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• W3135783072 abstract "We study here the Zakharov-Kuznetsov equation in dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=application/x-tex>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=3> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=application/x-tex>3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=4> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding=application/x-tex>4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the modified Zakharov-Kuznetsov equation in dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=application/x-tex>2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Those equations admit solitons, characterized by their velocity and their shift. Given the parameters of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> solitons <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R Superscript k> <mml:semantics> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>R^k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (with distinct velocities), we prove the existence and uniqueness of a multi-soliton <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding=application/x-tex>u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-vertical-bar u left-parenthesis t right-parenthesis minus sigma-summation Underscript k equals 1 Overscript upper K Endscripts upper R Superscript k Baseline left-parenthesis t right-parenthesis double-vertical-bar Subscript upper H Sub Superscript 1 Subscript Baseline right-arrow 0 as t right-arrow plus normal infinity period> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>K</mml:mi> </mml:munderover> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:msub> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:msub> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mn>0</mml:mn> <mml:mspace width=1em /> <mml:mtext>as</mml:mtext> <mml:mspace width=1em /> <mml:mi>t</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>| u(t) - sum _{k=1}^K R^k(t) |_{H^1} to 0 quad text {as} quad t to +infty .</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> The convergence takes place in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript s> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>H^s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with an exponential rate for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s greater-than-or-equal-to 0> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>s ge 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The construction is made by successive approximations of the multi-soliton. We use classical arguments to control of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript 1> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-norms of the errors (inspired by Martel [Amer. J. Math. 127 (2005), pp. 1103–1140]), and introduce a new ingredient for the control of the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript s> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>H^s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-norm in dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d greater-than-or-equal-to 2> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>dgeq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, by a technique close to monotonicity." @default.
• W3135783072 created "2021-03-15" @default.
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• W3135783072 date "2021-03-08" @default.
• W3135783072 modified "2023-09-25" @default.
• W3135783072 title "Asymptotic 𝐾-soliton-like solutions of the Zakharov-Kuznetsov type equations" @default.
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• W3135783072 doi "https://doi.org/10.1090/tran/8331" @default.
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