On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data
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Bibliographic record
Abstract
We consider the KdV equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript t Baseline u plus partial-differential Subscript x Superscript 3 Baseline u plus u partial-differential Subscript x Baseline u equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>x</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial _t u +\partial ^3_x u +u\partial _x u=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with quasi-periodic initial data whose Fourier coefficients decay exponentially and prove existence and uniqueness, in the class of functions which have an expansion with exponentially decaying Fourier coefficients, of a solution on a small interval of time, the length of which depends on the given data and the frequency vector involved. For a Diophantine frequency vector and for small quasi-periodic data (i.e., when the Fourier coefficients obey <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue c left-parenthesis m right-parenthesis EndAbsoluteValue less-than-or-equal-to epsilon exp left-parenthesis minus kappa 0 StartAbsoluteValue m EndAbsoluteValue right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>c</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo> ≤ </mml:mo> <mml:mi> ε </mml:mi> <mml:mi>exp</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:msub> <mml:mi> κ </mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>m</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">|c(m)| \le \varepsilon \exp (-\kappa _0 |m|)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> ε </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\varepsilon > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sufficiently small, depending on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa 0 greater-than 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> κ </mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\kappa _0 > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the frequency vector), we prove global existence and uniqueness of the solution. The latter result relies on our recent work [Publ. Math. Inst. Hautes Études Sci. <bold>119</bold> (2014) 217] on the inverse spectral problem for the quasi-periodic Schrödinger equation.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.006 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it