Papers by Tetsu Mizumachi
arXiv (Cornell University), Mar 14, 2013
We prove nonlinear stability of line soliton solutions of the KP-II equation with respect to tran... more We prove nonlinear stability of line soliton solutions of the KP-II equation with respect to transverse perturbations that are exponentially localized as x → ∞. We find that the amplitude of the line soliton converges to that of the line soliton at initial time whereas jumps of the local phase shift of the crest propagate in a finite speed toward y = ±∞. The local amplitude and the phase shift of the crest of the line solitons are described by a system of 1D wave equations with diffraction terms. 1. Introduction 2. The Miura transformation and resonant modes of the linearized operator 2.1. Resonant modes 2.2. Linearized Miura transformation 3. Semigroup estimates for the linearized KP-II equation 4. Preliminaries 5. Decomposition of the perturbed line soliton 6. Modulation equations 7. À priori estimates for c(t, y) and x y (t, y) 8. L 2 bound on v(t, z, y) 9. Low frequencies bound of v(t, x, y) in y 10. Virial estimates 11. Proof of Theorem 1.1 12. Proof of Theorem 1.2 13. Proof of Theorem 1.3 Appendix A. Proof of Lemma 6.1
Archive for Rational Mechanics and Analysis, Jul 1, 2009
We prove that the collision of two solitary waves of the BBM equation is inelastic but almost ela... more We prove that the collision of two solitary waves of the BBM equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space. We show precise estimates of the nonzero residue due to the collision. Moreover, we give a precise description of the collision phenomenon (change of size of the solitary waves and shifts in their trajectories). To prove these results, we extend the method introduced in [27] and for the generalized KdV equation, in particular in the quartic case. The main argument is the construction of an explicit approximate solution (in a certain sense) in the collision region.
Nonlinearity, 2008
We establish an asymptotic stability result for Toda lattice soliton solutions, by making use of ... more We establish an asymptotic stability result for Toda lattice soliton solutions, by making use of a linearized Bäcklund transformation whose domain has codimension one. Combining a linear stability result with a general theory of nonlinear stability by Friesecke and Pego for solitary waves in lattice equations, we conclude that all solitons in the Toda lattice are asymptotically stable in an exponentially weighted norm. In addition, we determine the complete spectrum of an operator naturally associated with the Floquet theory for these lattice solitons.
Archive for Rational Mechanics and Analysis, 2009
We prove that the collision of two solitary waves of the BBM equation is inelastic but almost ela... more We prove that the collision of two solitary waves of the BBM equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space. We show precise estimates of the nonzero residue due to the collision. Moreover, we give a precise description of the collision phenomenon (change of size of the solitary waves and shifts in their trajectories). To prove these results, we extend the method introduced in [27] and for the generalized KdV equation, in particular in the quartic case. The main argument is the construction of an explicit approximate solution (in a certain sense) in the collision region.
arXiv (Cornell University), Nov 26, 2010
Ground states of a L 2 -subcritical focusing nonlinear Schrödinger (NLS) equation are known to be... more Ground states of a L 2 -subcritical focusing nonlinear Schrödinger (NLS) equation are known to be orbitally stable in the energy class H 1 (R) thanks to its variational characterization. In this paper, we will show L 2 -stability of 1-solitons to a one-dimensional cubic NLS equation in the sense that for any initial data which are sufficiently close to a 1-soliton in L 2 (R), the solution remains in an L 2 -neighborhood of a nearby 1-soliton solution for all the time. The proof relies on the Bäcklund transformation between zero and soliton solutions of this integrable equation.
arXiv: Analysis of PDEs, 2017
In this paper, we study transverse linear stability of line solitary waves to the $2$-dimensional... more In this paper, we study transverse linear stability of line solitary waves to the $2$-dimensional Benney-Luke equation which arises in the study of small amplitude long water waves in $3$D. In the case where the surface tension is weak or negligible, we find a curve of resonant continuous eigenvalues near $0$. Time evolution of these resonant continuous eigenmodes is described by a $1$D damped wave equation in the transverse variable and it gives a linear approximation of the local phase shifts of modulating line solitary waves. In exponentially weighted space whose weight function increases in the direction of the motion of the line solitary wave, the other part of solutions to the linearized equation decays exponentially as $t\to\infty$.
arXiv (Cornell University), Mar 14, 2013
We prove nonlinear stability of line soliton solutions of the KP-II equation with respect to tran... more We prove nonlinear stability of line soliton solutions of the KP-II equation with respect to transverse perturbations that are exponentially localized as x → ∞. We find that the amplitude of the line soliton converges to that of the line soliton at initial time whereas jumps of the local phase shift of the crest propagate in a finite speed toward y = ±∞. The local amplitude and the phase shift of the crest of the line solitons are described by a system of 1D wave equations with diffraction terms. Contents 1. Introduction 2. The Miura transformation and resonant modes of the linearized operator 2.1. Resonant modes 2.2. Linearized Miura transformation 3. Semigroup estimates for the linearized KP-II equation 4. Preliminaries 5. Decomposition of the perturbed line soliton 6. Modulation equations 7.À priori estimates for c(t, y) and x y (t, y) 8. L 2 bound on v(t, z, y) 9. Low frequencies bound of v(t, x, y) in y 10. Virial estimates 11.
Discrete & Continuous Dynamical Systems - S, 2012
Asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation was earlie... more Asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation was earlier established for septic and higher-order nonlinear terms by using Strichartz estimate. We use here pointwise dispersive decay estimates to push down the lower bound for the exponent of the nonlinear terms.
International Mathematics Research Notices, 2011
Ground states of a L 2-subcritical focusing nonlinear Schrödinger (NLS) equation are known to be ... more Ground states of a L 2-subcritical focusing nonlinear Schrödinger (NLS) equation are known to be orbitally stable in the energy class H 1 (R) thanks to its variational characterization. In this paper, we will show L 2-stability of 1-solitons to a one-dimensional cubic NLS equation in the sense that for any initial data which are sufficiently close to a 1-soliton in L 2 (R), the solution remains in an L 2-neighborhood of a nearby 1-soliton solution for all the time. The proof relies on the Bäcklund transformation between zero and soliton solutions of this integrable equation.
arXiv (Cornell University), Apr 1, 2019
The 2D Benney-Luke equation is an isotropic model which describes long water waves of small ampli... more The 2D Benney-Luke equation is an isotropic model which describes long water waves of small amplitude in 3D whereas the KP-II equation is a unidirectional model for long waves with slow variation in the transverse direction. In the case where the surface tension is weak or negligible, linearly stability of small line solitary waves of the 2D Benney-Luke equation was proved by Mizumachi and Shimabukuro [Nonlinearity, 30 (2017), 3419-3465]. In this paper, we prove nonlinear stability of the line solitary waves by adopting the argument by Mizumachi ([
Memoirs of the American Mathematical Society, 2015
Mizumachi, Tetsu, 1969-Stability of line solitons for the KP-II equation in R 2 / Tetsu Mizumachi... more Mizumachi, Tetsu, 1969-Stability of line solitons for the KP-II equation in R 2 / Tetsu Mizumachi. pages cm.-(Memoirs of the American Mathematical Society 0065-9266 ; volume 238, number 1125) Includes bibliographical references.
Nonlinearity, 2017
The 2D Benney-Luke equation is an isotropic model which describes long water waves of small ampli... more The 2D Benney-Luke equation is an isotropic model which describes long water waves of small amplitude in 3D whereas the KP-II equation is a unidirectional model for long waves with slow variation in the transverse direction. In the case where the surface tension is weak or negligible, linearly stability of small line solitary waves of the 2D Benney-Luke equation was proved by Mizumachi and Shimabukuro [Nonlinearity, 30 (2017), 3419-3465]. In this paper, we prove nonlinear stability of the line solitary waves by adopting the argument by Mizumachi ([
Stability of Benney--Luke Line Solitary Waves in 2 Dimensions
SIAM Journal on Mathematical Analysis, 2020
The two-dimensional Benney--Luke equation is an isotropic model which describes long water waves ... more The two-dimensional Benney--Luke equation is an isotropic model which describes long water waves of small amplitude in 3 dimensions whereas the KP-II equation is a unidirectional model for long wav...
Kyoto Journal of Mathematics, 2008
We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinea... more We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinear Schrödinger equations iu t + u xx = V u ± |u| p−1 u for (x, t) ∈ R × R, in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai [18] in the 3-dimensional case using the endpoint Strichartz estimate. To prove asymptotic stability of solitary waves, we need to show that a dispersive part v(t, x) of a solution belongs to L 2 t (0, ∞; X) for some space X. In the 1-dimensional case, this property does not follow from the Strichartz estimate alone. In this paper, we prove that a local smoothing estimate of Kato type holds globally in time and combine the estimate with the Strichartz estimate to show (1 + x 2) −3/4 v L ∞ x L 2 t < ∞, which implies the asymptotic stability of a solitary wave.
Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2009
We transpose work by T.Mizumachi to prove smoothing estimates for dispersive solutions of the lin... more We transpose work by T.Mizumachi to prove smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schrödinger equation (NLS) in 1D. As an application we extend to dimension 1D a result on asymptotic stability of ground states of NLS proved by Cuccagna & Mizumachi for dimensions ≥ 3.
Kyoto Journal of Mathematics, 2007
We consider asymptotic stability of a small solitary wave to supercritical 2-dimensional nonlinea... more We consider asymptotic stability of a small solitary wave to supercritical 2-dimensional nonlinear Schrödinger equations iu t + ∆u = V u ± |u| p−1 u for (x, t) ∈ R 2 × R, in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai [14] in the n-dimensional case (n ≥ 3) by using the endpoint Strichartz estimate. Since the endpoint Strichartz estimate fails in 2dimensional case, we use a time-global local smoothing estimate of Kato type to prove the asymptotic stability of a solitary wave.
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Papers by Tetsu Mizumachi