Fredholm Determinants and the Camassa-Holm Hierarchy

McKean, Henry P.

doi:10.1007/978-3-319-22237-0_9

Fredholm Determinants and the Camassa-Holm Hierarchy

Henry P. McKean Jr.⁵

Chapter

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Part of the book series: Contemporary Mathematicians ((CM))

Abstract

The equation of Camassa and Holm [2]2 is an approximate description of long waves in shallow water.

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Notes

1.
See also Camassa, Holm, and Hyman [3].
2.
I recall H. Flaschka’s definition of integrability, the only honest one around: “You didn’t think I could integrate that, but I can!”
3.
0 is not in the spectrum.
4.
The indexing will be explained later.
5.
The computation can be found in McKean [12] I repeat it here for the reader’s convenience.
6.
The individual flows commute so the order of the product is immaterial.
7.
McKean and Trubowitz [13] encountered similar theta-like determinants in connection with the isospectral class of the quantum-mechanical oscillator − D ² + x ² − 1.
8.
m ₊∕m ₋ is the positive/negative part of m.
9.
\(f_{n}(x) \simeq\) a “norming constant” × e ^−x∕2 at \(+\infty.\)
10.
\(A^{\mathfrak{p}}\) inverse is \(A^{-\mathfrak{p}}\), in which \(-\mathfrak{p}\) is \(\mathfrak{p}\) with signature reversed.
11.
\(G = (1 - D^{2})^{-1} = \frac{1} {2}\exp (-\vert x - y\vert )\) as before.
12.
Here and below f _n is always normalized as in \(\|f_{n}\|^{2} =\int [(f_{n}^{{\prime}})^{2} + \frac{1} {4}f_{n}^{2}] = \lambda _{n}\int mf_{n}^{2} = 1.\)
13.
J = mD + Dm as before.
14.
mD + Dm is skew.
15.
The computation is the same.
16.
The preliminary evaluation of J from the previous display is used.
17.
\(\lambda _{0}m_{0}f_{0} = (\frac{1} {4} - D^{2})f_{0}\) is used to reduce the integrals to their final form.
18.
\(-\varLambda \int _{-\infty }^{+\infty }mf_{-}^{2} = 1\).
19.
f ₋(−1) = 0 implies f ₋ ≡ 0. which is not so.
20.
f ⊗ f is the matrix \([f_{i}f_{j}: i,j \in \mathbb{Z} - 0]\).
21.
t is the diagonal matrix \(t_{n}: n \in \mathbb{Z} - 0\), and similarly for \(\lambda\).
22.
1 − M(1 + CM)⁻¹ C = (1 + MC)⁻¹, (1 + CM)⁻¹ C being symmetric.
23.
Q ⁻¹(e ^t − 1) is symmetric and \(\eta = (e^{t} - 1)\xi\).
24.
See Sect. 9.5.2.
25.
Q ⁻¹ = 1 − (Q − 1)Q ⁻¹.
26.
PQ ⁻¹ is bounded.
27.
Q ⁻¹ = 1 − (Q − 1)Q ⁻¹.
28.
\(G = (1 - D^{2})^{-1} = \frac{1} {2}\exp (-\vert x - y\vert )\) as before.
29.
You may think line 3 is a little shaky but you can check line 4 directly from line 2.
30.
\(\lambda \int _{-\infty }^{\infty }mf \otimes f = \text{the identity}\).
31.
\(C\int _{-\infty }^{x}mf \otimes fC\) is compact with finite absolute trace; compare Sect. 9.5.3.
32.
\(\langle f_{1},f_{2}\rangle\) is the inner product \(\int _{-\infty }^{\infty }(f_{1}^{{\prime}}f_{2}^{{\prime}} + \frac{1} {4}f_{1}f_{2})\).
33.
The rest cancels.
34.
The rest cancels.
35.
\((1 +\xi \otimes \eta )^{-1} = 1 - (1+\xi \bullet \eta )^{-1}\xi \otimes \eta\).
36.
\(K\,K^{\dag } = e^{-\frac{1} {2} \vert x-y\vert }\).
37.
See [2] and/or [3].
38.
\(q(0) =\ln \mathop{ \mathrm{ch}}\nolimits (T\sqrt{H/2})\) is used; it comes from the second version of T.
39.
If you think of the two solitons as particles, you might prefer to say they “bounce off each other,” but all such descriptions are just a manner of speaking, not the thing itself.
40.
Change n to − n and use \(\lambda \int _{-\infty }^{\infty }mf \otimes f = 1\).
41.
m ≠ 0 on any interval, so the functions \(f_{n}: n \in \mathbb{Z} - 0\ \mathop{ \mathrm{span}}\nolimits H^{1}\); see Sect. 9.2.2.
42.
The spectrum is odd.
43.
\(q =\ln c^{2}\).
44.
[a, b] is the Wronskian \(a^{{\prime}}b - ab^{{\prime}}\).
45.
The scale \(\overline{x}\) must be adjusted by an additive \(-\ln (-\varLambda )\) before making \(\varLambda \uparrow 0\).
46.
\(\overline{x}^{{\prime}} = me^{x}/e^{\overline{x}}\).
47.
\(m/\overline{y}^{{\prime}} = -(e^{{\prime}2} -\frac{1} {4}e^{2})/\varLambda e^{2}\).
48.
McKean and Trubowitz [13] served as a model.
49.
\(\overline{x}^{{\prime}} =\vartheta ^{2}/\vartheta _{-}\vartheta _{+}\).
50.
(1 + cM)⁻¹ C is symmetric.
51.
See Holm and Staley [8] and Degasperis, Holm, and Hone [5].

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Courant Institute, 251 Mercer Street, New York, NY, 10012-1185, USA
Henry P. McKean Jr.

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Department of Mathematics, University of California, Berkeley, California, USA
F. Alberto Grünbaum
Institut de Recherche en Math_ematique at Physique, Department of Mathematics, Universit_e catholique de Louvain, Louvain-la-Neuve, Belgium, & Brandeis University, Waltham, Massachusetts, USA
Pierre van Moerbeke
Tulane University, New Orleans, Louisiana, USA
Victor H. Moll

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McKean, H.P. (2015). Fredholm Determinants and the Camassa-Holm Hierarchy. In: Grünbaum, F., van Moerbeke, P., Moll, V. (eds) Henry P. McKean Jr. Selecta. Contemporary Mathematicians. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-22237-0_9

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