Breather stability in Klein-Gordon equations Zoi Rapti Department - PowerPoint PPT Presentation

Breather stability in Klein-Gordon equations Zoi Rapti Department of Mathematics University of Illinois, Urbana-Champaign LENCOS July 09-12 2012 Seville, Spain

Motivation: DNA Modelling y 1 Peyrard-Bishop model originally introduced to study DNA denaturation Peyrard and Bishop, PRL 62 , 2755 (1989) Dauxois, Peyrard, and Bishop, PRE 47 , R44 (1993) y Similar model: Techera-Daeman-Prohofsky  1 n Techera, Daeman, and Prohofsky, PRA 40, 6636 (1989), y PRA 41 , 4543 (1990) n ( ) V y ( , ) W y n y  n 1 n y N Has been used to study  dynamical behavior  unzipping forces  melting curves

Morse Potential    a n y 2 ( ) ( 1 ) V y D e n n n Weak: AT Strong: GC y  1 n y n ( , ) W y n y  n 1 ( ) V y n Stacking interaction k       ( ) 2 b y y ( , ) ( 1 )( ) n W y y e y y  1 n n   1 1 n n n n 2

Extended Peyrard-Bishop model Coman and Russu, Biophys. J. 89 , 3285-3292 (2005) Varnai and Lavery, J. Am. Chem. Soc. 124 , 7272-7273 (2002) Rapti, Eur. Phys. J. E 32 , 209-216 (2010)   m k k k N           2 2 2 2 2   ( ) 1 ( ) 2 ( ) 3 ( ) H y V y y y y y y y      1 2 3 n n n n n n n n 2 2 2  n 1   k k k 1 2 3 Equations of motion:         m   ' ( ) ( 2 ) ( 2 ) y V y k y y y k y y y     1 1 1 2 2 2 n n n n n n n n    ( 2 ) 0 k y y y   3 3 3 n n n

Small coupling between oscillators After rescaling    N     ' ( )  0 , 1 ,..., y V y C y n N n n nm m  1 m         1 , , C C C    1 2 3 nn nn nn          2 ( 1 ), 1 C nn    0 , | | 3 C m n nm Use the notation  T ( ) [ ( ), ( ),..., ( )] y t y t y t y n t      1 2   ' ( ) 0 y V y Cy  T ( ) [ ( ), ( ),..., ( )] V y V y V y V y 1 2 n

Discrete breathers: time-periodic, localized oscillations in discrete systems due to a combination of nonlinearity and discreteness. MacKay and Aubry, Nonlinearity 7 , 1623-1643 (1994) (existence) Aubry, Physica D 103 , 201-250 (1997) (existence -stability) Archilla, Cuevas, Sánchez-Rey, and Alvarez , Physica D 180 , 235-255 (2003) (stability) Koukouloyiannis and Kevrekidis, Nonlinearity 22 , 2269-2285 (2009) Cuevas, Koukouloyiannis, Kevrekidis, Archilla , IJBC 21 , 2161-2177 (2011) Pelinovsky and Sakovich, submitted Koukouloyannis, Kevrekidis, Cuevas, Rothos, arXiv: 1204.5496v2 Dynamical equations       ' ( ) 0 y V y Cy (in compact form)     ( ) ( ), ( 0 ) 0 y t y t T y Linear stability of Newton operator N              ' ' ( ) V y C E  (eigenvalue problem) y(t) is periodic, hence so is the Newton operator. Then the evolution of the above equation can be studied by means of the Floquet matrix F E         T ( ) ( 0 ), ( ) [ ( ), ( )] T F t t t E

Aubry’s band theory for periodic problems Linear stability is equivalent to the Floquet matrix eigenvalues (Floquet  e i multipliers) lying on the unit circle:  Study of the Newton operator eigenvalue problem with . 0 E  ( , ) E The set of points has a band structure. The bands are symmetric with   / ( 0 ) 0 E dE d respect to the axis, so . 2 For a chain of length there are at most bands crossing any horizontal axes N N  in the space of coordinates . ( , ) E   2 2 Morse potential: / ( 0 ) 0 d E d

Linear stability condition for multi-breathers The condition for linear stability is equivalent to the existence of bands 2 N   crossing the axis . As the coupling strength changes the bands also move 0 E and they can lose crossing points (with E=0) bringing about instability: θ is not real anymore.   Key concept : consideration of the case . Consider time-reversible solutions 0 around the minimum of the Morse potential. There are three kinds of solutions:  ( ) 0 y n t 1. oscillators at rest 0 t  ( ) ( ) y n t y 2. excited oscillators with identical  3. excited oscillators with a phase difference of with the previous ones .   0 ( ) ( / 2 ) y n t y t T   Each site index is given a code which takes elements in depending on { 0 , 1 , 1 } n whether the oscillator is at rest, in-phase or out-of-phase, respectively. The vector       [ ,..., ] represents the state of the system at the limit . 0 1 N

Linear stability condition for multi-breathers N  M M Assume that there are oscillators at rest and excited ones. We will use the perturbation theory used in Archilla, Cuevas, Sánchez-Rey, and Alvarez , Physica D 180 , 235-255 (2003) to demonstrate the stability/instability of breathers of any code. N Perturbation theory establishes that if is a linear operator with a degenerate 0 { n } v eigenvalue with eigenvectors which are orthonormal with respect to E 0 ~ N  N  a scalar product and if is a perturbation of with small, then to a first 0 ~     N 0   N E order in the eigenvalues of the perturbed operator are , 0 i  Where are the eigenvalues of the perturbation matrix with elements Q ~ i  N  . , Q v v nm n m For a non-symmetric potential the perturbation matrix satisfies:     0 if 0 Q T T nm n m  0 0    ( ) ( ) y t y t dt 2     T if 1 Q C 2  nm nm n m    2        T if 1 Q C 0 2   ( ( )) y t dt nm nm n m 2 T     Q Q 2 nn nm  m n

Sturm-type theory for the perturbation matrix (Nearest Neighbors) In Archilla et al. (2003) it was stated that “Although, we have no mathematical proof, according to numerical calculations of the eigenvalues of the perturbation matrices corresponding to groups with different codes, the numbers of negative   i  and positive eigenvalues are equal to the numbers of - 1 and +1 in ”. 1 M { 1 }   1 i i     0 0 0 0 a a   1 1     0 0 0 a a a a 1 1 2 2      0 0 0 0 a a a   2 2 3           Q      0 0 0 0 a a a      2 1 1 M M M      0 0 0 a a a a   1 1 M M M M       0 0 0 0 a a M M      1 , for 1    1    1 i i  det( ) ( ) where and Q I f a      i M  , for -1  1 i i                  , 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) f a a f a f f a f a f    1 1 1   j j j j j j 1 1 M M M M M        for j=1,…,M -1 with ( ) ( ), ( ) 1 f a f 1 1 0

Sturm-type theory for the perturbation matrix (Nearest Neighbors)    j  By induction: ( 0 ) 0 and ( 0 ) ( 1 ) f f a a  1 1 M j j By a variation of the Sturm sequence property and since  ( 0 ) 1 f 0 The number of eigenvalues greater than   ( 0 ) ( 0 ) f a f 1 1 0 zero equals the number of agreements in sign of   ( 0 ) ( 0 ) f a f 2 2 1 consecutive members of the sequence on the left.   ( 0 ) ( 0 ) f a f 3 3 2  If some member is zero, then its sign is reassigned as the opposite of the preceding term.   ( 0 ) ( 0 ) f a f  1 M M M  ( 0 ) 0 f  1 M     1 M simple zero eigenvalue ' ( 0 ) ( 1 ) ( 1 ) ... f M a a a  1 1 2 M M  j   1 M M { 1 } M positive terms in correspond to sign agreements   0 0 j j 1 and positive zeros for . It can be shown that the zeros of M f f 0 M M and are separated. f  1 M