On the smooth periodic traveling waves of the Camassa-Holm equation - - PowerPoint PPT Presentation

On the smooth periodic traveling waves of the Camassa-Holm equation

Anna Geyer a and Jordi Villadelprat b

a Universitat Autonòma de Barcelona, Spain. b Universitat Rovira i Virgili, Tarragona, Spain.

Workshop on non-local dispersive equations NTNU Trondheim, Norway September 2015

The Camassa-Holm equation1 ut + 2k ux − utxx + 3 u ux = 2 uxuxx + u uxxx (CH) arises as shallow water approximations of the Euler equations. It may be written in non-local form as ut + uux + (1 − ∂2

x)−1∂x

• 2ku + u2 + 1

2u2

x

• = 0.
• 1R. Camassa and D. Holm. An integrable shallow water equation with

peaked solitons. Phys. Rev. Lett. (71) 1993.

Camassa-Holm equation ut + 2k ux − utxx + 3 u ux = 2 uxuxx + u uxxx (CH) Traveling Wave Solutions2 u(x, t) = ϕ(x − c t): ϕ′′(ϕ − c) + (ϕ′)2 2 + r + (c − 2k) ϕ − 3 2ϕ2 = 0, (1) where c is the wave speed and r ∈ R is an integration constant. We will concentrate on smooth periodic TWS. λ . . . wave length a . . . wave height

• 2J. Lenells. Traveling wave solutions of the Camassa-Holm equation. J.
• Differ. Equ. (217) 2005.

Proposition (Waves ← → Orbits)

◮ ϕ is a smooth periodic solution of (1) if and only if

(w, v) = (ϕ − c, ϕ′) is a periodic orbit of the planar system    w′ = v, v′ = −A′(w) + 1

2 v2

w , (2) where A(w):= αw + βw2 − 1

2w3. ◮ System (2) has the first integral

H(w, v):= wv2 2 + A(w).

◮ Every periodic orbit of (2) belongs to the period annulus P

• f a center, which exists if and only if −2β2 < 3α < 0.

Observations: periodic solution ϕ ← → periodic orbit γϕ wave lenght λ of ϕ = period T of γϕ wave height a of ϕ = ℓ(hϕ), where ℓ is an analytic diffeo with ℓ(h0) = 0. {ϕa}a∈(0,aM) ← → {γh}h∈(h0,h1) Consequence: λ(a) is a well-defined function λ : (0, aM) − → R+ λ(a) = wave length of ϕ T : (h0, h1) → R+ T(h) = period of γϕ Deduce qualitative properties of the function λ from those of the period function T. Result: λ(a) is either unimodal or monotonous.

Theorem (A.G. & J. Villadelprat, 2015)

Given c, k, c = −k, there exist real numbers r1 < rb1 < rb2 < r2 such that the Camassa-Holm equation

ut + 2k ux − utxx + 3 u ux = 2 uxuxx + u uxxx (CH)

has smooth periodic TWS ϕ(x − c t) satisfying

ϕ′′(ϕ − c) + (ϕ′)2 2 + r + (c − 2k) ϕ − 3 2ϕ2 = 0, if and only if the integration constant r ∈ (r1, r2). The set of smooth periodic TWS form a continous family {ϕa}a parametrized by the wave height a. The wave length λ = λ(a) of ϕa satisfies the following:

◮ If r ∈ (r1, rb1], then λ(a) is monotonous increasing. ◮ If r ∈ (rb1, rb2), then λ(a) has a unique critical point (maximum). ◮ If r ∈ [rb2, r2), then λ(a) is monotonous decreasing.

Criteria to bound the number of critical periods of a center for planar systems which have first integrals of a certain type:

◮ potential systems:

H(x, y) = V(x) + 1 2y2

[F. Mañosas, J. Villadelprat, JDE (2009)]

◮ systems with quadratic-like centers:

H(x, y) = A(x) + B(x)y + C(x)y2

[A. Garijo, J. Villadelprat, JDE (2014)]

Consider an analytic differential system ˙ x = p(x, y), ˙ y = q(x, y) satisfying these hypotheses: (H) The system has a center at the origin, an analytic first integral of the form H(x, y) = A(x) + B(x)y + C(x)y2 with A(0) = 0, and its integrating factor K depends only on x. The function V := 4AC−B2

4|C|

defines an involution σ which satisfies V ◦ σ = V. For any analytic function f one can define its σ-balance as Bσ(f)(x):= f(x) − f(σ(x)) 2 .

Recall the σ-balance Bσ(f)(x) = f(x)−f(σ(x))

2

• f the involution σ

satisfying V ◦ σ = V for the function V := 4AC−B2

4|C|

• n (xℓ, xr).

Theorem ([GaVi14]3, THM A (b))

Under hypotheses (H) let µ0 = −1 and define for i 1 µi :=

• 1

2 + 1 2i−3

• µi−1 +

√

|C|V (2i−3)K

• Kµi−1

√

|C|V ′

′ and ℓi :=

K

√

|C|V ′ µi

If the number of zeros of Bσ(ℓi) on (0, xr), counted with multiplicities, is n 0 and it holds that i > n, then the number of critical periods of the center at the origin, counted with multiplicities, is at most n.

• 3A. Garijo and J. Villadelprat, Algebraic and analytical tools for the study of

the period function, J. Differential Equations 257 (2014) 2464–2484.

The system    w′ = v, v′ = −A′(w) + 1

2 v2

w , (2) has the first integral H(w, v):= A(w) + w 2 v2. To apply the criterion of [GaVi14], our system has to satisfy the following hypotheses: (H) The system has a center at the origin, an analytic first integral of the form H(x, y) = A(x) + B(x)y + C(x)y2 with A(0) = 0, and its integrating factor K depends only on x.

Lemma (Move center to origin)

Let α and β satisfy −2β2 < 3α < 0. Then the transformation

• x = w−wc

2β √ ∆, y = v 2β √ ∆

• with ∆:= 4 + 6α

β2 , brings system (2) to

   x′ = y, y′ = −x − 3x2 + y2 2(x + ϑ) , (3) where ϑ:= 1

6

• 2

√ ∆ − 1

• is positive.

System (3) is analytic for x = −ϑ and satisfies hypotheses (H) with A(x) = 1

2x2 − x3, B(x) = 0, C(x) = x + ϑ, K(x) = 2(x + ϑ).

Period Annuli

   x′ = y, y′ = −A′(x) + y2 2(x + ϑ) , where A(x) = 1

2x2 − x3,

ϑ:= 1

6

• 2
• 4+ 6 α

β2

− 1

• .

Proposition (1)

If ϑ 1

6, then the period function of the center of system (3) is

monotonous increasing.

Proof.

◮ Apply the criterion in [GaVi14]

for n = 0 to deduce monotonicity. We study the number of roots of Bσ

• ℓ1) on
• 0, 1

3

• .

Proof.

We compute ℓ1(x) = 1 2 (6ϑ + 1)x − 4ϑ − 1 √ x + ϑ(3x − 1)3 . We are interested in the number of roots of Bσ

• ℓ1
• = 1

2 (ℓ1(x) − ℓ1(σ(x))) . There exist L, S ∈ R[x, y] such that L

• x, ℓ1(x)
• ≡ 0,

S

• x, σ(x)
• ≡ 0.

Let z = σ(x) and R = Resz(L(x, z), S(x, z)). We show that R = 0 on (0, 1

3) which implies that Bσ

• ℓ1
• = 0.

From ([GaVi14], THM A) we conclude that the period function is monotonous.

Proposition (2)

For ϑ < 1

6 the period function of the center of (3) is either

monotonous decreasing for ϑ ∈ (0, ϑ1] or unimodal ϑ ∈ (ϑ1, 1/6), where ϑ1 = − 1

10 + 1 15

√ 6.

Proof.

◮ Apply criterion in [GaVi14] to obtain an

upper bound for the critical periods.

◮ Compute the first period constants. ◮ Determine the sign of T ′(h) for h ≈ hm.

Proposition (2)

For ϑ < 1

6 the period function of the center of (3) is either

monotonous decreasing for ϑ ∈ (0, ϑ1] or unimodal ϑ ∈ (ϑ1, 1/6), where ϑ1 = − 1

10 + 1 15

√ 6. We study the number of roots of Bσ

• ℓi) on
• −ϑ, 0
• .

Intervals (0, ϑ0) (ϑ0, ϑ1) (ϑ1, 1/6) # roots of Bσ

• ℓ3
• 1

2

⇓ [GaVi14], THM A

Period function T(h) monot. ≤ 1 crit. per. ≤ 2 crit. per.

Lemma

Let an analytic differential system ˙ x = p(x, y), ˙ y = q(x, y) satisfy the hypothesis (H) with H(x, y) = A(x) + C(x)y2. Let T(h) be the period of the periodic orbit γh ⊂ {H = h}. Then T ′(h) = 1 h

• γh

R(x) dx y , where R =

1 2C

KA

A′

′ − K(AC)′

4A′C2 .

Proof.

We have dx

dt = Hy(x,y) K(x)

= 2C(x)y

K(x) , and so

T(h) =

• γh

dt =

• γh

K 2C

• (x)dx

y .

Proof of Lemma cont’.

We have T(h) =

• γh

K 2C

• (x)dx

y . Recalling that A(x) + C(x)y2 = h on γh we get 2hT(h) =

• γh

KA C

• (x) dx

y +

• γh

K(x)ydx =

• γh
• G + K
• (x) ydx,

with G:= 2 KA

A′

′ − KAC′

A′C , in view of [GrMaVi11]4, Lemma 4.1.

Now we apply Gelfand-Leray and obtain 2

• hT(h)

′ = 2hT ′(h) + 2T(h) =

• γh

G + K 2C

• (x)dx

y . This implies that T ′(h) =

1 2h

• γh

G−K

2C

• (x) dx

y , which proves the result.

• 4M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for

abelian integrals, Trans. Amer. Math. Soc 363 (2011) 109–129.

Lemma (Gelfand-Leray derivative5)

Let ω and η be two rational 1-forms such that dω = dH ∧ η and let γh ∈ H1(Lh, Z) be a continuous family of cycles on non-critical level curves Lh = {H = h} not passing through poles of neither ω nor η. Then d dh

• γh

ω =

• γh

η.

• 5Y. Ilyashenko and S. Yakovenko, Lectures on analytic differential

equations, Graduate Studies in Mathematics, AMS, Vol. 86 (2007)

Lemma ([GrMaVi11]6, Lemma 4.1)

Let γh be an oval inside the level curve {A(x) + C(x)y2 = h} and consider a function F such that F/A′ is analytic at x = 0. Then, for any k ∈ N,

• γh

F(x)yk−2dx =

• γh

G(x)ykdx, where G = 2

k

• CF

A′

′ − C′F

A′

• .
• 6M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for

abelian integrals, Trans. Amer. Math. Soc 363 (2011) 109–129.

Lemma

Let an analytic differential system ˙ x = p(x, y), ˙ y = q(x, y) satisfy the hypothesis (H) with B = 0. Let T(h) be the period of the periodic orbit γh inside the energy level {H = h}. Then T ′(h) = 1 h

• γh

R(x) dx y , where R =

1 2C

KA

A′

′ − K(AC)′

4A′C2 .

Lemma

If ϑ ∈ (0, 1

6), then the period function T(h) of the center of (3)

verifies that limh→hm T ′(h) = −∞, where hm = A(−ϑ).

Proposition (2)

For ϑ < 1

6 the period function of the center of (3) is either

monotonous decreasing for ϑ ∈ (0, ϑ1] or unimodal ϑ ∈ (ϑ1, 1/6), where ϑ1 = − 1

10 + 1 15

√ 6.

Proof.

We study the number of roots of Bσ

• ℓ3) for x ∈
• −ϑ, 0
• :

Intervals (0, ϑ0) (ϑ0, ϑ1) (ϑ1, 1/6) # roots of Bσ

• ℓ3
• 1

2

⇓ [GaVi14], THM A

Period function T(h) monot. ≤ 1 crit. per. ≤ 2 crit. per.

Proof.

Intervals (0, ϑ0) (ϑ0, ϑ1) (ϑ1, 1/6) # roots of Bσ

• ℓ3
• 1

2

⇓ [GaVi14], THM A

Period function T(h) monot. ≤ 1 crit. per. ≤ 2 crit. per. We know that

◮ T ′(h) −

→ −∞ as h → hm for ϑ ∈ (0, 1/6).

◮ First period constant ∆1 = 60ϑ2 + 12ϑ − 1 s.t. ∆1(ϑ1) = 0.

Hence T ′(h) ≶ 0 for ϑ ≶ ϑ1 near h = 0.

Proof.

Intervals (0, ϑ0) (ϑ0, ϑ1) (ϑ1, 1/6) # roots of Bσ

• ℓ3
• 1

2

⇓ [GaVi14], THM A

Period function T(h) monot. ≤ 1 crit. per. ≤ 2 crit. per.

⇓ h ≈ 0, h ≈ hm

Period function T(h)

• monot. decreasing

1 crit. period We know that

◮ T ′(h) −

→ −∞ as h → hm for ϑ ∈ (0, 1/6).

◮ First period constant ∆1 = 60ϑ2 + 12ϑ − 1 s.t. ∆1(ϑ1) = 0.

Hence T ′(h) ≶ 0 for ϑ ≶ ϑ1 near h = 0.

Sketch of the graph of the period function T(h) of the center of (3):

• Proposition (2)
• Proposition (1)

T0 = 2π √ 2ϑ, T1 = 2 ln

• (2ϑ + 1)(1 − 6ϑ)

1 + 6ϑ − 4

• ϑ(1 + 3ϑ)
• > 0

Summary

wave lenght λ of ϕ = period T of γϕ wave height a of ϕ = ℓ(hϕ) {ϕa}a∈(0,aM) ↔ {γh}h∈(h0,h1)

Theorem

Given c = −k, there exist real numbers r1 < rb1 < rb2 < r2 such that the Camassa-Holm equation has smooth periodic TWS if and only if r ∈ (r1, r2). The set of smooth periodic TWS form a continous family {ϕa}a parametrized by the wave height a. The wave length λ = λ(a) of ϕa satisfies the following:

◮ If r ∈ (r1, rb1], then λ(a) is monotonous increasing. ◮ If r ∈ (rb1, rb2), then λ(a) has a unique critical point (maximum). ◮ If r ∈ [rb2, r2), then λ(a) is monotonous decreasing.

The bifurcation values are r1 = − 2

3(k − 1 2c)2, r2 = c( 1 2c + 2k), and

rb1 = k(c − 1

2k), rb2 = √ 6−3 12 (3k

√ 6 + 2c + 8k)(k √ 6 − 2c − 2k)

Degasperis-Procesi equation ut + 2x − utxx + 4 u ux = 3 uxuxx + u uxxx (DP) Traveling Wave Solutions u(x, t) = ϕ(x − c t): ϕ′′(ϕ − c) + (ϕ′)2 + r + (c − 2k) ϕ − 2ϕ2 = 0, (4) where c is the wave speed and r ∈ R is an integration constant.

The corresponding planar system with center at the origin reads

   x′ = y, y′ = −−x + 2x2 − y2 x + ϑ , where ϑ:= 1

4

• 1
• 1+ 2α

β2

− 1

• > 0

with Hamiltonian H(x, y) = 1

2(x + ϑ)2y2 + A(x).

Sketch of the graph of the period function T(h): T0 = 2π √ ϑ, T1 = ln

• (ϑ + 1)(1 − 2ϑ)

1 + 4ϑ − 3

• ϑ(1 + 2ϑ)
• > 0

Theorem

Given c = − 2

3k, there exist real numbers r1 < rb1 < rb2 < r2 such that

the Degasperis-Procesi equation ut + 2kux − utxx + 4 u ux = 3 uxuxx + u uxxx, (DP) has smooth periodic TWS ϕ(x − c t) if and only if the integration constant r ∈ (r1, r2). The set of smooth periodic TWS form a continous family {ϕa}a parametrized by the wave height a. The wave length λ = λ(a) of ϕa satisfies the following:

◮ If r ∈ (r1, rb1], then λ(a) is monotonous increasing. ◮ If r ∈ (rb1, rb2), then λ(a) has a unique critical point (maximum). ◮ If r ∈ [rb2, r2), then λ(a) is monotonous decreasing.

The bifurcation values are r1 = − 1

8(c − 2k)2, r2 = c(c + 2k), and

rb1 = 2

9k(3c − 2k), rb2 = 2 81(3c + 14k)(3c − k).

b-family equation ut + 2k ux + (b + 1) u ux − utxx = b uxuxx + u uxxx b = 2: Camassa-Holm equation b = 3: Degasperis-Procesi equation Introduced by Holm and Staley as a model for fluid transport, where b describes the ratio between stretching and convection. It turns out that the planar system associated to periodic traveling wave solutions of the can be brought to the so-called Loud’s normal form

• ˙

x = −y + xy, ˙ y = x + Dx2 + Fy2, where F, D depend polynomially on the parameter b.

Thank you for your attention!