Multi-solitons pour des équations non-linéaires dispersives surcritiques

Combet, Vianney

Directeur de thèse : Robbiano, Luc ; Martel, Yvan

Membres du jury : Merle, Frank ; Gallay, Thierry ; Martel, Yvan ; Robbiano, Luc ; Saut, Jean-Claude ; Tzvetkov, Nikolay ; Tsai, Tai-Peng

Thèses d'Orsay, no. 800 (2010) , 148 p.

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     author = {Combet, Vianney},
     title = {Multi-solitons pour des \'equations non-lin\'eaires dispersives surcritiques},
     series = {Th\`eses d'Orsay},
     publisher = {Universit\'e de Paris-Sud Centre d'Orsay},
     number = {800},
     year = {2010},
     language = {fr},
     url = {http://archive.numdam.org/item/BJHTUP11_2010__0800__P0_0/}
}

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Combet, Vianney. Multi-solitons pour des équations non-linéaires dispersives surcritiques. Thèses d'Orsay, no. 800 (2010), 148 p. http://numdam.org/item/BJHTUP11_2010__0800__P0_0/

Bibliographie
Cité par

[1] J.L. Bona, P.E. Souganidis and W. A. Strauss. Stability and instability of solitary waves of Korteweg-de Vries type. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 411(1841) : 395-412, 1987. | MR | Zbl

[2] T. Cazenave and P.-L. Lions. Orbital stability of standing waves for some nonlinear Schrödinger equations. Communications in Mathematical Physics, 85(4) : 549-561, 1982. | MR | Zbl

[3] V. Combet. Construction and characterization of solutions converging to solitons for supercritical gKdV equations. Differential and Integral Equations, 23(5-6) : 513-568, 2010. | MR | Zbl

[4] V. Combet. Multi-soliton solutions for the supercritical gKdV equations. À paraître dans Communications in Partial Differential Equations. | MR | Zbl | DOI

[5] V. Combet. Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension. Arxiv preprint arXiv : 1008. 4613. | MR | Zbl | DOI

[6] R. Côte, Y. Martel and F. Merle. Construction of multi-soliton solutions for the $L^{2}$ -supercritical gKdV and NLS equations. À paraître dans Revista Matematica Iberoamericana. | MR | Zbl

[7] T. Duyckaerts and F. Merle. Dynamic of thresold solutions for energy-critical NLS. Geometric and Functional Analysis, 18(6) : 1787-1840, 2009. | MR | Zbl

[8] T. Duyckaerts and S. Roudenko. Threshold solutions for the focusing 3d cubic Schrödinger equation. À paraître dans Revista Matematica Iberoamericana. | MR | Zbl | DOI

[9] J. Ginibre and G. Velo. On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. Journal of Functional Analysis, 32(1) : 1-32, 1979. | MR | Zbl | DOI

[10] M. Grillakis, J. Shatah and W. Strauss. Stability theory of solitary waves in the presence of symmetry. I. Journal of Functional Analysis, 74(1) : 160-197, 1987. | MR | Zbl | DOI

[11] C.E. Kenig, G. Ponce and L. Vega. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Communications on Pure and Applied Mathematics, 46(4) : 527-620, 1993. | MR | Zbl

[12] C.E. Kenig, G. Ponce and L. Vega. On the concentration of blow up solutions for the generalized KdV equation critical in $L^{2}$ . Dans Nonlinear Wave Equations : A Conference in Honor of Walter A. Strauss on the Occasion of His Sixtieth Birthday, May 2-3, 1998, Brown University, 263 : 131-156, 2000. | MR | Zbl | DOI

[13] Y. Martel. Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. American Journal of Mathematics, 127(5) : 1103-1140, 2005. | MR | Zbl | DOI

[14] Y. Martel and F. Merle. Instability of solitons for the critical generalized Korteweg-de Vries equation. Geometric and Functional Analysis, 11(1) : 74-123, 2001. | MR | Zbl

[15] Y. Martel and F. Merle. Blow up in finite time and dynamics of blow up solutions for the $L^{2}$ -critical generalized KdV equation. Journal of the American Mathematical Society, 15(3) : 617-664, 2002. | MR | Zbl | DOI

[16] Y. Martel and F. Merle. Multi solitary waves for nonlinear Schrödinger equations. Dans Annales de l'institut Henri Poincaré/Analyse non linéaire, volume 23, pages 849-864. Elsevier, 2006. | MR | Zbl | Numdam

[17] Y. Martel and F. Merle. Asymptotic stability of solitons of the gKdV equations with general nonlinearity. Mathematische Annalen, 341(2) : 391-427, 2008. | MR | Zbl

[18] Y. Martel, F. Merle and T.-P. Tsai. Stability in $H^{1}$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations. Duke Mathematical Journal, 133(3) : 405-466, 2006. | MR | Zbl | DOI

[19] F. Merle. Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity. Communications in Mathematical Physics, 129(2) : 223-240, 1990. | MR | Zbl

[20] F. Merle. Existence of blow-up solutions in the energy space for the critical generalized KdV equation. Journal of the American Mathematical Society, 14(3) : 555-578, 2001. | MR | Zbl | DOI

[21] R.M. Miura. The Korteweg-de Vries equation : a survey of results. SIAM Review, 18(3) : 412-459, 1976. | MR | Zbl | DOI

[22] R.L. Pego and M.I. Weinstein. Eigenvalues, and instabilities of solitary waves. Philosophical Transactions : Physical Sciences and Engineering, 340(1656) : 47-94, 1992. | MR | Zbl

[23] M.I. Weinstein. Lyapunov stability of ground states of nonlinear dispersive evolution equations. Communications on Pure and Applied Mathematics, 39(1) : 51-67, 1986. | MR | Zbl

[1] T.B. Benjamin. The stability of solitary waves. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 328(1573):153-183, 1972. | MR

[2] J.L. Bona. On the stability theory of solitary waves. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 344(1638):363-374, 1975. | MR | Zbl

[3] J.L. Bona, P.E. Souganidis and W.A. Strauss. Stability and instability of solitary waves of Korteweg-de Vries type. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 411(1841):395-412, 1987. | MR | Zbl

[4] T. Cazenave. Blow up and scattering in the nonlinear Schrödinger equation. Instituto de Matemática, UFRJ, 1996.

[5] T. Cazenave and P.-L. Lions. Orbital stability of standing waves for some nonlinear Schrödinger equations. Communications in Mathematical Physics, 85(4):549-561, 1982. | MR | Zbl

[6] R. Côte, Y. Martel and F. Merle. Construction of multi-soliton solutions for the $L^{2}$ -supercritical gKdV and NLS equations. To appear in Revista Matematica Iberoamericana. | MR | Zbl

[7] T. Duyckaerts and F. Merle. Dynamic of thresold solutions for energy-critical NLS. Geometric and Functional Analysis, 18(6):1787-1840, 2009. | MR | Zbl

[8] T. Duyckaerts and S. Roudenko. Threshold solutions for the focusing 3d cubic Schrödinger equation. To appear in Revista Matematica Iberoamericana. | MR | Zbl | DOI

[9] M. Grillakis. Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system. Communications on Pure and Applied Mathematics, 43(3):299-333, 1990. | MR | Zbl

[10] M. Grillakis, J. Shatah and W. Strauss. Stability theory of solitary waves in the presence of symmetry. I. Journal of Functional Analysis, 74(1):160-197, 1987. | MR | Zbl | DOI

[11] C.E. Kenig and Y. Martel. Asymptotic stability of solitons for the Benjamin-Ono equation. Revista Matematica Iberoamericana, 25(3):909-970, 2009. | MR | Zbl

[12] C.E. Kenig, G. Ponce and L. Vega. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Communications on Pure and Applied Mathematics, 46(4):527-620, 1993. | MR | Zbl

[13] C.E. Kenig, G. Ponce and L. Vega. On the concentration of blow up solutions for the generalized KdV equation critical in $L^{2}$ . In Nonlinear Wave Equations: A Conference in Honor of Walter A. Strauss on the Occasion of His Sixtieth Birthday, May 2-3, 1998, Brown University, 263:131-156, 2000. | MR | Zbl | DOI

[14] Y. Martel. Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. American Journal of Mathematics, 127(5):1103-1140, 2005. | MR | Zbl | DOI

[15] Y. Martel and F. Merle. Instability of solitons for the critical generalized Korteweg-de Vries equation. Geometric and Functional Analysis, 11(1):74-123, 2001. | MR | Zbl

[16] Y. Martel and F. Merle. Blow up in finite time and dynamics of blow up solutions for the $L^{2}$ -critical generalized KdV equation. Journal of the American Mathematical Society, 15(3):617-664, 2002. | MR | Zbl | DOI

[17] Y. Martel and F. Merle. Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity, 18(1):55-80, 2005. | MR | Zbl

[18] Y. Martel and F. Merle. Asymptotic stability of solitons of the gKdV equations with general nonlinearity. Mathematische Annalen, 341(2):391-427, 2008. | MR | Zbl

[19] F. Merle. Existence of blow-up solutions in the energy space for the critical generalized KdV equation. Journal of the American Mathematical Society, 14(3):555-578, 2001. | MR | Zbl | DOI

[20] R.L. Pego and M.I. Weinstein. Eigenvalues, and instabilities of solitary waves. Philosophical Transactions : Physical Sciences and Engineering, 340(1656):47-94, 1992. | MR | Zbl

[21] M.I. Weinstein. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM Journal on Mathematical Analysis, 16(3):472-491, 1985. | MR | Zbl | DOI

[22] M.I. Weinstein. Lyapunov stability of ground states of nonlinear dispersive evolution equations. Communications on Pure and Applied Mathematics, 39(1):51-67, 1986. | MR | Zbl

[23] M.I. Weinstein. On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations. Communications in Partial Differential Equations, 11(5):545-565, 1986. | MR | Zbl

[1] J.L. Bona, P.E. Souganidis and W.A. Strauss. Stability and instability of solitary waves of Korteweg-de Vries type. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 411(1841): 395-412, 1987. | MR | Zbl

[2] T. Cazenave and P.-L. Lions. Orbital stability of standing waves for some nonlinear Schrödinger equations. Communications in Mathematical Physics, 85(4):549-561, 1982. | MR | Zbl

[3] V. Combet. Construction and characterization of solutions converging to solitons for supercritical gKdV equations. Differential and Integral Equations, 23(5-6):513- 568, 2010. | MR | Zbl

[4] R. Côte, Y. Martel and F. Merle. Construction of multi-soliton solutions for the $L^{2}$ -supercritical gKdV and NLS equations. To appear in Revista Matematica Iberoamericana. | MR | Zbl

[5] T. Duyckaerts and F. Merle. Dynamic of thresold solutions for energy-critical NLS. Geometric and Functional Analysis, 18(6):1787-1840, 2009. | MR | Zbl

[6] T. Duyckaerts and S. Roudenko. Threshold solutions for the focusing 3d cubic Schrödinger equation. To appear in Revista Matematica Iberoamericana. | MR | Zbl | DOI

[7] M. Grillakis, J. Shatah and W. Strauss. Stability theory of solitary waves in the presence of symmetry. I. Journal of Functional Analysis, 74(1):160-197, 1987. | MR | Zbl | DOI

[8] T. Kato. On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Studies in Applied Mathematics, 8:93-128, 1983. | MR | Zbl

[9] C.E. Kenig and Y. Martel. Asymptotic stability of solitons for the Benjamin-Ono equation. Revista Matematica Iberoamericana, 25(3):909-970, 2009. | MR | Zbl

[10] C.E. Kenig, G. Ponce and L. Vega. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Communications on Pure and Applied Mathematics, 46(4):527-620, 1993. | MR | Zbl

[11] Y. Martel. Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. American Journal of Mathematics, 127(5):1103-1140, 2005. | MR | Zbl

[12] Y. Martel and F. Merle. Blow up in finite time and dynamics of blow up solutions for the $L^{2}$ -critical generalized KdV equation. Journal of the American Mathematical Society, 15(3):617-664, 2002. | MR | Zbl

[13] Y. Martel and F. Merle. Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity, 18(1):55-80, 2005. | MR | Zbl

[14] Y. Martel and F. Merle. Multi solitary waves for nonlinear Schrödinger equations. In Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 23(6):849-864, 2006. | MR | Zbl | Numdam

[15] 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. Communications in Mathematical Physics, 231(2):347-373, 2002. | MR | Zbl

[16] F. Merle. Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity. Communications in Mathematical Physics, 129(2):223-240, 1990. | MR | Zbl

[17] F. Merle. Existence of blow-up solutions in the energy space for the critical generalized KdV equation. Journal of the American Mathematical Society, 14(3):555-578, 2001. | MR | Zbl

[18] R.M. Miura. The Korteweg-de Vries equation: a survey of results. SIAM Review, 18(3):412-459, 1976. | MR | Zbl | DOI

[19] R.L. Pego and M.I. Weinstein. Eigenvalues, and instabilities of solitary waves. Philosophical Transactions : Physical Sciences and Engineering, 340(1656):47-94, 1992. | MR | Zbl

[20] R.L. Pego and M.I. Weinstein. Asymptotic stability of solitary waves. Communications in Mathematical Physics, 164(2):305-349, 1994. | MR | Zbl

[21] G. Perelman. Some results on the scattering of weakly interacting solitons for non-linear Schrödinger equations. Spectral theory, microlocal analysis, singular manifolds, 14:78-137, 1997. | MR | Zbl

[22] I. Rodnianski, W. Schlag and A. Soffer. Asymptotic stability of N-soliton states of NLS. Arxiv preprint arXiv : math/0309114, 2003.

[23] M.I. Weinstein. Lyapunov stability of ground states of nonlinear dispersive evolution equations. Communications on Pure and Applied Mathematics, 39(1):51-67, 1986. | MR | Zbl

[1] T. Cazenave and F. Weissler. The Cauchy problem for the critical nonlinear Schrödinger equation in $H^{s}$ . Nonlinear Analysis, 14(10):807-836, 1990. | MR | Zbl

[2] V. Combet. Multi-soliton solutions for the supercritical gKdV equations. To appear in Communications in Partial Differential Equations. | MR | Zbl | DOI

[3] R. Côte, Y. Martel and F. Merle. Construction of multi-soliton solutions for the $L^{2}$ -supercritical gKdV and NLS equations. To appear in Revista Matematica Iberoamericana. | MR | Zbl

[4] T. Duyckaerts and F. Merle. Dynamic of threshold solutions for energy-critical NLS. Geometric and Functional Analysis, 18(6):1787-1840, 2009. | MR | Zbl

[5] T. Duyckaerts and S. Roudenko. Threshold solutions for the focusing 3d cubic Schrödinger equation. To appear in Revista Matematica Iberoamericana. | MR | Zbl | DOI

[6] J. Ginibre and G. Velo. On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. Journal of Functional Analysis, 32(1):1-32, 1979. | MR | Zbl | DOI

[7] M. Grillakis. Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system. Communications on Pure and Applied Mathematics, 43(3):299-333, 1990. | MR | Zbl

[8] M. Grillakis, J. Shatah and W.A. Strauss. Stability theory of solitary waves in the presence of symmetry. I. Journal of Functional Analysis, 74(1):160-197, 1987. | MR | Zbl | DOI

[9] Y. Martel. Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. American Journal of Mathematics, 127(5):1103-1140, 2005. | MR | Zbl | DOI

[10] Y. Martel and F. Merle. Multi solitary waves for nonlinear Schrödinger equations. In Annales de l'Institut Henri Poincaré/Analyse non linéaire, volume 23, pages 849-864. Elsevier, 2006. | MR | Zbl | Numdam

[11] Y. Martel, F. Merle and T.-R Tsai. Stability in $H^{1}$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations. Duke Mathematical Journal, 133(3):405- 466, 2006. | MR | Zbl | DOI

[12] F. Merle. Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity. Communications in Mathematical Physics, 129(2):223-240, 1990. | MR | Zbl

[13] G. Perelman. Some results on the scattering of weakly interacting solitons for nonlinear Schrödinger equations. Math. Top, 14: 78-137, 1997. | MR | Zbl

[14] G. Perelman. Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations. Communications in Partial Differential Equations, 29(7):1051-1095, 2004. | MR | Zbl

[15] I. Rodnianski, W. Schlag and A. Soffer. Asymptotic stability of N-soliton states of NLS. Preprint.

[16] M.I. Weinstein. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM Journal on Mathematical Analysis, 16(3):472-491, 1985. | MR | Zbl | DOI