Families of periodic solutions for some Hamiltonian PDEs ( with G. - - PowerPoint PPT Presentation

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Families of periodic solutions for some Hamiltonian PDEs

( with G. Arioli ) (1) The problem etc. (2) Main results (3) Numerical results (4) Proofs

ICERM, April 2016 (old-fashioned plain T E X)

1.1 – The problem etc. 2

We consider time-periodic solutions for the nonlinear wave equation (µ = 1) and the nonlinear beam equation (µ = 2) ∂2

t u(t, x) + (−1)µ∂2µ x u(t, x) = f(u(t, x)) ,

(t, x) ∈ R × (0, π) , with Dirichlet BCs. These PDEs are Hamiltonian with H(u, v) = π

• 1

2(∂µ xu)2 + 1 2v2 − F (u)

• dx ,

F ′ = f . From a period 2π one can get “related” periods via scaling. Changes f unless homogeneous. Our motivation:

• Observed instabilities in a bridge model [ Arioli, Gazzola 2000 ].
• CAP for Hamiltonian and/or parabolic PDEs with potential small denominator issues.

Existing relayed work: Variational methods for period 2π and related: µ = 1 [ Rabinowitz 1978; Rabinowitz 1981; . . . ] µ = 2 [ Lee 2000; Liu 2002; Liu 2004; . . . ] Perturbative methods for small u and positive-measure sets of periods near “special” values: µ = 1 [ Berti 2007; Gentile, Mastropietro, Procesi 2005; Gentile, Procesi 2009 ] µ = 2 [ Mastropietro, Procesi 2006; Gentile, Procesi 2009 ]

1.2 – The problem etc. 3

We restrict to f(u) = σu3 with σ = ±1. Setting u(t, x) = u(αt, x), where 2π

α is the desired period for u, we arrive at the equation

Lαu = σu3 , Lα = α2∂2

t + (−1)µ∂2µ x ,

where u = u(t, x) is 2π-periodic in t and satisfies Dirichlet boundary conditions at x = 0, π. µ = 1: Nonlinear wave equation α2∂2

t u − ∂2 xu = σu3 .

µ = 2: Nonlinear beam equation α2∂2

t u + ∂4 xu = σu3 .

Consider the vector space Ao of all real analytic functions u =

• n,k

un,kPn,k , Pn,k(t, x) = cos(nt) sin(kx) . We restrict our analysis to the subspace B consisting of all u ∈ Ao with the property that un,k = 0 only if n and k are both odd. Notice that LαPn,k = λn,kPn,k , λn,k = k2µ − (αn)2 = (kµ + αn)(kµ − αn) . We only consider α values for which λn,k = 0 for all odd n and k. This includes the set Qo of rationals α = p/q with p and q of opposite parity.

2.1 – Main results 4

• Definition. A solution u ∈ B of the equation Lαu = σu3 will be called a type (1, 1) solution

if |un,k| < |u1,1| whenever n > 1 or k > 1. First consider the nonlinear wave equation for some rational values of α. Our sample set: Q1 = 3

8, 5 12, 7 16, 9 20, 13 28, 1 2, 15 28, 11 20, 9 16, 7 12, 5 8, 9 14, 11 16, 7 10, 13 18, 3 4, 11 14, 5 6, 7 8, 9 10, 11 12, 13 14, 17 18

• .

Theorem 1. For each α ∈ Q1 the equation α2∂2

t u − ∂2 xu = u3 has a solution u ∈ B of type

(1, 1) with |u1,1| >

• 2(1 − α).
• Remark. Every solution u ∈ B of the equation α2∂2

t u−∂2 xu = u3 with α ∈ Qo yields a solution

˜ u ∈ B of the equation α2∂2

t ˜

u − ∂2

x˜

u = −˜ u3, and vice-versa. The functions u and ˜ u are related via ˜ u(t, x) = α−1u(x − π/2, t − π/2). Next we consider irrational values of α. Unfortunately we have to switch to the nonlinear beam equation. Still difficult to construct non-small solutions for specific α. Best known: α = 1/√c where c is an integer that is not the square of an integer. By Siegel’s theorem on integral points on algebraic curves of genus one, cλn,k = ck4 − n2 → ∞ as n ∨ k → ∞ . Unfortunately we have no useful bounds . . .

2.2 – Main results 5

So we make an assumption: Theorem 2. Let α = 1/ √

• 3. Assume that |3k4 − n2| ≥ 39 for all k ≥ 9 and all n ∈ N. Then

the equation α2∂2

t u + ∂4 xu = u3 has a solution u ∈ B of type (1, 1) with |u1,1| > 1.

We have verified the assumption minn |3k4 − n2| ≥ 39 for 9 ≤ k ≤ 1012. Our third result concerns irrational values of α that are close to the rationals in Q2 = 1

4, 3 10, 9 20, 1 2, 7 12, 5 8, 3 4, 5 6, 7 6, 5 4, 19 14, 17 12, 31 20, 13 8 , 31 18, 61 34

• .

Theorem 3. For each r ∈ Q2 there exists a set R ⊂ R of positive measure that includes r as a Lebesgue density point, such that for each α ∈ R, the equation α2∂2

t u + ∂4 xu = σu3 with

σ = sign(1 − α) has a solution u ∈ B of type (1, 1) with |u1,1| >

• 2|1 − α|.
• Remark. In all of the equations considered, other types of solutions can be obtained via scaling:

If u ∈ Ao satisfies the equation Lαu = σu3, and if we define ˜ u(t, x) = bµu(at, bx) , ˜ α = αbµ/a , (✡) with b and a nonzero integers, then ˜ u belongs to Ao and satisfies L˜

α˜

u = σ˜ u3.

3.1 – Numerical results 6

In our proofs we solve Lαu = σu3 via the fixed point equation u = Fα(u)

def

= L−1

α σu3 ,

σ = sign(1 − α) . For numerical experiments we use Fourier polynomials u =

• n≤N

k≤K

un,kPn,k , Pn,k(t, x) = cos(nt) sin(kx) . and truncate u3 to wavenumbers n ≤ N and k ≤ K. As N → ∞ the equation becomes Hamiltonian, even if K < ∞. Definition for the K < ∞ equation. The union of all smooth branches that include a solution

• f type (1, 1) will be referred to as the (1, 1) branch . Scaling each solution on the (1, 1) branch

via (✡) yields what we will call the (a, b) branch . In the following graphs we show the norm u0 =

• n,k

|un,k|

• f the numerical solution u as a function of α.

C o l o r s encode the index of u: the number of eigenvalues larger than 1 of DFα(u).

3.2 – Numerical results 7

The (1, 1) branch for the nonlinear wave equation α2utt − uxx = u3 truncated at N = K = 3, 5, 7, 9, 19, 39.

3.3 – Numerical results 8

The (1, 1) branch for the truncated nonlinear wave equation α2utt − uxx = u3 and some other (a, b) branches (thin lines). 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 α 0.5 1.0 1.5 2.0 ||u||

3.4 – Numerical results 9

The nonlinear wave equation, truncated at N ≫ K = 7. The (1, 1) branch undergoes a fold bifurcation at α ≃ 0.571 and a pitchfork bifurcation involving the (5, 3) branch at α ≃ 0.585.

0.57 0.58 0.59 0.60 0.61 0.62 α 0.5 1.0 1.5 ||u||

3.5 – Numerical results 10

The nonlinear beam equation α2utt + uxxxx = ±u3 truncated at K = 63 and N = 127.

0.2 0.4 0.6 0.8 1.0 α 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ||u|| 1.2 1.4 1.6 1.8 α 0.5 1.0 1.5 2.0 2.5 ||u||

1.80 1.82 1.84 1.86 α 2.0 2.5 3.0 ||u||

4.1 – Proofs 11

The proofs of Theorems 1,2,3 use the contraction mapping theorem. Given ρ = (ρ1, ρ2) with ρj > 0 denote by Ao

ρ the closure of . . . with respect to the norm

uρ =

• n,k

|un,k|̺n

1̺k 2 ,

̺j = 1 + ρj . Let Bρ = B ∩ Ao

ρ. Consider a quasi-Newton map associated with Fα,

Nα(h) = Fα(u0 + Ah) − u0 + (I − A)h , where u0 is an approximate fixed point and A an approximate inverse of I − DFα(u0). Denote by Bδ the open ball of radius δ in Bρ, centered at the origin. Theorem 3 is proved by verifying the following bounds. Lemma 4 For each r ∈ Q2 there exists a set R ⊂ R of positive measure that includes r as a Lebesgue density point, a pair ρ of positive real numbers, a Fourier polynomial u0 ∈ Bρ, a linear isomorphism A : Bρ → Bρ, and positive constants K, δ, ε satisfying ε + Kδ < δ, such that for every α ∈ R the map Nα defined as above is analytic on Bδ and satisfies Nα(0)ρ < ε , DNα(h)ρ < K , h ∈ Bδ .

4.2 – Proofs 12

Compactness of L−1

α

: Bρ → Bρ. Define |⌈s⌋| = dist(s, Z). A simple estimate on the eigenvalues of Lα is β2|λn,k| = (βkµ + n)|βkµ − n| ≥

• 2(βkµ ∨ n) − |⌈βkµ⌋|
• |⌈βkµ⌋| ,

β = α−1 . If α = p/q with p odd and q even: |⌈βkµ⌋| ≥ 1/p for k odd; so in this case L−1

α

is compact. For µ = 2 and irrational α we can use the following. Let (ψ1, ψ2, ψ3, . . .) be a summable sequence of nonnegative real numbers. Proposition 5. Let m ≥ 1. Consider an interval Jm of length m−2 ≤ |Jm| ≤ 1. Then

• β ∈ Jm :
• βk2
• ≥ ψk

for all k ≥ m

• has measure at least (1 − 4Ψm)|Jm|, where Ψm =

k≥m ψk.

Applying this with |Jm| = 1 and ψk = k−3/2 yields the Corollary 6. For almost every α ∈ R the operator Lα = α2∂2

t + ∂4 x has a compact inverse.

4.3 – Proofs 13

Subspaces for error terms: u ∈ Bρ,ν,κ iff u ∈ Bρ and un,k = 0 whenever n < ν or k < κ. Our enclosures for u ∈ Bρ consist of interval enclosures for each cn,k and for the norm of each Eν,κ in a representation u =

• n≤N

k≤K

cn,kPn,k +

• ν≤2N

κ≤2K

Eν,κ , Eν,κ ∈ Bρ,ν,κ . Estimating the map u → u3 on Bρ is “standard”. The operator norm of L−1

α

: Bρ,ν,κ → Bρ,ν,κ is bounded by β2/φ(ν, κ) where φ(ν, κ) = inf

n≥ν k≥κ

β2|λn,k| = inf

n≥ν k≥κ

(βkµ + n)|βkµ − n| , β = α−1 . Here ν, κ, n, k are odd positive integers. To prove Lemma 4 we use Lemma 7. Let r = p/q with p odd and q even. Given odd positive integers κ and ν, there exists a set R ⊂ R of positive measure that includes r as a Lebesgue density point, such that for all α ∈ R, φ(ν, κ) ≥ 7 4p

• βκ2 ∨ ν
• −

7 16p

• ,

β = α−1 . Idea of the proof: In the above inf distinguish between k ≥ m and k < m. For k ≥ m use Proposition 5 with Jm centered at r. And for k < m use that α is close to r. Do this for increasingly large m.

5 – References 12

Some references

• P.H. Rabinowitz, Free vibration for a semilinear wave equation, Comm. Pure Appl. Math.. 31, 31–68 (1978).
• P.H. Rabinowitz, On nontrivial solutions of a semilinear wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci.

8, 647–657 (1981).

• M. Berti, Nonlinear Oscillations of Hamiltonian PDEs, Birkh¨

auser Verlag, 2007.

• W.M. Schmidt, Metrical theorems on fractional parts of sequences, Trans. Amer. Math. Soc. 110, 493–518

(1964).

• L.H. Eliasson, B. Gr´

ebert, S. Kuksin, KAM for the non-linear beam equation, 2 Preprints.

• G. Gentile, V. Mastropietro, M. Procesi, Periodic solutions for completely resonant nonlinear wave equations

with Dirichlet boundary conditions, Commun. Math. Phys. 256, 437–490 (2005).

• V. Mastropietro, M. Procesi, Lindstedt series for periodic solutions of beam equations with quadratic and

velocity dependent nonlinearities, Commun. Pure Appl. Anal. 5, 1, 128 (2006).

• G. Gentile, M. Procesi, Periodic solutions for a class of nonlinear partial differential equations in higher dimen-

sion, Commun. Math. Phys. 289, 863–906 (2009).

• C. Lee, Periodic solutions of beam equations with symmetry, Nonlin. Anal. T.M.A. 42, 631–650 (2000)
• J.Q. Liu, Free vibrations for an asymmetric beam equation, Nonlin. Anal. T.M.A. 51, 487–497 (2002)
• J.Q. Liu, Free vibrations for an asymmetric beam equation, II, Nonlin. Anal. T.M.A. 56, 415–432 (2004)
• G. Arioli, F. Gazzola, On a nonlinear nonlocal hyperbolic system modeling suspension bridges, Milan J. Math.

83, 211–236 (2015).

• G. Arioli, F. Gazzola, Torsional instability in suspension bridges: the Tacoma Narrows Bridge case, Preprint

mp arc 15-83.

• G. Arioli, H. Koch, Non-symmetric low-index solutions for a symmetric boundary value problem, J. Differ.

Equations, 252 448–458 (2012).

• G. Arioli, H. Koch, Some symmetric boundary value problems and non-symmetric solutions, J. Differ. Equa-

tions 259, 796–816 (2015).

• G. Arioli, H. Koch, Computer-Assisted Methods for the Study of Stationary Solutions in Dissipative Systems,

Applied to the Kuramoto-Sivashinski Equation, Arch. Rat. Mech. Anal. 197, 1033–1051 (2010).

• G. Arioli, H. Koch, Integration of Dissipative Partial Differential Equations: A Case Study, SIAM J. Appl.
• Dyn. Syst. 9 1119–1133 (2010).