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Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations
, 2001
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Schur flows for orthogonal Hessenberg matrices. Hamiltonian and gradient flows, algorithms and control
 Algorithms and Control, Fields Inst. Commun
, 1994
"... Abstract We consider a standard matrix flow on the set of unitary upper Hessenberg matrices with nonnegative subdiagonal elements. The Schur parametrization of this set of matrices leads to ordinary differential equations for the weights and the parameters that are analogous with the Toda flow as id ..."
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Abstract We consider a standard matrix flow on the set of unitary upper Hessenberg matrices with nonnegative subdiagonal elements. The Schur parametrization of this set of matrices leads to ordinary differential equations for the weights and the parameters that are analogous with the Toda flow as identified with a flow on Jacobi matrices. We derive explicit differential equations for the flow on the Schur parameters of orthogonal Hessenberg matrices. We also outline an efficient procedure for computing the solution of Jacobi flows and Schur flows.
Why are solitons stable?
 BULL. AMER. MATH. SOC. (N.S
, 2009
"... The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, solitary wave or soliton solutions can be created, which can be st ..."
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The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, solitary wave or soliton solutions can be created, which can be stable enough to persist indefinitely. The construction of such solutions can be relatively straightforward, but the fact that they are stable requires some significant amounts of analysis to establish, in part due to symmetries in the equation (such as translation invariance) which create degeneracy in the stability analysis. The theory is particularly difficult in the critical case in which the nonlinearity is at exactly the right power to potentially allow for a selfsimilar blowup. In this article we survey some of the highlights of this theory, from the more classical orbital stability analysis of Weinstein and GrillakisShatahStrauss, to the more recent asymptotic stability and blowup analysis of MartelMerle and MerleRaphael, as well as current developments in using this theory to rigorously demonstrate controlled blowup for several key equations.
The Nonlinear Schrödinger Equation as Both a PDE and a Dynamical System
 IN HANDBOOK OF DYNAMICAL SYSTEMS
, 2000
"... Nonlinear dispersive wave equations provide excellent examples of innite dimensional dynamical systems which possess diverse and fascinating phenomena including solitary waves and wave trains, the generation and propagation of oscillations, the formation of singularities, the persistence of homoclin ..."
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Cited by 9 (1 self)
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Nonlinear dispersive wave equations provide excellent examples of innite dimensional dynamical systems which possess diverse and fascinating phenomena including solitary waves and wave trains, the generation and propagation of oscillations, the formation of singularities, the persistence of homoclinic orbits, the existence of temporally chaotic waves in deterministic systems, dispersive turbulence and the propagation of spatiotemporal chaos. Nonlinear dispersive waves occur throughout physical and natural systems whenever dissipation is weak. Important applications include nonlinear optics and long distance communication devices such as transoceanic optical bers, waves in the atmosphere and the ocean, and turbulence in plasmas. Examples of nonlinear dispersive partial dierential equations include the Korteweg de Vries equation, nonlinear Klein Gordon equations, nonlinear Schrödinger equations, and many others. In this survey article, we choose a class of nonlinear Schrödinger equa...
Refined asymptotics around solitons for gKdV equations
"... We consider the generalized Kortewegde Vries equation ∂tu + ∂x( ∂ 2 xu + f(u)) = 0, (t, x) ∈ [0, T) × R (0.1) with general C 2 nonlinearity f. Under an explicit condition on f and c> 0, there exists a solution in the energy space H 1 of (0.1) of the type u(t, x) = Qc(x − x0 − ct), called solito ..."
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Cited by 9 (6 self)
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We consider the generalized Kortewegde Vries equation ∂tu + ∂x( ∂ 2 xu + f(u)) = 0, (t, x) ∈ [0, T) × R (0.1) with general C 2 nonlinearity f. Under an explicit condition on f and c> 0, there exists a solution in the energy space H 1 of (0.1) of the type u(t, x) = Qc(x − x0 − ct), called soliton. Stability theory for Qc is wellknown. In [11], [14], we have proved that for f(u) = u p, p = 2, 3, 4, the family of solitons is asymptotically stable in some local sense in H 1, i.e. if u(t) is close to Qc (for all t ≥ 0), then u(t,. + ρ(t)) locally converges in the energy space to some Qc+ as t → +∞, for some c + ∼ c. The main improvement in [14] is a direct proof, based on a localized Viriel identity on the solution u(t). As a consequence, we have obtained an integral estimate on u(t,. + ρ(t)) − Qc+ as t → +∞. In [9] and [15], using the indirect approach of [11], we could extend the asymptotic
Stability of two soliton collision for nonintegrable gKdV
 equations, Comm. Math. Phys
"... We continue our study of the collision of two solitons for the subcritical generalized KdV equations ∂tu + ∂x( ∂ 2 xu + f(u)) = 0. (0.1) Solitons are solutions o the type u(t, x) = Qc0(x − x0 − c0t) where c0> 0. In [21], mainly devoted to the case f(u) = u4, we have introduced a new framework to ..."
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We continue our study of the collision of two solitons for the subcritical generalized KdV equations ∂tu + ∂x( ∂ 2 xu + f(u)) = 0. (0.1) Solitons are solutions o the type u(t, x) = Qc0(x − x0 − c0t) where c0> 0. In [21], mainly devoted to the case f(u) = u4, we have introduced a new framework to understand the collision of two solitons Qc1, Qc2 for (0.1) in the case c2 ≪ c1 (or equivalently, ‖Qc2‖H1 ≪ ‖Qc1‖H1). In this paper, we consider the case of a general nonlinearity f(u) for which Qc1, Qc2 are nonlinearly stable. In particular, since f is general and c1 can be large, the results are not pertubations of the ones for the power case in [21]. First, we prove that the two solitons survive the collision up to a shift in their trajectory and up to a small perturbation term whose size is explicitely controlled from above: after the collision, u(t) ∼ Q c
Comparison of quarterplane and twopoint boundaryvalue problems: the BBMequation
 Discrete & Cont. Dynamical Systems, Series A 13
, 2005
"... Abstract. The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude long waves in shallow water and other dispersive media. Interest will be turned to the twopoint boundary value problem wherein the wave motion is specified at both ends of a finit ..."
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Abstract. The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude long waves in shallow water and other dispersive media. Interest will be turned to the twopoint boundary value problem wherein the wave motion is specified at both ends of a finite stretch of the medium of propagation. The principal new result is an exact theory of convergence of the twopoint boundary value problem to the quarterplane boundary value problem in which a semiinfinite stretch of the medium is disturbed at its finite end. The latter problem has been featured in modeling waves generated by a wavemaker in a flume and in describing the evolution of long crested, deep water waves propagating into the near shore zone of large bodies of water. In addition to their intrinsic interest, our results provide justification for the use of the twopoint boundary value problem in numerical studies of the quarter plane problem. 1. Introduction. Considered
THE MIURA MAP ON THE LINE
, 2005
"... Abstract. We study relations between properties of the Miura map r ↦ → q = B(r) = r ′ + r2 and Schrödinger operators Lq = −d2 /dx2 + q where r and q are realvalued functions or distributions (possibly not decaying at infinity) from various classes. In particular, we study B as a map from L2 loc (R ..."
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Abstract. We study relations between properties of the Miura map r ↦ → q = B(r) = r ′ + r2 and Schrödinger operators Lq = −d2 /dx2 + q where r and q are realvalued functions or distributions (possibly not decaying at infinity) from various classes. In particular, we study B as a map from L2 loc (R) to the local Sobolev space H −1 loc (R) and the restriction of B to the Sobolev spaces Hβ (R) with β ≥ 0. For example, we prove that the image of B on L2 loc (R) consists exactly of those q ∈ H −1 loc (R) such that the operator Lq is positive. We also investigate mapping properties of the Miura map in these spaces. As an application we prove an existence result for solutions of the Kortewegde Vries equation in H−1 (R) for initial data in the range B(L2 (R)) of the Miura
INELASTIC INTERACTION OF NEARLY EQUAL SOLITONS FOR THE BBM EQUATION
"... Abstract. This paper is concerned with the interaction of two solitons of nearly equal speeds for the (BBM) equation. This work is an extension of [31] addressing the same question for the quartic (gKdV) equation. We consider the (BBM) equation, for λ ∈ [0,1), (1 − λ ∂ 2 x)∂tu + ∂x(∂2 x u − u + u2) ..."
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Abstract. This paper is concerned with the interaction of two solitons of nearly equal speeds for the (BBM) equation. This work is an extension of [31] addressing the same question for the quartic (gKdV) equation. We consider the (BBM) equation, for λ ∈ [0,1), (1 − λ ∂ 2 x)∂tu + ∂x(∂2 x u − u + u2) = 0. (BBM) Solitons are solutions of the form Rµ,x0 (t, x) = Qµ(x − µt − x0), for µ> −1, x0 ∈ R. For µ0> 0 small, let U(t, x) be the unique solution of (BBM) such that lim t→− ∞ ‖U(t) − Q−µ0 (. + µ0t) − Qµ0 (. − µ0t)‖H1 = 0. First, we prove that U(t) remains close to the sum of two solitons, for all time t ∈ R, U(t, x) = Q µ1(t)(x − y1(t)) + Q µ2(t)(x − y2(t)) + ε(t) where ‖ε(t) ‖ ≤ µ 2− 0, with y1(t) − y2(t)> 2  lnµ0  + O(1), which means that at the main order the