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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 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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 ... 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. .. 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... 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
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