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

Breather stability in Klein-Gordon equations

LENCOS July 09-12 2012 Seville, Spain Zoi Rapti Department of Mathematics University of Illinois, Urbana-Champaign

1  n

y

1

y

N

y

Motivation: DNA Modelling

Peyrard-Bishop model originally introduced to study DNA denaturation Peyrard and Bishop, PRL 62, 2755 (1989) Dauxois, Peyrard, and Bishop, PRE 47, R44 (1993) Similar model: Techera-Daeman-Prohofsky Techera, Daeman, and Prohofsky, PRA 40, 6636 (1989), PRA 41, 4543 (1990) Has been used to study

• dynamical behavior
• unzipping forces
• melting curves

) (

n

y V

n

y ) , (

1  n n y

y W

Morse Potential Stacking interaction

2

) 1 ( ) (  



n n y

a n n

e D y V

2 1 ) ( 1

) )( 1 ( 2 ) , (

1

   

  



n n y y b n n n

y y e k y y W

n n



Weak: AT Strong: GC

1  n

y ) (

n

y V

n

y ) , (

1  n n y

y W

Equations of motion:

Extended Peyrard-Bishop model

3 2 1

k k k   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)



   

             

N n n n n n n n n n

y y k y y k y y k y V y m H

1 2 3 3 2 2 2 2 1 1 2

) ( 2 ) ( 2 ) ( 2 ) ( 2  ) 2 ( ) 2 ( ) 2 ( ) ( '

3 3 3 2 2 2 1 1 1

          

      n n n n n n n n n n n

y y y k y y y k y y y k y V y m  

Small coupling between oscillators N n y C y V y

N m m nm n n

,..., 1 , ) ( '

1

   





  

After rescaling

3 | | , 1 ), 1 ( 2 , , 1

3 2 1

             

  

n m C C C C C

nm nn nn nn nn

     

Use the notation

T n t

y t y t y t y )] ( ),..., ( ), ( [ ) (

2 1



T n

y V y V y V y V )] ( ),..., ( ), ( [ ) (

2 1

 ) ( '     Cy y V y   

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      



E C y V      ) ( ' '   N

) ( '    Cy y V y   

Dynamical equations (in compact form) Newton operator (eigenvalue problem) Linear stability of

) ( ), ( ) (    y T t y t y 

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

T E

t t t F T )] ( ), ( [ ) ( ), ( ) (        

E

F

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

) , ( E  ) ( /   d dE

 E N N 2 ) , ( E 

 i

e ) ( /

2 2

  d E d

E

Linear stability condition for multi-breathers

The condition for linear stability is equivalent to the existence of bands crossing the axis . As the coupling strength changes the bands also move 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 around the minimum of the Morse potential. There are three kinds of solutions:

• 1. oscillators at rest
• 2. excited oscillators with identical
• 3. excited oscillators with a phase difference of with the previous ones

. Each site index is given a code which takes elements in depending on whether the oscillator is at rest, in-phase or out-of-phase, respectively. The vector represents the state of the system at the limit . N 2  E    ) (  t yn ) ( ) (

0 t

y t yn   ) 2 / ( ) ( T t y t yn  

n

 } 1 , 1 , {  ] ,..., [

1 N

     

Linear stability condition for multi-breathers

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. M N  M Perturbation theory establishes that if is a linear operator with a degenerate eigenvalue with eigenvectors which are orthonormal with respect to a scalar product and if is a perturbation of with small, then to a first

• rder in the eigenvalues of the perturbed operator are ,

Where are the eigenvalues of the perturbation matrix with elements . E  

i

E  

i

  

m n nm

v v Q N ~ ,

Q N ~  N } { n v N N 0 ~   N For a non-symmetric potential the perturbation matrix satisfies: 1 if  

m n nm nm

C Q  





 

n m nm nn

Q Q dt t y dt T t y t y

T T T T

 

 

  

2 2 2 2 2

)) ( ( ) 2 ( ) (     if  

m n nm

Q   1 if    

m n nm nm

C Q   

                               

     M M M M M M M M M

a a a a a a a a a a a a a a a a a a Q

1 1 1 1 2 3 2 2 2 2 1 1 1 1

            

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 and positive eigenvalues are equal to the numbers of -1 and +1 in ”.

1 1 1}

{

   M i i i

 where       

 

• 1

for , 1 for , 1

1 1 i i i i i

a      and

) ( ) det(

1 

  

 M

f I Q

) ( ) ( ) ( ) (

1 2 1

       

  M M M M M

f a f a f

, ) ( ) ( ) ( ) (

1 2 1 1

        

   j j j j j j

f a f a a f 1 ) ( ), ( ) (

1 1

       f a f for j=1,…,M-1 with

Sturm-type theory for the perturbation matrix (Nearest Neighbors)

By induction:

j j j M

a a f f 

1 1

) 1 ( ) ( and ) (   



By a variation of the Sturm sequence property and since ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 ) (

1 1 2 3 3 1 2 2 1 1

         

  M M M M

f f a f f a f f a f f a f f  The number of eigenvalues greater than zero equals the number of agreements in sign of consecutive members of the sequence on the left. If some member is zero, then its sign is reassigned as the opposite of the preceding term. simple zero eigenvalue

M M M

a a a M f ... ) 1 ( ) 1 ( ) ( '

2 1 1 1

  

 

positive terms in correspond to sign agreements and positive zeros for . It can be shown that the zeros of and are separated.

1 1 1}

{

   M j j j

 M M M

M

f

M

f

1  M

f

• The stability results for two-site breathers are the same as in the original model.

(in-phase) (out-of-phase)

• For 3-site breathers:

         1 1 1 1 Q              Q                 

3 2 3 2 3 3 1 1 2 1 2 1

a a a a a a a a a a a a Q ) ( 3 ) ( ) ( ,

3 1 3 2 2 1 2 3 2 1 3 2 1

a a a a a a a a a a a a             

Stability condition:

, ) (

3 1 3 2 2 1 3 2 1

      a a a a a a a a a Multi-breather stability in the extended Peyrard-Bishop model

Multi-breather stability in the extended Peyrard-Bishop model ] 1 , 1 , 1 [ , 2                             k k k k Q

To fix ideas: Eigenvalues:

           k 2 , 3 ,

Critical value of the coupling: depends on the onsite potential and the breather frequency

2  

cr

k

Morse potential:

b

  

cr

k k 

For there is one positive, one negative eigenvalue

Assume now that we have interactions with first and second neighbors. Multibreathers with all the oscillators in-phase are unstable in the case of harmonic coupling with first and second neighbors. Proof: for a>0, b>0 The perturbation matrix is symmetric; Λ=0 is an eigenvalue. Then, apply Gerschgorin’s Theorem.

] [

ij

q Q 







i j ij i

q r | | } | :| {

i ii i

r q z C z Z    

i

Z   Multi-breather stability in the extended Peyrard-Bishop model

Because of the symmetry, the discs become intervals.

) ( 2 2

1 1 22 11

b a q q b a q q b a q

nn M M MM

       

 

) ( 2 2

1 2 1

b a r r b a r r b a r

n M M

       



2 3    M n 2 3    M n )] ( 4 , [ b a   

Instability: one eigenvalue zero and the rest positive. The zero eigenvalue must be simple (otherwise first order perturbation does not suffice).

Multi-breather stability in the extended Peyrard-Bishop model

                                         b a a b a b a a b a b a b a a b a b a a b a b a Q              2 ) ( 2 . ) ( 2 2

Multi-breather stability in the extended Peyrard-Bishop model ) ( ) ( ) det( ) ( x xq x p xI Q x p

n n n

    

Characteristic polynomial

) ( ) ( '   

n n

q p

n n

nR b a a b a b a a b b a b a a b a b a b a a b a b a a b a b a b b b a n q                                                       2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 det ) (                 

Zero is a simple eigenvalue:

Multi-breather stability in the extended Peyrard-Bishop model

2 2 1

) 2 (

  

 

n n n

R b R b a R

Recurrence relation

 

1 1

   

      n n n

r r r r a R

Similarly, it can be shown that multibreathers with oscillators out-of-phase are stable in the case of harmonic coupling with first and second neighbors.

) 2 (

3 2

b a a R a R   

Initial condition

2 2

4 ) 2 ( 2 1 2 2 b b a b a r     



Multi-breather stability in the extended Peyrard-Bishop model

New solutions (without the time-symmetry) exist for next nearest coupling: Koukouloyiannis, Kevrekidis, Cuevas, Rothos, arXiv: 1204.5496v2 The phase differences between

• scillators 2 and 1 and 3 and 2

are:

3 4

• r

3 2

2 1 2 1

         

In this case we can still use the same ideas, except that now the symmetry coefficient needs to be modified:

dt t y dt t t y t y

T T m T T t m

 

 

  

2 2 2 2 2

)) ( ( ) ( ) (     3 , 3 2

• r

3 2 , 3

2 1 2 1

T t T t T t T t     ) cos( ) 1 ( 1 ) cos( ) 1 ( ) 1 (

2 2 m b b b m b b b b t

t t

m

              

Morse potential:

i ti

T T k k k k Q                                    with , 3 2 t , 3 t , 2

2 1 2 1 1 1 1 1 1 2 1 2 1

Multi-breather stability in the extended Peyrard-Bishop model

1 2 1

2 , 3 ,             k

Eigenvalues:

2 1

3 3

b b b

      

2 2

3 1 3 1

b b b

       ) 3 )( 3 1 ( 2 ) 3 1 )( 3 ( 2

2 2 2 1 b b b b cr

k              

Critical coupling value:

Assume now that we have interactions with first, second, and third neighbors. Multibreathers with all the oscillators in-phase are unstable in the case of harmonic coupling with up to third neighbors. Proof: proceed as previously.

Multi-breather stability in the extended Peyrard-Bishop model ) ( ) det( ) ( x xq xI Q x p

n n

    ) 2 ( ) 2 ( ) 2 ( ) 2 2 (

5 5 4 3 3 2 2 2 2 2 1

               

     n n n n n n

R c R c b a c R ac c bc b c R ac c bc b R c b a R

Assuming solutions of the form

n n

r R  ) ) 2 ( ) 2 ( ) 2 ( )( (

4 2 2 2 3 4

           c r c b a c r b ac r c b a r c r ) ( ) ( '   

n n

q p

n n

nR q  ) (

Need:

Multi-breather stability in the extended Peyrard-Bishop model

The order of the polynomial can be reduced by the transformation

2 4 1 1

2 2

          y y s ys s s s y

The palindromic polynomial has real coefficients: the roots come in real pairs that are reciprocal, in complex pairs that lie on the unit circle., or in quadruplets

Rescale:

c r s  ) 1 1 2 2 1 2 )( 1 (

2 2 3 4

                                     s c b c a s c a c b s c b c a s s

Palindromic

2 2 1 2

2 2

                           c a c b y c b c a y       x x x x 1 , 1 , ,

Multi-breather stability in the extended Peyrard-Bishop model ))] sin( ) cos( ( 2 [ ))] sin( ) )(cos( ( )) sin( ) )(cos( ( [

32 31 2 1 32 31 32 31 2 1

      n c n c r c r c c c n i n ic c n i n ic c r c r c c c R

n a n a n n a n a n n

             

 

For couplings that satisfy a>b>c : there are 2 real reciprocal roots and one pair on the unit circle: Bounded term The real root with absolute value>1 dominates for large n.

Other kinds of multibreathers When there are intervals not entirely contained in or then both positive and negative eigenvalues exist, hence the multibreathers are unstable.

] , ( ) , [ 

For the model with helicity and dipole-dipole interactions the same approach yields that unless the code is ±[1 1 1 1 … 1 1] the breather is unstable.

)) ( cos( ) ( '

2 2 3

    



   m N n N n m tw n n

y m n m n y V y    

Morse potential



tw



Other methods: reduction to the nonlinear Schrodinger case, when the coupling is strong, using (semi)-continuum limit approximations. Wattis, J. Phys. A, Math. Gen. 31, 3301-3323 (1998) (second neighbors and diagonal coupling) Techera, Daeman, and Prohofsky, PRA 41, 4543 (1990) (second neighbors) Peyrard and Bishop, PRL 62, 2755 (1989)

4 27 3 4 12 ) 9 4 ( ) ( '

3 2 1 4 3 2 1 2

             

xxxx xx tt

k k k h k k k h V    



   

2 2 2

) ( 2 2 1 ) (

xx x t

B A V t H     ) ( '    

xxxx xx tt

B A V     ,  B A

Strong coupling between oscillators

                                      

2 3 2 2 1 2 2 2 3 2 2 2 1 2 2

) 1 ) cos( 2 ( 2 cos 4 2 sin 4 2 2 3 sin 4 ) ( sin 4 2 sin 4 2 k k k k k k Da k k k k k k Da 

4 3 2 1 2 3 2 1 2 2

) 81 16 ( 12 1 ) 9 4 ( 2 k k k k k k k k Da        

Multiple scale analysis: leads to an NLS equation enough terms to capture the effect

• f the discreteness on the shape of the breather

. . ) 5 5 5 3 5 ( 5 ) 4 4 4 2 4 ( 4 ) 3 3 3 ( 3 ) 2 2 2 ( 2

1

c c L t i e J t i e F t i e K t i e H t i e G J t i e F t i e H t i e G F t i e                           

) 1 (

4 2 4 4 3 3 2 2 1 2

          

xxxx xx tt

           

Hierarchy of equations

,... , ,... , ,

3 3 1 4 4 2 2

x x x x t t t t t t          etc t x t x F F ), , , , (

4 3 2 1 1 1 

      ) cos( )) ( sech {( ) ( sech } 3 1 1 { ) )[cos( ( sech 2

2 2 2 1

t a a A t A           )}] 3 cos( ) ( sech ) 3 2 ( 24 1

2 2 2 1 2

t A     

Parameter depended

Effect of longer range interactions

NN 3NN

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