D EFORMATIONS OF O RTHOGONALF UNCTIONS : P EAKON AND T ODAuvx1x2x3x4x5

X IANG -K E C HANG→xx6

u = m1 e|x−x 1 | + m3 e|x−x 3 | + m5 e|x−x 5 |v = n2 e|x−x 2 | + n4 e|x−x 4 | + n6 e|x−x 6 |LSEC, ICMSEC, AMSS, C HINESE A CADEMY OF S CIENCES
SIDE13, F UKUOKA , J APAN

M OTIVATIONC ONTRIBUTIONS

A class of nonlinear integrable PDEsadmit some special weak solutions called“peakons”, which are characterised by ODEsystems, namely peakon lattices. The celebrated Toda lattice was originally obtained
as a simple model for describing a chainof particles with nearest neighbor exponential interaction. For some initial value problems, these lattices can be explicitly solved
by use of inverse spectral method involving certain “orthogonality", approximation
problems and continued fractions.
Interestingly, it is implied in [1, 6] that
there exists certain intimate connection between the Camassa-Holm (CH) peakon lattice (i.e. the ODE system describing the CHpeakons) and the finite Toda lattice, which
are both indeed related to some deformations of ordinary orthogonal polynomials.

CH PEAKONS &T ODA &OP S
Consider a family of polynomials{Pn (x)} orthogonal with respect to themeasure ν(x)dx satisfying
xPn (x) = Pn+1 (x) + bn Pn (x) + un Pn−1 (x).

Known ResultPeakon latticete x µ(x; 0)Toda latticetx
e µ(x; 0)
Lax x-part“τ -structure"
Refu̇k = uk (bk − bk−1 ),▶ If ν(x; t)dx = e ν(x; 0)dx, then⇒CH peakon lattice:u(x; t) =Novikov

DPA-Toda

K-vM

B-TodaC-TodaOPsdet((ci+j ))[1, 6]

sOPsdet((ci+j ))
[2]

PSOPs
P f ((ci,j ))[3, 4]

CBOPsdet((ci,j ))[5]• Partial-skew-orthogonality:∫∫
τ2n+2
mx − yP2n (x)yν(x)ν(y)dxdy =δ2n+1,m ,x+y
τ2n
∫∫
τ2n+2
mx − yP2n+1 (x)y
ν(x)ν(y)dxdy = −βm , 0 ≤ m ≤ 2n + 1.
x+y
τ2n+1• Four-term recurrence:z(Pn + un Pn−1 ) = Pn+1 + (bn + un )Pn − un (bn + un+1 )Pn−1 − u2n un−1 Pn−2 .
tx▶ If ν(x; t)dx = e ν(x; 0)dx, then▶ If ν(x; t)dx = e ν(x; 0)dx, thenxt⇒ Novikov peakon lattice:⇒ B-Toda lattice:ẋk = u(xk )2 ,Nonlinear form:ṁk = −mk u(xk )⟨ux ⟩(xk ),whereu(x; t) =N∑

mk (t)e−|x−xk (t)|u̇k = uk (bk − bk−1 ),ḃk = uk+1 (bk+1 + bk ) − uk (bk + bk−1 ).
Bilinear form (Hirota&Iwao&Tsujimotok=1

are peakon solutions of Novikov eq.
mt + mx u2 + 3muux = 0, m = u − uxx .

2001)τ̈k τk − (τ̇k ) = τ̇k−1 τk+1 − τ̇k+1 τk−1 .
2ṁk = −mk ⟨ux ⟩(xk ),ẋk = u(xk ),where2mCH

Partial-skew-orthogonal polynomials {Pn (x)}:ḃk = uk+1 − uk .
tx

CHN OVIKOV PEAKON &B-T ODA &PSOP S▶ If ν(x; t)dx = ext ν(x; 0)dx, then⇒Toda lattice:Our WorkN∑

mk (t)e−|x−xk (t)|k=1

are peakon solutions of CH eq.

DP PEAKON &C-T ODA &CBOP S
Cauchy biorthogonal polynomials {Φn (x)}:• Biorthogonality:∫∫mt + umx + 2mux = 0, m = u − uxx .
• Four-term recurrence:Φk (x)Φl (y)ν(x)ν(y)dxdy = hk δk,l .
x+yz (Ak Φk−1 (z) + Φk (z)) = Φk+1 (z) + Bk Φk (z) + Ck Φk−1 (z) + Dk Φk−2 (z)R EFERENCES[1] R. Beals, D. H. Sattinger, and J. Szmigielski.
Peakons, strings, and the finite Toda lattice. Commun. Pure Appl. Math., 54 (2001), 91–106.
[2] X. Chang, X. Hu, and J. Szmigielski. Multipeakons
of a two-component modified Camassa-Holm equation and the relation with the finite Kac-van Moerbeke lattice. Adv. Math., 299 (2016),1–35.
[3] X. Chang, X. Hu, S. Li, and J. Zhao. Application ofPfaffian in multipeakons of the Novikov equationand the finite Toda lattice of BKP type. Adv. Math.
338:1077–1118, 2018.
[4] X. Chang, Y. He, X. Hu, and S. Li. Partial-skeworthogonal polynomials and related integrable lattices with Pfaffian tau-functions. Commun. Math.
Phys. https://doi.org/10.1007/s00220-018-3273-y.
[5] X. Chang, X. Hu, and S. Li. Degasperis-Procesi
peakon dynamical system and finite Toda lattice ofCKP type. Nonlinearity 31:4746–4775, 2018.
[6] O. Ragnisco and M. M. Bruschi. Peakons, r-matrix
and Toda lattice. Physica A, 228:150–159, 1996.

with√

Ak =

√bk uk1, Bk = bk +bk−1
2

bk uk
1√
bk uk bk−1 , Dk = −uk−1, Ck = −uk −bk−1
2tx▶ If ν(x; t)dx = e ν(x; 0)dx, then⇒ DP peakon lattice:ẋk = u(xk ),whereu(x; t) =mk (t)ebk u k.
bk−1

▶ If ν(x; t)dx = ext ν(x; 0)dx, then
⇒ C-Toda lattice:ṁk = −2mk ⟨ux ⟩(xk ),N∑√−|x−xk (t)|k=1

are peakon solutions of DP eq.
mt + umx + 3mux = 0, m = u − uxx .

Nonlinear form:u̇k = uk (bk − bk−1 ),√√ḃk = 2( uk bk bk−1 − uk+1 bk+1 bk ).
Bilinear form:2τ̇k τk−1 − τk τ̇k−1 = (σk ) ,2τ̈k τk − (τ̇k ) = −2σk+1 σk .