October  2011, 4(5): 1047-1056. doi: 10.3934/dcdss.2011.4.1047

Sine-Gordon wobbles through Bäcklund transformations

1. 

Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Análisis Económico: Economía Cuantitativa, Universidad Autónoma de Madrid, Francisco Tomás y Valiente 5, 28049, Cantoblanco, Madrid, Spain

2. 

Departamento de Física Aplicada I, E. U. P., Universidad de Sevilla, Virgen de África 7, 41011, Sevilla, Spain

3. 

Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain

Received  September 2009 Revised  October 2009 Published  December 2010

In this work we construct the wobble exact solution of sine-Gordon equation by means of Bäcklund Transformations. We find the parameters of the transformations corresponding to the Bianchi diagram for the wobble as a particular $3$-soliton solutions. We show that this solution agrees with the wobbles obtained by Kälbermann and Segur by means of the Inverse Scattering Transform, and by Ferreira et al. using the Hirota method. The new formulation introduced allows to identify easily the parameters that define the building blocks of this solution -- a kink and a breather, and can be used in further studies of this solution in the perturbed sine-Gordon equation.
Citation: Sara Cuenda, Niurka R. Quintero, Angel Sánchez. Sine-Gordon wobbles through Bäcklund transformations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1047-1056. doi: 10.3934/dcdss.2011.4.1047
References:
[1]

M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Method for solving the sine-Gordon equation,, Phys. Rev. Lett., 30 (1973), 1262. doi: 10.1103/PhysRevLett.30.1262. Google Scholar

[2]

R. L. Anderson and N. H. Ibragimov, "Lie-Bäcklund Transformations in Applications,", SIAM, (1979). Google Scholar

[3]

A. V. Bäcklund, Om ytor med konstant negativ krökning,, Lunds Universitets Årsskrift Avd., 19 (1883), 1. Google Scholar

[4]

I. V. Barashenkov and B. S. Getmanov, Multisoliton solutions in the scheme for unified description of integrable massive fields,, Commun. Math. Phys., 112 (1987), 423. doi: 10.1007/BF01218485. Google Scholar

[5]

I. V. Barashenkov and O. F. Oxtoby, Wobbling kinks in $\phi^4$ theory,, Phys. Rev. E, 80 (2009), 026608. doi: 10.1103/PhysRevE.80.026608. Google Scholar

[6]

L. Bianchi, Sulla transformazione di Bäcklund per le superficie pseudosferiche,, Rend. Lincei, 5 (1892), 3. Google Scholar

[7]

R. Boesch and C. R. Willis, Existence of an internal quasimode for a sine-Gordon soliton,, Phys. Rev. B, 42 (1990), 2290. doi: 10.1103/PhysRevB.42.2290. Google Scholar

[8]

R. K. Bullough and R. K. Dodd, Solitons in mathematics: Brief history,, in, (1978). Google Scholar

[9]

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

[10]

O. V. Charkina and M. M. Bogdan, Internal modes of solitons and near-integrable highly-dispersive nonlinear systems,, Symm. Integr. and Geom., 2 (2006). Google Scholar

[11]

S. Cuenda and A. Sánchez, Length scale competition in nonlinear Klein-Gordon models: A collective coordinate approach,, Chaos, 15 (2005). doi: 10.1063/1.1876632. Google Scholar

[12]

S. Cuenda and A. Sánchez, Kink dynamics in spatially inhomogeneous media: The role of internal modes,, Phys. Rev. E, 75 (2007). doi: 10.1103/PhysRevE.75.036611. Google Scholar

[13]

L. Debnath., "Nonlinear Partial Differential Equations for Scientists and Engineers,", Birkhäuser, (1997). Google Scholar

[14]

P. G. Drazin, "Solitons,", London Math. Soc. Lecture Note Ser., 85 (1983). Google Scholar

[15]

L. A. Ferreira, B. Piette and W. Zakrzewski, Wobbles and other kink-breather solutions of the sine-Gordon model,, Phys. Rev. E, 77 (2008), 036613. Google Scholar

[16]

M. B. Fogel, S. E. Trullinger, A. R. Bishop and J. A. Krumhansl, Dynamics of sine-Gordon solitons in the presence of perturbations,, Phys. Rev. B, 15 (1977), 1578. doi: 10.1103/PhysRevB.15.1578. Google Scholar

[17]

C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-deVries equation,, Phys. Rev. Lett., 19 (1967), 1095. doi: 10.1103/PhysRevLett.19.1095. Google Scholar

[18]

D. R. Gulevich, F. V. Kusmartsev, Sergey Savel'ev, V. A. Yampol'skii and F. Nori, Shape and wobbling wave excitations in Josephson junctions: Exact solutions of the (2+1)-dimensional sine-Gordon model,, Phys. Rev. B, 80 (2009), 094509. doi: 10.1103/PhysRevB.80.094509. Google Scholar

[19]

G. Kälbermann, The sine-Gordon wobble,, J. Phys. A: Math. Gen., 37 (2004), 11603. Google Scholar

[20]

Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny and M. Peyrard, Internal modes of solitary waves,, Phys. Rev. Lett., 80 (1997), 5032. doi: 10.1103/PhysRevLett.80.5032. Google Scholar

[21]

Yu. S. Kivshar, F. Zhang and L. Vázquez, Resonant soliton-impurity interactions,, Phys. Rev. Lett., 67 (1991), 1177. doi: 10.1103/PhysRevLett.67.1177. Google Scholar

[22]

G. L. Lamb, "Elements of Soliton Theory,", John Wiley, (1980). Google Scholar

[23]

P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves,, Comm. Pure Appl. Math., 21 (1968), 467. doi: 10.1002/cpa.3160210503. Google Scholar

[24]

O. F. Oxtoby and I. V. Barashenkov, Resonantly driven wobbling kinks,, Phys. Rev. E, 80 (2009), 026609. doi: 10.1103/PhysRevE.80.026609. Google Scholar

[25]

M. Peyrard and D. K. Campbell, Kink-antikink interactions in a modified sine-Gordon model,, Physica D, 9 (1983), 33. doi: 10.1016/0167-2789(83)90290-7. Google Scholar

[26]

M. Peyrard and M. Remoissenet, Solitonlike excitations in a one-dimensional atomic chain with a nonlinear deformable substrate potential,, Phys. Rev. B, 26 (1982), 2886. doi: 10.1103/PhysRevB.26.2886. Google Scholar

[27]

N. R. Quintero and P. G. Kevrekidis, Nonequivalence of phonon modes in the sine-Gordon equation,, Phys. Rev. E, 64 (2001), 056608. doi: 10.1103/PhysRevE.64.056608. Google Scholar

[28]

N. R. Quintero, A. Sánchez and F. Mertens, Anomalous resonance phenomena of solitary waves with internal modes,, Phys. Rev. Lett., 84 (2000), 871. doi: 10.1103/PhysRevLett.84.871. Google Scholar

[29]

N. R. Quintero, A. Sánchez and F. Mertens, Existence of internal modes of sine-Gordon kinks,, Phys. Rev. E, 62 (2000). doi: 10.1103/PhysRevE.62.R60. Google Scholar

[30]

N. R. Quintero, A. Sánchez and F. Mertens, Reply to "Comment on 'Existence of internal modes of sine-Gordon kinks' ",, Phys. Rev. E, 73 (2006), 068602. doi: 10.1103/PhysRevE.73.068602. Google Scholar

[31]

C. Rogers and W. K. Schief, "Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory,", Cambridge University Press, (2002). doi: 10.1017/CBO9780511606359. Google Scholar

[32]

J. Rubinstein, Sine-Gordon equation,, J. Math. Phys., 11 (1970), 258. doi: 10.1063/1.1665057. Google Scholar

[33]

A. Sánchez and A. R. Bishop, Collective coordinates and length-scale competition in spatially inhomogeneous soliton-bearing equations,, SIAM Rev., 40 (1998), 579. doi: 10.1137/S0036144597317418. Google Scholar

[34]

A. Sánchez, A. R. Bishop and F. Domí nguez-Adame, Kink stability, propagation, and length scale competition in the periodically modulated sine-Gordon equation,, Phys. Rev. E, 49 (1994), 4603. doi: 10.1103/PhysRevE.49.4603. Google Scholar

[35]

A. C. Scott, F. Y. F. Chu and D. W. McLaughlin, The soliton - A new concept in applied science,, Proc. IEEE, 61 (1973), 1443. doi: 10.1109/PROC.1973.9296. Google Scholar

[36]

H. Segur, Wobbling kinks in $\varphi ^{4}$ and sine-Gordon theory,, J. Math. Phys., 24 (1983), 1439. doi: 10.1063/1.525867. Google Scholar

[37]

B. Yoon, Infinite sequence of conserved currents in the sine-Gordon theory,, Phys. Rev. D, 13 (1976), 3440. doi: 10.1103/PhysRevD.13.3440. Google Scholar

[38]

N. J. Zabusky and M. D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states,, Phys. Rev. Lett., 15 (1965), 240. doi: 10.1103/PhysRevLett.15.240. Google Scholar

[39]

V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,, Soviet Phys. JETP, 34 (1972), 62. Google Scholar

show all references

References:
[1]

M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Method for solving the sine-Gordon equation,, Phys. Rev. Lett., 30 (1973), 1262. doi: 10.1103/PhysRevLett.30.1262. Google Scholar

[2]

R. L. Anderson and N. H. Ibragimov, "Lie-Bäcklund Transformations in Applications,", SIAM, (1979). Google Scholar

[3]

A. V. Bäcklund, Om ytor med konstant negativ krökning,, Lunds Universitets Årsskrift Avd., 19 (1883), 1. Google Scholar

[4]

I. V. Barashenkov and B. S. Getmanov, Multisoliton solutions in the scheme for unified description of integrable massive fields,, Commun. Math. Phys., 112 (1987), 423. doi: 10.1007/BF01218485. Google Scholar

[5]

I. V. Barashenkov and O. F. Oxtoby, Wobbling kinks in $\phi^4$ theory,, Phys. Rev. E, 80 (2009), 026608. doi: 10.1103/PhysRevE.80.026608. Google Scholar

[6]

L. Bianchi, Sulla transformazione di Bäcklund per le superficie pseudosferiche,, Rend. Lincei, 5 (1892), 3. Google Scholar

[7]

R. Boesch and C. R. Willis, Existence of an internal quasimode for a sine-Gordon soliton,, Phys. Rev. B, 42 (1990), 2290. doi: 10.1103/PhysRevB.42.2290. Google Scholar

[8]

R. K. Bullough and R. K. Dodd, Solitons in mathematics: Brief history,, in, (1978). Google Scholar

[9]

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

[10]

O. V. Charkina and M. M. Bogdan, Internal modes of solitons and near-integrable highly-dispersive nonlinear systems,, Symm. Integr. and Geom., 2 (2006). Google Scholar

[11]

S. Cuenda and A. Sánchez, Length scale competition in nonlinear Klein-Gordon models: A collective coordinate approach,, Chaos, 15 (2005). doi: 10.1063/1.1876632. Google Scholar

[12]

S. Cuenda and A. Sánchez, Kink dynamics in spatially inhomogeneous media: The role of internal modes,, Phys. Rev. E, 75 (2007). doi: 10.1103/PhysRevE.75.036611. Google Scholar

[13]

L. Debnath., "Nonlinear Partial Differential Equations for Scientists and Engineers,", Birkhäuser, (1997). Google Scholar

[14]

P. G. Drazin, "Solitons,", London Math. Soc. Lecture Note Ser., 85 (1983). Google Scholar

[15]

L. A. Ferreira, B. Piette and W. Zakrzewski, Wobbles and other kink-breather solutions of the sine-Gordon model,, Phys. Rev. E, 77 (2008), 036613. Google Scholar

[16]

M. B. Fogel, S. E. Trullinger, A. R. Bishop and J. A. Krumhansl, Dynamics of sine-Gordon solitons in the presence of perturbations,, Phys. Rev. B, 15 (1977), 1578. doi: 10.1103/PhysRevB.15.1578. Google Scholar

[17]

C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-deVries equation,, Phys. Rev. Lett., 19 (1967), 1095. doi: 10.1103/PhysRevLett.19.1095. Google Scholar

[18]

D. R. Gulevich, F. V. Kusmartsev, Sergey Savel'ev, V. A. Yampol'skii and F. Nori, Shape and wobbling wave excitations in Josephson junctions: Exact solutions of the (2+1)-dimensional sine-Gordon model,, Phys. Rev. B, 80 (2009), 094509. doi: 10.1103/PhysRevB.80.094509. Google Scholar

[19]

G. Kälbermann, The sine-Gordon wobble,, J. Phys. A: Math. Gen., 37 (2004), 11603. Google Scholar

[20]

Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny and M. Peyrard, Internal modes of solitary waves,, Phys. Rev. Lett., 80 (1997), 5032. doi: 10.1103/PhysRevLett.80.5032. Google Scholar

[21]

Yu. S. Kivshar, F. Zhang and L. Vázquez, Resonant soliton-impurity interactions,, Phys. Rev. Lett., 67 (1991), 1177. doi: 10.1103/PhysRevLett.67.1177. Google Scholar

[22]

G. L. Lamb, "Elements of Soliton Theory,", John Wiley, (1980). Google Scholar

[23]

P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves,, Comm. Pure Appl. Math., 21 (1968), 467. doi: 10.1002/cpa.3160210503. Google Scholar

[24]

O. F. Oxtoby and I. V. Barashenkov, Resonantly driven wobbling kinks,, Phys. Rev. E, 80 (2009), 026609. doi: 10.1103/PhysRevE.80.026609. Google Scholar

[25]

M. Peyrard and D. K. Campbell, Kink-antikink interactions in a modified sine-Gordon model,, Physica D, 9 (1983), 33. doi: 10.1016/0167-2789(83)90290-7. Google Scholar

[26]

M. Peyrard and M. Remoissenet, Solitonlike excitations in a one-dimensional atomic chain with a nonlinear deformable substrate potential,, Phys. Rev. B, 26 (1982), 2886. doi: 10.1103/PhysRevB.26.2886. Google Scholar

[27]

N. R. Quintero and P. G. Kevrekidis, Nonequivalence of phonon modes in the sine-Gordon equation,, Phys. Rev. E, 64 (2001), 056608. doi: 10.1103/PhysRevE.64.056608. Google Scholar

[28]

N. R. Quintero, A. Sánchez and F. Mertens, Anomalous resonance phenomena of solitary waves with internal modes,, Phys. Rev. Lett., 84 (2000), 871. doi: 10.1103/PhysRevLett.84.871. Google Scholar

[29]

N. R. Quintero, A. Sánchez and F. Mertens, Existence of internal modes of sine-Gordon kinks,, Phys. Rev. E, 62 (2000). doi: 10.1103/PhysRevE.62.R60. Google Scholar

[30]

N. R. Quintero, A. Sánchez and F. Mertens, Reply to "Comment on 'Existence of internal modes of sine-Gordon kinks' ",, Phys. Rev. E, 73 (2006), 068602. doi: 10.1103/PhysRevE.73.068602. Google Scholar

[31]

C. Rogers and W. K. Schief, "Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory,", Cambridge University Press, (2002). doi: 10.1017/CBO9780511606359. Google Scholar

[32]

J. Rubinstein, Sine-Gordon equation,, J. Math. Phys., 11 (1970), 258. doi: 10.1063/1.1665057. Google Scholar

[33]

A. Sánchez and A. R. Bishop, Collective coordinates and length-scale competition in spatially inhomogeneous soliton-bearing equations,, SIAM Rev., 40 (1998), 579. doi: 10.1137/S0036144597317418. Google Scholar

[34]

A. Sánchez, A. R. Bishop and F. Domí nguez-Adame, Kink stability, propagation, and length scale competition in the periodically modulated sine-Gordon equation,, Phys. Rev. E, 49 (1994), 4603. doi: 10.1103/PhysRevE.49.4603. Google Scholar

[35]

A. C. Scott, F. Y. F. Chu and D. W. McLaughlin, The soliton - A new concept in applied science,, Proc. IEEE, 61 (1973), 1443. doi: 10.1109/PROC.1973.9296. Google Scholar

[36]

H. Segur, Wobbling kinks in $\varphi ^{4}$ and sine-Gordon theory,, J. Math. Phys., 24 (1983), 1439. doi: 10.1063/1.525867. Google Scholar

[37]

B. Yoon, Infinite sequence of conserved currents in the sine-Gordon theory,, Phys. Rev. D, 13 (1976), 3440. doi: 10.1103/PhysRevD.13.3440. Google Scholar

[38]

N. J. Zabusky and M. D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states,, Phys. Rev. Lett., 15 (1965), 240. doi: 10.1103/PhysRevLett.15.240. Google Scholar

[39]

V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,, Soviet Phys. JETP, 34 (1972), 62. Google Scholar

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