1 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 X) E
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 ∂ 2 µ x u ( t, x ) = f ( u ( t, x )) , ( t, x ) ∈ R × (0 , π ) , with Dirichlet BCs. These PDEs are Hamiltonian with � π � x u ) 2 + 1 2 v 2 − F ( u ) � F ′ = f . 1 2 ( ∂ µ H ( u , v ) = dx , 0 From a period 2 π 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 2 π 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 ) = σ u 3 with σ = ± 1. Setting u ( t, x ) = u ( αt, x ), where 2 π α is the desired period for u , we arrive at the equation L α u = σu 3 , L α = α 2 ∂ 2 t + ( − 1) µ ∂ 2 µ x , where u = u ( t, x ) is 2 π -periodic in t and satisfies Dirichlet boundary conditions at x = 0 , π . x u = σu 3 . α 2 ∂ 2 t u − ∂ 2 µ = 1: Nonlinear wave equation x u = σu 3 . Nonlinear beam equation α 2 ∂ 2 t u + ∂ 4 µ = 2: Consider the vector space A o of all real analytic functions � u = u n,k P n,k , P n,k ( t, x ) = cos( nt ) sin( kx ) . n,k We restrict our analysis to the subspace B consisting of all u ∈ A o with the property that u n,k � = 0 only if n and k are both odd. Notice that λ n,k = k 2 µ − ( αn ) 2 = ( k µ + αn )( k µ − αn ) . L α P n,k = λ n,k P n,k , We only consider α values for which λ n,k � = 0 for all odd n and k . This includes the set Q o 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 L α u = σu 3 will be called a type (1 , 1) solution if | u n,k | < | u 1 , 1 | whenever n > 1 or k > 1 . First consider the nonlinear wave equation for some rational values of α . Our sample set: � 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 � Q 1 = . 18 x u = u 3 has a solution u ∈ B of type Theorem 1 . For each α ∈ Q 1 the equation α 2 ∂ 2 t u − ∂ 2 � (1 , 1) with | u 1 , 1 | > 2(1 − α ). x u = u 3 with α ∈ Q o yields a solution Remark . Every solution u ∈ B of the equation α 2 ∂ 2 t u − ∂ 2 u ∈ B of the equation α 2 ∂ 2 u − ∂ 2 u 3 , and vice-versa. The functions u and ˜ u = − ˜ ˜ t ˜ x ˜ u are related u ( t, x ) = α − 1 u ( x − π/ 2 , t − π/ 2). via ˜ 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, cλ n,k = ck 4 − n 2 → ∞ n ∨ k → ∞ . as Unfortunately we have no useful bounds . . .
2.2 – Main results 5 So we make an assumption: √ 3 . Assume that | 3 k 4 − n 2 | ≥ 39 for all k ≥ 9 and all n ∈ N . Then Theorem 2 . Let α = 1 / x u = u 3 has a solution u ∈ B of type (1 , 1) with | u 1 , 1 | > 1 . the equation α 2 ∂ 2 t u + ∂ 4 We have verified the assumption min n | 3 k 4 − n 2 | ≥ 39 for 9 ≤ k ≤ 10 12 . Our third result concerns irrational values of α that are close to the rationals in � 1 3 9 1 7 5 3 5 7 5 19 17 31 13 31 61 � Q 2 = 4 , 10 , 20 , 2 , 12 , 8 , 4 , 6 , 6 , 4 , 14 , 12 , 20 , 8 , 18 , . 34 Theorem 3 . For each r ∈ Q 2 there exists a set R ⊂ R of positive measure that includes r x u = σu 3 with as a Lebesgue density point, such that for each α ∈ R , the equation α 2 ∂ 2 t u + ∂ 4 � σ = sign(1 − α ) has a solution u ∈ B of type (1 , 1) with | u 1 , 1 | > 2 | 1 − α | . Remark. In all of the equations considered, other types of solutions can be obtained via scaling: If u ∈ A o satisfies the equation L α u = σu 3 , and if we define u ( t, x ) = b µ u ( at, bx ) , α = αb µ /a , ˜ ˜ ( ✡ ) u belongs to A o and satisfies L ˜ u 3 . with b and a nonzero integers, then ˜ α ˜ u = σ ˜
3.1 – Numerical results 6 In our proofs we solve L α u = σu 3 via the fixed point equation α σu 3 , def = L − 1 u = F α ( u ) σ = sign(1 − α ) . For numerical experiments we use Fourier polynomials � u = u n,k P n,k , P n,k ( t, x ) = cos( nt ) sin( kx ) . n ≤ N k ≤ K and truncate u 3 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 of 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 � � u � 0 = | u n,k | n,k of 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 D F α ( u ).
3.2 – Numerical results 7 The (1 , 1) branch for the nonlinear wave equation α 2 u tt − u xx = u 3 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 α 2 u tt − u xx = u 3 and some other ( a, b ) branches (thin lines). || u || 2.0 1.5 1.0 0.5 α 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
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. || u || 1.5 1.0 0.5 α 0.57 0.58 0.59 0.60 0.61 0.62
3.5 – Numerical results 10 The nonlinear beam equation α 2 u tt + u xxxx = ± u 3 truncated at K = 63 and N = 127. || u || || u || 3.5 2.5 3.0 2.0 2.5 2.0 1.5 1.5 1.0 1.0 0.5 0.5 α α 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 || u || 3.0 2.5 2.0 α 1.80 1.82 1.84 1.86
4.1 – Proofs 11 The proofs of Theorems 1,2,3 use the contraction mapping theorem. Given ρ = ( ρ 1 , ρ 2 ) with ρ j > 0 denote by A o ρ the closure of . . . with respect to the norm � | u n,k | ̺ n 1 ̺ k � u � ρ = 2 , ̺ j = 1 + ρ j . n,k Let B ρ = B ∩ A o ρ . Consider a quasi-Newton map associated with F α , N α ( h ) = F α ( u 0 + Ah ) − u 0 + (I − A ) h , where u 0 is an approximate fixed point and A an approximate inverse of I − D F α ( u 0 ). Denote by B δ the open ball of radius δ in B ρ , centered at the origin. Theorem 3 is proved by verifying the following bounds. Lemma 4 For each r ∈ Q 2 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 u 0 ∈ B ρ , a linear isomorphism A : B ρ → B ρ , and positive constants K , δ , ε satisfying ε + Kδ < δ , such that for every α ∈ R the map N α defined as above is analytic on B δ and satisfies �N α (0) � ρ < ε , � D N α ( h ) � ρ < K , h ∈ B δ .
4.2 – Proofs 12 Compactness of L − 1 : B ρ → B ρ . α Define |⌈ s ⌋| = dist( s, Z ). A simple estimate on the eigenvalues of L α is β 2 | λ n,k | = ( βk µ + n ) | βk µ − n | ≥ 2( βk µ ∨ n ) − |⌈ βk µ ⌋| β = α − 1 . |⌈ βk µ ⌋| , � � If α = p/q with p odd and q even: |⌈ βk µ ⌋| ≥ 1 /p for k odd; so in this case L − 1 is compact. α For µ = 2 and irrational α we can use the following. Let ( ψ 1 , ψ 2 , ψ 3 , . . . ) be a summable sequence of nonnegative real numbers. Proposition 5 . Let m ≥ 1 . Consider an interval J m of length m − 2 ≤ | J m | ≤ 1 . Then � ≥ ψ k � βk 2 � � � � � β ∈ J m : k ≥ m for all � has measure at least (1 − 4Ψ m ) | J m | , where Ψ m = � k ≥ m ψ k . Applying this with | J m | = 1 and ψ k = k − 3 / 2 yields the Corollary 6 . For almost every α ∈ R the operator L α = α 2 ∂ 2 t + ∂ 4 x has a compact inverse.
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