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, 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, 149. Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511623998.  Google Scholar

[2]

M. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Studies in Applied Mathematics, 4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. x+425 pp.  Google Scholar

[3]

M. A. Alejo and C. Muñoz, Nonlinear Stability of mKdV breathers, Communications in Mathematical Physics, 324 (2013), 233-262. 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), 629-674. 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), 195-212. 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), 153-183. 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), 347-375. 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), 2611-2642. 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), 395-412. 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), 1235-1293. 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), 705-749 (electronic). doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

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V. Combet, Multi-soliton solutions for the supercritical gKdV equations, Comm. PDE, 36 (2011), 380-419. 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), 273-302. 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), 2581-2614. 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), 1204-1209. 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), 178-210. 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), 53-60. 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), 465-525. 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), 160-197. doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[22]

H. Grosse, Solitons of the modified KdV equation, Lett. Math. Phys., 8 (1984), 313-319. doi: 10.1007/BF00400502.  Google Scholar

[23]

H. Grosse, New solitons connected to the Dirac equation, Phys. Rep., 134 (1986), 297-304. 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), 187-216. 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), 349-367. 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), 351-361. 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), 527-620. 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), 617-633. 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), 213-245. 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), 467-490. 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), 867-901. 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), 1103-1140. 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), 219-254. 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), 55-80. 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), 757-857. 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), 39-79. 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), 563-648. 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), 391-427. 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), 347-373. 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), 735-753. 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), 1202-1204. 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), 1624-1719. doi: 10.1093/imrn/rnp204.  Google Scholar

[44]

C. Muñoz, $L^2$-stability of Multi-solitons, Séminaire EDP et Applications, École Polythecnique, France, Janvier 2011, http://www.dim.uchile.cl/~cmunoz. Google Scholar

[45]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349. 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), 357-384. 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), 9-74. doi: 10.1007/s002220050303.  Google Scholar

[48]

B. Thaller, The Dirac Equation, Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992. xviii+357 pp. 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), 51-67. 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), 472-491. doi: 10.1137/0516034.  Google Scholar

[51]

P. E. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Mathematics, vol. 1756, Springer-Verlag, Berlin, 2001.  Google Scholar

show all references

References:
[1]

M. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series, 149. Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511623998.  Google Scholar

[2]

M. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Studies in Applied Mathematics, 4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. x+425 pp.  Google Scholar

[3]

M. A. Alejo and C. Muñoz, Nonlinear Stability of mKdV breathers, Communications in Mathematical Physics, 324 (2013), 233-262. 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), 629-674. 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), 195-212. 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), 153-183. 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), 347-375. 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), 2611-2642. 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), 395-412. 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), 1235-1293. 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), 705-749 (electronic). 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), 380-419. 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), 273-302. 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), 2581-2614. 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), 1204-1209. 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), 178-210. 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), 53-60. 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), 465-525. 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), 160-197. doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[22]

H. Grosse, Solitons of the modified KdV equation, Lett. Math. Phys., 8 (1984), 313-319. doi: 10.1007/BF00400502.  Google Scholar

[23]

H. Grosse, New solitons connected to the Dirac equation, Phys. Rep., 134 (1986), 297-304. 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), 187-216. 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), 349-367. 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), 351-361. 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), 527-620. 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), 617-633. 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), 213-245. 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), 467-490. 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), 867-901. 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), 1103-1140. 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), 219-254. 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), 55-80. 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), 757-857. 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), 39-79. 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), 563-648. 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), 391-427. 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), 347-373. 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), 735-753. 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), 1202-1204. 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), 1624-1719. doi: 10.1093/imrn/rnp204.  Google Scholar

[44]

C. Muñoz, $L^2$-stability of Multi-solitons, Séminaire EDP et Applications, École Polythecnique, France, Janvier 2011, http://www.dim.uchile.cl/~cmunoz. Google Scholar

[45]

R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349. 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), 357-384. 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), 9-74. doi: 10.1007/s002220050303.  Google Scholar

[48]

B. Thaller, The Dirac Equation, Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992. xviii+357 pp. 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), 51-67. 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), 472-491. doi: 10.1137/0516034.  Google Scholar

[51]

P. E. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Mathematics, vol. 1756, Springer-Verlag, Berlin, 2001.  Google Scholar

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