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The bifurcations of solitary and kink waves described by the Gardner equation
Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion
1. | Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524048 |
References:
[1] |
D. Avitabile, D. J. B. Lloyd, J. Burke, E. Knobloch and B. Sandstede, To snake or not to snake in the planar Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 9 (2010), 704-733.
doi: 10.1137/100782747. |
[2] |
M. Beck, J. Knobloch, D. J. B. Lloyd, B. Sandstede and T. Wagenknecht, Snakes, ladders, and isolas of localized patterns, SIAM J. Math. Anal., 41 (2009), 936-972.
doi: 10.1137/080713306. |
[3] |
D. Blair, I. S. Aranson, G. W. Crabtree, V. Vinokur, L. S. Tsimring and C. Josserand, Patterns in thin vibrated granular layers: Interfaces, hexagons, and superoscillons, Phys. Rev. E, 61 (2000), 5600-5610.
doi: 10.1103/PhysRevE.61.5600. |
[4] |
B. Braaksma, G. Iooss and L. Stolovitch, Existence of quasipattern solutions of the Swift-Hohenberg equation, Arch. Ration. Mech. Anal., 209 (2013), 255-285.
doi: 10.1007/s00205-013-0627-7. |
[5] |
J. Burke, S. M. Houghton and E. Knobloch, Swift-Hohenberg equation with broken reflection symmetry, Phys. Rev. E, 80 (2009), 036202.
doi: 10.1103/PhysRevE.80.036202. |
[6] |
P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals For Engineers and Physicists, Springer-Verlag, Berlin, 1954. |
[7] |
P. Collet and J. P. Eckmann, Instabilities and Fronts in Extended Systems, Princeton University Press, Princeton, 1990.
doi: 10.1515/9781400861026. |
[8] |
S. Day, Y. Hiraoka, K. Mischaikow and T. Ogawa, Rigorous numerics for global dynamics: A study of the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 4 (2005), 1-31.
doi: 10.1137/040604479. |
[9] |
S. Deng and X. Li, Generalized homoclinic solutions for the Swift-Hohenberg equation, J. Math. Anal. Appl., 390 (2012), 15-26.
doi: 10.1016/j.jmaa.2011.11.074. |
[10] |
S. Deng and S. M. Sun, Multi-hump solutions with small oscillations at infinity for stationary Swift-Hohenberg equation, submitted. |
[11] |
J. P. Gaivão and V. Gelfreich, Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift-Hohenberg equation as an example, Nonlinearity, 24 (2011), 677-698.
doi: 10.1088/0951-7715/24/3/002. |
[12] |
P. Gandhi, C. Beaume and E. Knobloch, A new resonance mechanism in the Swift-Hohenberg rquation with time-periodic forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 860-892.
doi: 10.1137/14099468X. |
[13] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990. |
[14] |
M. Haragus and A. Scheel, Interfaces between rolls in the Swift-Hohenberg equation, Int. J. Dyn. Syst. Diff. Equ., 1 (2007), 89-97.
doi: 10.1504/IJDSDE.2007.016510. |
[15] |
G. Iooss and A. M. Rucklidge, On the existence of quasipattern solutions of the Swift-Hohenberg equation, J. Nonlinear Sci., 20 (2010), 361-394.
doi: 10.1007/s00332-010-9063-0. |
[16] |
J. Knobloch, M. Vielitz and T. Wagenknecht, Non-reversible perturbations of homoclinic snaking scenarios, Nonlinearity, 25 (2012), 3469-3485.
doi: 10.1088/0951-7715/25/12/3469. |
[17] |
N. A. Kudryashov and D. I. Sinelshchikov, Exact solutions of the Swift-Hohenberg equation with dispersion, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 26-34.
doi: 10.1016/j.cnsns.2011.04.008. |
[18] |
R. E. LaQuey, S. M. Mahajan, P. H. Rutherford and W. M. Tang, Nonlinear saturation of the trapped-ion mode, Phys. Rev. Lett., 34 (1975), 391-394.
doi: 10.1103/PhysRevLett.34.391. |
[19] |
L. Lee and H. Swinney, Lamellar structures and self-replicating spots in a reaction-diffusion system, Phys. Rev. E, 51 (1995), 1899-1915.
doi: 10.1103/PhysRevE.51.1899. |
[20] |
L. Lega, J. V. Moloney and A. C. Newell, Swift-Hohenberg equation for lasers, Phys. Rev. Lett., 73 (1994), 2978-2981.
doi: 10.1103/PhysRevLett.73.2978. |
[21] |
M. Lopez-Fernandez and S. Sauter, Fast and stable contour integration for high order divided differences via elliptic functions, Math. Comp., 84 (2015), 1291-1315.
doi: 10.1090/S0025-5718-2014-02890-1. |
[22] |
E. Makrides and B. Sandstede, Predicting the bifurcation structure of localized snaking patterns, Phys. D, 268 (2014), 59-78.
doi: 10.1016/j.physd.2013.11.009. |
[23] |
P. Mandel, Theoretical Problems in Cavity Nonlinear Optics, Cambridge University Press, Cambridge, 1997. |
[24] |
S. G. McCalla and B. Sandstede, Spots in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 12 (2013), 831-877.
doi: 10.1137/120882111. |
[25] |
A. Mielke, Instability and stability of rolls in the Swift-Hohenberg equation, Comm. Math. Phys., 189 (1997), 829-853.
doi: 10.1007/s002200050230. |
[26] |
D. Morgan and J. H. P. Dawes, The Swift-Hohenberg equation with a nonlocal nonlinearity, Phys. D, 270 (2014), 60-80.
doi: 10.1016/j.physd.2013.11.018. |
[27] |
L. A. Peletier and V. Rottschafer, Pattern selection of solutions of the Swift-Hohenberg equation, Phys. D, 194 (2004), 95-126.
doi: 10.1016/j.physd.2004.01.043. |
[28] |
L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 6 (2007), 208-235.
doi: 10.1137/050647232. |
[29] |
D. Smets and J. B. van den Berg, Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations, J. Diff. Eqns., 184 (2002), 78-96.
doi: 10.1006/jdeq.2001.4135. |
[30] |
J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328.
doi: 10.1103/PhysRevA.15.319. |
[31] |
J. B. van den Berg, L. A. Peletier and W. C. Troy, Global branches of multi-bump periodic solutions of the Swift-Hohenberg equation, Arch. Ration. Mech. Anal., 158 (2001), 91-153.
doi: 10.1007/PL00004243. |
[32] |
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Universitext, 1996.
doi: 10.1007/978-3-642-61453-8. |
show all references
References:
[1] |
D. Avitabile, D. J. B. Lloyd, J. Burke, E. Knobloch and B. Sandstede, To snake or not to snake in the planar Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 9 (2010), 704-733.
doi: 10.1137/100782747. |
[2] |
M. Beck, J. Knobloch, D. J. B. Lloyd, B. Sandstede and T. Wagenknecht, Snakes, ladders, and isolas of localized patterns, SIAM J. Math. Anal., 41 (2009), 936-972.
doi: 10.1137/080713306. |
[3] |
D. Blair, I. S. Aranson, G. W. Crabtree, V. Vinokur, L. S. Tsimring and C. Josserand, Patterns in thin vibrated granular layers: Interfaces, hexagons, and superoscillons, Phys. Rev. E, 61 (2000), 5600-5610.
doi: 10.1103/PhysRevE.61.5600. |
[4] |
B. Braaksma, G. Iooss and L. Stolovitch, Existence of quasipattern solutions of the Swift-Hohenberg equation, Arch. Ration. Mech. Anal., 209 (2013), 255-285.
doi: 10.1007/s00205-013-0627-7. |
[5] |
J. Burke, S. M. Houghton and E. Knobloch, Swift-Hohenberg equation with broken reflection symmetry, Phys. Rev. E, 80 (2009), 036202.
doi: 10.1103/PhysRevE.80.036202. |
[6] |
P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals For Engineers and Physicists, Springer-Verlag, Berlin, 1954. |
[7] |
P. Collet and J. P. Eckmann, Instabilities and Fronts in Extended Systems, Princeton University Press, Princeton, 1990.
doi: 10.1515/9781400861026. |
[8] |
S. Day, Y. Hiraoka, K. Mischaikow and T. Ogawa, Rigorous numerics for global dynamics: A study of the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 4 (2005), 1-31.
doi: 10.1137/040604479. |
[9] |
S. Deng and X. Li, Generalized homoclinic solutions for the Swift-Hohenberg equation, J. Math. Anal. Appl., 390 (2012), 15-26.
doi: 10.1016/j.jmaa.2011.11.074. |
[10] |
S. Deng and S. M. Sun, Multi-hump solutions with small oscillations at infinity for stationary Swift-Hohenberg equation, submitted. |
[11] |
J. P. Gaivão and V. Gelfreich, Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift-Hohenberg equation as an example, Nonlinearity, 24 (2011), 677-698.
doi: 10.1088/0951-7715/24/3/002. |
[12] |
P. Gandhi, C. Beaume and E. Knobloch, A new resonance mechanism in the Swift-Hohenberg rquation with time-periodic forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 860-892.
doi: 10.1137/14099468X. |
[13] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990. |
[14] |
M. Haragus and A. Scheel, Interfaces between rolls in the Swift-Hohenberg equation, Int. J. Dyn. Syst. Diff. Equ., 1 (2007), 89-97.
doi: 10.1504/IJDSDE.2007.016510. |
[15] |
G. Iooss and A. M. Rucklidge, On the existence of quasipattern solutions of the Swift-Hohenberg equation, J. Nonlinear Sci., 20 (2010), 361-394.
doi: 10.1007/s00332-010-9063-0. |
[16] |
J. Knobloch, M. Vielitz and T. Wagenknecht, Non-reversible perturbations of homoclinic snaking scenarios, Nonlinearity, 25 (2012), 3469-3485.
doi: 10.1088/0951-7715/25/12/3469. |
[17] |
N. A. Kudryashov and D. I. Sinelshchikov, Exact solutions of the Swift-Hohenberg equation with dispersion, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 26-34.
doi: 10.1016/j.cnsns.2011.04.008. |
[18] |
R. E. LaQuey, S. M. Mahajan, P. H. Rutherford and W. M. Tang, Nonlinear saturation of the trapped-ion mode, Phys. Rev. Lett., 34 (1975), 391-394.
doi: 10.1103/PhysRevLett.34.391. |
[19] |
L. Lee and H. Swinney, Lamellar structures and self-replicating spots in a reaction-diffusion system, Phys. Rev. E, 51 (1995), 1899-1915.
doi: 10.1103/PhysRevE.51.1899. |
[20] |
L. Lega, J. V. Moloney and A. C. Newell, Swift-Hohenberg equation for lasers, Phys. Rev. Lett., 73 (1994), 2978-2981.
doi: 10.1103/PhysRevLett.73.2978. |
[21] |
M. Lopez-Fernandez and S. Sauter, Fast and stable contour integration for high order divided differences via elliptic functions, Math. Comp., 84 (2015), 1291-1315.
doi: 10.1090/S0025-5718-2014-02890-1. |
[22] |
E. Makrides and B. Sandstede, Predicting the bifurcation structure of localized snaking patterns, Phys. D, 268 (2014), 59-78.
doi: 10.1016/j.physd.2013.11.009. |
[23] |
P. Mandel, Theoretical Problems in Cavity Nonlinear Optics, Cambridge University Press, Cambridge, 1997. |
[24] |
S. G. McCalla and B. Sandstede, Spots in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 12 (2013), 831-877.
doi: 10.1137/120882111. |
[25] |
A. Mielke, Instability and stability of rolls in the Swift-Hohenberg equation, Comm. Math. Phys., 189 (1997), 829-853.
doi: 10.1007/s002200050230. |
[26] |
D. Morgan and J. H. P. Dawes, The Swift-Hohenberg equation with a nonlocal nonlinearity, Phys. D, 270 (2014), 60-80.
doi: 10.1016/j.physd.2013.11.018. |
[27] |
L. A. Peletier and V. Rottschafer, Pattern selection of solutions of the Swift-Hohenberg equation, Phys. D, 194 (2004), 95-126.
doi: 10.1016/j.physd.2004.01.043. |
[28] |
L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 6 (2007), 208-235.
doi: 10.1137/050647232. |
[29] |
D. Smets and J. B. van den Berg, Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations, J. Diff. Eqns., 184 (2002), 78-96.
doi: 10.1006/jdeq.2001.4135. |
[30] |
J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328.
doi: 10.1103/PhysRevA.15.319. |
[31] |
J. B. van den Berg, L. A. Peletier and W. C. Troy, Global branches of multi-bump periodic solutions of the Swift-Hohenberg equation, Arch. Ration. Mech. Anal., 158 (2001), 91-153.
doi: 10.1007/PL00004243. |
[32] |
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Universitext, 1996.
doi: 10.1007/978-3-642-61453-8. |
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