Étude de quelques propriétés d'équations d'ondes non linéaires dispersives de type Schrödinger
Thèses d'Orsay, no. 304 (1992) , 102 p.

This thesis is divided into three parts, each one featuring the study of some properties of nonlinear Schrödinger type dispersive wave equations arising in several fields of physics.

In the first part, we study the Cauchy problem associated with a nonlinear Schrödinger equation with an external magnetic field. Under some growth restrictions on the potentials and the nonlinear term in the equation, we prove the local existence and the uniqueness of solutions for the Cauchy problem for this equation in a weighted Sobolev space. We also prove the conservation of the associated energy.

In the second part, we study the existence of smooth analytic solutions for a general nonlinear Schrödinger type equation. This equation contains some physical models arising in the context of water waves. In these models, the linear term may be a differential operator of order larger than two, and the nonlinear term may be nonlocal.

The third part is devoted to the study of the existence and the instability of some localised stationary solutions of a nonlinear Schrödinger equation with a general nonlinearity. These localised solutions have a nonzero limit when the space variable goes to infinity, and for some particular nonlinear terms, they have a definite physical interpretation. We prove, by linearizing the equation, that when these solutions exist, they are always unstable solutions of the evolution equation.

Mots-clés : Schrödinger equation, external magnetic fields, water waves, solitary waves, orbital stability 
@phdthesis{BJHTUP11_1992__0304__P0_0,
     author = {De Bouard, Anne},
     title = {\'Etude de quelques propri\'et\'es d'\'equations d'ondes non lin\'eaires dispersives de type {Schr\"odinger}},
     series = {Th\`eses d'Orsay},
     publisher = {Universit\'e de Paris-Sud Centre d'Orsay},
     number = {304},
     year = {1992},
     language = {fr},
     url = {http://archive.numdam.org/item/BJHTUP11_1992__0304__P0_0/}
}
TY  - BOOK
AU  - De Bouard, Anne
TI  - Étude de quelques propriétés d'équations d'ondes non linéaires dispersives de type Schrödinger
T3  - Thèses d'Orsay
PY  - 1992
IS  - 304
PB  - Université de Paris-Sud Centre d'Orsay
UR  - http://archive.numdam.org/item/BJHTUP11_1992__0304__P0_0/
LA  - fr
ID  - BJHTUP11_1992__0304__P0_0
ER  - 
%0 Book
%A De Bouard, Anne
%T Étude de quelques propriétés d'équations d'ondes non linéaires dispersives de type Schrödinger
%S Thèses d'Orsay
%D 1992
%N 304
%I Université de Paris-Sud Centre d'Orsay
%U http://archive.numdam.org/item/BJHTUP11_1992__0304__P0_0/
%G fr
%F BJHTUP11_1992__0304__P0_0
De Bouard, Anne. Étude de quelques propriétés d'équations d'ondes non linéaires dispersives de type Schrödinger. Thèses d'Orsay, no. 304 (1992), 102 p. http://numdam.org/item/BJHTUP11_1992__0304__P0_0/

[1] I.V. Barashenkov, V.G. Makhankov : Soliton like "bubbles" in a system of interacting bosons, Phys. Lett. A.128, 52-56 (1988) | MR | DOI

[2] J. Ginibre, G. Velo : On a class of nonlinear Schrödinger equations I, J. Funct. Anal., 32, 1-32 (1979) | MR | Zbl | DOI

[3] R. Glassey : On the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18, 1794-1797, (1977) | MR | Zbl | DOI

[4] N. Hayashi : Analytidty of solutions to the Korteweg-de Vries equation, Preprint | MR | Zbl

[5] N. Hayashi, S. Saitoh : Analyticity and global existence of small solutions to some nonlinear Schrödinger equations, Comm. Math. Phys. 129, 27-41 (1990) | MR | Zbl | DOI

[6] T. Kato : On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Phys. Théor. 46, 113-129 (1987) | MR | Zbl | Numdam

[7] K. Yajima : Schrödinger evolution equations with magnetic field, J. Anal. Math. 56, 29-76 (1991) | MR | Zbl | DOI

[1] K. Asada and D. Fujiwara, On some oscillatory integral transformation in L 2 ℝ n , Japan Jour. Math., 4 (1978), 299-361. | MR | Zbl

[2] T. Cazenave and F.B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, to appear. | MR | Zbl | DOI

[3] T. Cazenave and M. J. Esteban, On the evolution problem for nonlinear Schrödinger equations with an external magnetic field, preprint (1988).

[4] H.L. Cycon, R.G. Froese, W. Hirsch, and B. Simon, "Schrödinger Operators with Application to Quantum Mechanics and Global Geometry," Springer Verlag, Berlin-Heidelberg, 1987. | MR | Zbl

[5] M.J. Esteban and P.L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, to appear. | MR | Zbl

[6] J.M. Ghidaglia and J.C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506. | MR | Zbl | DOI

[7] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equation ; I : The Cauchy problem, J. Funct. Anal., 32 (1979), 1-32. | MR | Zbl | DOI

[8] J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. Henri Poincaré, Anal, non Linéaire, 2 (1985), 309-327. | MR | Zbl | Numdam | DOI

[9] E. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797. | MR | Zbl | DOI

[10] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Phys. Theor., 46 (1987), 113-129. | MR | Zbl | Numdam

[11] Y.G. Oh, Cauchy problem and Ehrenfest's Law for nonlinear Schrödinger equations with potentials, J. Diff. Eqns., 81 (1989), 255-274. | MR | Zbl | DOI

[12] M. Reed and B. Simon, "Methods of Modern Mathematical Physics ; Vol. II : Fourier Analysis and Selfadjointness," Academic Press, New York, 1975. | MR | Zbl

[13] Y. Tsutsumi, L 2 -solutions for nonlinear Schrödinger equations and nonlinear groups, Funk. Ekva, 30 (1987), 115-125. | MR | Zbl

[14] K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), 415-426. | MR | Zbl | DOI

[15] K. Yajima, Schrödinger evolution equations with magnetic field, preprint (1989). | MR | Zbl

[1] H. Aikawa, N. Hayashi, S. Saitoh, Analyticity of solutions for semi-linear heat equations in one space dimension II, preprint.

[2] P. Constantin, Decay estimates for Schrödinger equations, Com. Math. Phys. 127 (1990) 101-108. | MR | Zbl | DOI

[3] K.B. Dysthe, Note on a modification to the nonlinear Schrödinger equation for applications to deep water waves, Proc. Royal Soc. London A369 (1979) 105-114. | Zbl

[4] G.B. Folland, Lectures on partial differential equations, (Tata Institute), Berlin, Springer, 1983. | MR | Zbl | DOI

[5] J.M. Ghidaglia and J.C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlineaxity 3 (1990) 475-506. | MR | Zbl | DOI

[6] J.M. Ghidaglia and J.C. Saut, Nonelliptic Schrödinger equations, to appear. | MR | Zbl | DOI

[7] N. Hayashi, Analyticity of solutions to the Korteweg-de Vries equation, preprint. | MR | Zbl | DOI

[8] N. Hayashi, Global existence of small analytic solutions to nonlinear Schrödinger equations, Duke Math. J. 60 (1990) 717-727. | MR | Zbl | DOI

[9] N. Hayashi and S. Saitoh, Analyticity and global existence of small solutions to some nonlinear Schrödinger equations, Com. Math. Phys. 129 (1990) 27-41. | MR | Zbl | DOI

[10] S.J. Hogan, The fourth order evolution equation for deep water gravity capillary waves, Proc. Royal Soc. London A402 (1985) 359-372. | Zbl

[11] T. Kato and K. Masuda, Nonlinear evolution equations and analyticity I, Annales Institut Henri Poincaré, Analyse Non Linéaire 3 (1986) 455-467. | MR | Zbl | Numdam | DOI

[12] T. Kawahara, N. Sugimoto and T. Kakutani, Nonlinear interaction between short and long capillary-gravity waves, J. Phys. Soc. Japan 39 (1975) 1379-1386. | DOI

[13] E.I. Shulman, On the integrability of equations of Davey-Stewartson type, Theor. Math. Phys. 56 (1983) 131-136. | MR | Zbl

[14] E.M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, 1971. | MR

[15] V.E. Zakharov and E.I. Shulman, On additionnal motion invariants of classical hamiltonian wave systems, Physica 29D (1988) 283-320. | MR | Zbl

[1] I.V. Barashenkov, V.G. Makhankov : Soliton-like "Bubbles" in a System of Interating Bosons, Phys. Lett. A 128 (1988) 52-56. | MR | DOI

[2] I.V. Barashenkov, A.D. Gocheva, V.G. Makhankov, I.V. Puzynin, Stability of the Soliton-like "Bubbles" Phys. D 34 (1989) 240-254. | MR | Zbl

[3] I.V. Barashenkov, I.V. Puzynin, T. Zhanlav, T.L. Boyadjiev : Stability of the Moving "Bubbles" in a Bose Condensate, Proc. Workshop on Solitons and Applications, V.G. Makhankov, V.K. Fedyanin, O.K. Pashaev, World Scientific 1990, 281-297. | MR

[4] H. Berestycki, T. Cazenave : Instabilité des Etats Stationnaires dans les Equations de Schrödinger et de Klein-Gordon non linéaires, C.R. Acad Sc. Paris, Série I, 293 (1981) 489-492. | MR | Zbl

[5] H. Berestycki, T. Gallouët, O. Kavian : Equations de champs scalaires Euclidiens non linéaires dans le Plan. C.R. Acad Sc. Paris, Série I, 297 (1983) et Gallouët Thesis (1984), Univ. Pierre et Marie Curie, Paris, France. | MR | Zbl

[6] H. Berestycki, P.L. Lions : Nonlinear Scalar Field Equation I : Existence of a ground state, Arch. Rat. Mech. Anal. 82 (1983) 313-376. | MR | Zbl | DOI

[7] T. Cazenave, P.L. Lions : Orbital Stability of Standing Waves for some Nonlinear Schrödinger Equations, Comm. Math. Phys. 85 (1982) 549-561. | MR | Zbl | DOI

[8] L.E. Fraenkel : Formulae for High Derivatives of Composite Functions, Math. Proc. Camb. Phil. Soc. (1978) 83, 159-165. | MR | Zbl | DOI

[9] M. Grillakis : Linearized Instability for Nonlinear Schrödinger and Klein-Gordon Equations, Comm. Pure Appl. Math. 41 (1988) 747-774. | MR | Zbl | DOI

[10] M. Grillakis : Analysis of the Linearization around a Critical Point of an Infinité Dimensional Hamiltonian System, Comm. Pure Appl. Math 43 (1990) 299-333. | MR | Zbl | DOI

[11] M. Grillakis, J. Shatah, W. Strauss : Stability Theory of Solitary Waves in the Présence of Symmetry I, J. Funct. Anal. 74 (1987) 160-197. | MR | Zbl | DOI

[12] D.B. Henry, J. Fernando Perez, W.F. Wreszinski : Stability Theory for Solitary Waves solutions of Scalar Field Equations, Comm.Math. Phys. 85 (1982) 351-361. | MR | Zbl | DOI

[13] A. Pazy : Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New-York, 1983. | MR | Zbl

[14] M. Reed, B. Simon : Method of Modern Mathematical Physics, Vol II, IV, Academic Press, New-York, 1979.

[15] M. Weinstein : Modulational Stability of Ground States of Nonlinear Schrödinger Equations, Siam J. Math. Anal. 16 (1985) 472-491. | MR | Zbl | DOI

[16] M. Weinstein : Lyapunov Stability of Ground States of Nonlinear Dispersive Equations, Comm. Pure. Appl. Math. 39 (1986) 51-68. | MR | Zbl | DOI