<p>We consider the modified scattering problem for the nonlinear Klein–Gordon equation with cubic nonlinear interactions in one spatial dimension <Equation ID="Equ1"> <EquationNumber>1</EquationNumber> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} w_{tt}-\Delta w+w=\mu |w|^{2}w,\\ w\left( 0,x\right) =w_{0}\left( x\right) , w_{t}\left( 0,x\right) =w_{1}(x), \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>w</mi> <mrow> <mi mathvariant="italic">tt</mi> </mrow> </msub> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>w</mi> <mo>+</mo> <mi>w</mi> <mo>=</mo> <mi>μ</mi> <msup> <mrow> <mo stretchy="false">|</mo> <mi>w</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mi>w</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>w</mi> <mfenced close=")" open="("> <mn>0</mn> <mo>,</mo> <mi>x</mi> </mfenced> <mo>=</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> <mfenced close=")" open="("> <mi>x</mi> </mfenced> <mo>,</mo> <msub> <mi>w</mi> <mi>t</mi> </msub> <mfenced close=")" open="("> <mn>0</mn> <mo>,</mo> <mi>x</mi> </mfenced> <mo>=</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mu \in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>. We show that for any small real valued initial data <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$w_{0},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>w</mi> <mn>0</mn> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$w_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>w</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> low-regularity weighted Sobolev spaces there exists a unique modified final state <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$W_{+}\in \textbf{L}^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mo>+</mo> </msub> <mo>∈</mo> <msup> <mi mathvariant="bold">L</mi> <mi>∞</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> such that the corresponding solution exhibits a logarithmic phase correction as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$t\rightarrow \infty .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi>∞</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Our proof is based on the Factorization Technique. We present more simple and robust method for the proof of this result based on the Factorization Technique and on the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textbf{L}^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> - estimates for the transformed evolution operators, which considerably simplifies the proof of the asymptotics of solutions.</p>

Modified scattering for the nonlinear Klein–Gordon equation

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2026年5月26日

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Miguel Ballesteros,
Gerardo Franco Córdova,
Ivan Naumkin,
Alexis Vaed Vázquez

摘要

We consider the modified scattering problem for the nonlinear Klein–Gordon equation with cubic nonlinear interactions in one spatial dimension 1 $\begin{aligned} {\left\{ \begin{array}{ll} w_{tt}-\Delta w+w=\mu |w|^{2}w,\\ w\left( 0,x\right) =w_{0}\left( x\right) , w_{t}\left( 0,x\right) =w_{1}(x), \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} w_{tt} - Δ w + w = μ {| w |}^{2} w, \\ w (0, x) = w_{0} (x), w_{t} (0, x) = w_{1} (x), \end{matrix}) \end{matrix}$ for $\mu \in \mathbb {R}$ $μ \in R$ . We show that for any small real valued initial data $w_{0},$ $w_{0},$ $w_{1}$ $w_{1}$ low-regularity weighted Sobolev spaces there exists a unique modified final state $W_{+}\in \textbf{L}^{\infty }$ $W_{+} \in L^{\infty}$ such that the corresponding solution exhibits a logarithmic phase correction as $t\rightarrow \infty .$ $t \to \infty .$ Our proof is based on the Factorization Technique. We present more simple and robust method for the proof of this result based on the Factorization Technique and on the $\textbf{L}^{2}$ $L^{2}$ - estimates for the transformed evolution operators, which considerably simplifies the proof of the asymptotics of solutions.