×

Instability, index theorem, and exponential trichotomy for linear Hamiltonian PDEs. (English) Zbl 07455851

Memoirs of the American Mathematical Society 1347. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-5044-1/pbk; 978-1-4704-7013-5/ebook). v, 136 p. (2022).
Summary: Consider a general linear Hamiltonian system \(\partial_tu=JLu\) in a Hilbert space \(X\). We assume that \(L:X\rightarrow X^{\ast }\) induces a bounded and symmetric bi-linear form \(\left \langle L\cdot ,\cdot \right \rangle\) on \(X\), which has only finitely many negative dimensions \(n^-(L)\). There is no restriction on the anti-self-dual operator \(J:X^{\ast }\supset D(J)\rightarrow X\). We first obtain a structural decomposition of \(X\) into the direct sum of several closed subspaces so that \(L\) is blockwise diagonalized and \(JL\) is of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy of \(e^{tJL}\). In particular, \(e^{tJL}\) has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relate \(n^-\left ( L\right )\) and the dimensions of generalized eigenspaces of eigenvalues of \(JL\), some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostly \(J\) was assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schrödinger equations with nonzero conditions at infinity, where our general theory applies to yield stability or instability of some coherent states.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B35 Stability in context of PDEs
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35P05 General topics in linear spectral theory for PDEs
47A10 Spectrum, resolvent
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Alexander, J. C.; Sachs, R., Linear instability of solitary waves of a Boussinesq-type equation: a computer assisted computation, Nonlinear World, 2, 4, 471-507 (1995) · Zbl 0833.34046
[2] Angulo Pava, Jaime, Nonlinear dispersive equations, Mathematical Surveys and Monographs 156, xii+256 pp. (2009), American Mathematical Society, Providence, RI · Zbl 1202.35246
[3] Angulo Pava, Jaime; Bona, Jerry L.; Scialom, Marcia, Stability of cnoidal waves, Adv. Differential Equations, 11, 12, 1321-1374 (2006) · Zbl 1147.35079
[4] Bates, Peter W.; Jones, Christopher K. R. T., Invariant manifolds for semilinear partial differential equations. Dynamics reported, Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl. 2, 1-38 (1989), Wiley, Chichester
[5] Benjamin, T. B., The stability of solitary waves, Proc. Roy. Soc. (London) Ser. A, 328, 153-183 (1972)
[6] B\'{e}thuel, Fabrice; Gravejat, Philippe; Saut, Jean-Claude, Existence and properties of travelling waves for the Gross-Pitaevskii equation. Stationary and time dependent Gross-Pitaevskii equations, Contemp. Math. 473, 55-103 (2008), Amer. Math. Soc., Providence, RI · Zbl 1216.35132
[7] B\'{e}thuel, Fabrice; Gravejat, Philippe; Saut, Jean-Claude, Travelling waves for the Gross-Pitaevskii equation. II, Comm. Math. Phys., 285, 2, 567-651 (2009) · Zbl 1190.35196
[8] Benzoni-Gavage, S.; Mietka, C.; Rodrigues, L. M., Co-periodic stability of periodic waves in some Hamiltonian PDEs, Nonlinearity, 29, 11, 3241-3308 (2016) · Zbl 1362.35037
[9] Bona, J. L.; Souganidis, P. E.; Strauss, W. A., Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. London Ser. A, 411, 1841, 395-412 (1987) · Zbl 0648.76005
[10] Bona, Jerry L.; Sachs, Robert L., Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., 118, 1, 15-29 (1988) · Zbl 0654.35018
[11] Bronski, Jared; Johnson, Mathew A.; Kapitula, Todd, An instability index theory for quadratic pencils and applications, Comm. Math. Phys., 327, 2, 521-550 (2014) · Zbl 1301.35081
[12] Bronski, Jared C.; Hur, Vera Mikyoung; Johnson, Mathew A., Modulational instability in equations of KdV type. New approaches to nonlinear waves, Lecture Notes in Phys. 908, 83-133 (2016), Springer, Cham
[13] Bronski, Jared C.; Johnson, Mathew A.; Kapitula, Todd, An index theorem for the stability of periodic travelling waves of Korteweg-de Vries type, Proc. Roy. Soc. Edinburgh Sect. A, 141, 6, 1141-1173 (2011) · Zbl 1230.35118
[14] Caglioti, E.; Lions, P.-L.; Marchioro, C.; Pulvirenti, M., A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Comm. Math. Phys., 143, 3, 501-525 (1992) · Zbl 0745.76001
[15] Caglioti, E.; Lions, P.-L.; Marchioro, C.; Pulvirenti, M., A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. II, Comm. Math. Phys., 174, 2, 229-260 (1995) · Zbl 0840.76002
[16] Chow, Shui-Nee; Lin, Xiao-Biao; Lu, Kening, Smooth invariant foliations in infinite-dimensional spaces, J. Differential Equations, 94, 2, 266-291 (1991) · Zbl 0749.58043
[17] Chow, Shui-Nee; Lu, Kening, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74, 2, 285-317 (1988) · Zbl 0691.58034
[18] Chugunova, Marina; Pelinovsky, Dmitry, Count of eigenvalues in the generalized eigenvalue problem, J. Math. Phys., 51, 5, 052901, 19 pp. (2010) · Zbl 1310.35212
[19] Chiron, David; Mari\c{s}, Mihai, Traveling waves for nonlinear Schr\"{o}dinger equations with nonzero conditions at infinity, Arch. Ration. Mech. Anal., 226, 1, 143-242 (2017) · Zbl 1391.35351
[20] Chiron, David; Scheid, Claire, Travelling waves for the nonlinear Schr\"{o}dinger equation with general nonlinearity in dimension two, J. Nonlinear Sci., 26, 1, 171-231 (2016) · Zbl 1336.35318
[21] Cuccagna, Scipio; Pelinovsky, Dmitry; Vougalter, Vitali, Spectra of positive and negative energies in the linearized NLS problem, Comm. Pure Appl. Math., 58, 1, 1-29 (2005) · Zbl 1064.35181
[22] Cycon, H. L.; Froese, R. G.; Kirsch, W.; Simon, B., Schr\"{o}dinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, x+319 pp. (1987), Springer-Verlag, Berlin · Zbl 0619.47005
[23] Gurski, K. F.; Koll\'{a}r, R.; Pego, R. L., Slow damping of internal waves in a stably stratified fluid, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460, 2044, 977-994 (2004) · Zbl 1070.76020
[24] Ekeland, Ivar, Convexity methods in Hamiltonian mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 19, x+247 pp. (1990), Springer-Verlag, Berlin · Zbl 0707.70003
[25] Fan, Ky, Invariant subspaces of certain linear operators, Bull. Amer. Math. Soc., 69, 773-777 (1963) · Zbl 0118.10803
[26] Gesztesy, F.; Jones, C. K. R. T.; Latushkin, Y.; Stanislavova, M., A spectral mapping theorem and invariant manifolds for nonlinear Schr\"{o}dinger equations, Indiana Univ. Math. J., 49, 1, 221-243 (2000) · Zbl 0969.35123
[27] Grenier, Emmanuel, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., 53, 9, 1067-1091 (2000) · Zbl 1048.35081
[28] Grillakis, Manoussos, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math., 43, 3, 299-333 (1990) · Zbl 0731.35010
[29] Grillakis, Manoussos; Shatah, Jalal; Strauss, Walter, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74, 1, 160-197 (1987) · Zbl 0656.35122
[30] Grillakis, Manoussos; Shatah, Jalal; Strauss, Walter, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal., 94, 2, 308-348 (1990) · Zbl 0711.58013
[31] Guo, Yan; Strauss, Walter A., Instability of periodic BGK equilibria, Comm. Pure Appl. Math., 48, 8, 861-894 (1995) · Zbl 0840.45012
[32] H\v{a}r\v{a}gu\c{s}, Mariana; Kapitula, Todd, On the spectra of periodic waves for infinite-dimensional Hamiltonian systems, Phys. D, 237, 20, 2649-2671 (2008) · Zbl 1155.37039
[33] Hille, Einar; Phillips, Ralph S., Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, xii+808 pp. (1957), American Mathematical Society, Providence, R. I. · Zbl 0078.10004
[34] Hur, Vera Mikyoung; Johnson, Mathew A., Stability of periodic traveling waves for nonlinear dispersive equations, SIAM J. Math. Anal., 47, 5, 3528-3554 (2015) · Zbl 1327.35032
[35] Iohvidov, I. S., On the spectra of Hermitian and unitary operators in a space with indefinite metric, Doklady Akad. Nauk SSSR (N.S.), 71, 225-228 (1950)
[36] Jin, Jiayin; Liao, Shasha; Lin, Zhiwu, Nonlinear modulational instability of dispersive PDE models, Arch. Ration. Mech. Anal., 231, 3, 1487-1530 (2019) · Zbl 1410.37067
[37] Jin, Jiayin; Lin, Zhiwu; Zeng, Chongchun, Dynamics near the solitary waves of the supercritical gKDV equations, J. Differential Equations, 267, 12, 7213-7262 (2019) · Zbl 1423.35066
[38] Jin, Jiayin; Lin, Zhiwu; Zeng, Chongchun, Invariant manifolds of traveling waves of the 3D Gross-Pitaevskii equation in the energy space, Comm. Math. Phys., 364, 3, 981-1039 (2018) · Zbl 1406.35360
[39] C. A. Jones, S. J. Putterman, and P. Roberts, H., Motions in a Bose condensate V. Stability of solitary wave solutions of nonlinear Schrodinger equations in two and three dimensions. J. Phys. A, Math. Gen., 19 (1986), 2991-3011.
[40] C. A. Jones, and P. H. Roberts, Motions in a Bose condensate IV. Axisymmetric solitary waves. J. Phys. A, Math. Gen., 15 (1982), 2599-2619.
[41] Johnson, Mathew A., Stability of small periodic waves in fractional KdV-type equations, SIAM J. Math. Anal., 45, 5, 3168-3193 (2013) · Zbl 1282.35334
[42] Kato, Tosio, Perturbation theory for linear operators, Classics in Mathematics, xxii+619 pp. (1995), Springer-Verlag, Berlin · Zbl 0836.47009
[43] Kapitula, Todd; Deconinck, Bernard, On the spectral and orbital stability of spatially periodic stationary solutions of generalized Korteweg-de Vries equations. Hamiltonian partial differential equations and applications, Fields Inst. Commun. 75, 285-322 (2015), Fields Inst. Res. Math. Sci., Toronto, ON · Zbl 1331.35305
[44] Kapitula, Todd; Kevrekidis, Panayotis G.; Sandstede, Bj\"{o}rn, Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems, Phys. D, 195, 3-4, 263-282 (2004) · Zbl 1056.37080
[45] Kapitula, Todd; Kevrekidis, Panayotis G.; Sandstede, Bj\"{o}rn, Addendum: “Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems” [Phys. D {\bf 195} (2004), no. 3-4, 263-282; MR2089513], Phys. D, 201, 1-2, 199-201 (2005) · Zbl 1080.37070
[46] Kapitula, Todd; Stefanov, Atanas, A Hamiltonian-Krein (instability) index theory for solitary waves to KdV-like eigenvalue problems, Stud. Appl. Math., 132, 3, 183-211 (2014) · Zbl 1288.35422
[47] Koll\'{a}r, Richard; Miller, Peter D., Graphical Krein signature theory and Evans-Krein functions, SIAM Rev., 56, 1, 73-123 (2014) · Zbl 1300.47078
[48] Kre\u{\i }n, M. G., A new application of the fixed-point principle in the theory of operators in a space with indefinite metric, Dokl. Akad. Nauk SSSR, 154, 1023-1026 (1964)
[49] Lin, Zhiwu, Some stability and instability criteria for ideal plane flows, Comm. Math. Phys., 246, 1, 87-112 (2004) · Zbl 1061.76016
[50] Lin, Zhiwu, Nonlinear instability of ideal plane flows, Int. Math. Res. Not., 41, 2147-2178 (2004) · Zbl 1080.35085
[51] Lin, Zhiwu, Nonlinear instability of periodic BGK waves for Vlasov-Poisson system, Comm. Pure Appl. Math., 58, 4, 505-528 (2005) · Zbl 1067.35012
[52] Lin, Zhiwu, Instability of nonlinear dispersive solitary waves, J. Funct. Anal., 255, 5, 1191-1224 (2008) · Zbl 1157.35096
[53] Lin, Zhiwu; Wang, Zhengping; Zeng, Chongchun, Stability of traveling waves of nonlinear Schr\"{o}dinger equation with nonzero condition at infinity, Arch. Ration. Mech. Anal., 222, 1, 143-212 (2016) · Zbl 1457.35070
[54] Lin, Zhiwu; Xu, Ming, Metastability of Kolmogorov flows and inviscid damping of shear flows, Arch. Ration. Mech. Anal., 231, 3, 1811-1852 (2019) · Zbl 1426.76155
[55] Lin, Zhiwu; Yang, Jincheng; Zhu, Hao, Barotropic instability of shear flows, Stud. Appl. Math., 144, 3, 289-326 (2020) · Zbl 1445.76044
[56] Lin, Zhiwu; Zeng, Chongchun, Inviscid dynamical structures near Couette flow, Arch. Ration. Mech. Anal., 200, 3, 1075-1097 (2011) · Zbl 1229.35197
[57] Lin, Zhiwu; Zeng, Chongchun, Unstable manifolds of Euler equations, Comm. Pure Appl. Math., 66, 11, 1803-1836 (2013) · Zbl 1360.35160
[58] Liu, Yue, Instability of solitary waves for generalized Boussinesq equations, J. Dynam. Differential Equations, 5, 3, 537-558 (1993) · Zbl 0784.34048
[59] MacKay, R. S., Stability of equilibria of Hamiltonian systems. Nonlinear phenomena and chaos, Malvern, 1985, Malvern Phys. Ser., 254-270 (1986), Hilger, Bristol
[60] Mari\c{s}, Mihai, Traveling waves for nonlinear Schr\"{o}dinger equations with nonzero conditions at infinity, Ann. of Math. (2), 178, 1, 107-182 (2013) · Zbl 1315.35207
[61] Mari\c{s}, Mihai, Nonexistence of supersonic traveling waves for nonlinear Schr\"{o}dinger equations with nonzero conditions at infinity, SIAM J. Math. Anal., 40, 3, 1076-1103 (2008) · Zbl 1167.35518
[62] Nakanishi, Kenji; Schlag, Wilhelm, Invariant manifolds and dispersive Hamiltonian evolution equations, Zurich Lectures in Advanced Mathematics, vi+253 pp. (2011), European Mathematical Society (EMS), Z\"{u}rich · Zbl 1235.37002
[63] Pego, Robert L.; Weinstein, Michael I., Eigenvalues, and instabilities of solitary waves, Philos. Trans. Roy. Soc. London Ser. A, 340, 1656, 47-94 (1992) · Zbl 0776.35065
[64] Pontrjagin, L., Hermitian operators in spaces with indefinite metric, Bull. Acad. Sci. URSS. S\'{e}r. Math. [Izvestia Akad. Nauk SSSR], 8, 243-280 (1944) · Zbl 0061.26004
[65] Pelinovsky, Dmitry E., Spectral stability on nonlinear waves in KdV-type evolution equations. Nonlinear physical systems, Mech. Eng. Solid Mech. Ser., 377-400 (2014), Wiley, Hoboken, NJ · Zbl 1456.37079
[66] Renardy, Michael, On the linear stability of hyperbolic PDEs and viscoelastic flows, Z. Angew. Math. Phys., 45, 6, 854-865 (1994) · Zbl 0820.76008
[67] Shizuta, Yasushi, On the classical solutions of the Boltzmann equation, Comm. Pure Appl. Math., 36, 6, 705-754 (1983) · Zbl 0515.35002
[68] Stanislavova, Milena; Stefanov, Atanas, Linear stability analysis for travelling waves of second order in time PDE’s, Nonlinearity, 25, 9, 2625-2654 (2012) · Zbl 1259.35032
[69] Stanislavova, Milena; Stefanov, Atanas, On the spectral problem \(\mathcal{L}u=\lambda u'\) and applications, Comm. Math. Phys., 343, 2, 361-391 (2016) · Zbl 1427.47010
[70] Souganidis, P. E.; Strauss, W. A., Instability of a class of dispersive solitary waves, Proc. Roy. Soc. Edinburgh Sect. A, 114, 3-4, 195-212 (1990) · Zbl 0713.35108
[71] Tabeling, Patrick, Two-dimensional turbulence: a physicist approach, Phys. Rep., 362, 1, 1-62 (2002) · Zbl 1001.76041
[72] Vidav, Ivan, Spectra of perturbed semigroups with applications to transport theory., J. Math. Anal. Appl., 30, 264-279 (1970) · Zbl 0195.13704
[73] Yosida, K\={o}saku, Functional analysis, Classics in Mathematics, xii+501 pp. (1995), Springer-Verlag, Berlin · Zbl 0830.46001
[74] Wei, Dongyi; Zhang, Zhifei; Zhao, Weiren, Linear inviscid damping for a class of monotone shear flow in Sobolev spaces, Comm. Pure Appl. Math., 71, 4, 617-687 (2018) · Zbl 1390.35251
[75] Wei, Dongyi; Zhang, Zhifei; Zhao, Weiren, Linear inviscid damping and vorticity depletion for shear flows, Ann. PDE, 5, 1, Art. 3, 101 pp. (2019) · Zbl 1428.35336
[76] Zeidler, Eberhard, Nonlinear functional analysis and its applications. I, xxi+897 pp. (1986), Springer-Verlag, New York · Zbl 0583.47050
[77] Zillinger, Christian, Linear inviscid damping for monotone shear flows in a finite periodic channel, boundary effects, blow-up and critical Sobolev regularity, Arch. Ration. Mech. Anal., 221, 3, 1449-1509 (2016) · Zbl 1350.35147
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.