Third-order differential equations and local isometric immersions of pseudospherical surfaces
Bibliographic record
Abstract
The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and nonlocal) and the presence of associated linear problems. It thus contains many important known examples of integrable equations, like the sine-Gordon, Liouville, KdV, mKdV, Camassa–Holm and Degasperis–Procesi equations, and is also home to many new families of integrable equations. Our paper is concerned with the question of the local isometric immersion in E3 of the pseudospherical surfaces defined by the solutions of equations belonging to the class introduced by Chern and Tenenblat [Pseudospherical surfaces and evolution equations, Stud. Appl. Math. 74 (1986) 55–83]. In the case of the sine-Gordon equation, it is a classical result that the second fundamental form of the immersion depends only on a jet of finite order of the solution of the partial differential equation. A natural question is therefore to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudospherical surfaces. In a pair of earlier papers [N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local isometric immersions of pseudo-spherical surfaces, to appear in Comm. Anal. Geom., arXiv: 1308.6545; Local isometric immersions of pseudo-spherical surfaces and evolution equations, in Hamiltonian Partial Differential Equations and Applications, eds. P. Guyenne, D. Nichols and C. Sulem, Fields Institute Communications, Vol. 75 (Springer-Verlag, 2015), pp. 369–381], it was shown that this property fails to hold for all [Formula: see text]th-order evolution equations [Formula: see text] and all other second-order equations of the form [Formula: see text], except for the sine-Gordon equation and a special class of equations for which the coefficients of the second fundamental form are universal, that is functions of [Formula: see text] and [Formula: see text] which are independent of the choice of solution [Formula: see text]. In this paper, we consider third-order equations of the form [Formula: see text], [Formula: see text], which describe pseudospherical surfaces with the Riemannian metric given in [T. Castro Silva and K. Tenenblat, Third order differential equations describing pseudospherical surfaces, J. Differential Equations 259 (2015) 4897–4923]. This class contains the Camassa–Holm and Degasperis–Procesi equations as special cases. We show that whenever there exists a local isometric immersion in E3 for which the coefficients of the second fundamental form depend on a jet of finite order of [Formula: see text], then these coefficients are universal in the sense of being independent on the choice of solution [Formula: see text]. This result further underscores the special place that the sine-Gordon equations seem to occupy amongst integrable partial differential equations in one space variable.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".