Integrable discretisation of hodograph-type systems,
Abelian integrals and Whitham equations
byW.K. Schief
The University of New South Wales, Sydney[with B.G. Konopelchenko, Proc. R. Soc. London A (2014)]

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1. Systems of hydrodynamic type
Diagonal systems of hydrodynamic type:
uit = λi(u)uix,

i = 1, . . . , N + 1

λi: characteristic speeds

History.
• Dubrovin & Novikov ’83 – ’89: Theory of (more general) Hamiltonian quasi-linearsystems of hydrodynamic type
• Various authors: Relation to classical differential geometry• Tsarev ’91: The geometry of semi-Hamiltonian systems of hydrodynamic type. Generalised hodograph method

2. One-phase Whitham (averaged KdV) equations (1974)
1 (u1 , u2 , u3 )u1
u1=
λxt
2 (u1 , u2 , u3 )u2=
λu2xt
3 (u1 , u2 , u3 )u3 ,u3
=
λxt

where
1 + u2 + u3uK(s)λ1 =− (u2 − u1),2E(s)

u3 − u22s = 3u − u1

1 + u2 + u32 )K(s)u(1−sλ2 =+ (u3 − u2)
2E(s) − (1 − s2)K(s)u1 + u2 + u3
3− (u3 − u2)
λ =
2K(s)
,E(s) − K(s)where K and E are complete elliptic integrals of the first and second kind.

3. The geometric setup
Classical conjugate net equations:
ψuiuk = Aik ψui + Akiψuk ,i 6= k ∈ {1, . . . , N + 1}

Compatibility conditions (Darboux system):
Aikul = Akl Aik + Alk Ail − Aik AilCombescure transform ψ?:
ψ?ui = λiψui

Compatibility conditions (adjoint linear system):λiuk = Aik (λk − λi)

(ALS)

(DS)

4. Semi-Hamiltonian systems (Tsarev ’90)
Theorem. The system of diagonal hydrodynamic-type equations
uit1 = λi1(u)uix(SH)admits compatible flows
uit2 = λi2(u)uix,

uit3 = λi3(u)uix,...

iff the sets {λiα}, α = 1, 2, . . . constitute adjoint eigenfunctions, that is, obey (ALS)
for an underlying Darboux system.
Theorem (Tsarev ’90). The semi-Hamiltonian system (SH) may be linearized/implicitlysolved via
ω i(u) = x + λi1(u)t1,i = 1, . . . , N + 1,

where {ω i} is the general solution of (ALS). [Generalised hodograph method.]

5. Characteristic speeds
Fact. Special important characteristic speeds λi arise as• limits of PDEs without dispersion or dissipation• Whitham equations for slow modulations [Flaschka, Forest & McLaughlin ’80]

These are expressed in terms of (hyper)elliptic (or even general Abelian) integrals.
Q: Where do the hyperelliptic integrals come from?
A: Euler-Poisson-Darboux systems (one explanation)[Kudashev & Sharapov ’91, Gurevich, Krylov & El ’91, Tian ’94]

6. The Euler-Poisson-Darboux system
Consider the particular class of compatible hyperbolic equations
1k φ − i φ )φxixk = i(xixkx − xk(EPD)

The hyperelliptic integrals
ζkIqdζ,Q2g+1ibκ
i=1 (ζ − x )cycles bκ, κ = 1, . . . , g
k = 0, . . . , g − 1

are superpositions of separable solutions of (EPD) for i = 21 .
Example. Genus g=1. Then, the above elliptic integral is essentially given by
φO =2qπ x3 − x1v
u2u 3t x − x K,
31x −xx1 < x2 < x3

7. Generation of multi-phase flows
Fact [Kudashev ’91, Gurevich, Krylov & El ’91]. The characteristic speeds λ1, λ2, λ3
for the one-phase Whitham equations may be formulated (with ui = xi) asφO 0i0λ = φ − O φxi ,φxi

i = 1, 2, 3,1 1(x + x2 + x3)
2
constitutes the simplest non-constant solution of (EPD).

where

φ0 =Key observation. The right-hand side of λi is precisely of the form of the classical Levy
transform (Darboux-type transform) for conjugate nets with respect to xi generated
by φO and applied to φ0.
Theorem [Kodama, BGK & WKS ’15]. The characteristic speeds of multi-phase KdV
type may be generated by non-standard (!) applications of g Levy transformations.
The solution of the associated semi-Hamiltonian system may be obtained in the same
purely algebraic manner. [Cf. Ferapontov ’00 for the “standard” application]

8. Discretisation
Remark. The fact that the above method extends to the semi-Hamiltonian system isdue to the interpretation of both x and t in the context of the generalised hodographmethod as eigenfunctions in the sense of soliton theory.
Plan. Use standard discretisation of• conjugate net equations• adjoint linear systems• Darboux-type transformations (but non-standard application!)

to discretise the continuous theory.

9. The generalised hodograph transformation
Key idea. Consider commuting flows (nonlinear)uitα = λiα(u)uix,

i = 1, . . . , N + 1,

α = 1, . . . , N

and observe thatNX

dui ∧ dx +λiα(u) dui ∧ dtα = 0.

α=1This is equivalent to the linear (!) system of hodograph typexuk +

N
X
α=1

λiα(u)tα= 0,uki 6= kwhich, in turn, may be solved in terms of eigenfunctions (as indicated before).

10. Discrete hodograph equations (BGK & WKS ’14)
Discrete analogue:∆k x +NX

λiα[k](n)∆k tα = 0,

∆k f = f[k] − f,α=1

where the quantities λiα constitute adjoint eigenfunctions of an underlying standard
discrete Darboux system, that is,∆k λiα = Aik (λkα[i] − λiα[k]),Aik[l] = · · ·Interpretation. Integrable discretisation of (in)dependent variables ui, depending on
the point of view.
Main theorem [BGK & WKS ’14]. A canonical discretisation of the characteristic
speeds of multi-phase KdV type may be generated by non-standard (!) applications ofg discrete Levy transformations. The solution of the discrete generalised hodographsystem may be obtained in the same purely algebraic manner.

11. Discrete Euler-Poisson-Darboux system (BGK & WKS ’14)
Q: How does one define the discrete ingredients of the above theorem?
Compatible discrete Euler-Poisson-Darboux system:δ k k ∆iφ − δ ii∆k φ∆ik φ = i i
δ (n + ν i) − δ k (nk + ν k )ν i: give rise to “location” of discretisation points, δ i: lattice parameters

limit: xi = δ i(ni + ν i),

δ i → 0,i = 1, . . . , 2g − 1

Separable solutions:φsep =2g+1Y

ρi(ni; ζ),−iiiρ = (δ )i=1

Continuum limit: ρi → (ζ − xi)−

ias

Γ(ξ i − ni − ν i + 1),iiii
Γ(ξ − n − ν +  + 1)
δi → 0ζξi = iδ

12. Formal structure of “discrete” hyperelliptic integrals
As in the continuous case, we now make the choice i = 12 , δ i = δ .
Q: By analogy, we should consider contour integrals of the formIbκ

ζk2g+1Yρi(ζ) dζi=1

but what does the above mean? Importantly, we also require that
∆iIbκ

=Ibκ

∆i !

Denote the separable solution by
ϕ=2g+1Y
i=1p(ξ, ni, ν i),Γ(ξ − n − ν + 1),p(ξ, n, ν) = √3δ Γ(ξ − n − ν + 2 )with n1 < n2 < · · · < n2g < n2g+1.

ζ
ξ= ,
δ

13. Discretisation points
Key choice:12kν = ,2

ν 2k+1 = 0

corresponding to the discretisation points x2k = δ(n2k + 21 ) and x2k+1 = δn2k+1.
Result.
ggY)Γ(ξ − n2k+1 + 1) Y Γ(ξ − n2k + 12ϕ=.
1
32k2k+1+ 2 ) k=1 Γ(ξ − n + 1)
δ g+ 2 k=0 Γ(ξ − n1

Example. Genus g = 1:3 + 1))
1 Γ(ξ − n1 + 1) Γ(ξ − n2 + 1Γ(ξ
−n2
ϕ= 3.
33213
Γ(ξ
−n+1)Γ(ξ − n + 2 )δ 2 Γ(ξ − n + 2 )14. Cycles
Fact. The Gamma function is non-zero but has simple poles at non-positive integers.
Conclusion. Due to cancellation, ϕ(ξ) has no zeros or poles on
g+1[(n2k−1 − 1, n2k ),n2g+2 = ∞.

k=1

This guarantees the necessary property
∆iIbκ

=Ibκ

∆iin the following definitions.
Definition. The g cycles bκ (on the ξ -plane) are closed paths of counterclockwise
orientation passing through the pairs of intervals (n2κ−1 − 1, n2κ) and (n2g+1 −1, ∞) for κ = 1, . . . , g .

15. “Discrete” hyperelliptic integrals
Example. Genus g = 2:

Blue: Branch cuts in Riemann surface theory
Definition. “Discrete” hyperelliptic integrals are (well-)defined byIbκ

ζ k ϕ(ξ) dζ,

ζ
ξ= ,
δ

κ = 1, . . . , g,

k = 0, . . . , g − 1

and constitute “discrete” analogues of the hyperelliptic integrals
ζkIbκ

qdζ
Q2g+1ii=1 (ζ − x )16. Explicit form
Remark. Discrete hyperelliptic integrals may be evaluated explicitly in terms of theresidues of the meromorphic integrand ϕ.
Example. Genus g = 1:3nX−11res(ϕ(ξ), ξ = k)φ∆ =ϕ(ξ) dζ = δ2πi b12I

k=nso that
3nX−11 + 1) Γ(k − n2 + 1 ) res(Γ(η), η = k − n3 + 1)
1
Γ(k−n2φ∆ = √.
33213
δ k=n2 Γ(k − n + 2 ) Γ(k − n + 1)
Γ(k − n + 2 )17. Comparison

Comparison with the elliptic integral
φO =2qπ x3 − x1v
u2u 3t x − x K
31x −xevaluated at the points1)
x2 = δ(n2 + 2x3 = δn3x1 = δn1,Remark. The relation between the discrete and continuous elliptic integrals isunique in the sense that, as a functionof n, the ratio φ∆/φO is independentof the lattice parameter δ .

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18. Extension to even genus and Abelian integrals
Concluding remarks. The preceding may be adapted to the case of• hyperelliptic integrals associated with an even number 2g + 2 of branch points.
These hyperelliptic integrals arise, for instance, in connection with the multi-phaseaveraged nonlinear Schrödinger (NLS) equations.
• Abelian integrals such as those associated with the trigonal curvey3 =6Y(x − xk )k=1

which appear in the context of the Benney equations.

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