# American Institute of Mathematical Sciences

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 and 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), 1262-1264. doi: 10.1103/PhysRevLett.30.1262. [2] R. L. Anderson and N. H. Ibragimov, "Lie-Bäcklund Transformations in Applications," SIAM, Philadelphia, 1979. [3] A. V. Bäcklund, Om ytor med konstant negativ krökning, Lunds Universitets Årsskrift Avd., 19 (1883), 1-48. [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), 423-446. doi: 10.1007/BF01218485. [5] I. V. Barashenkov and O. F. Oxtoby, Wobbling kinks in $\phi^4$ theory, Phys. Rev. E, 80 (2009), 026608-1-026608-9. doi: 10.1103/PhysRevE.80.026608. [6] L. Bianchi, Sulla transformazione di Bäcklund per le superficie pseudosferiche, Rend. Lincei, 5 (1892), 3-12. [7] R. Boesch and C. R. Willis, Existence of an internal quasimode for a sine-Gordon soliton, Phys. Rev. B, 42 (1990), 2290-2306. doi: 10.1103/PhysRevB.42.2290. [8] R. K. Bullough and R. K. Dodd, Solitons in mathematics: Brief history, in "Solitons and Condensed Matter Physics" (edited by A. R. Bishop and T. Schneider), Springer-Verlag, 1978. [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), 1-32. doi: 10.1016/0167-2789(83)90289-0. [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), 047. [11] S. Cuenda and A. Sánchez, Length scale competition in nonlinear Klein-Gordon models: A collective coordinate approach, Chaos, 15 (2005), 023502. doi: 10.1063/1.1876632. [12] S. Cuenda and A. Sánchez, Kink dynamics in spatially inhomogeneous media: The role of internal modes, Phys. Rev. E, 75 (2007), 036611. doi: 10.1103/PhysRevE.75.036611. [13] L. Debnath., "Nonlinear Partial Differential Equations for Scientists and Engineers," Birkhäuser, Boston, 1997. [14] P. G. Drazin, "Solitons," London Math. Soc. Lecture Note Ser., 85, Cambridge University Press, 1983. [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), 036613-1-036613-9. [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), 1578-1592. doi: 10.1103/PhysRevB.15.1578. [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), 1095-1097. doi: 10.1103/PhysRevLett.19.1095. [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), 094509-1-094509-13. doi: 10.1103/PhysRevB.80.094509. [19] G. Kälbermann, The sine-Gordon wobble, J. Phys. A: Math. Gen., 37 (2004), 11603-11612. [20] Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny and M. Peyrard, Internal modes of solitary waves, Phys. Rev. Lett., 80 (1997), 5032-5035. doi: 10.1103/PhysRevLett.80.5032. [21] Yu. S. Kivshar, F. Zhang and L. Vázquez, Resonant soliton-impurity interactions, Phys. Rev. Lett., 67 (1991), 1177-1180. doi: 10.1103/PhysRevLett.67.1177. [22] G. L. Lamb, "Elements of Soliton Theory," John Wiley, New York, 1980. [23] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490. doi: 10.1002/cpa.3160210503. [24] O. F. Oxtoby and I. V. Barashenkov, Resonantly driven wobbling kinks, Phys. Rev. E, 80 (2009), 026609-1-026609-17. doi: 10.1103/PhysRevE.80.026609. [25] M. Peyrard and D. K. Campbell, Kink-antikink interactions in a modified sine-Gordon model, Physica D, 9 (1983), 33-51. doi: 10.1016/0167-2789(83)90290-7. [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), 2886-2899. doi: 10.1103/PhysRevB.26.2886. [27] N. R. Quintero and P. G. Kevrekidis, Nonequivalence of phonon modes in the sine-Gordon equation, Phys. Rev. E, 64 (2001), 056608-1-056608-4. doi: 10.1103/PhysRevE.64.056608. [28] N. R. Quintero, A. Sánchez and F. Mertens, Anomalous resonance phenomena of solitary waves with internal modes, Phys. Rev. Lett., 84 (2000), 871-874. doi: 10.1103/PhysRevLett.84.871. [29] N. R. Quintero, A. Sánchez and F. Mertens, Existence of internal modes of sine-Gordon kinks, Phys. Rev. E, 62 (2000), R60-R63. doi: 10.1103/PhysRevE.62.R60. [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), 068602-1-068602-3. doi: 10.1103/PhysRevE.73.068602. [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. [32] J. Rubinstein, Sine-Gordon equation, J. Math. Phys., 11 (1970), 258-266. doi: 10.1063/1.1665057. [33] A. Sánchez and A. R. Bishop, Collective coordinates and length-scale competition in spatially inhomogeneous soliton-bearing equations, SIAM Rev., 40 (1998), 579-615. doi: 10.1137/S0036144597317418. [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), 4603-4615. doi: 10.1103/PhysRevE.49.4603. [35] A. C. Scott, F. Y. F. Chu and D. W. McLaughlin, The soliton - A new concept in applied science, Proc. IEEE, 61 (1973), 1443-1483. doi: 10.1109/PROC.1973.9296. [36] H. Segur, Wobbling kinks in $\varphi ^{4}$ and sine-Gordon theory, J. Math. Phys., 24 (1983), 1439-1443. doi: 10.1063/1.525867. [37] B. Yoon, Infinite sequence of conserved currents in the sine-Gordon theory, Phys. Rev. D, 13 (1976), 3440-3445. doi: 10.1103/PhysRevD.13.3440. [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), 240-243. doi: 10.1103/PhysRevLett.15.240. [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), 62-69.

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), 1262-1264. doi: 10.1103/PhysRevLett.30.1262. [2] R. L. Anderson and N. H. Ibragimov, "Lie-Bäcklund Transformations in Applications," SIAM, Philadelphia, 1979. [3] A. V. Bäcklund, Om ytor med konstant negativ krökning, Lunds Universitets Årsskrift Avd., 19 (1883), 1-48. [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), 423-446. doi: 10.1007/BF01218485. [5] I. V. Barashenkov and O. F. Oxtoby, Wobbling kinks in $\phi^4$ theory, Phys. Rev. E, 80 (2009), 026608-1-026608-9. doi: 10.1103/PhysRevE.80.026608. [6] L. Bianchi, Sulla transformazione di Bäcklund per le superficie pseudosferiche, Rend. Lincei, 5 (1892), 3-12. [7] R. Boesch and C. R. Willis, Existence of an internal quasimode for a sine-Gordon soliton, Phys. Rev. B, 42 (1990), 2290-2306. doi: 10.1103/PhysRevB.42.2290. [8] R. K. Bullough and R. K. Dodd, Solitons in mathematics: Brief history, in "Solitons and Condensed Matter Physics" (edited by A. R. Bishop and T. Schneider), Springer-Verlag, 1978. [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), 1-32. doi: 10.1016/0167-2789(83)90289-0. [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), 047. [11] S. Cuenda and A. Sánchez, Length scale competition in nonlinear Klein-Gordon models: A collective coordinate approach, Chaos, 15 (2005), 023502. doi: 10.1063/1.1876632. [12] S. Cuenda and A. Sánchez, Kink dynamics in spatially inhomogeneous media: The role of internal modes, Phys. Rev. E, 75 (2007), 036611. doi: 10.1103/PhysRevE.75.036611. [13] L. Debnath., "Nonlinear Partial Differential Equations for Scientists and Engineers," Birkhäuser, Boston, 1997. [14] P. G. Drazin, "Solitons," London Math. Soc. Lecture Note Ser., 85, Cambridge University Press, 1983. [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), 036613-1-036613-9. [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), 1578-1592. doi: 10.1103/PhysRevB.15.1578. [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), 1095-1097. doi: 10.1103/PhysRevLett.19.1095. [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), 094509-1-094509-13. doi: 10.1103/PhysRevB.80.094509. [19] G. Kälbermann, The sine-Gordon wobble, J. Phys. A: Math. Gen., 37 (2004), 11603-11612. [20] Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny and M. Peyrard, Internal modes of solitary waves, Phys. Rev. Lett., 80 (1997), 5032-5035. doi: 10.1103/PhysRevLett.80.5032. [21] Yu. S. Kivshar, F. Zhang and L. Vázquez, Resonant soliton-impurity interactions, Phys. Rev. Lett., 67 (1991), 1177-1180. doi: 10.1103/PhysRevLett.67.1177. [22] G. L. Lamb, "Elements of Soliton Theory," John Wiley, New York, 1980. [23] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490. doi: 10.1002/cpa.3160210503. [24] O. F. Oxtoby and I. V. Barashenkov, Resonantly driven wobbling kinks, Phys. Rev. E, 80 (2009), 026609-1-026609-17. doi: 10.1103/PhysRevE.80.026609. [25] M. Peyrard and D. K. Campbell, Kink-antikink interactions in a modified sine-Gordon model, Physica D, 9 (1983), 33-51. doi: 10.1016/0167-2789(83)90290-7. [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), 2886-2899. doi: 10.1103/PhysRevB.26.2886. [27] N. R. Quintero and P. G. Kevrekidis, Nonequivalence of phonon modes in the sine-Gordon equation, Phys. Rev. E, 64 (2001), 056608-1-056608-4. doi: 10.1103/PhysRevE.64.056608. [28] N. R. Quintero, A. Sánchez and F. Mertens, Anomalous resonance phenomena of solitary waves with internal modes, Phys. Rev. Lett., 84 (2000), 871-874. doi: 10.1103/PhysRevLett.84.871. [29] N. R. Quintero, A. Sánchez and F. Mertens, Existence of internal modes of sine-Gordon kinks, Phys. Rev. E, 62 (2000), R60-R63. doi: 10.1103/PhysRevE.62.R60. [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), 068602-1-068602-3. doi: 10.1103/PhysRevE.73.068602. [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. [32] J. Rubinstein, Sine-Gordon equation, J. Math. Phys., 11 (1970), 258-266. doi: 10.1063/1.1665057. [33] A. Sánchez and A. R. Bishop, Collective coordinates and length-scale competition in spatially inhomogeneous soliton-bearing equations, SIAM Rev., 40 (1998), 579-615. doi: 10.1137/S0036144597317418. [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), 4603-4615. doi: 10.1103/PhysRevE.49.4603. [35] A. C. Scott, F. Y. F. Chu and D. W. McLaughlin, The soliton - A new concept in applied science, Proc. IEEE, 61 (1973), 1443-1483. doi: 10.1109/PROC.1973.9296. [36] H. Segur, Wobbling kinks in $\varphi ^{4}$ and sine-Gordon theory, J. Math. Phys., 24 (1983), 1439-1443. doi: 10.1063/1.525867. [37] B. Yoon, Infinite sequence of conserved currents in the sine-Gordon theory, Phys. Rev. D, 13 (1976), 3440-3445. doi: 10.1103/PhysRevD.13.3440. [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), 240-243. doi: 10.1103/PhysRevLett.15.240. [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), 62-69.
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