Orbital stability of periodic waves for the nonlinear Schroedinger equation

The nonlinear Schroedinger equation has several families of quasi-periodic travelling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.Comment: 34 pages, 7 figure

On the ill-posedness result for the BBM equation

We prove that the initial value problem (IVP) for the BBM equation is ill-posed for data in Hs(R), s < 0 in the sense that the ow-map u0 7! u(t) that associates to initial data u0 the solution u cannot be continuous at the origin from Hs(R) to even D0(R) at any _xed t > 0 small enough. This result is sharp.Fundação para a Ciência e a Tecnologia (FCT

Co-periodic stability of periodic waves in some Hamiltonian PDEs

International audienceThe stability theory of periodic traveling waves is much less advanced than for solitary waves, which were first studied by Boussinesq and have received a lot of attention in the last decades. In particular, despite recent breakthroughs regarding periodic waves in reaction-diffusion equations and viscous systems of conservation laws [Johnson–Noble–Rodrigues–Zumbrun, Invent math (2014)], the stability of periodic traveling wave solutions to dispersive PDEs with respect to 'arbitrary' perturbations is still widely open in the absence of a dissipation mechanism. The focus is put here on co-periodic stability of periodic waves, that is, stability with respect to perturbations of the same period as the wave, for KdV-like systems of one-dimensional Hamiltonian PDEs. Fairly general nonlinearities are allowed in these systems, so as to include various models of mathematical physics, and this precludes complete integrability techniques. Stability criteria are derived and investigated first in a general abstract framework, and then applied to three basic examples that are very closely related, and ubiquitous in mathematical physics, namely, a quasilinear version of the generalized Korteweg–de Vries equation (qKdV), and the Euler–Korteweg system in both Eulerian coordinates (EKE) and in mass Lagrangian coordinates (EKL). Those criteria consist of a necessary condition for spectral stability , and of a sufficient condition for orbital stability. Both are expressed in terms of a single function, the abbreviated action integral along the orbits of waves in the phase plane, which is the counterpart of the solitary waves moment of instability introduced by Boussinesq. However, the resulting criteria are more complicated for periodic waves because they have more degrees of freedom than solitary waves, so that the action is a function of N + 2 variables for a system of N PDEs, while the moment of instability is a function of the wave speed only once the endstate of the 1 solitary wave is fixed. Regarding solitary waves, the celebrated Grillakis–Shatah– Strauss stability criteria amount to looking for the sign of the second derivative of the moment of instability with respect to the wave speed. For periodic waves, stability criteria involve all the second order, partial derivatives of the action. This had already been pointed out by various authors for some specific equations, in particular the generalized Korteweg–de Vries equation — which is special case of (qKdV) — but not from a general point of view, up to the authors' knowledge. The most striking results obtained here can be summarized as: an odd value for the difference between N and the negative signature of the Hessian of the action implies spectral instability, whereas a negative signature of the same Hessian being equal to N implies orbital stability. Furthermore, it is shown that, when applied to the Euler–Korteweg system, this approach yields several interesting connexions between (EKE), (EKL), and (qKdV). More precisely, (EKE) and (EKL) share the same abbreviated action integral, which is related to that of (qKdV) in a simple way. This basically proves simultaneous stability in both formulations (EKE) and (EKL) — as one may reasonably expect from the physical point view —, which is interesting to know when these models are used for different phenomena — e.g. shallow water waves or nonlinear optics. In addition, stability in (EKE) and (EKL) is found to be linked to stability in the scalar equation (qKdV). Since the relevant stability criteria are merely encoded by the negative signature of (N + 2) × (N + 2) matrices, they can at least be checked numerically. In practice, when N = 1 or 2, this can be done without even requiring an ODE solver. Various numerical experiments are presented, which clearly discriminate between stable cases and unstable cases for (qKdV), (EKE) and (EKL), thus confirming some known results for the generalized KdV equation and the Nonlinear Schrödinger equation, and pointing out some new results for more general (systems of) PDEs

Nonlinear Stability Of Periodic Traveling Wave Solutions To The Schrödinger And The Modified Korteweg-de Vries Equations

This work is concerned with stability properties of periodic traveling waves solutions of the focusing Schrödinger equationi ut + ux x + | u |2 u = 0 posed in R, and the modified Korteweg-de Vries equationut + 3 u2 ux + ux x x = 0 posed in R. Our principal goal in this paper is the study of positive periodic wave solutions of the equation φ{symbol}ω″ + φ{symbol}ω3 - ω φ{symbol}ω = 0, called dnoidal waves. A proof of the existence of a smooth curve of solutions with a fixed fundamental period L, ω ∈ (2 π2 / L2, + ∞) → φ{symbol}ω ∈ Hper∞ ([0, L]), is given. It is also shown that these solutions are nonlinearly stable in the energy space Hper1 ([0, L]) and unstable by perturbations with period 2L in the case of the Schrödinger equation. © 2007 Elsevier Inc. All rights reserved.2351130Alvarez, B., Angulo, J., Existence and stability of periodic traveling wave for the Benjamin equation (2005) Comm. Pure Appl. Anal., 4, pp. 369-390Angulo, J., Stability of dnoidal waves to Hirota-Satsuma system (2005) Differential Integral Equations, 18, pp. 611-645Benjamin, T.B., The stability of solitary waves (1972) Proc. R. Soc. Lond. Ser. A, 338, pp. 153-183Bona, J.L., On the stability theory of solitary waves (1975) Proc. R. Soc. Lond. Ser. A, 344, pp. 363-374Bronski, J.C., Carr, L.D., Carretero-González, R., Deconinck, B., Kutz, J.N., Promislow, K., Stability of attractive Bose-Einstein condensates in a periodic potential (2001) Phys. Rev. E, 64 (5), p. 056615Bronski, J.C., Carr, L.D., Deconinck, B., Kutz, J.N., Bose-Einstein condensates in standing waves: The cubic nonlinear Schrödinger equation with a periodic potential (2001) Phys. Rev. Lett., 86 (8), pp. 1402-1405Bourgain, J., Global Solutions of Nonlinear Schrödinger Equations (1999) Amer. Math. Soc. Coll. Publ., 46. , American Mathematical Society, Providence, RIByrd, P.F., Friedman, M.D., (1971) Handbook of Elliptic Integrals for Engineers and Scientists. second ed., , Springer-Verlag, New YorkCazenave, T., An Introduction to Nonlinear Schrödinger Equations (1989) Textos Métodos Matemáticos. third ed., 26. , Universidade Federal do Rio de JaneiroColliander, J., Kell, M., Staffilani, G., Takaoka, H., Tao, T., Sharp global well-posedness for periodic and non-periodic KdV and mKdV on R and T (2003) J. Amer. Math. Soc. (16), 3Eastham, M.S.P., (1973) The Spectral Theory of Periodic Differential Equations, , Scottish Academic Press, LondonErdélyi, A., On Lamé functions (1941) Philos. Mag. (7), 31, pp. 123-130Grillakis, M., Linearized instability for nonlinear Schrödinger and Klein-Gordon equations (1988) J. Comm. Pure Appl. Math., XLI, pp. 747-774Grillakis, M., Shatah, J., Strauss, W., Stability theory of solitary waves in the presence of symmetry, I (1987) J. Funct. Anal., 74, pp. 160-197Grillakis, M., Shatah, J., Strauss, W., Stability theory of solitary waves in the presence of symmetry, II (1990) J. Funct. Anal., 94, pp. 308-348Ince, E.L., The periodic Lamé functions (1940) Proc. Roy. Soc. Edinburgh, 60, pp. 47-63Iorio Jr., R.J., Iorio de Magalhães, V., Fourier Analysis and Partial Differential Equations (2001) Cambridge Stud. Adv. Math., 70Kenig, C.E., Ponce, G., Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle (1993) Comm. Pure Appl. Math., 46, pp. 527-620Magnus, W., Winkler, S., Hill's Equation (1976) Tracts Pure Appl. Math., 20. , Wiley, New YorkReed, S., Simon, B., (1978) Methods of Modern Mathematical Physics: Analysis of Operator, vol. V, , Academic PressRussell, D.L., Sun, S.-M., Stability properties of non-constant stationary solutions of the periodic Korteweg-de Vries equation (1998) Appl. Anal., 68 (3-4), pp. 207-240Weinstein, M.I., Modulation stability of ground states of nonlinear Schrödinger equations (1985) SIAM J. Math., 16, pp. 472-490Weinstein, M.I., Lyapunov stability of ground states of nonlinear dispersive evolution equations (1986) Comm. Pure Appl. Math., 39, pp. 51-6