Bäcklund Transformations and Darboux
Integrability for Nonlinear Wave Equations
J. Clelland1
1 University
T. Ivey2
of Colorado, Boulder
2 College
of Charleston
May 2008
Outline
A Motivating Example
Definitions & Results
The method of Darboux
Bäcklund transformations
Main result
‘Backlund implies Darboux’
G-structure for BTs between MA equations
Implications of a BT to the wave equation
‘Darboux implies Bäcklund’
G-structure for Darboux-integrable PDE
EDS for BT to the wave equation
Solving for a BT for particular equations
Open Problems
Two ways to solve Liouville’s equation uxy = eu
1. Use Darboux integrability: For any solution,
uxx − 12 (ux )2 = f (x),
uyy − 12 (uy )2 = g(y)
for some functions f (x), g(y); conversely, given f , g, can solve these
Riccati ODEs for ux , uy , and integrate to get u.
2. Use a Bäcklund transformation (BT): Suppose u(x, y) and z(x, y)
satisfy the system
ux − zx = 2 exp((u + z)/2)
uy + zy = exp((u − z)/2);
then u satisfies Liouville iff z satisfies zxy = 0 (wave eqn.).
–set z = F(x) + G(y) and solve compatible ODEs for u in x- and
y-directions
–both explicitly determine u by solving ODE, given 2 functions
–what’s the relationship between these methods?
Darboux Integrability
Defn (Bryant-Griffiths-Hsu) A hyperbolic EDS of class k on M k+4 is
locally generated by 1-forms η0 , . . . , ηk−1 and two independent
decomposable 2-forms Ω1 , Ω2 (whose factors, together with the η’s,
form a coframe on M).
–special case for k = 1: a hyperbolic Monge-Ampère (MA) system on
M 5 , locally generated by Ω1 , Ω2 and a contact form η
–prolongation of class k hyperbolic EDS is hyperbolic of class k + 2
–define characteristic system Ki , i = 1, 2, spanned by the η’s and the
factors of Ωi
Defn A hyperbolic EDS is Darboux integrable if K1 and K2 both
contain Frobenius systems of rank 2, transverse to the span of the η’s,
and Darboux semi-integrable if only one of the Ki has this property.
–for k = 1, Monge-integrable means semi-integrable
–the only Darboux integrable hyperbolic MA system is the wave
equation (up to contact equiv.)
Darboux Example: Liouville’s equation uxy = eu
–use Monge notation p = ux , q = uy , r = uxx , s = uxy , t = uyy
–as a MA system on R5 , generators are
η = du−p dx−q dy,
Ω1 = (dp−eu dy)∧dx,
Ω2 = (dq−eu dx)∧dy
–1st prolongation on R7 generated by η0 = η and
η1 = dp−r dx−eu dy, η2 = dq−eu dx−t dy, Ω1 = (dr−peu dy)∧dx, Ω2 = (dt−qeu dx)∧dy.
–characteristic systems of the prolongation:
K1 = {η0 , η1 , η2 , dr − peu dy, dx} ⊃ {d(r − 21 p2 ), dx}
K2 = {η0 , η1 , η2 , dt − qeu dx, dy} ⊃ {d(t − 12 q2 ), dy}
–any integral surface is foliated by integrals of K1 and K2
–so, any solution has r − 21 p2 = f (x) and t − 21 q2 = g(y) for some f , g
–there are lots more examples; what about classification?
Classification of Darboux-Integrable MA Equations
Goursat (1898) classified the nonlinear MA equations of the form s = f (x, y, u, p, q) that are
Darboux-integrable at 2-jet level (i.e., after one prolongation), up to contact transformations
preserving the form:
(†means ‘Monge-integrable’)
√
(I)
(x + y)s = 2 pq,
p
p
2
2
us = 1 + p 1 + q ,
(II)
p
p
2
2
(sin u)s = 1 + p 1 + q ,
(III)
u s = ±φ(p)ψ(q),
(x + y)s = α(p)α(q),
where φ(x), ψ(x) satisfy df /dx ± f /x = K 6= 0,
where α satisfies α(x) − 1 = exp(x − α(x)),
p − u s/q = f (x, s/q),
p
s = eu 1 + p2 ,
p − y s = f (x, s),
s = eu ,
(IV)
(V)
(VI†)
(VII)
(VIII†)
(Liouville’s equation)
s = peu ,
“
”
s = (u − x)−1 + (u − y)−1 pq.
(IX)
(X†)
(XI)
Vessiot (1939-42) re-derived Goursat’s classification using Lie theory, showing that (VI) and
(VIII) are contact-equivalent to respectively (X) and
s = p/(x + y).
(VIII*)
Classification, continued
Vessiot also classified linear Darboux-integrable equations, obtaining
s = a(x, y)p + b(x, y)q − a(x, y)b(x, y)u,
(XII)
where h(x, y) = −ax and k(x, y) = −by must satisfy
(ln h)xy = 2h − k, (ln k)xy = 2k − h with h 6= k, and
s = 2u/(x + y)2 .
(XIII)
–Juráš (2000) showed that any hyperbolic MA eqn that’s Darboux
integrable at the 2-jet level must be contact-equivalent to the form
s = f (x, y, u, p, q), hence on the Goursat-Vessiot list
–Biesecker (2004) re-derived the Goursat-Vessiot list using method of
equivalence, characterizing each equation in terms of Laplace
invariants
Bäcklund transformations
Defn (Goursat, 1925) A Bäcklund transformation between two
hyperbolic MA systems, on M and M, is defined by a submfld
B6 ⊂ M × M such that:
B
I projections π : B → M and π : B → M are
π S π
submersions
SS
/
w
(M, I)
I pullbacks of generator 2-forms satisfy
(M, I)
η, Ω1 , Ω2
η, Ω1 , Ω2
{Ω1 , Ω2 } ≡ {Ω1 , Ω2 } ≡ {dη, dη} mod η, η
–corresponds to Goursat’s type B3 : if Σ2 ⊂ M is a integral of I, then
π ∗ I is Frobenius on π −1 (Σ), giving a 1-parameter family of integrals
for I, and vice-versa
–e.g., for Liouville’s equation on M and the wave equation on M (with
coords. X,Y,Z,P,Q), B6 ⊂ M × M is cut out by the equations
X = x,
Y = y,
p−P = 2 exp((u+Z)/2),
q+Q = − exp((u−Z)/2)
Constructing Bäcklund transformations
Remark Gardner (1978) shows how, for any class 3 hyperbolic
system, to construct BT’s of type B1 to some other PDE; e.g., from
s = eu (Liouville) to S = QZ, by
X = x,
Y = y,
Z = p,
Q = eu , P = r
Limit our attention to transformations of type B3 : for example,
Suppose Q4 carries a hyperbolic EDS J of class 0 (locally, a
quasilinear hyperbolic system of PDE for 2 fns. of 2 vars.), generated
by Ω1 , Ω2
–suppose M, M carry rank 1 integrable extensions of J:
B
–i.e., dη ≡ 0 mod η, ρ∗ Ω1 , ρ∗ Ω2 , & same for η
@
R
@
M
M
@
ρ @
R ? ρ
Q
–let B = {(m, m) ∈ M × M|ρ(m) = ρ(m)}
–such BTs are called holonomic; less interesting
Our main result & its inspiration
–an result in Goursat (credited to Darboux) says that a 2nd-order PDE
in the plane can be solved in terms of two arbitrary functions (and
finitely many derivatives) by integrating ODE iff the given PDE is
Darboux-integrable at some finite k-jet level
–Zvyagin (1991) published a classification of BTs between hyperbolic
MA eqns. and the wave equation, omitting holonomics (and proof!)
–obtained 6 examples besides Liouville’s eqn., without giving the
PDE in most cases (can identify 4 of them as (I), (II), (III) and (VII) )
Theorem (Clelland-I–) A hyperbolic MA equation has a BT to the
wave equation if and only if it is Darboux-integrable at the 2-jet level.
–local in the ‘Darboux implies Bäcklund’ direction
–proof is independent of Goursat-Vessiot and Zvyagin classifications
G-structure for Bäcklund transformations
Clelland (2002): assume I1 , I2 are hyperbolic MA systems on
M1 , M2 , linked by a BT; then locally on B6 ∃ a coframe θ1 ,θ2 ,
ω1 ,ω2 ,ω3 ,ω4 such that π1∗ η1 = θ1 and π2∗ η2 = θ2 up to multiple, and
dθ1 ≡ A1 ω1 ∧ ω2 + ω3 ∧ ω4
mod θ1 ,
A1 6= 0,
dθ2 ≡
mod θ2 ,
A2 6= 0,
ω1 ∧ ω2 + A2 ω3 ∧ ω4
dω1 ≡
B1 θ1 ∧ θ2 + C1 ω3 ∧ ω4 mod ω1 , ω2 ,
dω2 ≡
B2 θ1 ∧ θ2 + C2 ω3 ∧ ω4 mod ω1 , ω2 ,
dω3 ≡
B3 θ1 ∧ θ2 + C3 ω1 ∧ ω2 mod ω3 , ω4 ,
A1 A2 6= 1,
dω4 ≡ B4 θ1 ∧ θ2 + C4 ω1 ∧ ω2 mod ω3 , ω4 ,
–unique up to action of G = GL(2) × GL(2)
B
B
C
C
–invariants of the BT are A1 , A2 & vectors 1 , 3 , 1 , 3
B2
B4
C2
C4
–holonomic BTs have B1 = B2 = B3 = B4 = 0
–if C1 = C2 = 0 or C3 = C4 = 0, then both I1 and I2 encode the
wave eqn.
“Bäcklund implies Darboux”: sketch of proof
–assume M1 carries a MA system I1 contact-equivalent to the wave
equation; then
{ω1 − C1 θ1 , ω2 − C2 θ1 }
and
{ω3 − (C3 /A1 )θ1 , ω4 − (C4 /A1 )θ1 }
are Frobenius systems (i.e., spanned by exact 1-forms)
–can set A1 = 1, and assume that C2 6= 0, C4 6= 0; the above implies
d(C1 /C2 ) ∈ J1 = {θ1 , θ2 , ω1 , ω2 },
d(C3 /C4 ) ∈ J2 = {θ1 , θ2 , ω3 , ω4 }.
Case A: Both J1 , J2 contain rank 3 Frobenius systems; then
{θ2 , ω1 , ω2 } and {θ2 , ω3 , ω4 } contain rank 2 Frob. systems, and
(M2 , I2 ) encodes the wave equation as well.
Case B: J2 contains a rank 3 Frob. system but J1 doesn’t; again,
{θ2 , ω3 , ω4 } contains rank 2 Frob. system, so I2 is Monge-integrable.
–need to check that the prolongation of I2 is Darboux-integrable
“Bäcklund implies Darboux”
Checking Darboux-integrability for Case B
Can set C1 = 0; then θ2 , ω1 live on M2 but ω2 doesn’t.
Fake Argument: Pretend that EDS I2 on M2 is generated by θ2 ,
ω1 ∧ ω2 and ω3 ∧ ω4
–on any integral surface with ω1 6= 0, ∃ a function r such that
ω2 − rω1 = 0
–define (partial) prolongation of I2 , on M2 × R, generated by θ2 and
ψ = ω2 − rω1 (where r is new coord.) & derivs.
–to check for Darboux-integrability, find π = dr + . . . such that
dψ ≡ ω1 ∧ π mod θ2 , ψ
and see if K = {θ2 , ψ, π, ω1 } contains a rank 2 Frob. system
Real Argument: Carry out the above, but replace ω2 by modified form
ω
f2 = ea (ω2 − bω1 ) that is well-defined on M2 .
“Bäcklund implies Darboux”
Case C: Both J1 , J2 contain only rank 2 Frob. systems
–set C1 = C3 = 0; then θ2 , ω1 , ω3 well-defined on M2
–using modified forms, define (full) prolongation of I2 on M2 × R2
generated by
ψ1 = ω
f2 − rω1 ,
ψ2 = ω
f4 − tω3
–find π1 , π2 such that
dψ1 ≡ ω1 ∧ π1 ,
dψ2 ≡ ω3 ∧ π2
mod θ2 , ψ1 , ψ2
and check that the characteristic systems
K1 = {θ2 , ψ1 , π1 , ω1 },
K2 = {θ2 , ψ2 , π2 , ω3 }
each contain a rank 2 Frobenius system.
“Darboux implies Bäcklund”: sketch of proof
Let I be a hyperbolic MA system on M 5 , and Î its (full) prolongation
on N 7 ; assume Î is Darboux-integrable.
Locally on N, ∃ a coframe (θ0 , θ1 , θ2 , ω1 , π1 , ω2 , π2 ) such that
I
pullback under ρ : N → M of the contact form is θ0 , up to mult.
I
Î = hθ0 , θ1 , θ2 idiff , and
I
dθ0 ≡ ω1 ∧ θ1 + ω2 ∧ θ2
mod θ0
dθ1 ≡ ω1 ∧ π1
mod θ0 , θ1
dθ2 ≡ ω2 ∧ π2
mod θ0 , θ2
characteristic systems K1 , K2 of Î contain Frobenius systems
(∞)
K1
= {ω1 , π1 },
(∞)
K2
= {ω2 , π2 }
Main Case: Assume I is not Monge-integrable; then char. systems of
I each contain unique (up to mult.) integrable 1-forms.
–adapt ω1 , ω2 so that dω1 ≡ 0 mod ω1 , dω2 ≡ 0 mod ω2
–set of all such coframes gives a G-structure on N with 5-d. fibers
For the G-str. there are connection forms α1 , α2 , β1 , β2 , γ such that
3
2
θ0
γ
6 θ1 7
60
6 7
6
6 θ2 7
60
6 7
6
d 6ω1 7 = − 6 0
6 7
6
6π1 7
60
4ω 5
40
2
π2
0
2
0
γ + α1
0
0
0
0
0
0
0
γ + α2
0
0
0
0
0
0
0
−α1
β1
0
0
3 2 3
0
θ0
0
7 6 θ1 7
7 6 7
0
7 6 θ2 7
7 6 7
0
7 ∧ 6ω1 7
7 6 7
0
7 6 π1 7
5 4ω 5
0
2
−γ − 2α2
π2
3
ω1 ∧ θ1 + ω2 ∧ θ2
6ω1 ∧ π1 + θ0 ∧ (A2 θ2 + B2 ω2 )7
6
7
6ω2 ∧ π2 + θ0 ∧ (A1 θ1 + B1 ω1 )7
6
7
0
+6
7
6
7
C1 π1 ∧ θ1
6
7
4
5
0
C2 π2 ∧ θ2
0
0
0
0
−γ − 2α1
0
0
2
0
0
0
0
0
−α2
β2
–choose ω1 , ω2 to be exact, giving α1 = D1 ω1 , α2 = D2 ω2
–adjust π1 , π2 so that
dπ1 = (γ − C1 θ1 + E1 ω1 ) ∧ π1 ,
dπ2 = (γ − C2 θ2 + E2 ω2 ) ∧ π2
A BT to the wave equation ZXY = 0 would take the form of a bundle
B6 over M, with Z as fiber coordinate:
N7
ρ@
B6
π
R @
M5
@
R
@
R5
On the pullback of B to N, the contact form for the wave equation
would look like
Θ = dZ − P ω1 − Q ω2 ,
where P, Q are functions on B, such that
dP ≡ P1 θ1 mod θ0 , ω1 , Θ
dQ ≡ Q1 θ2 mod θ0 , ω2 , Θ,
P1 , Q1 6= 0.
–set up EDS on for functions P, Q on N × R (with Z as new coord.)
–torsion absorbtion requires computing identities among second
derivs. of Ai , Bi , Ci , Di , Ei !!
–involutive at first prolongation, with s1 = 2
Monge-integrable case: similar argument; BTs depend on s2 = 1
Alternate Approach: Solving for a Bäcklund transformation
For specific equations on the Goursat-Vessiot list, it’s possible to
solve PDE to get an explicit BT to the wave equation.
Example: Assume the BT for Liouville’s equation s = eu takes the
form p = f (u, x, y, Z, P, Q), q = g(u, x, y, Z, P, Q). Then
(dp − eu dy) ∧ dx ≡ ((fy + qfu + QfZ − eu )dy + fP dP + fQ dQ) ∧ dx
mod contact forms, so that fQ = 0, fy = eu − gfu − QfZ ;
similarly gP = 0 and gx = eu − fgu − PgZ .
Other PDEs arise from requiring that
r − 12 p2 = fx + ffu + PfZ + RfP − 21 f 2
t−
1 2
2q
= gy + ggu + QgZ + TgQ −
1 2
2g
= a function of x, P, R only
= a function of y, Q, T only,
implying that
fPu = fPZ = 0,
∂Z , ∂u , ∂y (fx + ffu + PfZ − 12 f 2 ) = 0,
gQu = gQZ = 0,
∂Z , ∂u , ∂x (gy + ggu + QgZ − 12 g2 ) = 0.
Solving for BT’s: Example, continued
Differentiating and equating mixed partials yields more equations:
fZ = −gQ fu ,
gZ = −fP gu ,
fPx = 0,
gQy = 0,
then
2gu fu = eu ,
gQ = −fP = constant.
–system is now involutive, again with s1 = 2
–setting fP = 1, solve these equations to get the BT
p = P + 2 exp((u + z + v(x) + w(y))/2) + v0 (x)
q = −Q + exp((u − z − v(x) − w(y))/2) − w0 (y)
where v(x) and w(y) are arbitrary functions
–using this technique shows BTs to the wave equation exist for (IV),
(V), (IX), (XI), (XII) and (XIII); can obtain some explicit examples
–for (XI), (XII) and (XIII) all such BTs are holonomic, with s1 = 2
–technique not feasible for Monge-integrable (VI), (VIII), (X)
Open Problems
I
Relate these Bäcklund transformations to nonlinear
superposition formulas (Anderson, Fels, Vassiliou)
I
Identify remaining Zvyagin examples using Biesecker’s
characterizations
I
Characterize special types of BTs: quasilinear, variational, and
especially auto-Bäcklund transformations
I
Completely integrable PDE (‘soliton’ equations) often have BTs
containing an arbitrary parameter, e.g., for sine-Gordon
λ
sin((u + v)/2)
2
λ 6= 0.
1
vy = −uy −
sin((u − v)/2).
2λ
–characterize those BTs where a parameter can be inserted (e.g.,
by lifting a symmetry from one side).
vx = ux +