Lecture Notes in Mathematics Editors: J.-M Morel, Cachan F Takens, Groningen B Teissier, Paris 1756 Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo Peter E Zhidkov Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory 123 Author Peter E Zhidkov Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research 141980 Dubna, Russia E-mail: zhidkov@thsun1.jinr.ru Cataloging-in-Publication Data applied for Mathematics Subject Classification (2000): 34B16, 34B40, 35D05, 35J65, 35Q53, 35Q55, 35P30, 37A05, 37K45 ISSN 0075-8434 ISBN 3-540-41833-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Violations are liable for prosecution under the German Copyright Law Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2001 Printed in Germany Typesetting: Camera-ready TEX output by the authors SPIN: 10759936 41/3142-543210 - Printed on acid-free paper Contents Page Introduction I Notation Chapter Evolutionary equations Results 1.1 The on existence (generalized) Korteweg-de Vries equation (KdVE) Schr6dinger equation (NLSE) blowing up of solutions 10 1.2 The nonlinear 26 1.3 On the 36 1.4 Additional remarks Chapter 37 39 Stationary problems 2.1 Existence of solutions An ODE 42 approach 2.2 Existence of solutions A variational method 49 2.3 The concentration- compactness method of P.L Lions 2.4 On basis properties of systems of solutions 56 2.5 Additional remarks 76 Chapter 3.1 3.2 3.3 Stability Stability of Stability of Stability of of solutions 79 soliton-like solutions 80 kinks for the KdVE solutions of the NLSE 90 nonvanishing as jxj 3.4 Additional remarks Chapter Invariant 62 94 103 105 measures 4.1 On Gaussian measures 4.2 An invariant measure in Hilbert spaces for the NLSE 4.3 An infinite series of invariant 4.4 Additional remarks oo measures 107 118 for the KdVE 124 135 Bibliography 137 Index 147 Introduction During differential large the last 30 years the theory equations (PDEs) possessing of solitons the - solutions of special a partial of nonlinear theory kind field that attracts the attention of both mathematicians and - has grown into physicists a in view important applications and of the novelty of the problems Physical problems leading to the equations under consideration are observed, for example, in the mono- of its graph by V.G Makhankov [60] One of the related mathematical discoveries is the possibility of studying certain nonlinear equations from this field by methods that these equations were developed to analyze the quantum inverse scattering problem; this subject, are called solvable by the method of the inverse scattering problem (on see, for example [89,94]) PDEs solvable At the by this method is time, the class of currently same sufficiently narrow and, on known nonlinear the other hand, there is The latter of differential called the qualitative theory equations of various probthe includes on well-posedness investigations particular approach such solutions of as the behavior stability or blowing-up, lems for these equations, approach, another in dynamical systems generated by these equations, etc., and this approach possible to investigate an essentially wider class of problems (maybe in a of properties makes it more general study) In the present book, the author qualitative theory are on about twenty years So, the selection of the material the existence of solutions for initial-value travelling problems or standing waves) of the stability of substituted in the are solitary are four main problems topics for these equations, special (for example, equations under consideration, waves, and the construction of invariant dynamical systems generated by the Korteweg-de is These kinds when solutions of problems arising studies of stationary for during related to the author's scientific interests There results and methods of the equations under consideration, both stationary and evolutionary, of that he has dealt with mainly problems some surveys Vries and nonlinear measures Schr6dinger equations We consider the following (generalized) Korteweg-de + Ut and the nonlinear f (U)U., + UXXX equation (KdVE) Schr6dinger equation (NLSE) iut + Au + f (Jul')u where i is the imaginary unit, and in the complex = Vries second), u u(x, t) = t E R, x is an = 0, unknown function E R in the case (real in the first of the KdVE and x E case R' for N the NLSE with a positive integer N, f (-) is a smooth real function and A = E k=1 P.E Zhidkov: LNM 1756, pp - 4, 2001 © Springer-Verlag Berlin Heidelberg 2001 82 aX2 k Laplacian Typical examples, important for physics, of the functions f (s) is the As 2) respectively, the are and following: as Isl" value initial-boundary for u travelling e `O(w, x) = the waves u in the equation (it what being be called the solitary (as JxJ for the NLSE, -+ oo dealt we wt) in the NLSE, is supplement with _ Loo some + A similar of existence and 0(k)(00) = (k = nontrivial solutions integer any argument r occurs Let kinds) In this case, the us typical Ix I, has can the 0, possessing limits Ej X = 00 as x + into the waves of the second order: le, be solitary waves solution of I roots on satisfying solved (see Chapter 2) (for example, on f interesting for our r > for functions the argument us problem which, the half-line proving of generally speaking, non-uniqueness depending only result for functions 1, == conditions of the sufficiently easily when such solutions exist exactly the method of the of consider solutions a into function, if necessary, expression for standing = In this case, We consider two methods of are real a waves notation, specifying, follows, the solutions of these kinds will f(1012)0 uniqueness I > there exists = standing boundary conditions, for example, 0, 1, 2) Difficulties arise when N > the above is Chapter expression arises for the KdVE For the KdVE and the NLSE with N problem problem type substitute the c R and w elliptic equation 061 the we bounded function a nonlinear following Cauchy problem and this just In what with) if of the of the KdVE and case where NLSE) Substituting A0 which - of the waves obtain the we It arises when problem O(w, x case positive constants) v are well-posedness is convenient to introduce is equation = the on and a for the KdVE and the NLSE used further In problems the stationary qonsider we (where contains results Chapter 2, e-a.,2 +S21 is the as a f of = W following: for r function of the the existence of solitary These waves qualitative theory of ordinary differential equations (ODEs) and the variational method As an example of the latter, concentration- compactness method of P.L Lions In touch upon recent results a on the property of being briefly we addition, basis in this consider the chapter (for example, in L2) we for systems of eigenfunctions of nonlinear one-dimensional Sturm-Liouville-type problems in finite intervals similar to those indicated above Chapter Lyapunov set is devoted to the sense X, equipped Omitting with a some distance stability of solitary waves, which is understood details, this R(., ), means there exists that, a if for unique an arbitrary solution u(t), uo in the from t > 0, a of to X for any fixed t > equation under consideration, belonging the to X for any fixed T(t), belonging R, if for any > satisfying R(T (0), u(O)) < b, one Probably the historically first result on the stability that obtained A.N by for all u(t), belonging to R(T (t), u(t)) < C for has Kolmogorov, the one-dimensional of solitary stability a solitary case a of kink for a is called wave in our nonlinear diffusion a kink if a waves was [48]: I.G Petrovskii and N.S Piskunov terminology, they proved (in particular) equation (in solution a called stable with respect to the distance X for any fixed t > and all t 0, then there exists b > such that for any solution > e t > is 0' (w, x) 0 X x) Let introduce us functions of the in the real Sobolev space H1 special distance a argument of consisting by the following rule: x, p(u,v)= JJu(-)-v(-+,r)JJHi inf ,ERN If we for call two functions some 7- E and u from v H1, satisfying set of classes of R, equivalent, then the a stability of solitary waves smooth family of any two solutions t in the = solitary O(wl, x sense W2 At the same distance p, then T.B Benjamin stability of they in his the many authors and For be solitary taken in the and wit) same sense time, can if two O(W2, X - the parameter f (s) form we = of solitary paper first, because usually possesses (a, b) Therefore E have close at t = velocities wi in the to be close for all t > in the has proved the stability of solitary wave Later, his approach point Sobolev spaces, or non-equal sense solitary was of the same sense with respect to the distance p He called this s a [7] w Lebesgue as they waves are with the close to each other at the L02t), for all t > if easily verified be pioneering the usual KdVE with the - second, on of standard functional spaces such cannot be close in the and depending waves the KdVE r) - investigate the of the KdVE with respect to this distance p; the KdVE is invariant up to translations in x; v(x =_ equivalent functions metric space For several reasons, it is natural to distance p becomes a u(x) the condition of waves stability developed by shall consider their results waves of the following NLSE, the distance p should be modified It should form: d(u,v)=infllu(.)-e"yv(. r)IIHI (u,vEH') T"Y where H' is only now the complex space, -r R' and E E R To clarify remark here that the usual one-dimensional cubic NLSE with two-parameter family ob (x, where w > t) and b family, arbitrary = this f (s) = fact, s we has a of solutions V-2-w exp I i [bx are - (b real parameters close at t = in the _ W)t] Therefore, sense cosh[v/w-(x two - 2bt)] arbitrary solutions from this of the distance p, cannot be close for all t > in the any two standing close in the of the waves close at t NLSE, parameter of the distance p for all t sense above family 40(x, t) > in the = At the to each cannot be other, time, the functions of same in the stability By analogy, W - of the distance p and sense nonequal w, the definition of satisfy V to different values of they correspond to two values of the corresponding the if same sense of the distance sense d In the two cases of the KdVE and the condition for the necessary") stability of NLSE, present we solitary a sufficient lim satisfying waves (and O(x) "almost and = 1XI-00 O(x) for a is called the 0, that > nonlinearity of general type Next, consider the Confirming this the function f of point of prove the for non-vanishing of our Chapter 4, JxJ as -+ oo We present theory physics waves of a equations If For the NLSE, energy and, for we construct higher that kinks many our always are stable assumptions on we of have a recurrence an present case an a new constructing attention theorem on such one means the the theory application con- is well-known in corresponding measure this measures in the stability according phenomenon explains measure interesting invariant phenomenon which it of the waves and important applications bounded invariant invariant solitary remain open in this direction the Fermi-Pasta-Ularn we of stability of dynamical system generated by the KdVE in the scattering problem, a Roughly speaking, By computer simulations, system, then the Poincar6 with problem We concentrate trajectories many "soliton" is satisfied > of kinks under stability questions It is the Fermi-Pasta-Ulam of nonlinear Poisson of all tion deal with the we dynamical systems the many equations These objects have nected with stability is devoted to the Chapter type It should be said however that In dw of kinks for the KdVE with respect to the distance widespread opinion a we d Q (0) to the distance general type The last part of NLSE view, if the condition NLSE) stability there is Among physicists p physical literature Roughly speaking, solitary wave is stable (with respect a p for the KdVE and to d for the we in the Q-criterion was for to equa- observed for our dynamical phenomenon partially associated with the conservation of when it is solvable by the method of the inverse infinite sequence of invariant measures associated conservation laws The author wishes to thank all his colleagues and friends for the useful scien- tific contacts and discussions with them that have contributed appearance of the present book importantly to the AN INFINITE SERIES OF INVARIANT MEASURES FOR THE KDVE 133 4.3 (to) z (uo, ei),,-,, = i H(z) Ej(zoeo + + -2 2me2m) and J is _2irk matrix (i e J* -A (J)2k-1,2k A 1,2, ,m)and (J)k,l for all other values where :::::: - = Let Theorem det( ) (1 (jrk)2n-2) + measure =: A of the indexes k,1 for all t azo,3 IV.1.3, the Lebesgue (2m + 1) -(J)2k,2k-I (k skew-symmetric (2m + 1) a t 2au prove that us (IV.3.8) 2m, 0, = x 0,1, ,2m = Indeed, according to ij=5_,_2m f dzo orm(Q) dZ2,,, is invariant an mea- n sure for the with the dynamical system phase L,, generated by the problem space (IV.3.7),(IV.3.8) Therefore, orm (hm (Q, t)) I = dzo Vdzo dZ2,rn dZ2m dzo dZ2m h- (O,t) for arbitrary Borel an this set C R 2m+1 that immediately implies Let us arguments, take V =-: continuity of the function 17, arbitrary closed an In view of the bounded set Q C In view of the above Hp'e-r'(A) get: we ym (hn (9, eEn (Pmu) -E,, (hm (u,t)) dyn (u) t)) Further, ym (Q) - therefore, according integrand respect K C proof to in the m jn(P u)-En(hm(u,t)) I dym (u), t)) Proposition IV.3.11 and to right-hand integer Hpne-r'(A) pm (hm (Q, side of this and > such that E Q u p(Q \ K) of Theorem IV.1.8 equality < c, Take Lemma is an a IV.3.6, we obtain that the function bounded arbitrary c > the existence of which and can be uniformly with a compact proved as set in the By Proposition IV.3.12, [ttm (K lim n Q) - tt,, (h,,, (K n Q, t))] = 0, M-00 hence, by Proposition IV.3.11, we get the relation lim sup [ftm (Q) - ym (hm (Q, t))] < C, c, M-00 which, in view of the arbitrariness of Corollary c > 0, yields the statement of Lemma IV.3.13.0 IV.3.14 For any bounded open set c lim M-00 I tt,, (Q) - p,,, (h (Q, t)) I Hpn,-'(A) = and for any t E R 134 CHAPTER Lemma IV.3.15 Let Q C ,,n(Q) ,n (h = Proof n-1 Take too dist(A, B) v E Let K, = =: u is bounded open a exists compact a set hn-1 (K, t) = By Proposition IV.3.10 for Hpn,,-r'(A) Then, there > c - uEA, vEB Then c \ K) Then, If, is a compact t) f2j Let a minf dist(K, ffl); dist(KI, aQ,)}, where JJU Vjjn-j and aA is a boundary of a set A C Hpne-,'(A) inf > arbitrary < n-1 C h (Q, = an ,n (0 set, too, and K, a bounded open set and t E R a (Q, t)) K C Q such that Clearly, be By Theorem IV.3.1 and Lemma IV.3.4 h'-'(Q, t) Hpn,,-,'(A), set in Hpn,,-,'(A) INVARIANT MEASURES < v - r} of E K there exists u any positive radius a r a ball B,(u) such that a dist (h,, (u, for all v B, (u) and all E Let m B1, ., t); h,,, (v, t)) Bi be a < finite covering of the compact set K by I these balls Let also Qq Qi E v : dist(v, ffli) where 0, and > B U Bi i=1 Since in view of for any Proposition IV.3.3 h,, (u, t) nE H r'(A), u we P sufficiently large ,,n(f2) ,n (B) + < m < c Further, by Lemma IV.1.11 and lim inf y,,,, (B) + c Un(Q) < ,n Let us unbounded) as m -4 oo = lim inf ym (h Corollary (B, t)) + e < IV.3.14 tz'(f2j) + e c > < (Q) Hence, ,,n(gj) and Lemma IV.3.15 is Hpn,-,,' (A) M-00 in view of the arbitrariness of By analogy /.,n(Q,) in Qa c M-00 Therefore, (U, t) get that hm(B,t) for all h n-1 = ,n(Qj)' proved n prove Theorem IV.3.2 First, let Q C Hpn,,-,'(A) be jjujj.,,-j kj, an open (generally set We set f2k =fuEQ: 11h n-1 (U, t) I _j + < 00 where k > U Qk Then Q , and each set f2k is open and bounded; in addi- k=1 00 tion, hn-1 (n, t) U h n-1 (Qk, t) and /,,n(gk) ,n (h n-1 (Qk' t)) by Lemma IV.3.15 k=1 Therefore, ,,n (h n-1 (n, t)) = liM Un (h n-1 (Qk' k-00 t)) = liM k-oo ttn (f2k) = ,n (f2) 135 ADDITIONAL REMARKS 4.4 Let hn-1 (A, t) is now can a be Hpn,,-,'(A) A C now an Borel subset of the space be obtained arbitrary Borel Hpn,,-,' (A) The of the set A by approximations By Proposition IV.1.1, set equality in Eo, ,E,,_1 set from Hpn,-' (A) 4.4 Additional remarks First of all, of r the on Concerning Some of them [105] not consider these we invariant measures, there is of the invariant an NLSE) In [4,24,55], In satisfactory a our sense [30], an nonlinear papers are some are In [67] equation with quite different an problem the NLSE with and A measure wave Another (see for are example in detail differential partial con- equations conditions for the from a a [108], weak the measures invariant author, a paper we as seem for [4] an to be not others, are abstract wave equa- completely constructed for in these two explicitly more and [109] is connected with the f(X, JU12)U investigate this = case are the where p AluIPu following: > of nonzero and I_L(B) < 00 -C(l+s di.) : < such that > and case and obtain sufficient y similar to those from Theorem IV.2.2 to be finite for any ball B C X The obtained conditions (as- proved However, cubic nonlinear measures besides measures nonlinearities such [107], is not result of the paper abstract construction from measure case nonlinearity Methods exploited related to invariant our measures of the KdVE and E in the [4,11,18,19,24,30,55,65,67,86,107- is constructed for and [11], In superlinear constants In in NLSE Similar a important details of the proof in this paper carefully reestablishes example, easily follows invariant Unfortunately, tion for the invariance of these 109,111-1131 equation properties number of papers devoted to their nonlinear is constructed for measure considered, are the invariance in theorem investigations sociated with the energy conservation law El in the case recurrence the KdVE and NLSE which indicated earlier One of the first results in this direction is obtained are where explain the to recurrence a dynamical systems generated by struction for in of the Poincar6 application Here [16,20,53,66]) approaches are dynamical systems generated by of trajectories not based note that there we bounded on anY proved El completely Thus, Theorem IV.3.2 is (h n-1 (A, t)) IV.3.4, the continuity of and from their boundedness Hpn,-'(A) n = open sets from outside The by last two statements of Theorem IV.3.2 follow from Lemma the functionals ,n (A) d, > 0, d2 E (0, 1) AJuJP this implies: p > if A < and f(X, S) :5 C(l + S d2) for all x, s For f(JU12) this in A for > Unfortunately, paper an important question remains open p E (0, 2) about the well-posedness of the initial-boundary value problem (IV.2.1)-(IV.2.4) with for any ball B C X if there exist C = N = in this superlinear result is obtained this problem for by an J case with initial data from a space like L2_ The required Bourgain [16,17] who proved the well-posedness arbitrary in a sense of A This allowed the author of this paper to construct an invariant measure for the one-dimensional NLSE with the power nonlinearity and (0, 5) (see [18]) [65] A result in this to show its boundedness in the above direction for the cubic NLSE is also [67], In the paper law sense for p E presented in the paper invariant measure, associated with an IV.3.2 the existence of on periodic in the a spatial may be posed: Eo, El, E2 Eo In this comments a for the KdVE case, our for the measures problem = 0, (or the NLSE)? question space Hn-' corresponds phase of the For example, consider the However, we can main difficulties in the way of an to the nth conservation law should prove a required measure, evolution problem for the KdVE with initial to construct corresponding H-1 invariant constructing we at least from (or not know any results like that In we associated with the lowest is not answered yet space similar to the Sobolev space Unfortunately, measures it One could observe that in Theorems IV.2.2 and IV.3.2 hypothesize that we can well-posedness data from on the measure on Therefore, analogous \IU12U Ux + there invariant are conservation law some con- variable for the usual cubic NLSE conservation laws invariant is function, constant, is presented in the paper [112] In this connection, the follow- ing question make conservation higher to the result of Theorem infinite sequence of invariant an iUt + where \ is a the square of the second derivative of the unknown containing structed for the sinh-Gordon equation A result the INVARIANT MEASURES C11APTER 136 H-2-', > 0) opinion, this is one of the our measures corresponding to the above conservation law in the paper Finally, [113], the following Cauchy problem for the NLSE written in the real form ' I U - t UX X + V (X) U, + U2+ U'X t - X V(X)U' (U')' + (U2)2)U2 = 0, x,t E R, (IV.4.1) f(X, (U')' + (U2)2)Ul = 0, X,t E R, (IV.4.2) f (X, - u'(x, to) where V(x) is is assumed that the function The main Theorem IV.2.2 eigenvalues to +oo of the operator v,, E x (IV.4-3) 1, 2, = R, is considered In this paper, it satisfies conditions similar to those introduced in hypothesis IxI as f i u', real-valued function of a positive, tends = oo (_ d2 d on the potential V and increases + V(X) as IxI Yi satisfy * - is that this function is oo so rapidly the condition E v,,, that the < +oo n Under phase like the above- described, in this paper we construct an invariant dynamical system generated by the problem (IV.4.1)-(IV.4.3) on the hypotheses measure for a space X this paper some obtained in the = L2(R) results on L2(R) where L2(R) is the real space the boundedness of the supercritical case f (x, s) =: A Is IP measure In addition, under consideration in are Bibliography [1] [2] A Ambrosetti, P.H Rabinowitz Dual theory and applications, J Funct Anal Amirkhanov, E.P Zhidkov A LV particle-like Part II: [3] LV for for some sufficient condition of the existence of a JINR Commun., Part I: P5-80-479, models, field P5-80-585, Dubna (1980) (in Russian) the existence f o a solution particle-like Dubna Commun., P5-11705, A.A Arsen'ev On invariant an point 14, 349-381 (1973) Amirkhanov, E.P Zhidkov, G.I Makarenko A sufficient condition JINR [4] solution variational methods in critical infinite-dimensional phase of a nonlinear scalar field equation, (1978) (in Russian) measures for classical dynamical systems space, Matem Sbornik with 121, 297-309 (1983) (in Russian) [5] N.K Bary On 383-386 [6] N.K bases in Hilbert space, Doklady Akad Nauk SSSR 54, No 5, (1946) (in Russian) Bary Biorthogonal systems and bases in Hilbert space, Moskov Gos Univ U6enye Zapiski 148, Matematika 4, 69-107 (1951) (in Russian); Math Rev 14, 289 [7] (1953) T.B Benjamin The stability of solitary 153-183 [81 T.B [10] H Royal Soc London A328, (1972) Benjamin Lectures tiod', A Newell (ed.), [9] waves, Proc Berestycki, on Lect nonlinear wave motion, In: "Nonlinear Wave Mo- Appl Math 15, AMS, Providence, 1974, P.L Lions Existence d'ondes solitaires dans des lin6aires du type Klein-Gordon, C.R Acad Sci AB288, H P.L Berestycki, of positive No solutions Lions, for 1, 141-157 (1981) L.A Peletier An ODE semilinear problems No 7, p 3-48 proWmes 395-398 approach non (1979) to the existence in RN, Indiana Univ Math J 30, 138 [11] [12] BIBLIOGRAPHY B Bidegaray ica D82, Ph Invariant J Blanchard, classical scalar measures for some partial differential equations, Phys- (1995) 340-364 L Stube, fields, the Vazques On Ann Inst H stability of solitary Theor Poincar6, Phys 47, No waves for 309-336 3, (1987) [13] N.N Bogoliubov, N.M Krylov General measure theory in nonlinear mechanics, In: "Selected papers of N.N Bogoliubov", Kiev, Naukova Durnka, 1969, p 411-463 [14] J.L Bona On the No [15] J.L tary J stability of solitary waves, Proc Royal Soc London A344, 1638, 363-374 (1975) Bona, waves 395-412 [161 (in Russian) P.E Stability and instability of soli- Royal Soc London A411, Proc No 1841, (1987) Fourier applications (1993) W.A Strauss of Iforteweg-de Vries type, Bourgain and Souganidis, Part 1: transform restriction to nonlinear evolution phenomena for equations, Schridinger equations, 107-156; certain lattice subsets Georn Funct Anal 3, No Part 2: The KDV-equation, 209- 262 [17] J Bourgain Global Solutions of Nonlinear SchrJdinger Equations, AMS, Prov- idence,1999 [18] J Bourgain Periodic nonlinear Commun Math [19] J Bourgain equation, [20] J Phys 166, Invariant [22] T Anal.: Theory, Cazenave, Meth & solutions No 3, 155-168 (1988) T Cazenave, ear the with an SchrJdinger (1996) of Hamiltonian perturbations of 2D linear 1127-1140 2, 363-439 (1998) Nonlin- (1983) stability of stationary external P.L Lions Orbital Schn5dinger equations, nonlinear 2D-defocusing 421-445 of the logarithmic Schr6dinger equation, Appl 7, M.J Esteban On the Schr,5dinger equations [23] for Ann of Math 148, No T Cazenave Stable solutions and invariant measures, (1994) Phys 176, Bourgain Quasi-periodic ear Schn5dinger equation measures Commun Math Schn5dinger equations, [21] 1-26 states for nonlinear magnetic field, Mat Aplic Comput 7, stability of standing Commun Math Phys 85, waves No for some nonlin- 4, 549-561 (1982) BIBLIOGRAPHY [24] 139 Equilibrium statistical solutions for dynamical systems with I.D Chueshov infinite number of degrees of freedom, Math USSR Sbornik an 58, No 2, 397-406 (1987) [25] C.V Coffman and variational characterization a A Cohen, equation, [27] Yu.L N 'Rans Amer Math Soc Daletskii, S.V in Hilbert its generalizations Moscow State Univ [30] [31] Space, Differential Equations Interscience Cauchy problem for " In: Trudy Semin Publ, 1988, measure (1985) Phys 98, I.M Gelfand Sur Matem un A.M Ser u 4, in Infinite- Spectral theory Self New York - London, 1963 Korteweg-de Viies equation and Petrovskogo" 13, Moscow, (in Russian) for the equation 13, 35-40 Utt _ UXX + U3 = lin6aires, Kharkov, 0, Za- (1936) Yaglom Integration quantum physics, Uspekhi cations in the 2: lemme de la th6orie des espaces Obshestva, Gelfand, 1-16 Publ., imeni L G p 56-105 Commun Math I.M 46, + u (in Russian) 1983 L F riedlander An invariant piski [32] Fomin Measures and Dunford, J.T Schwartz Linear operators, Part A.V Faminskii The for -o Rat Mech Anal - 312, No 2, 819-840 (1989) Spaces, Nauka, Moscow, Adjoint Operators [29] of other solutions, Arch for Au Kappeler Nonuniqueness for solutions of the Korteweg-de Viies T Dimensional [28) state solution (1972) 81-95 [26] Uniqueness of the ground in functional spaces and its appli- Matem Nauk 11, No 1, 77-114 (1956) (in the nonlinear Russian) [33] A.B Shabat Schr6dinger equation, Different A.V Giber, On the Cauchy problem for Uravnenija 6, No 1, 137-146 (1970) (in Rus- sian) [34] B Gidas, Ni the maximum [351 B principle, elliptic equations pt 1", New York, D e.a., Nirenberg Symmetry and related properties Commun Math Gidas, Ni Wei-Ming, nonlinear [36] L Wei-Ming, L Phys 68, No via (1979) Nirenberg Symmetry of positive solutions of in RN , In: "Mathematical 1981, 3, 209-243 Analysis and Applications, p 369-402 Gilbarg, N.S Mrudinger Elliptic partial differential equations of second order, Springer, Berlin, 1983 140 [37] BIBLIOGRAPHY J Ginibre, G Velo On Cauchy problem, general [381 [39] Glassey On linear Schr,5dinger equations, I.C Gohberg, of blowing up M Grillakis, presence J Stability theory of solitary Funct Anal 74, J.F Henri, Perez, New York, of scalarfield equation, Commun Iliev, E.Kh Mathematics [45] 73, T Kato On the [46] for [47] Kenig, the I.T Cauchy problem for G Ponce, Korteweg-de field theory Vries Stability theory for solitary Math Phys 85, No 3, Spectral Methods and Surveys 351-361 in Soliton in Pure and wave (1982) Equa- Applied Suppl the (generalized) Korteweg-de (1983) Vries equa- Stud." 8, 93-128 Schn5dinger equations, L Ann Inst H Vega Well-posedness of Poincar6, Phys In: the initial value problem equation, J Amer Math Soc 4, No 2, 323-347 (1991) B.L Shekhter On Moscow, Nauka, 1987, [48] in the waves 1, 160-197 (1987) 1, 113-129 (1987) Kiguradze, linear Self- 1994 T Kato On nonlinear C.E non 1950 K.P Kirchev Khristov, In: "Advances in Math Theor 46, No No W.F Wreszinski tions, Longman Publ., Pitman Monographs tion, Linear of symmetry I, J D.B [44] Theory of non- (in Russian) 1965 W Strauss [42] I.D to the Cauchy problem for Phys 18, 1794-1797 (1977) J Math Theory, I The (1979) Shatah, P.H Halmos Measure [43] 1-32 M.S Krein Introduction to the [41] solubons Schn5dinger equations 32, of solutions Adjoint Operators, Moscow, Nauka, [40] nonlinear case, J Funct Anal R.T the class a "Itogy Nauki i p 183-201 boundary problem arising in the nonTekhniki", Sovrem Probl Matem 30, a (in Russian) A.N Kolmogorov, I.G Petrovskii, N.S Piskunov An investigation of an equation of diffusion with the increase of the quantity of matter and its applica- tion to a biological problem, Bull Moscow Gos Univ 1, No 6, 1-26 (1937) (in Russian) [49] V.P Kotlyarov, E.Ya Khruslov Numerical Methods in Mathematical 103-107 (in Russian) Asymptotic solitons In: Physics", Kiev, Naukova "Functional and Dumka, 1988, p BIBLIOGRAPHY [50] M.D equation and polynomial N.S for Miura, C.S Gardner, N.Z Zubusky Korteweg- R.M Kruskal, de Vries [51] 141 Kruzhkov, generalizations V Uniqueness and its conservation laws, J Math Matem Sbornik nonexistence of (1970) 3, 952-960 A.V Farninskii Generalized solutions Korteweg-de Wies equation, the Phys 11, No of the Cauchy problem 120, No 3, 396-425 (1983) (in Russian) [521 N.M Kryloff, N.N a l'6tude des Application Ann of Math 38, No [53] Bogoluboff S.B Kuksin On 2, La Th6orie Generale de la Mesure dans Syst mes Dynamiques de la M6canique non son lin6aire, (1937) 65-113 Long- Time Behavior Solutions of Nonlinear Wave Equations, "Proceedings of the XIth International Congress of Mathematical PhysicZ, Paris, July 18-23, 1994 International Press, Boston, 1995, p 273-277 In: [54) Kwong Uniqueness of positive solutions of Au M.K Rat Mech Anal 105, No 3, 243-266 [55] J.L Lebowitz, H.A Rose, Schrddinger [56] P.L Lions Part 1: = in RN, Arch Speer Statistical mechanics of the nonlinear No 3-4, 657-687 (1988) and compactness in Sobolev spaces, J Funct Anal 49, locally compact 109-145; (The (1985) - case.) non Lineaire 1, (1984) Part 2: 223-283 limit compactness method in the calculus of varia- case, Ann Inst H Poincar6, Anal P.L Lions The concentration ations + uP (1989) Phys 50, P.L Lions The concentration tions The [58] Symmetry u (1982) 315-334 [57] E.R equation, J Statist - - compactness principle Revista Matem in the calculus lberoamericana, 1, No 1: of vari- 145-201; No 2: 45-121 [59] J.W No [60] Macky A singular nonlinear boundary value problem, Pacif J Math 78, 2, 375-383 (1978) V.G Makhankov Soliton Phenomenology, Kluwer Acad Publ., Dordrecht, 1990 [61] V.G structures: construction 164, V.I Makhankov, Yu.P Rybakov, No 2, 121-148 of solutions and (1994) (in Russian) - Sanyuk Localized non-topological stability problems, Uspekhi Fiz Nauk 142 [62] BIBLIOGRAPHY A.P Makhmudov Fundamentals baijanian Cos [63] Baryakhtar, V.E Zakharov Commun Math H.P H of infinitely K in K McLeod, Rn, Arch solutions 4, of 2, V [73] Perspectives nonlinear Naukova Cubic wave equations (4): 3, 479-491 (1995) hyperelliptic function theory points, Commun Pure Appl Math 29, to a of nonlinear Applied Mathematics", wave equa- L Sirovich (ed.) p 239-264 Uniqueness of positive radial solutions of Au+f (u) = 99, 115-145 (1987) stability, destruction, delaying Schr,5dinger equation, Doklady optimal nonlinear and self-channelling of Akad Nauk SSSR, 285, constants in some Sobolev Schr,5dinger equation, Doklady inequalities and their Akad Nauk SSSR 307, (1989) (in Russian) a Nemytskii, - in (1985) (in Russian) nonlinear Irish Acad 62A, V Leningrad, R.S Palais 339-403 [74] a 538-542 Moscow No Statistical mechanics York, 1994, J Serrin Z Nehari On Royal many branch Rat Mech Anal 807-811 application [72] New S.A Nasibov On No Phys 168, Vaninsky In: "Trends and S.A Nasibov On the No (eds.), Kiev, (1976) McKean, tions, [71] Azer- nonlinear of E T ubowitz Hill's operator and McKean, Springer-Verlag, [70] and V.M Chernousenko H.P McKean Statistical mechanics 143-226 [69] spectral analysis, Baku, (in Russian) p 168-212 in the presence [681 nonlinear Cauchy problem for the Korteweg-de Vries equation with non-vanishing initial data, In: "Integrability and kinetic equations for solitons", Schn6dinger, [67] of (in Russian) V.A Marchenko The Durnka, 1990, [66] 1984 Completeness of eigenelements for some non-linear operaequations, Doklady Akad Nauk SSSR 263, No 1, 23-27 (1982) (in Russian) V.G [651 Publ., A.P Makhmudov tor [64] Univ differential equation arising 117-135 in nuclear physics, Proc (1963) Stepanov Qualitative Theory of Differential Equations, 1949 (in Russian) The symmetries of solitons, Bull Amer Math Soc 34, No 4, (1997) S.I Pohozaev Eigenfunctions of the equation Nauk SSSR 165, No Au + 1, 36-39 (1965) (in Russian) Af (u) = 0, Doklady Akad 143 BIBLIOGRAPHY [75] S.I Pohozaev 07) [76] S.I Pohozaev On of fibering for solving nonlinear boundary-value method P.H Rabinowitz Variational methods lems, [78] (1979) (in Russian) MIAN imeni V.A Steklova 192, 146-163 problems, Trudy [77] a Indiana Univ Math J Reed, New York R.D [81] H.A B Simon Methods - l [83] linear Physics, Acad Press, Advanced Mathematical value tions, Pacif J Math 22, No Math 6, Acad Shatah, problems for 3, 477-503 Part Physics, 1, 173-190 A.V Skorokhod 1-2, class a nonlinear 207-218 Schn5dinger (1988) of nonlinear differential lineare della non London and New Press, W.A Strauss Phys 100, No of the equa- (1967) un'equazione differenziale G Sansone Su states potential, Physica D30, Ryder Boundary Math [85] a G.H J Modern Mathematical Rose, M.1 Weinstein On the bound Symposia [84] of 1978 equation with [82] AMS, London, Part 1: 1972; Part 2: 1975 Richtmyer Principles of Springer, Regional Theory with Applica- Conf Ser in Math 65, 1988 Providence, [801 for nonlinear elliptic eigenvalue prob- P.H Rabinowitz Minimax Methods in Critical Point M (1990) (in Russian) 23, No 8, 729-754 (1974) tions to Differential Equations, CBMS [79] equations, Doklady Akad Nauk to nonlinear approach an SSSR 247, No 6, 1327-1331 fisica nucleare, York, 1971, In: 3-139 Instability of nonlinear bound states, Commun (1985) Integration in Hilbert Space, Moscow, Nauka, 1975 (in Rus- sian) [86] S.I Sobolev On an invariant measure for nonlinear a Trudy Petrozavodskogo Gos Universiteta, Matematika, Schn5dinger equation, No 2, 113-124 (1995) (in Russian) [871 W.A Strauss Existence Math [88] Phys 55, 149-162 of solitary in higher dimensions, Commun (1977) W.A Strauss Nonlinear Wave 73, AMS, Providence, waves 1989 Equations, CBMS Regional Conf Ser in Math 144 [891 BIBLIOGRAPHY L.A Takhtajan, Moscow, Nauka, [901 L.D Faddeev A Hamiltonian 1986 approach in soliton theory, (in Russian) M Tsutsurni Nonexistence of global solutions damped nonlinear Schn5dinger equations, to the Cauchy problem for the SIAM J Math Anal 15, No 2, 357-366 (1984) [911 Y Tsutsurni [92] (1987) M.1 Weinstein evolution [93] L2-solutions for nonlinear Schr,5dinger equations, Funkcial Ek- 115-125 30, vac Lyapunov stability of ground states of nonlinear dispersive equations, Commun Pure Appl Math 39, 51-68 (1986) M.1 Weinstein Existence and in equations arising long wave dynamic stability of solitary wave solutions of propagation, Commun Part Diff Equat 12, No 10, 1133-1173 (1987) [94] V.E Theory [95] E.P Method Zhidkov, of Manakov, Inverse S.P Novikov, Problem, Moscow, Nauka, L.P Pitaevskii Soliton 1980 (in Russian) K.P Kirchev Stability of soliton solutions of some nonlinear equations of mathematical physics, Soviet J of Particles and Nuclei 16, No 3, 259-279 [961 S.V Zakharov, E.P (1985) Zhidkov, V.P Shirikov On ferential equations of the second boundary-value problem for ordinary dif- a order, J Vichislit Matern i Matem Fiz 4, No 5, 804-816 (1964) (in Russian) [97] E.P Zhidkov, V.P Shirikov, boundary-value problem for ond [981 order, Commun., nonlinear No Zhidkov, P.E Zhidkov An some models the nonlinear of Puzynin Cauchy problem and a ordinary differential equation of the sec- 2005, Dubna (1965) (in Russian) E.P Dubna [99] JINR a I.V investigation of particle-like solutions in physics, JINR Commun., P5-12609, P5-12610, (1979) (in Russian) P.E Zhidkov On the existence in R' probl6,n for a nonlinear elliptic of a positive solution of the Dirichlet equation, JINR Commun Dubna, 5-82-69 (1992) (in Russian) [100] P.E Zhidkov On the stability of the soliton solution Schn5dinger equation, Differentsial Uravnenija 22, Russian) No of the nonlinear 6, 994-1004 (1986) (in 145 BIBLIOGRAPHY [101] P.E Zhidkov eralized Stability of solutions of the for wave a gen- equation, JINR Commun., P5-86-800, Dubna (1986) Vries Iforteweg-de of the solitary kind (in Russian) [102] P.E Zhidkov A Cauchy problem for Commun., P5-87-373, [103] P.E Zhidkov On the solutions 155-160 [104] of a nonlinear nonlinear a Schn5dinger equation, JINR (1987) (in Russian) Dubna solvability of the Cauchy problem and stability of some a Matem Modelirovanie Schr6dinger equation, 1, No 10, (1989) (in hussian) P.E Zhidkov On the Cauchy problem for a Vries generalized Iforteweg-de equation with periodic initial data, Differentsial Uravnenija 26, No 5, 823-829 (1990) (in Russian) [105] P.E Zhidkov On an invariant measure Soviet Math Dokl 43, No 2, 431-434 [106] P.E Zhidkov Existence kink-solutZons of a nonlinear Schn5dinger equation, (1991) of solutions the nonlinear for to the Cauchy problem Schro5dinger equation, Siberian and stability of Math J 33, No 2, 239-246 (1992) [107] P.E Zhidkov An invariant JINR [108] Commun., P5-94-199, P.E Zhidkov An invariant P.E Zhidkov Or invariant the nonlinear for Schr6dinger equation, (1994) (in Russian) measure Theory, Meth & Appl 22, Anal.: [109] measure Dubna for 319-325 measures a nonlinear wave equation, Nonlinear (1994) for some infinite- dimensional dynamical systems, Ann Inst H Poincar6, Phys Theor 62, No 3, 267-287 [110] P.E tion, [111] Zhidkov, V Zh Sakbaev On Matem Zametki 55, No P.E Zhidkov Invariant the Korteweg-de Vries 4, 25-34 measures a nonlinear (1995) ordinary differential equa- (1994) (in Russian) generated by higher equation, Sbornik: Mathematics conservation laws for 187, No 6, 803-822 (1996) [112] P.E Zhidkov On linear [113] P.E an infinite series of invariant measures for the Schn5dinger equation, Prepr JINR, E5-96-77, Dubna (1996) Zhidkov tems with Invariant applications to a measures nonlinear Geometric Methods in Mathematical Marchenko (eds.), Kluwer Acad cubic for infinite-dimensional dynamical Schr,6dinger equation, Physics", Publ., 1996, In: "Algebraic non- sys- and A Boutet de Monvel and V p 471-476 146 BIBLIOGRAPHY [1141 P.E Zhidkov Completeness of systems of eigenfunctions for the Sturmpotential depending on the spectral parameter and for non-linear problem, Sbornik: Mathematics 188, No 7, 1071-1084 (1997) Liouville operator with one [1151 [116] P.E Zhidkov Eigenfunction expansions associated with Schr,6dinger equation, JINR Commun., E5-98-61, Dubna (1998) P.E Zhidkov On the property functions of a 191, No 3, 43-52 [117) [118] problems, tions of [119] P.E a Riesz basis Electronic J of Differential nonlinear Meth & problem, for the system Sbornik: of eigen- Mathematics, properties of eigenfunctions of nonlinear Sturm P.E Zhidkov On the property Theory, a (2000) (in Russian) P.E Zhidkov Basis ville of being nonlinear Sturm-Lio uville- type nonlinear a a basis for a No No 4, 471-483 Eigenfunction Schn5dinger equation on a Liou- 1-13 (2000) denumerable set of solu- Schridinger-type boundary-value problem, Appl 43, Zhidkov of being Equations 2000, - 28, Nonlinear Anal.: (2001) expansions associated half-line, Prepr JINR, E5-99-144, with a Dubna nonlinear (1999) hadex Gronwell's lemma admissible functional 63 algebra additive Bary Bary Hn-,Olti,, of the KdVE 10,11 107 measure HI-solution of the NLSE 27 107 basis 63 invariant theorem 63 kink 3,43 basis 62 measure 106 Korteweg de, Vries equation linearly independent system 62 - blow up for the NLSE 36-37 Borel sigma-algebra 107 lower semicontinuous functional 63 bounded nonlinear measure centered Gaussian 106 measure 111-112 concentration- compactness method 56 Pohozaev countably additive measure 107 cylindrical set III dynamical system 106 eigenvalue 62 eigenfunction 62 Fermi-Pasta-Ulam phenomenon Poincare Gaussian measure in Rn 108 Gaussian measure in trace class 111 identity 41 recurrence theorem 106 of the Q-criterion stability quadratically close systems 63 Riesz basis 62 sigma-algebra 107 solitary wave soliton-like solution 43 of the KdVE 10,11 stability in the Lyapunov sense 2-3 stability according to Poisson 106 stability of the form 3,79 of the NLSE 27 weak convergence of a Hilbert space 111-112 Gelfand theorem 63 generalized solution generalized solution global solution 10 Schri5dinger equation operator of measures 115 ... [16] (see also [17]) where superlinear of the blow up of we well-posedness of problem L2-solutions periodic in phenomenon data, to review the whole literature devoted to the mention the paper of... http://www .springer. de © Springer- Verlag Berlin Heidelberg 2001 Printed in Germany Typesetting: Camera-ready TEX output by the authors SPIN: 10759936 41/3142-543210 - Printed on acid-free paper... Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo Peter E Zhidkov Korteweg- de Vries and Nonlinear Schrödinger Equations: Qualitative Theory 123 Author Peter E Zhidkov