Solitons and cavitons in a nonlocal Whitham equation
SSolitons and cavitons in a nonlocal Whitham equation
N. Kulagin , L. Lerman , , A. Malkin , Institute of Physical Chemistry and Electrochemistry,Russian Academy of Science, Moscow, Russia, Dept. of Fund. Math. of Higher School of Economics, and Lobachevsky State University of Nizhny Novgorod, Russia
Abstract
Solitons and cavitons (localized solutions with singularities) for the nonlocal Whithamequations are studied. The equation of a fourth order with a parameter in front of fourthderivative for traveling waves is reduced to a reversible Hamiltonian system defined on a two-sheeted four-dimensional space. When this parameter is small we get a slow-fast Hamiltoniansystem. Solutions of the system which stay on one sheet represent smooth solutions ofthe equation but those which perform transitions through the branching plane representsolutions with jumps. They correspond to solutions with singularities – breaks of the firstand third derivatives but continuous even derivatives. The system has two types of equilibriaon different sheets, they can of saddle-center or saddle-foci. Using analytic and numericalmethods we found many types of homoclinic (and periodic as well) orbits to these equilibriaboth with a monotone asymptotics and oscillating ones. They correspond to solitons andcavitons of the initial equation. When we deal with homoclinic orbits to a saddle-centerthe values of the second parameter (physical wave speed) is discrete but for the case of asaddle-center it is continuous. The presence of majority such solutions displays the verycomplicated dynamics of the system.
Nonlinear nonlocal Whitham equation ∂V∂t + V ∂V∂x = ∂∂x ∞ (cid:90) −∞ dx (cid:48) R ( x − x (cid:48) ) V ( x (cid:48) , t ) (1)represents a wide class of equations which are of great interest for nonlinear wave theory. Itcombines the typical hydrodynamic nonlinearity and an integral term descriptive of dispersionof the linear theory. The kernel of the integral term is conventionally defined by the dispersionrelation ω = k (cid:101) R ( k ) with R ( x ) = ∞ (cid:90) −∞ (cid:101) R ( k ) e − ikx dk. (2)1 a r X i v : . [ n li n . C D ] J un q.(1) with (cid:101) R = (1 + k ) − was proposed by G. Whitham instead of Korteweg-de Vries equationin order to describe sharp crests of the water waves of a greatest height [1].The usage of relatively simple Whitham type equations appeared to be very fruitful forvarious physical applications. A number of special cases of Eq.(1) were examined in detail.Among them are the Benjamin-Ono [2, 3] and Joseph [4] equations describing internal waves instratified fluids of infinite and finite depth. These equations appeared to be integrable by theinverse scattering technique and the behavior of their solutions has been studied rather well.The Benjamin-Ono and Joseph equations are however the only representatives of the Whithamequations possessing this property [5]. Another widely known equations of that class were studiednot so exhaustively, although the literature on the subject is quite extensive. A list of well-knownWhitham equations involves the Leibovitz one for the waves in rotating fluid [6], the Klimontovichequation for magnetohydrodynamic waves in non-isothermal collision-less plasma [7], equationsfor shallow water waves [1], capillary [10] and hydroelastic [11] waves. The review on nonlinearnonlocal equations in theory of waves is presented in detailed monograph [8].The characteristic feature of conservative Whitham equations is the existence of solitary wavesolutions. For all just listed equations these solutions are smooth except some limiting cases ofpeaking for the waves of greatest amplitude. Besides, the amplitudes and velocity spectra ofsolitons can be bounded or not. But in any case the spectra are continuous. These propertiesare believed to be typical, but, as will be shown below, they are not essential for the solitons ofWhitham equations.Here we examine a particular case of the Whitham equation with a resonance dispersionrelation (cid:101) R ( k ) = 11 − k + D k . (3)That equation has been proposed for nonlinear acoustic waves in simple peristaltic systems [45].With small D it is also applicable to the waves in a medium with internal oscillators [12]. Atentative analysis of some peculiarities of solitons to that equation has been performed in a shortcommunication [13].We study here specific features of solitary wave solutions to Eqs.(1)-(3). It is shown that thisequation possesses both smooth and singularity involving solitons with exponential asymptotics,bound states of solitons and solitary waves with oscillating asymptotics. The velocity spectra ofexponentially localized solitons turn out to be discrete ones. Hereafter we shall study an ordinary differential equation that obtained by the inversion ofintegral operator defined by Eqs (1)-(3) and transferring to the traveling wave solutions. Theresult takes the form of the fourth order differential equation D S ( IV ) + S (cid:48)(cid:48) + S = V, S = λV + 12 V , (4)2ith traveling coordinate y = x + λt , boundary conditions lim S ( y ) = 0 , as | y | → ∞ , andparameters D << , ≤ λ ≤ . Thus, physically treated, the problem of searching forsolutions of this type can be thought as a nonlinear boundary value problem for the parameter λ. This equation for very small D is of the singularly perturbed type similar to many suchequations, see, for instance, [10, 30, 14, 10, 32, 37, 21, 16]. Another feature of this equation is itsrelation with the class of implicit differential equations [9]. To see this, let us introduce, similarto [38], new variables u = S, u = S (cid:48) , v = − S (cid:48) − D S (cid:48)(cid:48)(cid:48) , v = DS (cid:48)(cid:48) . Then the equation isreduced to the Hamiltonian system u (cid:48) = u , v (cid:48) = u − V ( u ) , Du (cid:48) = v , Dv (cid:48) = − u − v , (5)where V ( u ) is given by solving the equation u = 2 λV + V w.r.t. V . This system is also re-versible w.r.t. the involution L : ( u , v , u , v ) → ( u , − v , − u , v ) , i.e. if ( u ( y ) , v ( y ) , u ( y ) , v ( y )) is a solution to the system, then ( u ( − y ) , − v ( − y ) , − u ( − y ) , v ( − y )) is as well [19]. The fixedpoint set F ix ( L ) of L is 2-plane v = u = 0 . The quadratic equation for V ( u ) has generally either two or no real solutions, so function V is two-valued. To keep this into account in more convenient way, let us consider the space R with coordinates ( u , v , u , v , V ) and its smooth 4-dimensional submanifold M given by realsolutions of the equation V + 2 λV − u = 0 . This is a two-sheeted submanifold with respect tothe projection π : ( u , v , u , v , V ) → ( u , v , u , v ) , its image is half-space u ≥ − λ / . Bothsheets are glued along 3-dimensional branching 3-plane u = − λ / , V = − λ. The shape of thissubmanifold is the direct product of a parabola and a 3-plane.On each sheet (upper one V ( u ) = − λ + √ λ + 2 u and lower one V = − λ − √ λ + 2 u )the system generates its own differential system. In the half-space u ≥ − λ / on every sheetthe Peano theorem on the existence of solutions is valid [31], but in the open half-space u > − λ / the usual theorem of existence and uniqueness of solutions works, so, despite of the non-smoothness of the function √ λ + 2 u on the boundary, through every point in the closed sheetof the open half-space an orbit passes and the only orbit through the point on a sheet overthe open half-space. There are two questions here: 1) how does one need to adjoin orbits fromdifferent sheets in order to preserve continuity of S ( y ) , when the related orbits hit the boundaryof a sheet, i.e. they satisfy the equality u ( y ) = − λ / and for this y we have u ( y ) (cid:54) = 0 , and2) about possible non-uniqueness for orbits through boundary points where u = − λ / (and V = − λ ).The inequality u ( y ) (cid:54) = 0 means that the related orbit must leave a sheet or enter to a sheet,in dependence on the sign u ( y ) . Indeed, the restrictions of both vector fields to the branchingplane u = − λ / coincide. Hence, if u ( y ) > at the boundary point, then the orbit looksinward the region u > − λ / on both sheets and should enter to both of them (we remind u (cid:48) = u ). But it is impossible, if u ( y ) (i.e. S ( y ) ) varies continuously in y near this value y .If, on the contrary, one gets u ( y ) < , then the orbits look outward in both sheets andshould leave an either sheet. Thus, in order to preserve continuity of u we need to use the3eversibility of vector fields and make jumps on the boundary of both sheets in accordance tothe action of L. How does this save the situation, one can see as follows. Suppose u ( y ) > atthe boundary point X = ( u ( y ) , v ( y ) , u ( y ) , v ( y )) , u ( y ) = − λ / , the vector field looksinward. For u (cid:54) = 0 , the boundary point does not belong to the set F ix ( L ) and its L -image isanother boundary point X (1)0 = L ( X ) with coordinates ( u ( y ) , − v ( y ) , − u ( y ) , v ( y )) . Thus,the vector field at X (1)0 looks outward the region u > − λ / . So, suppose we move for y < y , as y increases to y , in the lower sheet along the orbit which hits the branching plane at the point X (1)0 . Next we jump to the point L ( X (1)0 ) = X and move further in the upper sheet along theorbit through the point X . In this motion the value of u = S varies continuously, but u = S (cid:48) undergoes a jump at y = y . There is a pairing composed orbit, where we move first for y < y on the upper sheet till the point X (1)0 , then jump to X and go further on the lower sheet alongthe orbit through X . Observe that if an orbit in a sheet does not cross the branching pointits behavior is defined by the smooth (in fact – analytic) vector field and any tool working inthis case can be used. Our main concern below will be on solutions to (5) which are homoclinicorbits to equilibria that exist in the system. As we shall see, these solutions are symmetric butthey can be either smooth or with singularities (they cross the branching plane several times).Smooth homoclinic orbits we call sometimes solitons and those with singularities cavitons. Herewe follow our terminology in [13].In order to facilitate simulations, we can eliminate jumps as follows. The submanifold M isthe graph of a smooth function u = λV + V / in variables ( V, v , u , v ) . Let us rewrite thesystem in variables ( r, v , u , v ) , r = V + λrr (cid:48) = u , v (cid:48) = λ (1 − λ/ − r + r / , Du (cid:48) = v , Dv (cid:48) = − ( u + v ) . The system obtained has singularities along 3-plane r = 0 (it is not defined). For upper sheetwe get r = V + λ = √ λ + 2 u > , but for lower sheet the sign is opposite r = V + λ = −√ λ + 2 u < . In order to eliminate the singularity, we multiply equations 2-4 at r andchange the “time” to s, ds = dy/r, obtaining a smooth differential system. The orbits anddirection of moving along orbits of this smooth system coincide on the upper sheet r > withthose for orbits of the initial system (5), but on the lower sheet r < the true direction of movingalong orbits is opposite. In particular, this approach allows one to assert that through any pointof the boundary r = 0 , if this point is not singular, a unique orbit can pass on either sheet. Thissmooth system looks as follows drds = u , dv ds = λ (1 − λ/ r − r + r / , D du ds = rv , D dv ds = − r ( u + v ) . (6)Additional equilibria of the system, appeared due to the change of “time”, fill the plane r = 0 , u = 0 . The system (6) is Hamiltonian and reversible w.r.t. the involution L : ( r, v , u , v ) → ( r, − v , − u , v ) with the smooth Hamiltonian H = u v + 12 ( u + v ) − λ (1 − λ/ r / r − r . JJ = − rD − − rD − , ˙ X = J ∇ H. It is seen that this structure degenerates at r = 0 . The transition to the smooth system sets the question: what are interrelations between orbitsof the smooth system (6) and orbits of the initial system on the upper and lower sheets? Theanswer is the same as was done above: for orbits not crossing the plane r = 0 the behavior isthe same as for its counterpart. For r > direction in y and direction in s are the same, for r < orbits are the same but direction of motion is opposite. Situation for orbits crossing r = 0 appears as before: if an orbit from upper sheet ( r > ) hits the plane r = 0 at a point X thetrue motion is described as above. We make jump by the action of L , X = L ( X ) and after thatwe continue by the orbit through X in the lower sheet ( r < ). If we start from the lower sheet,the procedure is similar. At such transformation the function S varies continuously, as well asits S yy , S yyyy but S y and S yyy change their signs. This leads to the solutions with a singularities(sharpening). The shape of a dependence S ( y ) , its smoothen variant is plotted on Fig. 1a,b andthe projection on ( r, u ) -plane is on Fig. 2. The explanations on the orbit behavior will be givenbelow. Remark 1
It is worth remarking that when we deal with symmetric orbits (invariant w.r.t. theaction of L ), even though they cross the plane r = 0 , the orbits (curves in the phase space) withbe the same as for related orbits of the smooth system, since, due to symmetry, the passage tothe symmetric point occurs on the same orbit. We use this under simulations. a) b)Figure 1: (a) Graph of a true 2-round soliton-caviton S ( y ) ; (b) and its smoothing r ( s ) . Let us demonstrate this approach for the limiting system as D = 0 (slow system [9]). Then thirdand fourth equation give the representation for the so-called slow manifold v = 0 , u = − v . ( u , r ) -planeInserting them into the first and second equations we get the slow system in the half-plane u ≥ − λ / u (cid:48) = − v , v (cid:48) = u + λ ∓ (cid:112) λ + 2 u , (7)where upper sign corresponds to the upper system and lower sign does to the lower system. Firstwe investigate the systems on the upper and lower sheets separately and compare their behaviorwith the smooth system obtained by the change of “time” y. Both systems are Hamiltonian ones and reversible w.r.t. the symmetry ( u , v ) → ( u , − v ) . The upper system has two equilibria (0 , and − λ ) , being a saddle and a center. The lowersystem has not equilibria in the half-plane u ≥ − λ / and all their orbits go from the points u = − λ / , | v | < ∞ from negative v to hit this line at points with positive v . An orbit onthe upper sheet hitting, as y increases, the line u = − λ / at the point ( − λ / , v ∗ ) , has tobe continued from the point ( − λ / , − v ∗ ) with the further increasing y . The similar is donefor orbits on the lower sheet. Under this procedure we get either periodic orbits (smooth, if itbelongs to the upper sheet, or with a sharpening, if this orbit intersects the line u = − λ / ).The phase portraits are the following (see, Fig. 3) where solid lined represent orbits of the uppersystem and dashed line do those for the lower system.Now let us go to the smooth system in variables ( r, v ) , here r can take any sign. r (cid:48) = − v , v (cid:48) = λ (1 − λ/ r − r + r / . (8)The Hamiltonian of the system is h = v λ (2 − λ ) r − r r . The system is also reversible with respect to the involution L : ( r, v ) → ( r, − v ) . This impliesthat if ( r ( s ) , v ( s )) is a solution, then ( r ( − s ) , − v ( − s )) as well.6igure 3: Slow manifold dynamics with jumps.The equilibria of the system are v = 0 , r = 0 , λ, − λ , of them the first and third are centers,the second is a saddle with two separatrix loops. The left center corresponds to the point of theglued system on the line u = − λ / where v = 0 (the point of tangency for orbits of the line).Here all orbits, except for two loops and equilibria, are periodic ones. Thus we get the followingplot (Fig. 4). Figure 4: Phase portrait for smoothen system (8). λ = 0 Here we consider first the degenerated limiting case λ = 0 (in fact, physically not relevant).This consideration is useful to compare with results for values λ > . Thus, we get two systems7efined in the half-space u ≥ , on the upper sheet u (cid:48) = u , v (cid:48) = u − √ u , Du (cid:48) = v , Dv (cid:48) = − u − v , (9)with equilibria (0 , , , and (2 , , , and on the lower sheet u (cid:48) = u , v (cid:48) = u + √ u , Du (cid:48) = v , Dv (cid:48) = − u − v (10)with the only equilibrium (0 , , , . Smoothing the system in variables ( r, v , u , v ) with changing time gives the system ( r isarbitrary) r (cid:48) = u , v (cid:48) = − r + r / , Du (cid:48) = rv , Dv (cid:48) = − r ( u + v ) . (11)with integral H = u v + ( u + v ) / r / − r / . Also the system is reversible w.r.t. theinvolution L : ( r, v , u , v ) → ( r, − v , − u , v ) , its fixed points plane is given as v = u = 0 . The system has a plane P of equilibria r = u = 0 . Only a curve from this plane belongsto the level H = c . In particular, in the level H = 0 where orbits asymptotic to equilibrium atthe origin r = v = u = v = 0 lie, the intersection of P with this level is given in coordinates v , v as straight line ( v , . The system obtained has a complex equilibrium O at the origin: all four its eigenvalues arezeroth. Hence the study of its local orbit behavior is a rather complicated problem. In the firstturn we are interested in its orbits that enter and leave the equilibrium O as s → ±∞ . Suchorbits, if they exist, have power asymptotics in s → ∞ . Let us find these asymptotics using thefollowing ansatz r = As − α (1 + o (1)) , v = Bs − β (1 + o (1)) , u = Cs − γ (1 + o (1)) , v = Es − δ (1 + o (1)) , (12)with unknown coefficients A, B, C, E and exponents α, β, γ, δ to be found. Inserting these func-tions into differential equations (11) we get exponents α = 4 / , β = 5 / , γ = 7 / , δ = 2 = 6 / and coefficients A = 2 √ D / , B = 415 √ D / , C = − √ D / , E = 289 D. (13)The same calculation with the change s = − s to get orbits tending O as s → −∞ givesnaturally the same exponents but reverses signs for B and C staying signs of A, E the same.This calculation shows that the system have to possess by one orbit entering the equilibrium O as s → ∞ and one orbit entering O as s → −∞ . Both these orbits belong to the half-space r > since A > in both cases. In particular, the dependence u in r is of power type u ∼ r / . This is seen in the Fig. 5a where a homoclinic orbit found is shown (soliton). It is clearly seenthe tangency at the entrance to the equilibrium. The related homoclinic orbits for < λ < enter and leave the equilibrium as s → ±∞ with different asymptotics that can be seen on theFig. 9a. This becomes clear below.Sometimes our simulations performed with the initial equation (4) not with the system.Then, to find symmetric homoclinic and periodic solution we sought for orbits which intersect8ither one or two times the fixed point set F ix ( L ) . For initial differential equation (4) this planedefines by relations S (cid:48) = 0 , S (cid:48)(cid:48)(cid:48) = 0 . The 3-plane S (cid:48) = 0 is a cross-section for the majority of itspoints, so zeroes of the graph S (cid:48)(cid:48)(cid:48) ( D ) give values of D for which homoclinic orbits exist. As anexample, such plot is presented on Fig. 5b.a) b)Figure 5: (a) The projection of the homoclinic orbit on the plane ( r, u ) ; (b) Graph S (cid:48)(cid:48)(cid:48) ( D ) . As we shall see, the orbit behavior of the system reminds that for the case of a saddle-centerequilibrium in a Hamiltonian system in two degrees of freedom [39, 35, 28, 47] (see details below).Again, for small positive D system (11) is slow-fast with slow variables ( r, v ) and fast variables ( u , v ) . In this case it is informative to investigate slow and fast systems [21] separately. Slowsystem is derived if we set D = 0 in the third and fourth equations, then we have either r = 0 or v = 0 , u = − v . 3-plane r = 0 divides the phase space into two half-spaces r > and r < .We shall work in the half-space r > , here we get slow 2-plane v = 0 , u = − v . On this planethe slow system is given as ˙ r = − v , ˙ v = − r + r / . Its structure is plotted on Fig.2.The fast systems is derived, if one changes s → s/D = τ , as a result small parameter D arises as a multiplier in the first two equations. Then one sets D = 0 that makes variables r, v be parameters. The third and fourth equation with parameters r , v have integral h =( u + v ) + v . The periods π/r of these linear systems depend on r . The motions in thefull system, as
D > small, in some thin neighborhood of the slow manifold is a combinationof two motions: the slow motion along the orbits of the slow system and fast rotation aroundthe slow orbit. This follows from results of [21]. In particular, if one moves along the homoclinicorbit of the slow system, then the orbit behavior looks very similar to that which is observednear a homoclinic orbit to a saddle-center in a Hamiltonian system with two degrees of freedom[39, 35]. 9 λ (cid:54) = 0 : equilibria Now we turn to the case
D > small enough. One of our main concern is to find solitonsolutions to the equation (5). This corresponds to homoclinic orbits for equilibrium that existson the upper sheet (see below). In fact, there are two equilibria on this sheet but only one ofthem has outgoing and ingoing orbits (separatrices). The situation under consideration dependsheavily on the value of parameter λ and changes at the ends of the segment λ ∈ [0 , .For positive λ in the half-space u > − λ / both systems on 4-dimensional sheets V > − λ and V < − λ are analytic Hamiltonian ones u (cid:48) = u , v (cid:48) = u + λ ∓ (cid:112) λ + 2 u , Du (cid:48) = v , Dv (cid:48) = − u − v , (14)with related Hamiltonians H = u v − u − λu + u + v ±
13 ( λ + 2 u ) / . So, all available methods can be applied to the study, in particular, it concerns existence ofhomoclinic orbits and nearby dynamics. The equilibrium at the origin O (0 , , , on the uppersheet for < λ < is a saddle-center, its eigenvalues are a pair of pure imaginary numbersand two reals. Indeed, linearizing at O gives a linear Hamiltonian system, its characteristicpolynomial is D σ + σ − (1 − λ ) /λ with roots ± iD − (cid:113) (1 + (cid:112) D (1 − λ ) /λ ) / , ± (cid:115) − λ ) λ + (cid:112) λ + 4 D (1 − λ ) λ . Coordinates of the second equilibrium on the upper sheet are (2(1 − λ ) , , , , its characteristicpolynomial is D σ + σ + (1 − λ ) / (2 − λ ) with pure imaginary roots (the elliptic point) σ = − − (cid:112) − D (1 − λ ) / (2 − λ )2 D , σ = − − λ ) / (2 − λ )1 + (cid:112) − D (1 − λ ) / (2 − λ ) . As λ approach to − , both equilibria coalesce into one equilibrium with non semi-simple doublezero and two pure imaginary eigenvalues. The lower sheet does not contain equilibria at all. λ > For λ > two equilibria exist on the upper sheet, O and another one P + = (2(1 − λ ) , , , ,the latter exists for < λ < . Their characteristic equations are D σ + σ + λ − λ = 0 , D σ + σ + 1 − λ − λ = 0 .
10n the lower sheet there is an only equilibrium P − = (2(1 − λ ) , , , which exists for λ > with the characteristic equation D σ + σ + λ − λ − = 0 .The types of these equilibria depend on the value of D . These are as follows. For the uppersheet we have1. if < D < / , then O is an elliptic point;2. if D > / , then O is an elliptic point for < λ < λ , λ = 4 D / (4 D − , and O is asaddle-focus for λ > λ ; P + exists for < λ < , within this interval it is a saddle-center.For the lower sheet at P − P − exists for λ > , within this interval: if < D < / it is a saddle-focus for < λ < λ , λ = 1 + 1 / (1 − D ) , and is an elliptic point for > λ > λ ;2. if D > / , then P − is a saddle-focus for λ > . λ Negative values of the parameter λ also have physical sense. Let us first investigate the type ofthe equilibrium at O . Here we have another distribution of equilibria on sheets in comparisonwith λ > . On the upper sheet we have the only equilibrium S + = (2(1 − λ ) , , , and on the lowersheet the unique equilibrium is O = (0 , , , . Their types are as follows. The characteristicequation for S + is D σ + σ + (1 − λ ) / (2 − λ ) = 0 and for O is D σ + σ − (1 − λ ) /λ = 0 . For S + we get the following.1. If < D < / , then S + is an elliptic point for any λ <
2. if / < D < / , then S + is an elliptic point for λ + < λ < and is a saddle-focus pointfor −∞ < λ < λ + , λ + = − D − − D ;3. if D > / , then S + is a saddle-focus point for any λ < . For O we have1. If < D < / , then O is a saddle-focus for λ − < λ < , λ − = 4 D / (4 D − , and O isan elliptic point for −∞ < λ < λ − ;
2. if D > / , then O is a saddle-focus for any λ < .Suppose some solution of the upper system hits the branching plane u = − λ / at a finitevalue y of “time” y and its tangent vector at this point is directed outward, i.e. to the half-space11 < − λ / , that is, ˙ u ( y ) = u ( y ) < . To continue this solution by means of the lowersystem we apply to the point m + = ( − λ / , v ( y ) , u ( y ) , v ( y )) the involution m − = L ( − λ / , v ( y ) , u ( y ) , v ( y )) = ( − λ / , − v ( y ) , − u ( y ) , v ( y )) . At this symmetric point coordinates of ˙ u and ˙ v of the vector field change signs but coordinatesof ˙ v and ˙ u are the same. The lower vector field on the 3-plane u = − λ / coincides with theupper vector field. Now we proceed the orbit from the point m − using the lower vector field for y > y , this trajectory enters to the half-space u > − λ / . If its continuation reach again thebranching plane, we do the same using upper vector field. As was said above, the value of S does not change at these switchings but S (cid:48) and S (cid:48)(cid:48)(cid:48) do. In this way we can get cavitons beingnon-smooth homoclinic orbits which represent orbits joining stable and unstable separatrices ofthe saddle-center and crossing 3-plane u = − λ / under their journey.In what follows, we perform simulations with the smooth system (6). If some solution to thissystem stay all time in the half-space r > , then this solution corresponds to the upper sheetsystem. In particular, solitons correspond to homoclinic orbits of the equilibrium O ( λ, , , . Homoclinic orbits to O , which spend part time in the half-space r < , correspond to cavitons. The existence of homoclinic loops to a saddle-center is a rather delicate problem, since one needsto find the merge of one-dimensional stable and unstable manifolds of the saddle-center within 3-dimensional singular level of the Hamiltonian. This level is singular (it is not a smooth manifoldat any its point) because this level has a cone-type singularity at the equilibrium. Thus, sucha problem should be studied in a two-parameter unfolding generally. The task becomes easier,if one considers reversible Hamiltonian systems and searches for symmetric homoclinic orbits.Then generally an unfolding has to be one-parametric (in fact, this depends on the type of anaction of the reversible involution near a saddle-center, [39, 47]). If we investigate 2-parameterfamilies of reversible Hamiltonian systems, then one expects a possibility to construct curvesin the parameter plane along which the systems has homoclinic orbits to the related saddle-centers. Remind that existence of a saddle-center for Hamiltonian systems is a structurallystable phenomenon.Since we are of interest with spectra on parameter λ , for which the system has homoclinicorbits to the saddle-center, we recall the result proved first in [47]. There a general one-parameterunfolding of reversible Hamiltonian systems was studied under an assumption that it unfolds asystem with a homoclinic orbit to a saddle-center and for this orbit a genericity condition holdsfound first in [39]. Then it was proved that the set of parameter values which correspond tosystems with symmetric homoclinic orbits to the saddle-center (not obligatory 1-round ones)is self-limiting and self-similar: each point is an accumulation point for this set. It is worthnoting that our system can have both solitons and cavitons. On the mathematical language thiscorresponds to the case when both unstable separatrices of the saddle-center can merge with12table ones forming one or even two homoclinic loops. In case of one loop this can be impossible,if one deals with the case B as was discovered in [39] and was indicated in [47].The orbit structure of an analytic Hamiltonian system near a homoclinic orbit to a saddle-center was studied first in [39, 34] and then this was extended to different situations includingreversible systems [35, 47, 28, 29, 53, 27]. The study is based on the reduction of instead ofstudying the flow to the investigation of Poincar´e map and its orbit structure generated by theflow on some cross-section to a homoclinic orbit. This is heavily facilitated by the usage of alocal normal form near a saddle-center due to Moser [49]: there exists an analytic symplecticlocal coordinates ( x , y , x , y ) such that the Hamiltonian H in these coordinates casts in theform H = h ( ξ, η ) , where h is an analytic function in variables ξ = x y , η = ( x + y ) / , h = σξ + ωη + · · · , σω (cid:54) = 0 . Such normal form is integrable, local functions ξ, η , as functionsin ( x , y , x , y ) , are local integrals of the flow generated by Hamiltonian H . This easily allowsone to construct local map from a cross-section to stable separatrix to a cross-section to unstableseparatrix. This map has a singularity at the trace of of the stable separatrix but can beredefined to get a continuous map everywhere and analytic at all points except for the trace ofthe separatrix. Orbits of the system correspond to orbits of Poincar´e map. so its studying givesa complete information concerning orbit behavior of the flow. Principal elements of this picturewere found in [39, 47, 28, 29]. In particular, suppose a homoclinic orbit to the saddle-center existand some genericity condition holds for it, then each Lyapunov periodic orbit possesses within itslevel of H four transverse homoclinic Poincar´e orbits [39] implying the existence of complicated(chaotic) dynamics nearby [52, 51].Separatrices of the saddle-center are orbits (different from the equilibrium itself) on twoinvariant analytic curves through the equilibrium, in Moser coordinates they are x = x = y = 0 and y = x = y = 0 (strong stable W s and strong unstable W u local manifolds). A two-dimensional center manifold near the saddle-center is given as x = y = 0 , it is filled withLyapunov saddle periodic orbits lying each in its own level of the Hamiltonian. These periodicorbits are saddle ones in the related level of H .The continuation of an unstable separatrix by the flow within the singular level can lead toits merge with one of two stable separatrices forming a homoclinic orbit to the saddle-center.The local orbit structure of the flow near such orbit is rather well known since [39] (see also[35, 28, 47, 28, 29, 53]). The orbit behavior depends essentially on the case which is realized oftwo possible ones here [39]. To explain this, let us remind the local structure of the Hamiltoniannear a saddle-center (see, Fig. 6-8). We present here only related pictures (see details in [41]).On these pictures it is seen the local behavior of orbits, as well.The cases mentioned depend on how the homoclinic orbit connects cutting disks of two solidcylinders: the orbit can connect disks from the same solid cylinder (case 1) or two different ones(case 2). As simulations show that we deal with the case 2 for the system under study. Thus, twoseparatrices going to the half-space r > may form homoclinic orbits corresponding to solitons,and two remaining going to the half-space r < may form homoclinic orbits corresponding tocavitons. Related orbits have been found, as an example, they are plotted in Fig.9a,b. They13igure 6: c = 0 , observe two glued points – saddle-centerFigure 7: c < represent both solitons and solitons.If the genericity condition mentioned above holds for a homoclinic orbit of the saddle-center,then in the level H = 0 containing the equilibrium and the homoclinic orbit there exist alsocountably many saddle long periodic orbits accumulating at the homoclinic orbit to the saddle-center [39, 35]. As an example, such periodic orbit is shown on Fig. 17 but its fact there aremany of them. For the system we study all this picture takes place at fixed values of parameters D ∗ , λ ∗ for which a homoclinic orbit to a saddle-center exists.When varying parameters D, λ , the orbit structure of the flow varies. In particular, a homo-clinic orbit to the saddle-center generically fails to exist (it is destroyed). Instead, multi-roundhomoclinic orbits to O can arise [36]. Because saddle periodic orbits accumulate to the formerhomoclinic loop, a situation may occur, when an orbit on an one-dimensional unstable manifoldof the saddle-center (which persists under small changes D, λ ) gets lie on the stable manifold ofsome saddle periodic orbit γ in the same level of H . Since the system under consideration is,in addition, reversible, and if saddle-center O and saddle periodic orbit γ are symmetric, thenpairing orbit of the stable manifold of the saddle-center gets lie by symmetry on the unstablemanifold of γ . Thus, in this case a heteroclinic connection is made up of two heteroclinic orbits,a symmetric saddle-center and a symmetric saddle periodic orbit γ .Such heteroclinic connection can be of two types in dependence of how two heteroclinic orbitsare displaced with respect to two local solid cylinders H = H ( p ) , described above. Namely, theycan either intersect both the same cylinder (case 1) or one heteroclinic orbit intersects one14igure 8: c > a) b)Figure 9: (a) 1-round homoclinic, λ = 0 . , D = 0 . ; (b) Unfolding of this 1-round soliton. cylinder, but another one does another cylinder (case 2). Our simulations show that we dealhere with the case 2. This implies that orbits leaving the unstable manifold of γ can return to lieon its stable manifold only making at least one passage near two remaining stable and unstablemanifolds of the saddle-center. For our case this means that such orbits have to intersect theplane r = 0 before they return to the stable manifold of γ . Existence of a heteroclinic connectionof the type indicated is shown in the Fig.10. Studying an orbit behavior near this connectionwas performed recently [40].Another feature of the system under varying parameters ( D, λ ) is the appearance of newhomoclinic orbits to the saddle-center. Due to reversibility of the systems and the type of actionof the reversor L locally (the intersection of the fixed point set of L with the singular level H = 0 is the curve through p ) both homoclinic orbits (solitons) and cavitons are usually symmetricorbits (i.e. invariant w.r.t. L ). So, at a fixed D , λ , < λ < , only one symmetric soliton canexist and only one symmetric caviton. Under varying parameters these homoclinic orbits usuallyare destroyed, but can exist multi-round homoclinic orbits which before closing make severalexcursions near the former 1-round homoclinic orbits. Moreover, for a reversible system in theplane of parameters ( D , λ ) there are usually countably many bifurcation curves accumulatingto the curve of 1-round homoclinic orbits [47, 29]. Our calculations show just this behavior, see,Figs 12-16. 15igure 10: Heteroclinic connection in the singular level of H .One more situation that can arise under varying parameters ( D , λ ) , is the existence of non-symmetric homoclinic orbits. By symmetry, if such an orbit exists, there is another homoclinicorbit being the symmetric counterpart of the former. The existence of such nonsymmetric ho-moclinic orbit requires the 2-parameter analysis, they exist at selected points (see Fig.11) a,b.a) b)Figure 11: (a) Nonsymmetric soliton at ( λ = 0 . , D = 0 . ); (b) Its unfolding. − λ Let us now study the problem for small positive − λ near the point (0 , , , on the uppersheet. As was said above, this equilibrium is degenerate at λ = 1 with double zero eigenvalueand two imaginary eigenvalues ± iω. Let us scale the initial equation (5): λ = 1 − ε , τ = εy, ddy = ε ddτ , S = ε X, D = κε .
16) b)Figure 12: (a) Simplest caviton; (b) Its unfolding. a) b)Figure 13: (a) 2-round homoclinic orbit, λ = 0 . , D = 0 . ; (b) Its unfolding – 2-hump soliton. a) b)Figure 14: (a) 2-round homoclinic orbit with sharpening, λ = 0 . , D = 0 . ; (b) Its unfolding – 2-humpcaviton.
17) b)Figure 15: (a) 3-round homoclinic orbit, λ = 0 . , D = 0 . ; (b) Its unfolding – 3-hump soliton. a) b)Figure 16: (a) 3-round homoclinic orbit with sharpening, λ = 0 . , D = 0 . ; (b) Its unfolding – 3-humpcaviton. As a result of these transformations we come to the following equation κ d Xdτ + d Xdτ − X + 12 X − ε X (1 − X + 12 X ) + ε ( − X + 3 X − X + 58 X ) + · · · , (15)that defines the behavior of solutions as λ → − . As above, let us reduce the equation to theHamiltonian system by means of the change of variables u = X, u = X (cid:48) , v = − X (cid:48) − κ X (cid:48)(cid:48)(cid:48) ,v = κX (cid:48)(cid:48) . The equation is reduced to the slow-fast Hamiltonian system with respect to thesymplectic form dv ∧ du + κdv ∧ du u (cid:48) = u , v (cid:48) = − u + u − ε u (1 − u + u ) + ε ( − u + 3 u − u + u ) + · · · ,κu (cid:48) = v , κv (cid:48) = − ( u + v ) . (16)The Hamiltonian of the system is H = u v + 12 ( u + v ) + 1 + O ( ε )2 u − O ( ε )6 u + ε (1 + O ( ε ))8 u − ε (1 + O ( ε ))8 u + · · · . H .In this form we get a problem about an orbit behavior near a ghost separatrix on an almostinvariant elliptic manifold where a saddle equilibrium with a homoclinic orbit exists. Suchproblem was studied partially in [22].The system (16) is also reversible with respect to the involution L : ( u , v , u , v ) → ( u , − v , − u , v ) with its fixed point set F ix ( L ) = { v = 0 , u = 0 . } In order to find a homoclinic orbit to the saddle-center it is important to keep in mind thatwe have a reversible slow-fast Hamiltonian system whose fast system is a fast rotation. Indeed,to get the fast system, we do the scaling of the independent variable τ /κ = ξ , then the smallmultiplier κ appears in the right hand sides of the first and second differential equations. Then,setting κ = 0 we get u , v as parameters of the system of two remaining equations. They arelinear and have an equilibrium – center – on any leaf u = u , v = v . Thus, all assumptionsof the theorem 1 from [21] hold and therefore there is a neighborhood U of a compact region inthe slow plane u = − v , v = 0 , where analytic Hamiltonian by an analytic symplectic changeof variables is transformed to the function H = H ( I, u, v, κ ) + R ( x, y, u, v, κ ) , I = ( x + y ) / , | R | = O ( ε [ − c/κ ]) . Thus, in U the Hamiltonian is exponentially close to an integrable Hamiltonian H with I being an additional integral. In particular, this theorem works for a region U whichcontains on the slow plane the separatrix loop of the saddle. Also, in this case a theorem from[22] holds which asserts the validity of the Moser normal form [49] for H in some neighborhoodof saddle-center of the size O ( Cε ) . These two theorems allows two prove the following theorembeing an analog of the theorem 1 from [14]
Theorem 1
For a small positive κ in the whole phase space a neighborhood of the order κ ofthe homoclinic orbit on the slow manifold exist such that two branches of stable and unstableseparatrices of the saddle-center which cut the cross-section x = 0 first time are displaced onthe distance of the order R exp[ − c/κ ] with some positive constants R, c.
For ε = 0 the equation above is the well studied, it also models the form of stationary water19aves on the surface of a liquid with the surface tension [10, 30, 14], if we eliminate terms ofthe order ε and higher. It was proved for small κ [14] this equation to have not localizedsolutions, or, in other terms, no homoclinic solutions to the corresponding saddle-center existfor the related slow-fast Hamiltonian system. Nevertheless, our simulations have shown theexistence of homoclinic orbits under varying ε, κ (see, as a hint, Fig.5b). More exactly, thefollowing hypothesis seems to be valid Hypothesis . There is a neighborhood of the point (0 , on the parameter plane ( κ, ε ) suchthat a countable set of bifurcation curve exists which correspond to the existence of homoclinicorbits of any roundness. The calculations of equilibria and their types show, in particular, that if D > / , then forpositive λ > λ the equilibrium O on the upper sheet is the saddle-focus. The simulationsdiscovered the abundance of symmetric homoclinic orbits to this equilibrium (see Fig.18-21).These homoclinic orbits are usually transverse in the following sense. The related singular3-dimensional level of the Hamiltonian (containing the saddle-focus) includes both smooth 2-dimensional stable and unstable manifolds of the saddle-focus and their intersection along thehomoclinic orbit is transverse within this level. The theory of the complicated orbit behavior neara saddle-focus loop was developed by Shilnikov [50] for general systems and later adapted [18]to cover the case of Hamiltonian systems (see also [19, 24, 17] where some elements of complexdynamics were proved for reversible systems. An overview of these results can be found in [25]).It says that near a transverse homoclinic orbit there exists multi-pulse homoclinic orbits and acomplicated behavior of nearby orbits (hyperbolic subsets) [18]. Moreover, varying levels of theHamiltonian leads to many bifurcations of hyperbolic sets, creations of elliptic periodic orbits,etc. [42, 43]. Also, for the system under study there are those symmetric homoclinic orbits whichintersect during their travel the branching plane u = − λ / . Such homoclinic orbits can alsobe named cavitons with oscillating asymptotics at infinity. Near them multi-pulse cavitons alsoexist as well as a complicated orbit structure.Let us prove the existence of two symmetric homoclinic orbits to the equilibrium O on theupper sheet for small enough λ − λ . To do this, we use results of studying the HamiltonianHopf bifurcation [46] and their realization for the Swift-Hohenberg equation [23, 26]. Recall thatthe Hamiltonian Hopf bifurcation is the bifurcation in an one-parameter family of Hamiltoniansystems in two degrees of freedom having equilibria for all values of a parameter and at somecritical value of a parameter the related equilibrium has two double pure imaginary eigenvalueseach with the 2-dimensional Jordan box (non-semisimple case). The type of bifurcation thatoccurs under transition through this critical value of the parameter depends on the sign of somecoefficient in the normal form of the Hamiltoniian near this equilibrium. In particular, if thiscoefficient is positive, then for those values of the parameter, when the equilibrium is a saddle-focus, the system, if it is, in addition, reversible, the saddle-focus gives the birth of two small20) b)Figure 18: (a) Saddle-focus homoclinics, D = 0 . , λ = − ; (b) and its unfolding. a) b)Figure 19: (a) Saddle-focus homoclinics, D = 0 . , λ = − , symmetric homoclinic orbits. The reversibility here guarantees their existence, otherwise, to findsuch orbits is a very delicate problem related (for analytic systems) with exponentially smallsplitting of stable and unstable manifolds of the saddle-focus [20].We prove the result reducing our problem to that being similar to the problem as for theSwift-Hohenberg equation. To that end, let us scale the traveling coordinate y = γξ , γ = √ D, in the initial equation (4). After scaling and dividing at D we get the equation u ( IV ) + 2 u (cid:48)(cid:48) + u = (1 − D + 4 D λ ) u − D λ u + 2 D λ u + · · · . In notations of [23] we get α = 1 − D + D λ , β = − D λ . The change u → − u allows one to make β positive as in [23]. Thus we get the criterion of the birth of homoclinic orbit when crossing21) b)Figure 20: (a) Saddle-focus homoclinics, D = 0 . , λ = 3 . , and (b) its unfolding. a) b)Figure 21: (a) Saddle-focus homoclinics, D = 0 . , λ = 3 . , α = 0 , i.e. we just get λ = λ . Nevertheless, the equation differs from [23], since coefficientat the term with u is positive. So, we need to calculate the needed coefficient in the normalform directly. We perform this calculation using the averaging. This was done long ago [44] butunpublished.We calculate the coefficient, we remark that saddle-foci appear as λ > λ as D > / . Denote − ν = 1 − D + D λ and consider ν as small positive parameter. After scaling u = − κu with κ = √ D/λ / we come to the equation of the form (we preserve old notations) u ( IV ) + 2 u (cid:48)(cid:48) + u = − νu + βu + u + · · · . (17)22et us write the equation in the form of two second order equations u (cid:48)(cid:48) + u = v, v (cid:48)(cid:48) + v = − νu + βu + u + · · · . After scaling u → √ ν, v → νv and denoting µ = √ ν , we get the system u (cid:48)(cid:48) + u = µv, v (cid:48)(cid:48) + v = βu − µ ( u − u ) + O ( µ ) . At µ = 0 we have the system whose solutions are of the form u = A exp[ iξ ] + ¯ A exp[ − iξ ] , v = B exp[ iξ ] + ¯ B exp[ − iξ ] − β A exp[2 iξ ] + ¯ A exp[ − iξ ]) + 2 β | A | . We add here new variables u (cid:48) = p, v (cid:48) = q and differentiation of above equalities under anassumption that A, B are constant gives the relations for p, q through A, ¯ A, B, ¯ B . We considerthese relations as the change of variables ( u, v, p, q ) → ( A, ¯ A, B, ¯ B ) . Observe that this change ofvariables depend π -periodically in ξ. Performing this change of variables, we come to the system of four first order differentialequations in variables ( A, ¯ A, B, ¯ B ) which is the π -periodic system in the so-called standardform of the averaging method (see, [15]) X (cid:48) = µF ( X, ξ ) . Averaging this system in ξ gives theaverage system Y (cid:48) = µF ( Y ) , F ( Y ) = 12 π π (cid:90) F ( Y, ξ ) dξ. For our case we have A (cid:48) = − iµ B , B (cid:48) = iµ A −
27 + 2 β | A | ] , ¯ A (cid:48) = c.c., ¯ B (cid:48) = c.c. (18)The coefficient we sought for is β . It is positive that means the existence of the homoclinicskirt in the system (18) which is integrable and the existence of two symmetric homoclinic orbitsin the initial system due to its reversibility [33]. The structure of the averaged system is easilyrestored if introduce real variables ( a, b, c, d ) , A = a + ib, B = c + id. In these variables we havea Hamiltonian system a (cid:48) = c, c (cid:48) = a − L ( a + b )) , b (cid:48) = d, d (cid:48) = b − L ( a + b )) , L = 27 + 2 β with Hamiltonian H = c + d − a + b L
16 ( a + b ) and an additional integral K = ad − bc . The common level H = K = 0 gives the homoclinicskirt, i.e. one-parameter family of homoclinic orbits to the equilibrium O of a saddle type withmerged 2-dimensional stable and unstable manifolds.In the similar way one can check that zero equilibrium O on the lower sheet for negative λ also gives the birth of homoclinic orbits to a saddle-focus as < D < / at λ = λ − . As we have seen above, saddle-foci in the system exist both for negative λ and for positive λ. Results of our simulations in these cases are plotted in Fig. 18,19 for negative λ . For positive λ we get the following plots, Fig. 20,21. 23 Conclusion
In this work we have studied localized traveling wave solutions of the nonlocal Whitham equationby means of the reduction to a Hamiltonian system. This initial equation is of the fourth orderwith a nonlinearity being double-valued. The reduction allows to derive a two degrees of freedomHamiltonian system but it defined on the two-sheeted space due to the type of nonlinearity. Inaddition, the system is reversible with respect to some involution. This permitted to obtain aclear geometric description of both smooth solutions and solutions with singularities and applyto the problem of developed methods of the theory of Hamiltonian dynamics, in particular,theory of homo- and heteroclinic orbits. The search for homolinic and heteroclinic orbits indynamical systems is a very nontrivial problem, being global in its own nature. Therefore,numerical methods with the sharp set up allow to solve this problem for the concrete equationlike that under study. But the numerical search can be made much more rigorous if we have somepoints in the parameter space (our ( D , λ ) ) at which the system has degenerate equilibria. Thenbifurcation methods allows one to find homoclinic orbits through the bifurcation. We do thisusing Hamiltonian Hopf bifurcation and calculation the needed coefficients in the local normalform to determine the type of the bifurcation. All this together allowed one to investigate thesystem with many details.
10 Acknowledgement
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