<p>We consider the generalized Benjamin-Ono equation: <Equation ID="Equ260"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3113_Article_Equ260.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="226" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \partial _tu+\partial _x(-|D|u+|u|^{p-1}u)=0, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>+</mo> <msub> <mi>∂</mi> <mi>x</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mo stretchy="false">|</mo> <mi>D</mi> <mo stretchy="false">|</mo> <mi>u</mi> <mo>+</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3113_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-supercritical power <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3113_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p>3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3113_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-subcritical power <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3113_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$2<p<3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We will construct strongly interacting multi-solitary wave of the form: <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3113_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="153" /> </InlineMediaObject> <EquationSource Format="TEX">$\sum _{i=1}^nQ(\cdot -t-x_i(t))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <mi>Q</mi> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo>-</mo> <mi>t</mi> <mo>-</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3113_Article_IEq6.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and the parameters <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3113_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$x_i(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3113_Article_IEq8.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </InlineMediaObject> <EquationSource Format="TEX">$x_{i}(t)-x_{i+1}(t)\sim \sqrt{t}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>∼</mo> <msqrt> <mi>t</mi> </msqrt> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3113_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$t\rightarrow +\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. We will also prove the uniqueness of such solutions in the case of <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3113_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_3113_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p>3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Strongly interacting multi-solitons for generalized Benjamin-Ono equations

期刊论文
发表日期：

2025年8月20日

查看全文

Yang Lan,
Zhong Wang

摘要

We consider the generalized Benjamin-Ono equation: $\begin{aligned} \partial _tu+\partial _x(-|D|u+|u|^{p-1}u)=0, \end{aligned}$ $\begin{matrix} \partial_{t} u + \partial_{x} {(- | D | u + | u |}^{p - 1} u) = 0, \end{matrix}$ with $L^2$ $L^{2}$ -supercritical power $p>3$ $p > 3$ or $L^2$ $L^{2}$ -subcritical power $2<p<3$ $2 < p < 3$ . We will construct strongly interacting multi-solitary wave of the form: $\sum _{i=1}^nQ(\cdot -t-x_i(t))$ $\sum_{i = 1}^{n} Q (\cdot - t - x_{i} (t))$ , where $n\ge 2$ $n \geq 2$ , and the parameters $x_i(t)$ $x_{i} (t)$ satisfying $x_{i}(t)-x_{i+1}(t)\sim \sqrt{t}$ $x_{i} (t) - x_{i + 1} (t) \sim \sqrt{t}$ as $t\rightarrow +\infty $ $t \to + \infty$ . We will also prove the uniqueness of such solutions in the case of $n=2$ $n = 2$ and $p>3$ $p > 3$ .