by studying the one-dimensional case without the assumption of relative high speeds. The main tools used to establish our results combine several techniques including energy estimates, bootstrap, modulation and localization arguments.</p>

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On the asymptotic behavior of solutions for a nonlinear Schrödinger system

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月25日

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  • Vicente C. P. Alvarez,
  • Ademir Pastor

摘要

We deal with the construction of some special solutions for a two-component nonlinear Schrödinger system. More precisely, let \(R_1\) R 1 and \(R_2\) R 2 be two traveling waves of a scalar Schrödinger equation; our goal is to construct a solution of the system which behave at infinity like the pair \((R_1,R_2)\) ( R 1 , R 2 ) . Initially, we prove the existence of such solutions in dimensions \(1\le d\le 3\) 1 ≤ d ≤ 3 with the assumption of relative high speeds between the solitary waves. We then finish our results by studying the one-dimensional case without the assumption of relative high speeds. The main tools used to establish our results combine several techniques including energy estimates, bootstrap, modulation and localization arguments.