LEADER 04917nam 22006735 450 001 9910299786203321 005 20221121171536.0 010 $a3-319-19935-8 024 7 $a10.1007/978-3-319-19935-1 035 $a(CKB)3710000000463330 035 $a(EBL)3568246 035 $a(SSID)ssj0001546711 035 $a(PQKBManifestationID)16141464 035 $a(PQKBTitleCode)TC0001546711 035 $a(PQKBWorkID)14796546 035 $a(PQKB)11007190 035 $a(DE-He213)978-3-319-19935-1 035 $a(MiAaPQ)EBC3568246 035 $a(PPN)188459987 035 $a(EXLCZ)993710000000463330 100 $a20150808d2015 u| 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aDynamics of partial differential equations /$fby C. Eugene Wayne, Michael I. Weinstein 205 $a1st ed. 2015. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2015. 215 $a1 online resource (90 p.) 225 1 $aFrontiers in Applied Dynamical Systems: Reviews and Tutorials,$x2364-4532 ;$v3 300 $aDescription based upon print version of record. 311 $a3-319-19934-X 320 $aIncludes bibliographical references at the end of each chapters. 327 $aPreface to the Series; Preface; Contents; 1 Dynamical Systems and the Two-Dimensional Navier-Stokes Equations; 1 Introduction; 2 The ?-limit set of the two-dimensional Navier-Stokes equation; 3 Metastable states, pseudo-spectrum and intermediate time scales; 4 Finite Dimensional Attractors for the Navier-Stokes equations; References; 2 Localized States and Dynamics in the Nonlinear Schro?dinger/Gross-Pitaevskii Equation; 1 Introduction; 1.1 Outline; 2 NLS and NLS/GP; 3 Bound States - Linear and Nonlinear; 3.1 Linear bound states; 3.2 Nonlinear bound states 327 $a3.3 Orbital stability of nonlinear bound states3.4 The free soliton of focusing NLS: V0 and g=-1; 3.5 V(x), a simple potential well; model of a pinned nonlinear defect mode; 3.6 NLS/GP: Double-well potential with separation, L; 3.7 NLS/GP: V(x) periodic and the bifurcations from the spectral band edge; 4 Soliton/Defect Interactions; 5 Resonance, radiation damping and infinite time dynamics; 5.1 Simple model - part 1: Resonant energy exchange between an oscillator and a wave-field; 5.2 Simple model - part 2: Resonance, Effective damping, and Perturbations of Eigenvalues in Continuous Spectra 327 $a6 Ground state selection and energy equipartition in NLS/GP6.1 Linearization of NLS/GP about the ground state; 6.2 Ground state selection and energy equipartition; 7 A nonlinear toy model of nonlinearity-induced energy transfer; 8 Concluding remarks; References 330 $aThis book contains two review articles on the dynamics of partial differential equations that deal with closely related topics but can be read independently. Wayne reviews recent results on the global dynamics of the two-dimensional Navier-Stokes equations. This system exhibits stable vortex solutions: the topic of Wayne's contribution is how solutions that start from arbitrary initial conditions evolve towards stable vortices. Weinstein considers the dynamics of localized states in nonlinear Schrodinger and Gross-Pitaevskii equations that describe many optical and quantum systems. In this contribution, Weinstein reviews recent bifurcations results of solitary waves, their linear and nonlinear stability properties, and results about radiation damping where waves lose energy through radiation.   The articles, written independently, are combined into one volume to showcase the tools of dynamical systems theory at work in explaining qualitative phenomena associated with two classes of partial differential equations with very different physical origins and mathematical properties. 410 0$aFrontiers in Applied Dynamical Systems: Reviews and Tutorials,$x2364-4532 ;$v3 606 $aDifferential equations, Partial 606 $aDynamics 606 $aErgodic theory 606 $aPartial Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12155 606 $aDynamical Systems and Ergodic Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M1204X 615 0$aDifferential equations, Partial. 615 0$aDynamics. 615 0$aErgodic theory. 615 14$aPartial Differential Equations. 615 24$aDynamical Systems and Ergodic Theory. 676 $a515.353 700 $aWayne$b C. Eugene$4aut$4http://id.loc.gov/vocabulary/relators/aut$0755613 702 $aWeinstein$b Michael I$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910299786203321 996 $aDynamics of Partial Differential Equations$92512147 997 $aUNINA