American Institute of Mathematical Sciences

Advanced
October  2011, 4(5): 995-1006. doi: 10.3934/dcdss.2011.4.995

Nonsymmetric moving breather collisions in the Peyrard-Bishop DNA model

Azucena Álvarez 1, , Francisco R. Romero 1, , José M. Romero 2, and  Juan F. R. Archilla 2,

1. 

Grupo de Física No Lineal. Facultad de Física. Universidad de Sevilla, Avda. Reina Mercedes, s/n. 41012-Sevilla, Spain, Spain
2. 

Grupo de Física No Lineal. ETSII. Universidad de Sevilla, Avda. Reina Mercedes, s/n. 41012-Sevilla, Spain, Spain

Received  October 2009 Revised  February 2010 Published  December 2010

We study nonsymmetric collisions of moving breathers (MBs) in the Peyrard-Bishop DNA model. In this paper we have considered the following types of nonsymmetric collisions: head-on collisions of two breathers traveling with different velocities; collisions of moving breathers with a stationary trapped breather; and collisions of moving breathers traveling with the same direction. The various main observed phenomena are: one moving breather gets trapped at the collision region, and the other one is reflected; breather fusion without trapping, with the appearance of a new moving breather; and breather generation without trapping, with the appearance of new moving breathers traveling either with the same or different directions. For comparison we have included some results of a previous paper concerning to symmetric collisions, where two identical moving breathers traveling with opposite velocities collide. For symmetric collisions, the main observed phenomena are: breather generation with trapping, with the appearance of two new moving breathers with opposite velocities and a stationary breather trapped at the collision region; and breather generation without trapping, with the appearance of new moving breathers with opposite velocities. A common feature for all types of collisions is that the collision outcome depends on the internal structure of the moving breathers and the exact number of pair-bases that initially separates the stationary breathers when they are perturbed. As some nonsymmetric collisions result in the generation of a new stationary trapped breather of larger energy, the trapping phenomenon could play an important part of the complex mechanisms involved in the initiation of the DNA transcription processes.
Keywords: Peyrard-Bishop DNA moedel, Moving breathers., Klein-Gordon chains, DNA.
Mathematics Subject Classification: Primary: 70Kxx, 82C22; Secondary: 92D2.
Citation: Azucena Álvarez, Francisco R. Romero, José M. Romero, Juan F. R. Archilla. Nonsymmetric moving breather collisions in the Peyrard-Bishop DNA model. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 995-1006. doi: 10.3934/dcdss.2011.4.995
References:
[1]

Focus issue edited by S. Flach and R. S. Mackay, Localization in nonlinear lattices, Physica D, 119 (1999), 1.

[2]

Focus issue edited by Yu S. Kivshar and S. Flach, Nonlinear localized modes: Physics and applications, Chaos, 13 (2003), 586.

[3]

Focus issue edited by T. Dauxois, R. S. Mackay and G. P. Tsironis, Condensed matter, dynamical systems and biophysics, Physica D, 216 (2006), 1.

[4]

A. Alvarez, F. R. Romero, J. F. R. Archilla, J. Cuevas and P. V. Larsen, Breather trapping and breather tansmission in a DNA model with an interface, Eur. Phys. J. B, 51 (2006), 119. doi: doi:10.1140/epjb/e2006-00191-0.

[5]

A. Alvarez, F. R. Romero, J. Cuevas and J. F. R. Archilla, Discrete moving breather collisions in a Klein-Gordon chain of oscillators, Phys. Lett. A, 372 (2008), 1256. doi: doi:10.1016/j.physleta.2007.09.035.

[6]

A. Alvarez, F. R. Romero, J. Cuevas and J. F. R. Archilla, Moving breather collisions in Klein-Gordon chains of oscillators, Eur. Phys. J. B, 70 (2009), 543. doi: doi:10.1140/epjb/e2009-00256-6.

[7]

S. Aubry, Breathers in nonlinear lattices: Existence, linear stability and quantization, Physica D, 103 (1997), 201. doi: doi:10.1016/S0167-2789(96)00261-8.

[8]

S. Aubry and T. Cretegny, Mobility and reactivity of discrete breathers, Physica D, 119 (1998), 34. doi: doi:10.1016/S0167-2789(98)00062-1.

[9]

D. Campbell, J. Schonfeld and C. Wingate, Resonance structure in kink-antikink interactions in $\phi^4$ theory, Physica D, 9 (1983), 1. doi: doi:10.1016/0167-2789(83)90289-0.

[10]

D. Chen, S. Aubry and G. P. Tsironis, Breather mobility in discrete $\varphi^4$ nonlinear lattices, Phys. Rev. Lett., 77 (1996), 4776. doi: doi:10.1103/PhysRevLett.77.4776.

[11]

J. Cuevas, J. F. R. Archilla, Yu. B. Gaididei and F. R. Romero, Moving breathers in a DNA model with competing short and long range dispersive interactions, Physica D, 163 (2002), 106. doi: doi:10.1016/S0167-2789(02)00351-2.

[12]

J. Cuevas and J. C. Eilbeck, Soliton collisions in a waveguide array with saturable nonlinearity, Phys. Lett. A, 358 (2006), 15. doi: doi:10.1016/j.physleta.2006.04.095.

[13]

J. Cuevas, F. Palmero, J. F. R. Archilla and F. R. Romero, Moving breathers in a bent DNA-related model, Phys. Lett. A, 299 (2002), 221. doi: doi:10.1016/S0375-9601(02)00731-4.

[14]

J. Cuevas, F. Palmero, J. F. R. Archilla and F. R. Romero, Moving discrete breathers in a Klein-Gordon chain with an impurity, J. Phys. A: Math. and Gen., 35 (2002), 10519. doi: doi:10.1088/0305-4470/35/49/302.

[15]

T. Dauxois and M. Peyrard, Energy localization in nonlinear lattices, Phys. Rev. Lett., 70 (1993), 3935. doi: doi:10.1103/PhysRevLett.70.3935.

[16]

T. Dauxois, M. Peyrard and C. R. Willis, Localized breather-like solutions in a discrete Klein-Gordon model and application to DNA, Physica D, 57 (1992), 267. doi: doi:10.1016/0167-2789(92)90003-6.

[17]

S. Dmitriev, P. Kevrekidis and Y. Kivshar, Radiationless energy exchange in three-soliton collisions, Phys. Rev. E, 78 (2008), 046604. doi: doi:10.1103/PhysRevE.78.046604.

[18]

Y. Doi, Energy exchange in collisions of intrinsic localized modes, Phys. Rev. E, 68 (2003), 066608. doi: doi:10.1103/PhysRevE.68.066608.

[19]

S. Flach and C. R. Willis, Discrete breathers, Phys. Rep., 295 (1998), 181. doi: doi:10.1016/S0370-1573(97)00068-9.

[20]

K. Forinash, T. Cretegny and M. Peyrard, Local modes and localization in amulticomponent nonlinear lattice, Phys. Rev. E, 55 (1997), 4740. doi: doi:10.1103/PhysRevE.55.4740.

[21]

K. Forinash, M. Peyrard and B. A. Malomed, Interaction of discrete breathers with impurity modes, Phys. Rev. E, 49(1994), 3400. doi: doi:10.1103/PhysRevE.49.3400.

[22]

P. G. Kevrekidis, K. Ø. Rasmussen and A. R. Bishop, The discrete nonlinear Schrödinger equation: A survey of recent results, Int. J. Mod. Phys. B, 15 (2001), 2833. doi: doi:10.1142/S0217979201007105.

[23]

R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7 (1994), 1623. doi: doi:10.1088/0951-7715/7/6/006.

[24]

J. L. Marín and S. Aubry, Breathers in nonlinear lattices: Numerical calculation from the anticontinuous limit, Nonlinearity, 9 (1996), 1501. doi: doi:10.1088/0951-7715/9/6/007.

[25]

M. Meister and L. M. Floría, Bound states of breathers in the Frenkel-Kontorova model, Eur. Phys. J. B, 37 (2004), 213. doi: doi:10.1140/epjb/e2004-00049-5.

[26]

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed and D. J. Frantzeskakis, Soliton collisions in discrete nonlinear Schrödinger equation, Phys. Rev. E, 68 (2003), 046604. doi: doi:10.1103/PhysRevE.68.046604.

[27]

M. Peyrard and A. R. Bishop, Statistical mechanics of a nonlinear model for DNA denaturation, Phys. Rev. Lett., 62 (1989), 2755. doi: doi:10.1103/PhysRevLett.62.2755.

[28]

B. Sánchez-Rey, G. James, J. Cuevas and J. F. R. Archilla, Bright and dark breathers in Fermi-Pasta-Ulam lattices, Phys. Rev. B, 70 (2004), 014301. doi: doi:10.1103/PhysRevB.70.014301.

[29]

M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Chapman and Hall, 1994.

[30]

A. J. Sievers and S. Takeno, Intrinsic localized modes in anharmonic crystals, Phys. Rev. Lett., 61 (1988), 970. doi: doi:10.1103/PhysRevLett.61.970.

show all references

References:
[1]

Focus issue edited by S. Flach and R. S. Mackay, Localization in nonlinear lattices, Physica D, 119 (1999), 1.

[2]

Focus issue edited by Yu S. Kivshar and S. Flach, Nonlinear localized modes: Physics and applications, Chaos, 13 (2003), 586.

[3]

Focus issue edited by T. Dauxois, R. S. Mackay and G. P. Tsironis, Condensed matter, dynamical systems and biophysics, Physica D, 216 (2006), 1.

[4]

A. Alvarez, F. R. Romero, J. F. R. Archilla, J. Cuevas and P. V. Larsen, Breather trapping and breather tansmission in a DNA model with an interface, Eur. Phys. J. B, 51 (2006), 119. doi: doi:10.1140/epjb/e2006-00191-0.

[5]

A. Alvarez, F. R. Romero, J. Cuevas and J. F. R. Archilla, Discrete moving breather collisions in a Klein-Gordon chain of oscillators, Phys. Lett. A, 372 (2008), 1256. doi: doi:10.1016/j.physleta.2007.09.035.

[6]

A. Alvarez, F. R. Romero, J. Cuevas and J. F. R. Archilla, Moving breather collisions in Klein-Gordon chains of oscillators, Eur. Phys. J. B, 70 (2009), 543. doi: doi:10.1140/epjb/e2009-00256-6.

[7]

S. Aubry, Breathers in nonlinear lattices: Existence, linear stability and quantization, Physica D, 103 (1997), 201. doi: doi:10.1016/S0167-2789(96)00261-8.

[8]

S. Aubry and T. Cretegny, Mobility and reactivity of discrete breathers, Physica D, 119 (1998), 34. doi: doi:10.1016/S0167-2789(98)00062-1.

[9]

D. Campbell, J. Schonfeld and C. Wingate, Resonance structure in kink-antikink interactions in $\phi^4$ theory, Physica D, 9 (1983), 1. doi: doi:10.1016/0167-2789(83)90289-0.

[10]

D. Chen, S. Aubry and G. P. Tsironis, Breather mobility in discrete $\varphi^4$ nonlinear lattices, Phys. Rev. Lett., 77 (1996), 4776. doi: doi:10.1103/PhysRevLett.77.4776.

[11]

J. Cuevas, J. F. R. Archilla, Yu. B. Gaididei and F. R. Romero, Moving breathers in a DNA model with competing short and long range dispersive interactions, Physica D, 163 (2002), 106. doi: doi:10.1016/S0167-2789(02)00351-2.

[12]

J. Cuevas and J. C. Eilbeck, Soliton collisions in a waveguide array with saturable nonlinearity, Phys. Lett. A, 358 (2006), 15. doi: doi:10.1016/j.physleta.2006.04.095.

[13]

J. Cuevas, F. Palmero, J. F. R. Archilla and F. R. Romero, Moving breathers in a bent DNA-related model, Phys. Lett. A, 299 (2002), 221. doi: doi:10.1016/S0375-9601(02)00731-4.

[14]

J. Cuevas, F. Palmero, J. F. R. Archilla and F. R. Romero, Moving discrete breathers in a Klein-Gordon chain with an impurity, J. Phys. A: Math. and Gen., 35 (2002), 10519. doi: doi:10.1088/0305-4470/35/49/302.

[15]

T. Dauxois and M. Peyrard, Energy localization in nonlinear lattices, Phys. Rev. Lett., 70 (1993), 3935. doi: doi:10.1103/PhysRevLett.70.3935.

[16]

T. Dauxois, M. Peyrard and C. R. Willis, Localized breather-like solutions in a discrete Klein-Gordon model and application to DNA, Physica D, 57 (1992), 267. doi: doi:10.1016/0167-2789(92)90003-6.

[17]

S. Dmitriev, P. Kevrekidis and Y. Kivshar, Radiationless energy exchange in three-soliton collisions, Phys. Rev. E, 78 (2008), 046604. doi: doi:10.1103/PhysRevE.78.046604.

[18]

Y. Doi, Energy exchange in collisions of intrinsic localized modes, Phys. Rev. E, 68 (2003), 066608. doi: doi:10.1103/PhysRevE.68.066608.

[19]

S. Flach and C. R. Willis, Discrete breathers, Phys. Rep., 295 (1998), 181. doi: doi:10.1016/S0370-1573(97)00068-9.

[20]

K. Forinash, T. Cretegny and M. Peyrard, Local modes and localization in amulticomponent nonlinear lattice, Phys. Rev. E, 55 (1997), 4740. doi: doi:10.1103/PhysRevE.55.4740.

[21]

K. Forinash, M. Peyrard and B. A. Malomed, Interaction of discrete breathers with impurity modes, Phys. Rev. E, 49(1994), 3400. doi: doi:10.1103/PhysRevE.49.3400.

[22]

P. G. Kevrekidis, K. Ø. Rasmussen and A. R. Bishop, The discrete nonlinear Schrödinger equation: A survey of recent results, Int. J. Mod. Phys. B, 15 (2001), 2833. doi: doi:10.1142/S0217979201007105.

[23]

R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7 (1994), 1623. doi: doi:10.1088/0951-7715/7/6/006.

[24]

J. L. Marín and S. Aubry, Breathers in nonlinear lattices: Numerical calculation from the anticontinuous limit, Nonlinearity, 9 (1996), 1501. doi: doi:10.1088/0951-7715/9/6/007.

[25]

M. Meister and L. M. Floría, Bound states of breathers in the Frenkel-Kontorova model, Eur. Phys. J. B, 37 (2004), 213. doi: doi:10.1140/epjb/e2004-00049-5.

[26]

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed and D. J. Frantzeskakis, Soliton collisions in discrete nonlinear Schrödinger equation, Phys. Rev. E, 68 (2003), 046604. doi: doi:10.1103/PhysRevE.68.046604.

[27]

M. Peyrard and A. R. Bishop, Statistical mechanics of a nonlinear model for DNA denaturation, Phys. Rev. Lett., 62 (1989), 2755. doi: doi:10.1103/PhysRevLett.62.2755.

[28]

B. Sánchez-Rey, G. James, J. Cuevas and J. F. R. Archilla, Bright and dark breathers in Fermi-Pasta-Ulam lattices, Phys. Rev. B, 70 (2004), 014301. doi: doi:10.1103/PhysRevB.70.014301.

[29]

M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Chapman and Hall, 1994.

[30]

A. J. Sievers and S. Takeno, Intrinsic localized modes in anharmonic crystals, Phys. Rev. Lett., 61 (1988), 970. doi: doi:10.1103/PhysRevLett.61.970.

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