Bifurcations of Solitary Waves

The paper provides a brief review of the recent results devoted to bifurcations of solitary waves. The main attention is paid to the universality of soliton behavior and stability of solitons while approaching supercritical bifurcations. Near the transition point from supercritical to subcritical bi...

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Дата:	2008
Автори:	Kuznetsov, E.A., Agafontsev, D.S., Dias, F.
Формат:	Стаття
Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2008
Назва видання:	Журнал математической физики, анализа, геометрии
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/106521
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Цитувати:	Bifurcations of Solitary Waves / E.A. Kuznetsov, D.S. Agafontsev, F. Dias // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 4. — С. 529-550. — Бібліогр.: 42 назв. — англ.

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spelling	irk-123456789-1065212016-09-30T03:03:05Z Bifurcations of Solitary Waves Kuznetsov, E.A. Agafontsev, D.S. Dias, F. The paper provides a brief review of the recent results devoted to bifurcations of solitary waves. The main attention is paid to the universality of soliton behavior and stability of solitons while approaching supercritical bifurcations. Near the transition point from supercritical to subcritical bifurcations, the stability of two families of solitons is studied in the frame-work of the generalized nonlinear Schrodinger equation. It is shown that one-dimensional solitons corresponding to the family of supercritical bifurcations are stable in the Lyapunov sense. The solitons from the subcritical bifurcation branch are unstable. The development of this instability results in the collapse of solitons. Near the time of collapse, the pulse amplitude and its width exhibit a self-similar behavior with a small asymmetry in the pulse tails due to self-steepening. 2008 Article Bifurcations of Solitary Waves / E.A. Kuznetsov, D.S. Agafontsev, F. Dias // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 4. — С. 529-550. — Бібліогр.: 42 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106521 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The paper provides a brief review of the recent results devoted to bifurcations of solitary waves. The main attention is paid to the universality of soliton behavior and stability of solitons while approaching supercritical bifurcations. Near the transition point from supercritical to subcritical bifurcations, the stability of two families of solitons is studied in the frame-work of the generalized nonlinear Schrodinger equation. It is shown that one-dimensional solitons corresponding to the family of supercritical bifurcations are stable in the Lyapunov sense. The solitons from the subcritical bifurcation branch are unstable. The development of this instability results in the collapse of solitons. Near the time of collapse, the pulse amplitude and its width exhibit a self-similar behavior with a small asymmetry in the pulse tails due to self-steepening.
format	Article
author	Kuznetsov, E.A. Agafontsev, D.S. Dias, F.
spellingShingle	Kuznetsov, E.A. Agafontsev, D.S. Dias, F. Bifurcations of Solitary Waves Журнал математической физики, анализа, геометрии
author_facet	Kuznetsov, E.A. Agafontsev, D.S. Dias, F.
author_sort	Kuznetsov, E.A.
title	Bifurcations of Solitary Waves
title_short	Bifurcations of Solitary Waves
title_full	Bifurcations of Solitary Waves
title_fullStr	Bifurcations of Solitary Waves
title_full_unstemmed	Bifurcations of Solitary Waves
title_sort	bifurcations of solitary waves
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2008
url	http://dspace.nbuv.gov.ua/handle/123456789/106521
citation_txt	Bifurcations of Solitary Waves / E.A. Kuznetsov, D.S. Agafontsev, F. Dias // Журнал математической физики, анализа, геометрии. — 2008. — Т. 4, № 4. — С. 529-550. — Бібліогр.: 42 назв. — англ.
series	Журнал математической физики, анализа, геометрии
work_keys_str_mv	AT kuznetsovea bifurcationsofsolitarywaves AT agafontsevds bifurcationsofsolitarywaves AT diasf bifurcationsofsolitarywaves
first_indexed	2025-07-07T18:36:11Z
last_indexed	2025-07-07T18:36:11Z
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fulltext	Journal of Mathemati al Physi s, Analysis, Geometry2008, vol. 4, No. 4, pp. 529�550Bifur ations of Solitary WavesE.A. KuznetsovP.N. Lebedev Physi al Institute53 Leninsky Ave., Mos ow, 119991, RussiaE-mail:kuznetso�itp.a .ruD.S. AgafontsevL.D. Landau Institute for Theoreti al Physi s2 Kosygin Str., Mos ow, 119334, RussiaE-mail:dmitry�itp.a .ruF. DiasCMLA, ENS Ca han, CNRS, PRES UniverSud61 Av. President Wilson, F-94230 Ca han, Fran eRe eived June 25, 2008The paper provides a brief review of the re ent results devoted to bi-fur ations of solitary waves. The main attention is paid to the universalityof soliton behavior and stability of solitons while approa hing super riti- al bifur ations. Near the transition point from super riti al to sub riti albifur ations, the stability of two families of solitons is studied in the frame-work of the generalized nonlinear S hr�odinger equation. It is shown thatone-dimensional solitons orresponding to the family of super riti al bifur- ations are stable in the Lyapunov sense. The solitons from the sub riti albifur ation bran h are unstable. The development of this instability resultsin the ollapse of solitons. Near the time of ollapse, the pulse amplitudeand its width exhibit a self-similar behavior with a small asymmetry in thepulse tails due to self-steepening.Key words: stability, riti al regimes, wave ollapse.Mathemati s Subje t Classi� ation 2000: 37K50, 70K50.1. Introdu tionA ording to the usual de�nition, solitons are nonlinear lo alized obje ts pro-pagating uniformly with a onstant velo ity (see, for example, [1, 2℄). The solitonvelo ity V represents the main soliton hara teristi s whi h often de�nes the soli-ton shape, in parti ular its amplitude and width. E.A. Kuznetsov, D.S. Agafontsev, and F. Dias, 2008 E.A. Kuznetsov, D.S. Agafontsev, and F. DiasIt is well known that if the velo ity V of a moving obje t is su h that theequation !k = k �V (1)has a nontrivial solution where ! = !k is the dispersion law for linear wavesand k is the wave ve tor, then this obje t will lose energy due to Cherenkovradiation. This also pertains, to a large extent, to solitons as lo alized stationaryentities. They annot exist if the resonan e ondition (1) is satis�ed. Hen efollows the �rst, and simplest, sele tion rule for solitons: the soliton velo ity mustbe either less than the minimum phase velo ity of linear waves or greater thanthe maximum phase velo ity. The boundary separating the region of existen eof solitons from the resonan e region (1) determines the riti al soliton velo ityV r. As it is easily seen, this velo ity oin ides with the group velo ity of linearwaves at the tou hing point where the straight line ! = kV is tangent to thedispersion urve ! = !k (in the multidimensional ase � the point of tangen y ofthe plane ! = k �V to the dispersion surfa e). If tou hing o urs from below, thenthe riti al velo ity determines the maximum soliton velo ity for this parameterrange and, onversely, for tou hing from above V r oin ides with the minimumphase velo ity. Two regimes are possible in rossing this boundary orrespondingto super riti al or sub riti al bifur ations (soft or rigid ex itation regimes).While approa hing the super riti al bifur ation point from below or abovethe soliton amplitude vanishes smoothly a ording to the same � Landau � law(/ jV � V rj1=2) as for the phase transitions of the se ond kind (see, for instan e,[3℄). The behavior of solitons in this ase is ompletely universal, both for theiramplitudes and their shapes. As V ! V r solitons transform into os illating wavetrains with the arrying frequen y orresponding to the extremal phase velo ityof linear waves V r. The shape of the wave train envelope oin ides with thatfor the soliton of the standard � ubi � nonlinear S hr�odinger equation (NLS).The soliton width happens to be proportional to jV � V rj�1=2.Bifur ations of solitons were �rst observed for gravity- apillary waves in nu-meri al simulations by Longuet�Higgins [4℄ and explained later in [5�9℄. Thenthe bifur ation � a transition from periodi solutions to a soliton solution � wasstudied in [5℄ and [6℄ using normal forms. The stationary NLS for gravity- apillarywave solitons was derived in [8℄. In [10℄ it was shown that this me hanism an beextended to opti al solitons. In fa t, this paper provided the �rst demonstrationof the universality of soliton behavior near a super riti al bifur ation for waves ofarbitrary nature. It is worth noting that the universal hara ter of solitons allowsnot only to �nd their shapes, but also to study their stability. This analysis, asstated in [11℄, shows that near super riti al bifur ation the solitons are stable onlyin the one-dimensional ase.The question of whether the bifur ation is super riti al or sub riti al dependson the hara ter of nonlinear intera tion. The super riti al bifur ation o urs for530 Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 Bifur ations of Solitary Wavesa fo using nonlinearity when the produ t !00T < 0, where !00 = �2!=�k2 is these ond derivative of the frequen y with respe t to the wave number, taken at thetou hing point k = k0, and T is the value of the matrix element Tk1k2k3k4 of thefour-wave intera tion for ki = k0. If !00T > 0, whi h orresponds to a defo usingnonlinearity, then there are no solitons � lo alized solutions � with amplitudevanishing gradually as V ! V r. In the theory of phase transitions this orre-sponds to a �rst-order phase transition, and in the theory of turbulen e, usingLandau's terminology [12℄, it orresponds to a rigid regime of ex itation. Thetransition through the riti al velo ity is a ompanied by a jump in the solitonamplitude. The magnitude of the jump is determined by the next higher-orderterms in the expansion of the Hamiltonian. Like for the �rst-order phase transi-tions, the universality of soliton behavior is no longer guaranteed in this situation.When the amplitude jump at this transition is small, it is enough to keep a �nitenumber of next order terms in the Hamiltonian expansion to des ribe su h a bifur- ation. In the phase transitions this orresponds to a �rst-order phase transition lose to a se ond-order transition, whi h o urs, for example, near a tri- riti alpoint. As shown in [13℄, this situation arises for one-dimensional internal-wavesolitons propagating along the interfa e between two ideal �uids with di�erentdensities in the presen e of both gravity and apillarity. A ording to [13℄ thematrix element T in this ase vanishes for density ratio �1=�2 = (21 � 8p5)=11.This type of bifur ations an also be met for gravity water waves with �nite depthwhen the matrix element T = 0 at � r = k0h � 1:363. In nonlinear opti s, asshown in [11℄, a de rease of T (Kerr onstant) an be provided by the intera tionof light pulses with a ousti waves (Mandelstamm�Brillouin s attering). If thejump in soliton amplitude is of order one, then we need to keep all the remainderterms in the Hamiltonian expansion.In this paper we give a brief review of the re ent results devoted to thissubje t. The main attention will be paid to the universality of soliton behaviorand stability of solitons while approa hing the super riti al bifur ation point.The paper is organized as follows. The next se tion is devoted to stationarysolitons for arbitrary nonlinear wave media and their properties near the super- riti al bifur ation. Se tion 2 deals with the stability of solitons based on theLyapunov theorem and the Hamiltonian approa h. It is shown by means of inte-gral estimates of Sobolev type in their multipli ative variant (Gagliardo�Nirenberginequalities) that only one-dimensional solitons are Lyapunov stable. It is worthnoting that, in ontrast to the method of normal forms, whi h is extensively usedin [5, 6, 13, 14℄ for studying bifur ations of solitons, the Hamiltonian approa h isfundamental for investigating soliton stability. In the method of normal forms, theintrodu tion of envelopes is not unique. Consequently, the Hamiltonian equationsof motion lose their initial Hamiltonian stru ture after their averaging. In thisse tion it is shown that near the bifur ation point the multi-dimensional solitonsJournal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 531 E.A. Kuznetsov, D.S. Agafontsev, and F. Diasare unstable due to the modulational instability. In the last se tion we onsiderwhi h nonlinear e�e ts must be taken into a ount near the transition from super- riti al to sub riti al bifur ations and how they hange the shape of solitons andtheir stability. 2. Super riti al Bifur ationsLet us onsider a purely onservative nonlinear wave medium whi h an bedes ribed by the HamiltonianH = Z !kjakj2dk+Hint; (2)where !k is the dispersion law of small-amplitude waves, ak are normal amplitudesof the waves, and the Hamiltonian Hint des ribes the nonlinear intera tion of thewaves.The equations of motion of the medium an be written in terms of the ampli-tudes ak in the standard manner�ak�t + i!kak = �iÆHintÆa�k ; (3)so that in the absen e of an intera tion the system onsists of a olle tion ofnonintera ting os illators (waves):ak(t) = ak(0)e�i!kt:Equation (3) des ribes the dynami s in the wave number spa e. To go ba k tothe physi al spa e one needs to perform the inverse Fourier transform (x; t) = 1(2�)d=2 Z ak(t)eik�rdk: (4)Originally, the fun tion (x; t) is related to the hara teristi s of the medium(�u tuations of the density and velo ity of the medium, ele tri and magneti �elds, and so on) by a linear transformation (see, for example, [15℄). It is impor-tant that if (x; t) is a periodi fun tion of the oordinates, then its spe trumak(t) onsists of a sum of Æ-fun tions. For lo alized distributions (x; t) ! 0 asjxj ! 1. The Fourier amplitude ak(t), being a lo alized fun tion of k, does not ontain Æ-fun tion singularities.Let us now onsider the solution of (3) in the form of a soliton propagatingwith the onstant velo ity V: (x; t) = (x�Vt):532 Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 Bifur ations of Solitary WavesIn this ase the entire dependen e of ak on time t is ontained in the os illatingexponent: ak(t) = ke�ik�Vt;where by virtue of (3) the amplitude k will satisfy the equation(!k � k �V) k = �ÆHÆ �k � fk: (5)The di�eren e !k � k �V appearing in this equation will be positive for all k ifthe soliton velo ity is less than the minimum phase velo ityjVj < min(!k=k): (6)Conversely, the di�eren e will be negative for all k if the soliton velo ity is greaterthan the maximum phase velo ityjVj > max(!k=k): (7)We will show that a soliton solution is possible if the ondition (6) (or (7)) issatis�ed. Let us assume the opposite to be true � let the onditions (6) and (7)be violated, i.e., the equation (1) has a solution. For simpli ity, we will assumethat it is unique: k = k0. Then, sin e xÆ(x) = 0, the homogeneous linear equation(!k � k �V)Ck = 0possesses a nontrivial solution in the form of a mono hromati waveCk = AÆ(k � k0):In this ase (5) an be written (by virtue of the Fredholm alternative) k = AÆ(k� k0) + fk!k � k �V with fk0 = 0: (8)This equation, in ontrast to (5), ontains a free parameter � the omplexamplitude A. It an be solved, for example, by iterations, taking AÆ(k � k0) asthe zeroth term. It is important that be ause of the nonlinearity as a result ofiterations one will obtain multiple harmoni s with k = nk0 where n is integer.The solution will onsist of a olle tion of Æ-fun tions. Correspondingly, in phy-si al spa e the solution will be a periodi fun tion of the oordinates, i.e., it willbe nonlo alized. Hen e follows the �rst sele tion rule for solitons: the di�eren e!k � k �V must be sign-de�nite, whi h is equivalent to the requirements (6) or(7). In other words, it means the absen e of Cherenkov radiation.Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 533 E.A. Kuznetsov, D.S. Agafontsev, and F. DiasIn this entire s heme, however, there is an important ex eption. Havingrepresented (5) in the form (8), we have in fa t assumed that the singularity inthe expression fk!k � k �V (9)is nonremovable. This may not be the ase � the singularity in the denominatorin (9) ould be an elled with the numerator, i.e., it ould be removable [10℄.For example, this happens for the lassi soliton of the KdV equation, for equa-tions whi h are generalizations of the KdV equation [16℄ for a ombination of theone-dimensional NLS and the MKdV equations [10, 17℄ integrated by the sameZakharov�Shabat operator [18℄ and so on. In all of these ases an ellation o ursas a result of the k dependen e of the matrix elements. However, even in these ases, the sele tion rule for solitons remains the same after the resonan e (1) isremoved � the part remaining in the denominator must be sign-de�nite.In what follows the singularities in (9) are assumed to be nonremovable in theforbidden region, and we study the behavior of the soliton solution as the solitonvelo ity approa hes the riti al value. For de�niteness, it is assumed that theplane ! = k �V is tangent to the dispersion surfa e ! = !k from below, i.e., the riterion (6) holds. Let tou hing o ur at the point k = k0. Then, instead of (8),in the allowed region k = fk!k � k �V :As the velo ity V approa hes the riti al value V r, the denominator inthis expression be omes small near the tou hing point, so that k gets a sharppeak at this point k = �12!�� + k0(V r � V )��1 fk: (10)Here !�� = �2!=�k��k� is a symmetri , positive-de�nite, tensor of the se ondderivatives, evaluated at k = k0, and � = k� k0.It is evident from (10) that as V approa hes the riti al velo ity, the width ofthe peak narrows as pV r � V , and the distribution orresponding to the mainpeak k = k0 approa hes a mono hromati wave. A ounting for nonlinearity, thespe trum ontains harmoni s whi h are multiples of k = k0. If it is assumed thatthe amplitude of the soliton vanishes gradually as V ! V r (whi h would orre-spond to a se ond-order phase transition), then the solution (x) (or, equivalently, k) an be sought as an expansion in terms of harmoni s: (x0) = 1Xh=�1 n(X)eihk0�x0 ; x0 = x�Vt: (11) 534 Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 Bifur ations of Solitary WavesHere the small parameter � =p1� V=V r (12)and the �slow� oordinate X = �x0 are formally introdu ed, so that n(X) is theamplitude of the envelope of n-th harmoni . The assumption that the solitonamplitude vanishes ontinuously at V = V r means that the leading term of theseries in (11) orresponds to the �rst harmoni , and all other harmoni s are smallin the parameter �. This is the ondition under whi h the nonlinear S hr�odingerequation is derived (see, for example, [10, 19, 2℄). In the present ase, we arriveat the stationary NLS�k0V r�2 1 + 12!�� 2 1�X��X� +Bj 1j2 1 = 0 (13)at leading order in �, where B is related to the matrix ~Tk1k2k3k4 of four-waveintera tions as B = �(2�)d ~Tk0k0k0k0 : (14)In this approximation the leading term in the intera tion Hamiltonian has theformHint = ~Tk0k0k0k02 Z �k1 �k2 k3 k4Æk1+k2�k3�k4dk1dk2dk3dk4 = �B2 Z j 1j4dx;(15)and the tilde means renormalization of the vertex due to the three-wave intera -tion � in the present ase the intera tion with the zeroth and se ond harmoni s.As we have already noted, !�� in (13) is a symmetri positive-de�nite tensor.For this reason, performing a rotation to its prin ipal axes and arrying out the orresponding extensions along ea h axis, (13) an be transformed into the stan-dard form ��2 +� � �j j2 = 0; (16)where � = sign( ~T!��). Hen e it follows, in the �rst pla e, that soli-tons are possible only if � is negative (fo using nonlinearity when the produ t~T!�� is negative) and, in the se ond pla e, that the amplitude of the solitons isproportional to � =p1� V=V r;i.e., the amplitude vanishes a ording to a square-root law, the size of the solitonin reases as 1=� as the velo ity approa hes the riti al value.In order to illustrate how this me hanism works onsider the simplest example,i.e., the time-dependent one-dimensional nonlinear S hr�odinger equationi� �t + xx + 2j j2 = 0: (17)Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 535 E.A. Kuznetsov, D.S. Agafontsev, and F. DiasAs is well known, this equation, unlike the general equation (3), has one additionalsymmetry, namely the gradient symmetry ! ei�. To �nd the orrespondingsolution one should put (x; t) = ei�t (x� V t), where obeys the equationL(�i�x) � iV x + � � xx = 2j j2 : (18)For the present ase, in a ordan e with (1), the ondition for Cherenkov radiationwill be written as follows: kV = (k) or L(k) = 0 ; (19)where the dispersion law for the equation (18) takes the form (k) = � + k2: (20)Hen e one an see that for � < 0 the resonan e ondition (19) is satis�ed for anyvalue of the velo ity V ! Consequently solitons do not exist in this ase. This anbe he ked dire tly by solving equation (18): for � < 0 all solutions are periodi or quasi-periodi . Soliton solutions are possible for positive �. Their velo itieslie in the range �2p� � V � 2p�. At the points k = �p� the dispersive urve = (k) tou hes the straight line = kV r. At these points the solution mustvanish in agreement with the general onsiderations. It dire tly follows from theexa t solution of (18): = ei�t eiV x0=2�k osh(�kx0) ; x0 = x� V t; �k =p� � V 2=4: (21)Solitons exist only for � > V 2=4: The upper boundary in this inequality de�nesthe riti al velo ity V r = 2p�:It is important to note also that for � > V 2=4 the operator L in the equation (18)is positive de�nite. 3. Stability of SolitonsTo in lude the time dependen e in the averaged equations the amplitudes n in the expansion (11) must be assumed to depend not only on the �slow� oordinate X but also on the slow time T = �2t. Then a multis ale expansiongives the nonstationary analog of the NLSEi t � �2 +� � �j j2 = 0 (22)instead of the stationary NLSE (16). The soliton stability problem for this equa-tion has been well studied (see, for example, [10℄ and [21℄). We re all the basi 536 Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 Bifur ations of Solitary Wavespoints in the investigation of stability. The equation (22) as an equation forenvelopes inherits the anoni al Hamiltonian form (3)i� �t = Æ ~HÆ � ; (23)where the Hamiltonian~H = �2N + Z (jr j2 � j j4)dr; (� = �1); (24)arises as a result of averaging the initial Hamiltonian. The equation (22) preserves,besides ~H, the total number N of parti les (adiabati invariant), so that solitonsare stationary points of the energy fun tional E = ~H��2N with the �xed numberof parti les Æ(E + �2N) = 0: (25)The number of parti les (or intensity) in a soliton solution as a fun tion of � hasthe form Ns = Z j j2dx = �2�d Z jg(�)j2d�; (26)where d is the dimension of the spa e, and g(�) satis�es the equation�g +�g + jgj2g = 0:In the one-dimensional ase g = p2 se h � and, orrespondingly, Ns = 4�. In thetwo-dimensional ase Ns is independent of � for the entire family of solitons, whilein the three-dimensional ase Ns de reases with in reasing �. The dependen e ofNs on �2 is ru ial from the standpoint of soliton stability. It is obvious that themost dangerous disturban es will be those having wave numbers lose to k = k0moving together with the soliton, i.e., modulation-type disturban es.A ording to (25) the envelope solitons are stationary points of the energy Efor a �xed number of waves N . Therefore su h solutions will be stable in theLyapunov sense if they realize a minimum (or a maximum) of the energy for�xed N .Consider �rst the s aling transformations leaving N un hanged, (x) = 1ad=2 s �xa� ; (27)where s is the solitoni solution. The energy E under this transformationbe omes a fun tion of the s aling parameter a:E(a) = I1sa2 � I2sad ; (28)Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 537 E.A. Kuznetsov, D.S. Agafontsev, and F. Diaswhere I1s = R jr sj2dx, I2s = R j sj4dx and � = �1. Hen e it is easy to seethat in the one-dimensional ase the energy (28) is bounded from below and hasa minimum at a = 1 orresponding to the soliton solution. In this aseEs = �2�33 and 2I1s = I2s = 4�33 :The soliton also realizes a minimum of E with respe t to another simple trans-formation, i.e., the gauge one, 0(x)! 0(x) exp[i�(x)℄, whi h also preserves N ,E = Es + Z (�x)2 20dx:Thus, for d = 1 both simple transformations yield a minimum for the energy, thusindi ating soliton stability for the one-dimensional geometry.Now we give an exa t proof of this fa t. The ru ial point of this proof is basedon integral estimations of Sobolev type. These inequalities arise as the sequen esof general imbedding theorem. The (Sobolev) theorem says that a spa e Lp anbe imbedded into the Sobolev spa e W 12 if the spa e dimensionD < 2p(p+ 4):This means that between the normskukp = �Z jujpdDx�1=p ; p > 0;kukW 12 = �Z (�2juj2 + jruj2)dDx�1=2 ; �2 > 0;there exists the following inequality (see, e.g., [23℄):kukp �MkukW 12 ; (29)where M is some positive onstant. For D = 1 and p = 4, the inequality (29) is1Z�1 j j4dx �M1 24 1Z�1 (�2j j2 + j xj2)dx352 : (30)Here it is straightforward to get a multipli ative variant of the Sobolev inequality,the so- alled Gagliardo�Nirenberg inequality (GNI) [24℄ (see also [23, 25, 26℄).Use the s aling transform x!�x in (29). Instead of (30) we have1Z�1 j j4dx �M1 24�2 1Z�1 j j2dx � �+ 1Z�1 j xj2dx � 1�352 :538 Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 Bifur ations of Solitary WavesThis inequality holds for any (positive) � in luding a minimal value for the r.h.s.The omputing of its minimum yields the GNI:I2�CN3=2I1=21 ; (31)where I1 = R j xj2dx, I2 = R j j4dx, and C is a new onstant. Then this inequa-lity an be improved by �nding the best (minimal possible) onstant C.To �nd Cbest onsider all extrema of the fun tionalJf g = I2N3=2I1=21 : (32)The latter problem redu es to the solution of stationary NLSE (16):��2 + xx + 2j j2 = 0:Hen e we �nd that the best onstant Cbest is a value of Jf g on the solitonsolution: Cbest = I2sN3=2I1=21s = 2I1=21sN3=2 : (33)This inequality allows us to obtain immediately a proof of 1D soliton stability.Substituting (31) into (24) results the following estimation for the energy:E � I1 � CbestI1=21 N3=2 = Es + (I1=21 � I1=21s )2: (34)The inequality be omes pre ise on the soliton solution, thus proving its stability.Note that this provides the stability of solitons not only with respe t to smallperturbations, but also against �nite perturbations.In the three-dimensional ase, in ontrary, the fun tion E(a) in (28) hasa maximum, orresponding to the soliton solution, and is unbounded from be-low as a! 0. The gauge transformation gives a minimum of E and therefore allsoliton solutions at d = 3 represent saddle points of the energy. That indi ates apossible instability of solitons in this ase.The fa t of (linear) instability of three-dimensional solitons follows from theVakhitov-Kolokolov riterion [20℄. It is as follows: if�Ns��2 > 0; (35)then the soliton is stable and, respe tively, unstable if this derivative is negative.This riterion has a simple physi al meaning. The value ��2 for the stationarynonlinear S hr�odinger equation (16) an be interpreted as the energy of the boundstate�soliton. If we add one �parti le� to the system and the energy of this boundJournal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 539 E.A. Kuznetsov, D.S. Agafontsev, and F. Diasstate de reases, then we have a stable situation. If by adding one �parti le�the level ��2 is pushed towards the ontinuous spe trum, then su h a soliton isunstable.At d = 3 the derivative �Ns=��2 < 0 and therefore 3D solitons are unsta-ble (the modulational instability). For the two-dimensional ase the Vakhitov�Kolokolov riterion (35) gives an absen e of linear exponential instability. A moredetailed analysis in this ase yields the power type instability (for details see thesurvey [21℄ and [22℄).Thus, the solitons are stable only in the one-dimensional ase, while in thetwo-dimensional ( riti al) and three-dimensional ases they are unstable and anbe onsidered as separatrix solutions separating ollapsing solutions from the dis-persive ones [27℄.This is probably the simplest method for explaining the well-known empiri alfa t that solitons, as a rule, exist only in one-dimensional systems. For multidi-mensional systems the stable solitons are rare and an only appear as a result oftopologi al onstraints or of a me hanism that removes Cherenkov singularities(whi h is dis ussed in the present paper). The latter, as an be easily understood,is due to the existen e of a ertain lass of symmetry.4. From Super riti al to Sub riti al Bifur ationsFor sub riti al bifur ation at the riti al velo ity the soliton undergoes a jumpin its amplitude. In this ase the orresponding theory an be developed near thetransition point between sub riti al and super riti al bifur ations (in analogy withthe tri- riti al point for phase transitions). In the series of papers [11, 13, 28�30℄it was shown that in this ase the soliton behavior an be des ribed by means ofthe generalized nonlinear S hr�odinger equation (NLSE) for the envelope , whi hin the one-dimensional ase is as follows:i� �t � �2!0 + !0002 xx � �j j2 + 4i�j j2 x + bkj j2 + 3Cj j4 = 0; (36)where !0 � !(k0) and k0 are the arrying frequen y and the wave number, res-pe tively, �2 = (V r � V )=V r � 1, !000 the se ond derivative of !(k) taken atk = k0. Here the four-wave oupling oe� ient � is assumed to have additionalsmallness hara terizing the proximity to the transition from super riti al to sub- riti al bifur ations. The transition point is de�ned from the equation � = 0.For example, for the interfa ial deep-water waves propagating along the interfa ebetween two ideal �uids in the presen e of apillarity [28, 29℄� = k301 + � �A2 r �A2� ; 540 Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 Bifur ations of Solitary Waveswhere � is the density ratio, A = (1 � �)=(1 + �) the Atwood number, A2 r =5=16 and � r = (21 � 8p5)=11, as it was shown in [13℄. For � < � r, the four-wave oupling oe� ient � is negative, and the orresponding nonlinearity is ofthe fo using type. In this ase, the solitary waves near the riti al velo ity V rare des ribed by the stationary (�=�t = 0) NLSE and undergo a super riti albifur ation at V = V r [13℄. For � > � r, the oupling oe� ient hanges sign and,as a result, the bifur ation be omes sub riti al. For water waves in �nite depthh the oe� ient � hanges its sign at � r = k0h � 1:363 [31℄ while !000 is alwaysnegative. Thus the nonlinearity belongs to the fo using type for �(= kh) > � r andrespe tively be omes defo using in the region � < � r [31, 32℄. In nonlinear opti s,as shown in [11℄, a de rease of � (Kerr onstant) an be provided by the intera tionof light pulses with a ousti waves (Mandelstamm�Brillouin s attering).Be ause of smallness of � we keep in (36) the following order nonlinear terms:the gradient term (� �) responsible for self-steepening of the pulse (analog ofthe Lifshitz term in phase transitions), the nonlo al term (due to presen e ofthe integral operator bk, the Fourier transform of its kernel is equal to jkj) andthe six-wave nonlinear term with oupling oe� ient C. Two additional 4-waveintera tion terms, both lo al and nonlo al, appear as a result of expansion of thefour-wave matrix element Tk1k2k3k4 in powers of the small parameters �i = ki�k0:Tk1k2k3k4 = �2� + �2� (�1 + �2 + �3 + �4) (37)� 8� (j�1 � �3j+ j�2 � �3j+ j�2 � �4j+ j�1 � �4j):The existen e of nonlo al ontribution in the expansion is onne ted with non-analyti al dependen e of the matrix element T in its arguments. For interfa ialdeep-water waves (IW) this nonanalyti ity originates from the solution of Lapla eequation for the hydrodynami potential and its redu tion to the moving inter-fa e. For instan e, for water waves (WW) with a �nite depth the nonlo al termis absent [32℄ as well as for ele tromagneti waves in nonlinear diele tri s [11℄be ause of the analyti ity of matrix elements with respe t to frequen ies, whi his a onsequen e of ausality (see, for example, Refs. [11, 33℄). In the latter asethe spatial dispersion e�e ts are relativisti ally small and an be negle ted.For both IW and WW near the transition point, !000C is positive; moreover, is also positive for IW, and therefore the orresponding nonlinearities are fo using,thus providing the existen e of lo alized solutions. Depending on the sign of �there exist two bran hes of solitons. For IW they were found numeri ally [28, 29℄using the Petviashvili s heme [34℄. Expli it solutions for both kinds of IW solitons an be obtained in the limiting ase only when V ! V r. For negative � theseare the lassi al NLS solitons with a se h shape. For the sub riti al bifur ationat V = V r the soliton amplitude remains �nite with algebrai de ay (� 1=jxj)Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 541 E.A. Kuznetsov, D.S. Agafontsev, and F. Diasat in�nity [28, 29℄. When a nonlo al nonlinearity is absent ( = 0), the solitonsolutions an be found expli itly. For both bran hes at large � the number ofwaves N = R j j2dx approa hes from below and above the same value N r whi h oin ides with the number of waves N on the solitons with � = 0. For the solitonsin �bers this property means that the energy of opti al pulse saturates tendingto the onstant value with a de rease of the pulse duration.On the other hand, all solitons of (36) are stationary points of the energy Efor �xed number of waves : Æ(E + �2N) = 0, where the energy in dimensionlessvariables is given byE = Z �j xj2 + �2 j j4 + i�( �x � x �)j j2 � 2 j j2k̂j j2 � Cj j6� dx: (38)This allows one to use the Lyapunov theorem in the analysis of their stability.Here, for the IW, � � (V r � V )1=2j�� rj�1, C = 319=1281 and� = sign(�� r); � = 6=p427; = 32=p427; (39)and for the WW ase ( ompare with [32℄)� = sign(� r � �); � � �0:397; C � 0:176: (40)As shown in [11, 28, 29, 32℄, for N < N r solitons orresponding to the super riti- al bran h realize the minimum values of the energy and therefore they are stablein the Lyapunov sense, i.e., stable with respe t to not only small perturbationsbut also against �nite ones. In parti ular, the boundedness of E from below anbe viewed if one onsiders the s aling transformation = (1=a)1=2 s(x=a) retain-ing the number of waves N , where = s(x) is the soliton solution. Under thistransform E be omes a fun tion of the s aling parameter a:E (a) = �1a � 12a2� �2 Z j sj4 dx: (41)It is worth noting that the dispersion term and all nonlinear terms in E, ex eptR �2 j j4dx, have the same s aling dependen e / a�2. The latter means that at� = 0 (36) an be related to the riti al NLS equation like the two-dimensional ubi NLS equation. From (41) it is also seen that for � < 0 E(a) has a min-imum orresponding to the soliton. Unlike in the super riti al ase, the s alingtransformation for another soliton bran h with � > 0 gives a maximum of E(a)on solitons and unboundedness of E as a ! 0. Under the gauge transformation = sei�, on the ontrary, the energy rea hes a minimum on soliton solutionsand onsequently the solitons with � > 0 represent saddle points. This indi atesa possible instability of solitons for the whole sub riti al bran h, at least withrespe t to �nite perturbations.542 Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 Bifur ations of Solitary WavesWe onsider this question in detail and emphasize a nonlinear stage of insta-bility following to our re ent paper [30℄. This problem, indeed, is not trivial inspite of a lose similarity with the riti al NLSE. It is worth noting that (36) at� = = C = 0 represents an integrable model (the so- alled derivative NLSE)[35℄, and exponentially de aying solitons in this model are stable. It is more orless evident also that small oe� ients ; C annot break the stability of soli-tons. This means that in the spa e of parameters we may expe t the existen e ofa threshold. Above this, the threshold solitons must be unstable and the deve-lopment of this instability may lead to ollapse, i.e., the formation of a singularityin �nite time.Consider the energy (38) written in terms of amplitude r and phase ' ( =rei'): E = Z �r2x + �2 r4 � 2 r2bkr2 � 13r6 + r2 �'x + �r2�2� dx; (42)where by an appropriate hoi e of the new dimensionless variables the renormal-ized onstant ~C = C + �2 an be taken equal to 1/3. Hen e one an see that theenergy takes its minimum value when the last term in (42) vanishes, i.e. when'x + �r2 = 0: (43)The integration of this equation gives an x�dependen e for the phase that is alled the hirp in nonlinear opti s. It is interesting to note that the remainingpart of the energy does not ontain the phase at all.First, study the lo al model when = 0. Let the energy be negative insome region : E < 0. Then, following [36, 26℄, one an establish that dueto radiation of small amplitude waves E < 0 an only de rease, be oming moreand more negative, but the maximum value of j j, a ording to the mean valuetheorem, an only in rease: maxx2 j j4 � 3jE jN : (44)This pro ess is possible only for the energies whi h are unbounded from below.In a ordan e with (41) su h a situation is realized when � > 0. In this ase theradiation leads to the appearan e of in�nitely large amplitudes r. However, it isimpossible to on lude that the singularity formation develops in �nite time.For > 0 the estimations on the maximum value of j j are not as transparentas they are for the lo al ase. Instead of (44), it is possible to obtain a similarestimate maxx j j4 � 3jEjN :However, it is expressed through the total energy E and the total number ofwaves N . Besides, two inequalities must be satis�ed: E < 0 and N < 2N2 .Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 543 E.A. Kuznetsov, D.S. Agafontsev, and F. DiasFor interfa ial waves, N2 � 1:39035 > N r � 1:3521. Thus, the maximum am-plitude in this ase is bounded from below by a onservative quantity and thismaximum an never disappear during the nonlinear evolution.Now we onsider the situation where the self-steepening pro ess anbe negle ted (� = 0). In this ase (36) be omesi t + xx � �2 � �j j2 + bkj j2 + 3Cj j4 = 0:It is possible to obtain a riterion of ollapse using the virial equation (for details,see [37, 36, 38℄). This equation is written for the positive de�nite quantityR = Z x2j j2dx;whi h, up to the multiplier N , oin ides with the mean square size of the distri-bution. The se ond derivative of R with respe t to time is de�ned by the virialequation Rtt = 8�E � �4 Z j j4dx� : (45)Hen e, for � > 0 one an easily obtain the following inequality:Rtt < 8E;whi h yields, after double integration, R < 4Et2+�1t+�2. Here �1;2 are onstantswhi h are obtained from the initial onditions. Hen e, it follows that for the stateswith negative energy, E < 0, there always exists su h a moment of time t0 whenthe positive de�nite quantity R vanishes. At this moment of time the amplitudebe omes in�nite. Therefore the ondition E < 0 represents a su� ient riterion of ollapse ( ompare with [37, 36℄). However, it is ne essary to add that this riterion an be improved in the same way as in [39, 27℄ for the three-dimensional ubi NLS equation. From (45) one an see that for the stationary ase (on the solitonsolution) Es = �4 R j sj4dx, in agreement with (41). As it was shown before, for� > 0 the soliton realizes a saddle point of E for �xed N . It follows from (41) thatfor small a the energy be omes unbounded from below, but for a > 1 it de reases(this orresponds to spreading). Therefore in order to a hieve a blow-up regimethe system should pass through the energeti barrier equal to Es. Thus, for this ase the riterion E < 0 must be hanged into the sharper riterion: E < Es.This riterion an be obtained rigorously using step by step the s heme presentedin [39, 27℄ and therefore we skip its derivation. 544 Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 Bifur ations of Solitary Waves −30 −20 −10 0 10 20 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xFig. 1: Initial (solid line) and �nal (dashed line) at t = 1:18 distributions for j j,interfa ial waves, selfsimilar variables. The soliton amplitude was in reased by1%, � = 1, � = 1. The ratio between �nal and initial soliton amplitudes in thephysi al variables is about 11.In order to verify all the theoreti al arguments about the formation of ollapsepresented above we performed a numeri al integration of the NLSE (36) for � > 0by using the standard 4th order Runge�Kutta s heme. The initial onditions were hosen in the form of solitons but with larger amplitudes than for the stationarysolitons. In reasing in the initial amplitude varied in the interval from 0.1% up to10%. The initial phase was given by means of (43). In all runs with theses initial onditions we observed a high in rease of the soliton amplitude up to a fa tor 14with a shrinking of its width. In a peak region the pulses for both IW and WW ases behaved similarly. Near the maximum the pulse peak was almost symmetri :anisotropy was not visible. The di�eren e was observed in the asymptoti regionsfar from the pulse ore, where the pulses had di�erent asymmetries for IW andWW be ause of the opposite sign for � (see (39), (40)). For the given values of �we did not observe the simultaneous formation of two types of singularities withblowing-up amplitudes and sharp gradients as it was demonstrated in the re entnumeri al experiments for the three-dimensional ollapse of short opti al pulsesdue to self-fo using and self-steepening in the framework of the generalized NLSequation [40℄ and equations of the Kadomtsev�Petvishvili type [41℄. Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 545 E.A. Kuznetsov, D.S. Agafontsev, and F. Dias −80 −60 −40 −20 0 20 40 60 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xFig. 2: Initial (solid line) and �nal (dashed line) at t = 2:7192 distributions forj j, WW solitons, selfsimilar variables. The soliton amplitude was in reased by1%, � = 1, � = 1. The ratio between �nal and initial soliton amplitudes in thephysi al variables is about 11.In our numeri al omputations we found that the amplitude and its spatial ollapsing distribution developed in a selfsimilar manner. Near the ollapse pointin the equation (with � > 0) one an negle t the term proportional to �. In thisasymptoti regime (36) admits selfsimilar solutions,r(x; t) = (t0 � t)�1=4f � x(t0 � t)1=2� ; (46)where t0 is the ollapse time.To verify that we approa hed the asymptoti behavior given by (46), at ea hmoment of time we normalized the -fun tion by the maximum (in x) of itsmodulus max j j �M and introdu ed new self-similar variables, (x; t) =M (�; �); � =M2(x� xmax); � = lnM: (47)Here xmax is the point orresponding to the maximum of j j. In omparisonwith those given by (46), new variables are more onvenient be ause they do notrequire the determination of the ollapsing time t0.Fig. 1 and Fig. 2 show typi al dependen es of j j as a fun tion of the self-similar variable � at t = 0 (solid line) and at the �nal time (dashed line) for546 Journal of Mathemati al Physi s, Analysis, Geometry, 2008, vol. 4, No. 4 Bifur ations of Solitary Waves 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 tFig. 3: Dependen e of 1=max j j4 on time. Interfa ial waves.both the IW and WW ases. In both �gures one an see a fairly good oin iden ebetween the initial soliton distribution and the �nal one at the entral ( ollapsing)part of the pulse and asymmetry of the pulse at its tails due to self-steepening.The latter demonstrates that the ollapse has a selfsimilar behavior. The form ofthe entral part of the pulse approa hes the soliton shape be ause asymptoti allythe NLS model (36) tends to the riti al NLS system. It should be mentioned thatthis has been well known for the lassi al two-dimensional NLS equation sin e thepaper by Fraiman [42℄.Fig. 3 shows how 1=max j j4 depends on time. This dependen e is almostlinear in the orresponden e with the selfsimilar law (46). If the initial amplitudeswere less than the stationary soliton values, then the distribution would spreadin time dispersively, in full orresponden e with qualitative arguments based onthe s aling transformations (41).A knowledgments. The authors thank A.I. Dya henko for valuable dis us-sions on erning the numeri al simulations. We a knowledge support from CNRSunder the framework of PICS No. 4251 and RFBR under Grant 07-01-92165.The work of DA and EK was also supported by RFBR (Grant 06-01-00665),the Program of RAS "Fundamental problems in nonlinear dynami s" and GrantNSh 7550.2006.2. 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Bifurcations of Solitary Waves

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