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Stability and Well-Posedness in Integrable Evolution Equations STABILITY AND WELL-POSEDNESS IN INTEGRABLE NONLINEAR EVOLUTION EQUATIONS STABILITY AND WELL-POSEDNESS IN INTEGRABLE EVOLUTION EQUATIONS By YUSUKE SHIMABUKURO, B.Sc., M.Sc. A Thesis Submitted to the Department of Mathematics and Statistics and the School of Graduate Studies of McMaster University in Partial Fulfilment of the Requirement for the Degree of Doctor of Philosophy McMaster University c Copyright by Yusuke Shimabukuro DOCTOR OF PHILOSOPHY (2016) McMaster University Department of Mathematics and Statistics Hamilton, Ontario, Canada TITLE: Stability and well-posedness in integrable evolution equations AUTHOR: Yusuke Shimabukuro B.Sc. (University of Arizona) M.Sc. (University of Arizona) SUPERVISOR: Dr. Dmitry Pelinovsky NUMBER OF PAGES: IX, 153 II Abstract This dissertation is concerned with analysis of orbital stability of solitary waves and well-posedness of the Cauchy problem in the integrable evolution equations. The analysis is developed by using tools from integrable systems, such as higher- order conserved quantities, B¨acklund transformation, and inverse scattering trans- form. The main results are obtained for the massive Thirring model, which is an integrable nonlinear Dirac equation, and for the derivative NLS equation. Both equations are related with the same Kaup-Newell spectral problem. Our studies rely on the spectral properties of the Kaup-Newell spectral problem, which convey key information about solution behavior of the nonlinear evolution equations. III Acknowledgements First and foremost I would like to express my deepest gratitude to my advisor, Professor Dmitry Pelinovsky. It has been truly an exciting journey under his guidance. His vast knowledge, ideas, and insights have inspired and helped me to complete this dissertation. I cannot thank him enough for his constant support and patience throughout these four years of my PhD study. I also owe my gratitude to his family for having us for dinner several times. I always enjoyed the Russian tea time after dinner. I would like to thank Professor Jean-Pierre Gabardo. He was generous enough to meet me and discuss math problems every week for past three years. It was a wonderful opportunity to learn and explore various areas in analysis. I enjoyed every minute of it and feel very lucky to have met him. I would like to thank Professor Lia Bronsard. She has given me great guidance in academics and personal matters. Her summer funding greatly helped me during the prime time of my PhD study. She took us to the cherry tree garden that was my first discovery of a little of Japanese in Hamilton. She took us ice skating on Lake Ontario, which was my first experience to walk out onto a lake. She gave me wonderful escapes from studies as well as great encouragement for my studies. I would like to express my gratitude to Professor Walter Craig. His keen comments and insights when I gave talks (or just when I was in the same seminar room) helped me understand about the subjects better. Also, his course on special topics in mathematical physics during my first semester at McMaster was my thrill, IV listening about Hamiltonian PDEs, the water wave equations and so on. My thanks extend to professors at external universities. I would like to thank Professor Tetsu Mizumachi for his hospitality during three weeks of my stay at Hiroshima University. He introduced me to the project on the 2D Benney-Luke equation which I truly enjoy working on. It is an honor to collaborate with him. I would like to thank Professor Catherine Sulem for introducing us the problem on the derivative NLS. Without our communication with her, one of Chapters in this dissertation would not be present. I would like to thank Professor Vladimir Zakharov for introducing me to inte- grable systems. For personal support during my graduate study, I would like to thank all my colleagues at McMaster for their support, and in particular, Alessandro for fun math discussions. I would like to thank Gleb, Toby, Austin, Diogo, Chengwei, Nadeem and Adrien for their friendship and uncountable good memories from hiking in Arizona's canyons, dancing to country music, staying at Coffee Exchange all night with me, enjoying temporary arts in Art Crawl in Hamilton, playing card games, and having the warmest times at Portuguese bar. I thank Wesley and Peter for being always cheerful housemates. I am thankful to my oldest friends, Atsushi and Takahiro, in Okinawa who have given me the warmest friendship and support. My special thanks are to my family, Masao, Hisako, Taeko, and my love, Hannah, for their unconditional love and support. I would not be here where I am today without all of your support. To my family and Hannah Table of Contents 1 Introduction 1 1.1 General Background . .1 1.2 Orbital stability of Dirac soliton by energy method (Chapter 2) . .4 1.3 Orbital stability of Dirac soliton by B¨acklund transform (Chapter 3)6 1.4 Global well-posedness of the derivative NLS equation (Chapter 4) .7 1.5 Transverse instability of Dirac line soliton (Chapter 5) . .9 2 Orbital Stability of Dirac Soliton by Energy Method 10 2.1 MTM orbital stability result . 10 2.2 Spectrum of the linearized operator . 13 2.3 Positivity of the Hessian operator . 19 2.4 Proof of orbital stability . 22 2.5 Conserved quantities by the inverse scattering method . 25 3 Orbital Stability Theory by B¨acklund transformation 29 3.1 Main result . 29 3.2 B¨acklund transformation for the MTM system . 30 3.3 From a perturbed one-soliton solution to a small solution . 34 3.4 From a small solution to a perturbed one-soliton solution . 47 3.5 Proof of the main result . 55 VII 4 Global well-posedness of the derivative NLS by the inverse scat- tering transform 57 4.1 Main result . 57 4.2 Direct scattering transform . 62 4.2.1 Jost functions . 63 4.2.2 Scattering coefficients . 73 4.3 Formulations of the Riemann{Hilbert problem . 79 4.3.1 Reformulation of the Riemann{Hilbert problem . 81 4.3.2 Two transformations of the Riemann-Hilbert problem . 84 4.4 Inverse scattering transform . 85 4.4.1 Solution to the Riemann{Hilbert problem . 86 4.4.2 Estimates on solutions to the Riemann-Hilbert problem . 92 4.4.3 Reconstruction formulas . 100 4.5 Proof of the main result . 108 5 Transverse instability of line solitary waves in massive Dirac equa- tions 111 5.1 Background . 111 5.2 Massive Dirac equations . 113 5.2.1 Periodic stripe potentials . 114 5.2.2 Hexagonal potentials . 116 5.3 Transverse of line solitary waves . 117 5.3.1 Perturbation theory for the massive Thirring model . 123 5.3.2 Perturbation theory for the massive Gross{Neveu model . 129 5.4 Numerical approximations . 136 5.4.1 Eigenvalue computations for the massive Thirring model . 137 5.4.2 Eigenvalue computations for the massive Gross{Neveu model 141 VIII Chapter 1 Introduction 1.1 General Background The Hamiltonian H with n degrees of freedom is completely integrable in the Liouville sense if there exist n independent first integrals I1 = H, I2; ··· ;In in involution, i.e., fIi;Ijg = 0. These integrals are used as new coordinates in which corresponding dynamics is linear in time. Concept of Liouville integrability can be extended to an infinite dimensional Hamiltonian with a countable set of first integrals in involution. A new theory of completely integrable Hamiltonian systems was stimulated by Gardner, Kruskal and Miura [40] who found that the eigenvalues of the Schr¨odinger operator 2 L = −@x + u(x; t) are invariant with respect to t if u(x; t) evolves according to the KdV equation ut − 6uux + uxxx = 0: (1.1) Peter Lax [65] formulated a Lax representation of the KdV equation in the form: Lt = [A; L]; (1.2) where [·; ·] is a Lie bracket and A is a skew symmetric operator which is given by d3 d A = −4 + 6u + 3u : dx3 dx x The KdV equation (1.1) is associated with the linear equations defined by the operators L and A, Lφ = λφ, Aφ = φt: If a spectral parameter λ is independent of space x and time t, a simple computa- 1 Ph.D. Thesis -Yusuke Shimabukuro Mathematics - McMaster University tion shows (Lφ)t = Ltφ + Lφt = (Lt + LA)φ, (Lφ)t = (λφ)t = λAφ = ALφ from which the Lax equation (1.2) is derived. From the spectral problem Lφ = λφ, solution behavior of the KdV equation can be studied in great detail by the inverse scattering transform [40] where the Gelfand-Levitan-Marchenko equation Z 1 K(x; y; t) + F (x + y; t) + K(x; z; t)F (z + y; t)dz = 0 (1.3) x is crucial to express the KdV solution u(x; t) as d u(x; t) = −2 K(x; x; t): dx The inhomogeneous part F (x) distinguishes two important parts of the KdV so- lution, N Z X 3 1 3 F (x) = c e−κnx+8κnt + r(k)e8ik t+ikxdk; (1.4) n 2π n=1 R where the first term is related to N solitons and the second term is related to dispersive wave packets. The explicit pure N-solitons are derived by setting r(k) = 0 in (1.4). The study of linear differential equations dates back to the nineteenth century when Sturm and Louville studied spectral property for the second-order ordinary differential equations. At the same time, transformation methods, related to lin- ear and nonlinear equations, were investigated by Darboux and Backlund. For example, Darboux [25] showed that the linear equation y00 = my + f(x)y; m = constant (1.5) is related to another linear equation d2 1 w00 = mw + θ w (1.6) dx2 θ 0 θ0 00 through w = y − θ y, where θ = f(x)θ.
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