Multi-site breathers in Klein-Gordon lattices: bifurcations, stability, and resonances Dmitry Pelinovsky, Anton Sakovich Department of Mathematics and Statistics, McMaster University, Ontario, Canada Workshop on Localization in Lattices; Seville, Spain, July 11, 2012 Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 1 / 22
Klein-Gordon lattice Klein-Gordon (KG) lattice models a chain of coupled anharmonic oscillators with a nearest-neighbour interactions u n + V ′ ( u n ) = ǫ ( u n − 1 − 2 u n + u n + 1 ) , ¨ where { u n ( t ) } n ∈ Z : R → R Z , dot represents time derivative, ǫ is the coupling constant, and V : R → R is an on-site potential. V u n u n +1 u V Applications: dislocations in crystals (e.g. Frenkel & Kontorova ’1938) oscillations in biological molecules (e.g. Peyrard & Bishop ’1989) Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 2 / 22
Anharmonic oscillator We make the following assumptions: V ′ ( u ) = u ± u 3 + O ( u 5 ) , where + / − corresponds to hard/soft potential; 0 < ǫ ≪ 1: oscillators are weakly coupled. In the anti-continuum limit ( ǫ = 0), each oscillator is governed by 1 ϕ 2 + V ( ϕ ) = E , ϕ + V ′ ( ϕ ) = 0 , ¨ ⇒ 2 ˙ where ϕ ∈ H 2 per ( 0 , T ) . 4 The period of the oscillator is 3.5 3 T, π � a ( E ) √ dx 2.5 T ( E ) = 2 , � E − V ( x ) 2 − a ( E ) 1.5 0 0.1 0.2 0.3 where a ( E ) , the amplitude, is the E smallest root of V ( a ) = E . Figure: Period versus energy in hard (magenta) and soft (blue) V . Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 3 / 22
Multi-breathers in the anti-continuum limit Breathers are spatially localized time-periodic solutions to the Klein-Gordon lattice. Multi-breathers are constructed by parameter continuation in ǫ from ǫ = 0. For ǫ = 0 we take u ( 0 ) ( t ) = � l 2 ( Z , H 2 σ k ϕ ( t ) e k ∈ per ( 0 , T )) , k ∈ S where S ⊂ Z is the set of excited sites and e k is the unit vector in l 2 ( Z ) at the node k . The oscillators are in phase if σ k = + 1 and out-of-phase if σ k = − 1. σ n 1 − 1 1 a ( E ) Z − a ( E ) Figure: An example of a multi-site discrete breather at ǫ = 0. Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 4 / 22
Persistence of multi-breathers Theorem (MacKay & Aubry ’1994) Fix the period T � = 2 π n, n ∈ N and the T-periodic solution ϕ ∈ H 2 per ( 0 , T ) of the anharmonic oscillator equation for T ′ ( E ) � = 0 . There exist ǫ 0 > 0 and C > 0 such that ∀ ǫ ∈ ( − ǫ 0 , ǫ 0 ) there exists a solution u ( ǫ ) ∈ l 2 ( Z , H 2 per ( 0 , T )) of the Klein–Gordon lattice satisfying � u ( ǫ ) − u ( 0 ) � � l 2 ( Z , H 2 ( 0 , T )) ≤ C ǫ. � � � The proof is based on the Implicit Function Theorem and uses invertibility of the linearization operators ∂ 2 t + 1 : H 2 per ( 0 , T ) → L 2 L 0 = per ( 0 , T ) , T � = 2 π n , ∂ 2 t + V ′′ ( ϕ ( t )) : H 2 per , even ( 0 , T ) → L 2 T ′ ( E ) � = 0 . L e = per , even ( 0 , T ) , Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 5 / 22
Stability of discrete breathers Multibreathers in Klein–Gordon lattices: Morgante, Johansson, Kopidakis, Aubry ’2002 - numerical results Archilla, Cuevas, Sánchez-Rey, Alvarez ’2003 - Aubry’s spectral band theory Koukouloyannis, Kevrekidis ’2009 - MacKay’s action-angle averaging In this project: no restriction to small-amplitude approximation multi-site breathers with “holes” Similar works: Pelinovsky, Kevrekidis, Franzeskakis ’2005 - discrete NLS lattice Youshimura ’2011 - Fermi-Pasta-Ulam bi-atomic lattice Youshimura ’2012 - KG unharmonic lattice Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 6 / 22
Floquet Multipliers Linearize about the breather solution to the dKG by replacing u with u + w , where w : R → R Z is a small perturbation, and collect the terms linear in w : w n + V ′′ ( u n ) w n = ǫ ( w n − 1 − 2 w n + w n + 1 ) , ¨ n ∈ Z . In the anti-continuum limit , it is easy to find the Floquet multipliers: on “holes", n ∈ Z \ S , � cos T � w n ( T ) � � � w n ( 0 ) � sin T w n + w n = 0 , ¨ = , w n ( T ) ˙ − sin T cos T w n ( 0 ) ˙ Floquet multipliers are µ 1 , 2 = e ± iT on excited sites, n ∈ S , � � � 1 0 � � � w n ( T ) w n ( 0 ) w n + V ′′ ( ϕ ) w n = 0 , ¨ = , T ′ ( E ) ( V ′ ( a )) 2 w n ( T ) ˙ w n ( 0 ) ˙ 1 Floquet multipliers are µ 1 , 2 = 1 of geometric multiplicity 1 and algebraic multiplicity 2. Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 7 / 22
Splitting of the unit Floquet multiplier Introduce a limiting configuration u ( 0 ) ( t ) that has M excited sites with N − 1 “holes" in between them: M u ( 0 ) ( t ) = � σ j ϕ ( t ) e jN M = 3 , N = 2 j = 1 For ǫ > 0, Floquet multipliers split as follows: Im µ Im µ 1 ǫ = 0 1 ǫ > 0 e iT e iT Re µ Re µ 1 e − iT e − iT Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 8 / 22
Floquet exponents A Floquet multiplier µ can be written as µ = e λ T . Lemma For small ǫ > 0 the linearized stability problem has 2 M small Floquet exponents , where ˜ λ = ǫ N / 2 Λ + O ǫ ( N + 1 ) / 2 � � λ is determined from the eigenvalue problem T ( E ) 2 2 T ′ ( E ) K N Λ 2 c = S c , c ∈ C M . − Here S ∈ R M × M is a tridiagonal matrix with elements S i , j = − σ j ( σ j − 1 + σ j + 1 ) δ i , j + δ i , j − 1 + δ i , j + 1 , 1 ≤ i , j ≤ M , and K N is defined by � T � ∂ 2 � K N = ϕ ( t ) ˙ ˙ ϕ N − 1 ( t ) dt , t + 1 ϕ k = ϕ k − 1 , ϕ 0 = ϕ. 0 Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 9 / 22
Stability of multibreathers Sandstede (1998) showed that the matrix S has exactly n 0 positive and M − 1 − n 0 negative eigenvalues in addition to the simple zero eigenvalue, where n 0 = # ( sign changes in { σ n } ) . Hence, stability of multibreathers is determined by the sign of T ′ ( E ) K N ( T ) and the phase parameters { σ k } M − 1 k = 1 . Theorem If T ′ ( E ) K N ( T ) > 0 the linearized problem for the multibreathers has exactly n 0 pairs of “stable” Floquet exponents and M − 1 − n 0 pairs of “unstable” Floquet exponents counting their multiplicities. If T ′ ( E ) K N ( T ) < 0 the conclusion changes to the opposite. Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 10 / 22
Stable configurations of multibreathers T ′ ( E ) K N ( T ) > 0: anti-phase T ′ ( E ) K N ( T ) < 0: in-phase breathers, n 0 = M − 1 breathers, n 0 = 0 4 3.5 T ′ ( E ) < 0 if V ′ ( u ) = u + u 3 3 T, π (hard potential). 2.5 2 T ′ ( E ) > 0 if V ′ ( u ) = u − u 3 1.5 0 0.1 0.2 0.3 (soft potential). E Figure: Period versus energy in hard (magenta) and soft (blue) V . Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 11 / 22
Resonances of multibreathers Let ϕ ( t ) be expanded in the Fourier series, � 2 π nt � � ϕ ( t ) = c n cos T n ∈ N odd Then, we compute explicitly T 2 N − 3 ( E ) n 2 | c n | 2 K N ( T ) = 4 π 2 � [ T 2 − ( 2 π n ) 2 ] N − 1 . n ∈ N odd Hard potentials: T ( E ) < 2 π ; K N ( T ) > 0 for odd N and K N ( T ) < 0 for even N . Soft potentials: T ( E ) > 2 π ; resonances occur for T ( E ) = 2 π ( 1 + 2 n ) , n ∈ N . N odd N even V ′ ( u ) = u + u 3 in-phase anti-phase anti: 2 π < T < T ∗ V ′ ( u ) = u − u 3 N anti-phase in: T ∗ N < T < 6 π where K N ( T ) changes sign at T ∗ N , e.g., T ∗ 2 = 5 . 476 π . Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 12 / 22
Three-site KG lattice Consider a three-site KG lattice with a soft potential and Dirichlet boundary conditions, u 0 + u 0 − u 3 ¨ 0 = 2 ǫ ( u 1 − u 0 ) u 1 + u 1 − u 3 ¨ 1 = ǫ ( u 0 − 2 u 1 ) u − 1 = u 1 , Two limiting configurations are of interest: u ( 0 ) ( t ) = ϕ ( t ) e 0 u ( 0 ) ( t ) = ϕ ( t )( e − 1 + e 1 ) Fundamental breather ( M = 1) Breather with a “hole” ( M = 2, N = 2) Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 13 / 22
Breather solutions Periodic solutions are computed with the shooting method. 7 7 6 6 6.2 6 5 5 6 T/ π T/ π 5.8 5.8 4 4 5.6 5.6 0.95 1 0 0.05 0.1 3 3 2 2 −0.2 0 0.2 0.4 0.6 0.8 1 −0.2 0 0.2 0.4 0.6 0.8 1 a 0 a 1 ǫ = 0 . 01: u 0 ( 0 ) = a 0 ( T ) , ˙ u 0 ( 0 ) = 0; u 1 ( 0 ) = a 1 ( T ) , ˙ u 1 ( 0 ) = 0 Solid – fundamental breather ( M = 1) Dashed – breather with a “hole” ( M = 2, N = 2). Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 14 / 22
Breather with a “hole” ( M = 2, N = 2) The breather u ( 0 ) ( t ) = ϕ ( t )( e − 1 + e 1 ) is unstable for T ∈ ( 2 π, T ∗ 2 ) . It then remains stable until the symmetry-breaking bifurcation occurs. 10 8 6 4 Re( λ ) 2 0 −2 −4 5 5.2 5.4 5.6 5.8 6 T, pi Figure: Real part of the Floquet multipliers versus T . Dmitry Pelinovsky (McMaster University) Breathers in Klein-Gordon lattices Seville, Spain 15 / 22
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