# American Institute of Mathematical Sciences

July  2016, 36(7): 3811-3843. doi: 10.3934/dcds.2016.36.3811

## The Gardner equation and the stability of multi-kink solutions of the mKdV equation

 1 CNRS and Departamento de Ingeniería Matemática y CMM, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile

Received  May 2015 Revised  November 2015 Published  March 2016

Multi-kink solutions of the defocusing, modified Korteweg-de Vries equation (mKdV) found by Grosse [22,23] are shown to be globally $H^1$-stable, and asymptotically stable. Stability in the one-kink case was previously established by Zhidkov [51] and Merle-Vega [41]. The proof uses transformations linking the mKdV equation with focusing, Gardner-like equations, where stability and asymptotic stability in the energy space are known. We generalize our results by considering the existence, uniqueness and the dynamics of generalized multi-kinks of defocusing, non-integrable gKdV equations, showing the inelastic character of the kink-kink collision in some regimes.
Citation: Claudio Muñoz. The Gardner equation and the stability of multi-kink solutions of the mKdV equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3811-3843. doi: 10.3934/dcds.2016.36.3811
##### References:
 [1] M. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering,, London Mathematical Society Lecture Note Series, (1991). doi: 10.1017/CBO9780511623998. Google Scholar [2] M. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform,, SIAM Studies in Applied Mathematics, (1981). Google Scholar [3] M. A. Alejo and C. Muñoz, Nonlinear Stability of mKdV breathers,, Communications in Mathematical Physics, 324 (2013), 233. doi: 10.1007/s00220-013-1792-0. Google Scholar [4] M. A. Alejo and C. Muñoz, Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers,, Analysis and PDE, 8 (2015), 629. doi: 10.2140/apde.2015.8.629. Google Scholar [5] M. A. Alejo and C. Muñoz, On the variational structure of breather solutions,, preprint, (). Google Scholar [6] M. A. Alejo, C. Muñoz and L. Vega, The Gardner equation and the $L^2$-stability of the $N$-soliton solutions of the Korteweg-de Vries equation,, Transactions of the AMS., 365 (2013), 195. doi: 10.1090/S0002-9947-2012-05548-6. Google Scholar [7] T. B. Benjamin, The stability of solitary waves,, Proc. Roy. Soc. London A, 328 (1972), 153. doi: 10.1098/rspa.1972.0074. Google Scholar [8] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rat. Mech. Anal., 82 (1983), 347. doi: 10.1007/BF00250555. Google Scholar [9] F. Béthuel, P. Gravejat, J.-C. Saut and D. Smets, Orbital stability of the black soliton to the Gross-Pitaevskii equation,, Indiana Univ. Math. J., 57 (2008), 2611. doi: 10.1512/iumj.2008.57.3632. Google Scholar [10] J. L. Bona, P. Souganidis and W. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type,, Proc. Roy. Soc. London, 411 (1987), 395. doi: 10.1098/rspa.1987.0073. Google Scholar [11] M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations,, Amer. J. Math., 125 (2003), 1235. doi: 10.1353/ajm.2003.0040. Google Scholar [12] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$,, J. Amer. Math. Soc., 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar [13] V. Combet, Multi-soliton solutions for the supercritical gKdV equations,, Comm. PDE, 36 (2011), 380. doi: 10.1080/03605302.2010.503770. Google Scholar [14] R. Côte, Y. Martel and F. Merle, Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations,, Rev. Mat. Iberoam., 27 (2011), 273. doi: 10.4171/RMI/636. Google Scholar [15] S. Cuccagna, On asymptotic stability in 3D of kinks for the $\phi^4$ model,, Trans. Amer. Math. Soc., 360 (2008), 2581. doi: 10.1090/S0002-9947-07-04356-5. Google Scholar [16] T. Dauxois and M. Peyrard, Physics of Solitons,, Cambridge University Press, (2006). Google Scholar [17] C. S. Gardner, M. D. Kruskal and R. Miura, Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion,, J. Math. Phys., 9 (1968), 1204. doi: 10.1063/1.1664701. Google Scholar [18] P. Gérard and Z. Zhang, Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation,, J. Math. Pures Appl. (9), 91 (2009), 178. doi: 10.1016/j.matpur.2008.09.009. Google Scholar [19] F. Gesztesy and B. Simon, Constructing solutions of the mKdV-equation,, J. Funct. Anal., 89 (1990), 53. doi: 10.1016/0022-1236(90)90003-4. Google Scholar [20] F. Gesztesy, W. Schweiger and B. Simon, Commutation methods applied to the mKdV-equation,, Trans. AMS, 324 (1991), 465. doi: 10.1090/S0002-9947-1991-1029000-7. Google Scholar [21] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, I,, J. Funct. Anal., 74 (1987), 160. doi: 10.1016/0022-1236(87)90044-9. Google Scholar [22] H. Grosse, Solitons of the modified KdV equation,, Lett. Math. Phys., 8 (1984), 313. doi: 10.1007/BF00400502. Google Scholar [23] H. Grosse, New solitons connected to the Dirac equation,, Phys. Rep., 134 (1986), 297. doi: 10.1016/0370-1573(86)90053-0. Google Scholar [24] J. Holmer, J. Marzuola and M. Zworski, Fast soliton scattering by delta impurities,, Comm. Math. Phys., 274 (2007), 187. doi: 10.1007/s00220-007-0261-z. Google Scholar [25] J. Holmer, J. Marzuola and M. Zworski, Soliton splitting by external delta potentials,, J. Nonlinear Sci., 17 (2007), 349. doi: 10.1007/s00332-006-0807-9. Google Scholar [26] D. B. Henry, J. F. Perez and W. F. Wreszinski, Stability theory for solitary-wave solutions of scalar field equations,, Comm. Math. Phys., 85 (1982), 351. doi: 10.1007/BF01208719. Google Scholar [27] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,, Comm. Pure Appl. Math., 46 (1993), 527. doi: 10.1002/cpa.3160460405. Google Scholar [28] C. E. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations,, Duke Math. J., 106 (2001), 617. doi: 10.1215/S0012-7094-01-10638-8. Google Scholar [29] E. Kopylova and A. I. Komech, On asymptotic stability of kink for relativistic ginzburg-landau equations,, Arch. Ration. Mech. Anal., 202 (2011), 213. doi: 10.1007/s00205-011-0415-1. Google Scholar [30] M. Kowalczyk, Y. Martel and C. Muñoz, Kink dynamics in the $\varphi^4$ model: Asymptotic stability for odd perturbations in the energy space,, preprint, (). Google Scholar [31] 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 [32] J. H. Maddocks and R. L. Sachs, On the stability of KdV multi-solitons,, Comm. Pure Appl. Math., 46 (1993), 867. doi: 10.1002/cpa.3160460604. Google Scholar [33] Y. Martel, Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations,, Amer. J. Math., 127 (2005), 1103. doi: 10.1353/ajm.2005.0033. Google Scholar [34] Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations,, Arch. Ration. Mech. Anal., 157 (2001), 219. doi: 10.1007/s002050100138. Google Scholar [35] Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gKdV equations revisited,, Nonlinearity, 18 (2005), 55. doi: 10.1088/0951-7715/18/1/004. Google Scholar [36] Y. Martel and F. Merle, Description of two soliton collision for the quartic gKdV equation,, Ann. of Math. (2), 174 (2011), 757. doi: 10.4007/annals.2011.174.2.2. Google Scholar [37] Y. Martel and F. Merle, Stability of two soliton collision for nonintegrable gKdV equations,, Comm. Math. Phys., 286 (2009), 39. doi: 10.1007/s00220-008-0685-0. Google Scholar [38] Y. Martel and F. Merle, Inelastic interaction of nearly equal solitons for the quartic gKdV equation,, Invent. Math., 183 (2011), 563. doi: 10.1007/s00222-010-0283-6. Google Scholar [39] Y. Martel and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity,, Math. Ann., 341 (2008), 391. doi: 10.1007/s00208-007-0194-z. Google Scholar [40] Y. Martel, F. Merle and T. P. Tsai, Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations,, Comm. Math. Phys., 231 (2002), 347. doi: 10.1007/s00220-002-0723-2. Google Scholar [41] F. Merle and L. Vega, $L^2$ stability of solitons for KdV equation,, Int. Math. Res. Not., (2003), 735. doi: 10.1155/S1073792803208060. Google Scholar [42] R. M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation,, J. Math. Phys., 9 (1968), 1202. doi: 10.1063/1.1664700. Google Scholar [43] C. Muñoz, On the inelastic 2-soliton collision for gKdV equations with general nonlinearity,, Int. Math. Research Notices, 2010 (2010), 1624. doi: 10.1093/imrn/rnp204. Google Scholar [44] C. Muñoz, $L^2$-stability of Multi-solitons,, Séminaire EDP et Applications, (2011). Google Scholar [45] R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves,, Comm. Math. Phys., 164 (1994), 305. doi: 10.1007/BF02101705. Google Scholar [46] G. Perelman, Two soliton collision for nonlinear Schrödinger equations in dimension 1,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 357. doi: 10.1016/j.anihpc.2011.02.002. Google Scholar [47] A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations,, Invent. Math., 136 (1999), 9. doi: 10.1007/s002220050303. Google Scholar [48] B. Thaller, The Dirac Equation,, Texts and Monographs in Physics. Springer-Verlag, (1992). doi: 10.1007/978-3-662-02753-0. Google Scholar [49] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations,, Comm. Pure. Appl. Math., 39 (1986), 51. doi: 10.1002/cpa.3160390103. Google Scholar [50] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations,, SIAM J. Math. Anal., 16 (1985), 472. doi: 10.1137/0516034. Google Scholar [51] P. E. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory,, Lecture Notes in Mathematics, (1756). Google Scholar

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##### References:
 [1] M. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering,, London Mathematical Society Lecture Note Series, (1991). doi: 10.1017/CBO9780511623998. Google Scholar [2] M. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform,, SIAM Studies in Applied Mathematics, (1981). Google Scholar [3] M. A. Alejo and C. Muñoz, Nonlinear Stability of mKdV breathers,, Communications in Mathematical Physics, 324 (2013), 233. doi: 10.1007/s00220-013-1792-0. Google Scholar [4] M. A. Alejo and C. Muñoz, Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers,, Analysis and PDE, 8 (2015), 629. doi: 10.2140/apde.2015.8.629. Google Scholar [5] M. A. Alejo and C. Muñoz, On the variational structure of breather solutions,, preprint, (). Google Scholar [6] M. A. Alejo, C. Muñoz and L. Vega, The Gardner equation and the $L^2$-stability of the $N$-soliton solutions of the Korteweg-de Vries equation,, Transactions of the AMS., 365 (2013), 195. doi: 10.1090/S0002-9947-2012-05548-6. Google Scholar [7] T. B. Benjamin, The stability of solitary waves,, Proc. Roy. Soc. London A, 328 (1972), 153. doi: 10.1098/rspa.1972.0074. Google Scholar [8] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rat. Mech. Anal., 82 (1983), 347. doi: 10.1007/BF00250555. Google Scholar [9] F. Béthuel, P. Gravejat, J.-C. Saut and D. Smets, Orbital stability of the black soliton to the Gross-Pitaevskii equation,, Indiana Univ. Math. J., 57 (2008), 2611. doi: 10.1512/iumj.2008.57.3632. Google Scholar [10] J. L. Bona, P. Souganidis and W. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type,, Proc. Roy. Soc. London, 411 (1987), 395. doi: 10.1098/rspa.1987.0073. Google Scholar [11] M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations,, Amer. J. Math., 125 (2003), 1235. doi: 10.1353/ajm.2003.0040. Google Scholar [12] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$,, J. Amer. Math. Soc., 16 (2003), 705. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar [13] V. Combet, Multi-soliton solutions for the supercritical gKdV equations,, Comm. PDE, 36 (2011), 380. doi: 10.1080/03605302.2010.503770. Google Scholar [14] R. Côte, Y. Martel and F. Merle, Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations,, Rev. Mat. Iberoam., 27 (2011), 273. doi: 10.4171/RMI/636. Google Scholar [15] S. Cuccagna, On asymptotic stability in 3D of kinks for the $\phi^4$ model,, Trans. Amer. Math. Soc., 360 (2008), 2581. doi: 10.1090/S0002-9947-07-04356-5. Google Scholar [16] T. Dauxois and M. Peyrard, Physics of Solitons,, Cambridge University Press, (2006). Google Scholar [17] C. S. Gardner, M. D. Kruskal and R. Miura, Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion,, J. Math. Phys., 9 (1968), 1204. doi: 10.1063/1.1664701. Google Scholar [18] P. Gérard and Z. Zhang, Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation,, J. Math. Pures Appl. (9), 91 (2009), 178. doi: 10.1016/j.matpur.2008.09.009. Google Scholar [19] F. Gesztesy and B. Simon, Constructing solutions of the mKdV-equation,, J. Funct. Anal., 89 (1990), 53. doi: 10.1016/0022-1236(90)90003-4. Google Scholar [20] F. Gesztesy, W. Schweiger and B. Simon, Commutation methods applied to the mKdV-equation,, Trans. AMS, 324 (1991), 465. doi: 10.1090/S0002-9947-1991-1029000-7. Google Scholar [21] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry, I,, J. Funct. Anal., 74 (1987), 160. doi: 10.1016/0022-1236(87)90044-9. Google Scholar [22] H. Grosse, Solitons of the modified KdV equation,, Lett. Math. Phys., 8 (1984), 313. doi: 10.1007/BF00400502. Google Scholar [23] H. Grosse, New solitons connected to the Dirac equation,, Phys. Rep., 134 (1986), 297. doi: 10.1016/0370-1573(86)90053-0. Google Scholar [24] J. Holmer, J. Marzuola and M. Zworski, Fast soliton scattering by delta impurities,, Comm. Math. Phys., 274 (2007), 187. doi: 10.1007/s00220-007-0261-z. Google Scholar [25] J. Holmer, J. Marzuola and M. Zworski, Soliton splitting by external delta potentials,, J. Nonlinear Sci., 17 (2007), 349. doi: 10.1007/s00332-006-0807-9. Google Scholar [26] D. B. Henry, J. F. Perez and W. F. Wreszinski, Stability theory for solitary-wave solutions of scalar field equations,, Comm. Math. Phys., 85 (1982), 351. doi: 10.1007/BF01208719. Google Scholar [27] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,, Comm. Pure Appl. Math., 46 (1993), 527. doi: 10.1002/cpa.3160460405. Google Scholar [28] C. E. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations,, Duke Math. J., 106 (2001), 617. doi: 10.1215/S0012-7094-01-10638-8. Google Scholar [29] E. Kopylova and A. I. Komech, On asymptotic stability of kink for relativistic ginzburg-landau equations,, Arch. Ration. Mech. Anal., 202 (2011), 213. doi: 10.1007/s00205-011-0415-1. Google Scholar [30] M. Kowalczyk, Y. Martel and C. Muñoz, Kink dynamics in the $\varphi^4$ model: Asymptotic stability for odd perturbations in the energy space,, preprint, (). Google Scholar [31] 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 [32] J. H. Maddocks and R. L. Sachs, On the stability of KdV multi-solitons,, Comm. Pure Appl. Math., 46 (1993), 867. doi: 10.1002/cpa.3160460604. Google Scholar [33] Y. Martel, Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations,, Amer. J. Math., 127 (2005), 1103. doi: 10.1353/ajm.2005.0033. Google Scholar [34] Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations,, Arch. Ration. Mech. Anal., 157 (2001), 219. doi: 10.1007/s002050100138. Google Scholar [35] Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gKdV equations revisited,, Nonlinearity, 18 (2005), 55. doi: 10.1088/0951-7715/18/1/004. Google Scholar [36] Y. Martel and F. Merle, Description of two soliton collision for the quartic gKdV equation,, Ann. of Math. (2), 174 (2011), 757. doi: 10.4007/annals.2011.174.2.2. Google Scholar [37] Y. Martel and F. Merle, Stability of two soliton collision for nonintegrable gKdV equations,, Comm. Math. Phys., 286 (2009), 39. doi: 10.1007/s00220-008-0685-0. Google Scholar [38] Y. Martel and F. Merle, Inelastic interaction of nearly equal solitons for the quartic gKdV equation,, Invent. Math., 183 (2011), 563. doi: 10.1007/s00222-010-0283-6. Google Scholar [39] Y. Martel and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity,, Math. Ann., 341 (2008), 391. doi: 10.1007/s00208-007-0194-z. Google Scholar [40] Y. Martel, F. Merle and T. P. Tsai, Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations,, Comm. Math. Phys., 231 (2002), 347. doi: 10.1007/s00220-002-0723-2. Google Scholar [41] F. Merle and L. Vega, $L^2$ stability of solitons for KdV equation,, Int. Math. Res. Not., (2003), 735. doi: 10.1155/S1073792803208060. Google Scholar [42] R. M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation,, J. Math. Phys., 9 (1968), 1202. doi: 10.1063/1.1664700. Google Scholar [43] C. Muñoz, On the inelastic 2-soliton collision for gKdV equations with general nonlinearity,, Int. Math. Research Notices, 2010 (2010), 1624. doi: 10.1093/imrn/rnp204. Google Scholar [44] C. Muñoz, $L^2$-stability of Multi-solitons,, Séminaire EDP et Applications, (2011). Google Scholar [45] R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves,, Comm. Math. Phys., 164 (1994), 305. doi: 10.1007/BF02101705. Google Scholar [46] G. Perelman, Two soliton collision for nonlinear Schrödinger equations in dimension 1,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 357. doi: 10.1016/j.anihpc.2011.02.002. Google Scholar [47] A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations,, Invent. Math., 136 (1999), 9. doi: 10.1007/s002220050303. Google Scholar [48] B. Thaller, The Dirac Equation,, Texts and Monographs in Physics. Springer-Verlag, (1992). doi: 10.1007/978-3-662-02753-0. Google Scholar [49] M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations,, Comm. Pure. Appl. Math., 39 (1986), 51. doi: 10.1002/cpa.3160390103. Google Scholar [50] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations,, SIAM J. Math. Anal., 16 (1985), 472. doi: 10.1137/0516034. Google Scholar [51] P. E. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory,, Lecture Notes in Mathematics, (1756). Google Scholar
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