Travelling solitons in the externally driven nonlinear Schrödinger equation
aa r X i v : . [ n li n . PS ] J u l Travelling solitons in the externally driven nonlinear Schr¨odinger equation
I. V. Barashenkov ∗ Department of Mathematics, University of Cape Town, Rondebosch 7701; andNational Institute for Theoretical Physics, Stellenbosch, South Africa
E. V. Zemlyanaya † Joint Institute for Nuclear Research, Dubna, 141980 Russia (Dated: September 18, 2018)We consider the undamped nonlinear Schr¨odinger equation driven by a periodic external force.Classes of travelling solitons and multisoliton complexes are obtained by the numerical continuationin the parameter space. Two previously known stationary solitons and two newly found localisedsolutions are used as the starting points for the continuation.We show that there are two families of stable solitons: one family is stable for sufficiently lowvelocities while solitons from the second family stabilise when travelling faster than a certain criticalspeed. The stable solitons of the former family can also form stably travelling bound states.
PACS numbers: 05.45.Yv
I. INTRODUCTION
The damped nonlinear Schr¨odinger equation driven bya time-periodic external force, iu t + u xx + 2 | u | u + δu = ae i Ω t − iβu, (1a)and its parametrically driven counterpart model twofundamental energy supply mechanisms in a nearly-conservative spatially distributed system. While the un-perturbed Schr¨odinger is an archetypal equation for theslowly varying envelope of a group of dispersive waves,the damped-driven equations arise whenever the resonantforcing of small amplitude is used to compensate weakdissipative losses.The simplest (and perhaps the most visually appeal-ing) realisation of Eq.(1a) is that of the amplitude equa-tion for a strongly coupled pendulum array with the hori-zontal sinusoidal driving [1], taken in its continuum limit.Here a and Ω are the driving strength and driving fre-quency, respectively; δ is the detuning of the driving fre-quency from the continuum of linear waves in the array,and β is the damping coefficient.The array of torsionally coupled pendula can serve asa prototype model for the whole variety of systems incondensed matter physics. Accordingly, Eq.(1a) was em-ployed to study systems as diverse as the ac-driven longJosephson junctions [2] and charge-density-wave conduc-tors with external electric field [3]; double-layer quantumHall (pseudo)ferromagnets [4] and easy-axis ferromagnetsin a rotating magnetic field [5]. Eq.(1a) arises in the the-ory of rf-driven waves in plasma [6, 7] and shear flowsin nematic liquid crystals [8]; the same equation governsthe amplitude of the slowly varying π -mode in the forcedFermi-Pasta-Ulam lattice [9]. ∗ Email: [email protected] † Email: [email protected]
A closely related equation is the one with the spatially periodic forcing, iu t + u xx + 2 | u | u + δu = ae iKx − iβu, (1b)and, more generally, the one driven by the harmonic wave[10–12]: iu t + u xx + 2 | u | u + δu = ae i ( Kx +Ω t ) − iβu. (1c)A discrete version of Eq.(1b) describes an array ofcoupled-waveguide resonators excited by a driving field[13] whereas Eq.(1c) models pulse propagation in anasymmetric twin-core optical fiber [10].Equation (1c) includes (1a) and (1b) as particularcases. The transformation u ( x, t ) = Ψ( X, t ) e i ( Kx +Ω t ) , X = x − Kt takes (1c) to i Ψ t + Ψ XX + 2 | Ψ | Ψ − κ Ψ = a − iβ Ψ , (2)with κ = K + Ω − δ . The equation in this form has ahistory of applications of its own — in particular, in thephysics of optical cavities. Originally, it was introducedas the Lugiato - Levefer model [14] of the diffractive cav-ity driven by a plane-wave stationary beam. Later it wasemployed to describe a synchronously pumped ring laserwith a nonlinear dispersive fiber [15, 16]. More recentlythe same equation was shown to govern the envelopes ofshort baroclinic Rossby waves in the two-layer model ofthe atmosphere, or the ocean [17].Equation (2) has undergone an extensive mathematicalanalysis. Topics covered included existence [18, 20, 21],stability [20, 22] and bifurcation [7, 19] of nonpropagat-ing solitons and their bound states [16, 23–25]; statisticalmechanics of soliton creation and annihilation [26]; soli-ton autoresonance phenomena [12, 27]; regular [28] andchaotic [29] attractors on finite spatial intervals. Hereand below we use the word “soliton” simply as a syn-onym for “localised travelling wave”.The recent paper [30] studied solitons of the undamped( β = 0) equation (2) travelling with constant or oscillat-ing velocities. Summarising results of their direct numer-ical simulations of Eq.(2), the authors formulated an em-pirical stability criterion of the soliton against small andlarge perturbations. So far, this criterion has not beengiven any mathematical proof or physical justification.Despite being tested on a variety of initial conditions, itstill has the status of conjecture.In order to verify the validity of the empirical stabil-ity criterion at least for infinitesimal perturbations, oneneeds to have the travelling soliton existence and lin-earised stability domains accurately demarcated. Theclassification of bifurcations occurring when stability islost would also be a useful step towards the justificationof the criterion. This is what we shall concern ourselveswith in this paper.Here, we study travelling solitons of Eq.(2) by path-following them in the parameter space. One advantageof this approach over simulations is that it furnishes all soliton solutions moving with a given velocity — all sta-ble and all unstable. This, in turn, allows one to under-stand the actual mechanisms and details of the solitontransformations.The outline of this paper is as follows. In the nextsection, we give a brief classification of space- and time-independent solutions of Eq.(2) which may serve as thebackgrounds for the solitons. In particular, we show thatthere is only one stable background and determine thevalue of the limit speed of the soliton propagating overit. In section III we describe insights one can draw fromthe analysis of the eigenvalues of the symplectic linearisedoperator and its hermitian counterpart. These pertain tothe stability and bifurcation of the solitons.In section IV we present four nonpropagating directlydriven solitons. Two of these have already been availablein literature while the other two have not been knownbefore. In sections V and VI, we report on the contin-uation of these stationary solitons to nonzero velocities.Our results on the existence and stability of the trav-elling solitons and their complexes, are summarised insection VII. In particular, Fig.8 gives a chart of “stable”velocities for each value of the driving strength. II. FLAT SOLUTIONS
Assuming that κ > t ′ = κ t , x ′ = κX ,and Ψ = κψ , equation (2) becomes iψ t ′ + ψ x ′ x ′ + 2 | ψ | ψ − ψ = − h − iγψ, where h = − a/κ , γ = β/κ . (In what follows, we omitprimes above x and t for notational convenience.)In this paper we study the above equation with zerodamping: γ = 0. Without loss of generality we can as-sume that h >
0. Since we shall be concerned with soli- tons travelling at nonzero velocities, it is convenient totransform the equation to a co-moving frame: iψ t − iV ψ ξ + ψ ξξ + 2 | ψ | ψ − ψ = − h, (3)where ξ = x − V t .Flat solutions are roots of the cubic equation2 | ψ | ψ − ψ = − h ; (4)these have been classified in [20]. If 0 < h < (2 / / ,there are 3 roots, of which two ( ψ and ψ ) are positive,and one ( ψ ) is negative. Here ψ < < ψ < < ψ < . If h > (2 / / , there is only one (negative) solution ψ , with ψ > .Let ψ denote a root of equation (4) — one of the threeroots ψ , ψ and ψ . The value ψ does not depend on V : the flat solution has the same form in any frame ofreference. However the spectrum of small perturbationsof the flat solution does include a dependence on V . Let-ting ψ = ψ + [ u ( ξ ) + iv ( ξ ))] e λt in (3), linearising in u and v , and, finally, taking u, v ∝ e ikξ , we obtain( λ − ikV ) = − ( k + a )( k + b ) , (5)where we have introduced a = q − ψ , b = q − ψ . (6)To determine whether ψ can serve as a backgroundto a stationary localised solution of (3), consider a time-independent perturbation — that is, set λ = 0: k V = ( k + a )( k + b ) . (7)The only flat solution that is a priori unsuitable as abackground for localised solutions is such ψ whose as-sociated quadratic equation (7) has two nonnegative realroots, ( k ) ≥ k ) ≥ ψ has two nonnegative roots for any choice of h and V .This disqualifies ψ as a possible soliton background. Wealso conclude that travelling solitons may not exist for h greater than (2 / / .Next, if V ≤ c , where c = a + b, (8)the smaller positive solution ψ will have either two com-plex or two negative roots ( k ) , , whereas for velocitiesgreater than c , both roots are nonnegative. Hence the ψ solution can serve as a background only for V ≤ c .When V < b − a , the decay to the background is mono-tonic (both roots are negative), while when V > b − a ,the decay is by ondulation (the roots are complex). Thisflat solution admits a simple explicit expression: ψ = r
23 cos (cid:18) α − π (cid:19) , where α = arccos − r h ! , π ≤ α ≤ π. Finally, the larger positive solution ψ has two realroots of opposite signs (for all V and 0 ≤ h ≤ (2 / / ).This flat solution may also serve as a soliton background.Next, one can readily check that a flat solution ψ isstable if ψ < . Therefore, even if there are solitonsasymptotic to the flat solution ψ as x → ∞ or x → −∞ ,these will be of little physical interest as the background ψ is always unstable.In summary, only the small positive flat solution (theone with ψ < ) is stable. It may serve as a backgroundfor solitons only if V < c ; that is, the soliton propagationspeed is limited by c .The inequality V ≤ c limiting the soliton propagationspeed, has a simple physical interpretation. Indeed, onecan easily check that c gives the lower bound for the phasevelocity of radiation waves [in the original ( x, t ) referenceframe]. Therefore, a soliton travelling faster than c wouldbe exciting resonant radiation. This is inconsistent withthe asymptotic behaviour ψ x → | x | → ∞ ; neithercould it be reconciled with the energy conservation. III. INSIGHTS FROM LINEARISATION
Travelling wave solutions depend on x and t only incombination ξ = x − V t . For these, the partial differentialequation (3) reduces to an ordinary differential equation − iV ψ ξ + ψ ξξ + 2 | ψ | ψ − ψ = − h. (9)It is this equation that we will be solving numerically inthe following sections.Let ψ s ( ξ ) be a localised solution of (9). In order torepresent results of continuation graphically, we will needto characterise the function ψ s ( ξ ) by a single value. Aconvenient choice for such a bifurcation measure is themomentum integral P = i Z ( ψ ∗ ξ ψ − ψ ξ ψ ∗ ) dξ. (10)One advantage of this choice is that the momentum is anintegral of motion for equation (3); hence P is a physi-cally meaningful characteristic of solutions. Another use-ful property of the momentum is that in some cases itsextrema mark the change of the soliton stability proper-ties (see below). A. The hermitian and symplectic operator
Many aspects of the soliton’s bifurcation diagram canbe explained simply by the behaviour of the eigenvaluesof the operator of linearisation about the travelling-wave solution in question. Therefore, before proceeding to thenumerical continuation of travelling waves, we introducethe linearised operator and discuss some of its properties.Consider a perturbation of the solution of Eq.(9) ofthe form ψ = ψ s + [ u ( ξ ) + iv ( ξ )] e λt , with small u and v .Substituting ψ in Eq. (3) and linearising in u and v , weget a symplectic eigenvalue problem H ~y = λJ~y. (11)Here ~y is a two-component vector-function ~y ( ξ ) = (cid:18) uv (cid:19) , and H is a hermitian differential operator acting on suchfunctions: H = (cid:18) − ∂ ξ + 1 − R + I ) − V ∂ ξ − RI V ∂ ξ − RI − ∂ ξ + 1 − I + R ) (cid:19) , with R and I denoting the real and imaginary part ofthe solution ψ s ( ξ ): ψ s = R + i I . Finally, J is a constantskew-symmetric matrix J = (cid:18) −
11 0 (cid:19) . Assume that ψ s ( ξ ) is a localised solution decaying to ψ as x → ±∞ , where ψ < . The continuous spec-trum of the hermitian operator H occupies the posi-tive real axis with a gap separating it from the origin: E ≥ E >
0. Discrete eigenvalues E n satisfy E n < E .On the other hand, the continuous spectrum of the sym-plectic eigenvalues (that is, the continuous spectrum ofthe operator J − H ) occupies the imaginary axis of λ out-side the gap ( − iω , iω ). The gap width here is given by ω = q ( k + a )( k + b ) − V k > , (12)where k is the positive root of the bicubic equation V ( k + a )( k + b ) = k (2 k + a + b ) . Discrete eigenvalues of the operator J − H may includepairs of opposite real values λ = ± ρ ; pure imaginarypairs λ = ± iω , with 0 ≤ ω ≤ ω ; and, finally, complexquadruplets λ = ± ρ ± iω .We routinely evaluate the spectrum of symplecticeigenvalues as we continue localised solutions in V . Ifthere is at least one eigenvalue λ with Re λ >
0, the solu-tion ψ s is considered linearly unstable. Otherwise (thatis, if all eigenvalues have Re λ ≤ B. Zero eigenvalues
While the eigenvalues of the operator J − H (thatis, the eigenvalues of the symplectic eigenvalue problem(11)) determine stability or instability of the solution ψ s ,the eigenvalues of the operator H are significant for thecontinuability of this solution. Of particular importanceare its zero eigenvalues.At a generic point V , the operator H has only onezero eigenvalue, with the translational eigenvector ~ Ψ ξ ≡ ( R ξ , I ξ ). This is due to the fact that the stationary equa-tion (9) has only one continuous symmetry. For a given V , the solution ψ s ( ξ ) is a member of a one -parameterfamily of solutions ψ s ( ξ − θ ), where θ is an arbitrarytranslation.On the other hand, the nonhermitian operator J − H has two zero eigenvalues at a generic point. The reasonis that the equation (3) as well as its linearisation, arehamiltonian systems. Real and imaginary eigenvalues ofoperators which generate hamiltonian flows always comein pairs: If µ is an eigenvalue, so is − µ [33]. The twozero eigenvalues of the operator J − H reflect the factthat the function ψ s ( ξ ), considered as a solution of thepartial differential equation (3), is a member of a two-parameter family. One parameter is the translation; theother one is the velocity V .For generic V , the repeated zero eigenvalue of J − H is defective: there is only one eigenvector ~ Ψ ξ associatedwith it. There is also a generalised eigenvector ~ Ψ V , where ~ Ψ V ≡ (cid:18) ∂ R ∂V , ∂ I ∂V (cid:19) . This vector-function is not an eigenvector of J − H ; in-stead, differentiating (9) in V one checks that ~z = ~ Ψ V satisfies the nonhomogeneous equation H ~z = − J ~ Ψ ξ . (13)[That is, ~ Ψ V is an eigenvector of the square of the sym-plectic operator: ( J − H ) ~ Ψ V = 0.]As we continue in V , a pair of opposite pure-imaginarysymplectic eigenvalues may collide at the origin on the λ -plane and cross to the positive and negative real axis,respectively. The algebraic multiplicity of the eigenvalue λ = 0 increases from 2 to 4 at the point V = V c ; howeverif the hermitian operator H does not acquire the secondeigenvalue E = 0 at this point, the geometric multiplicityremains equal to 1. The change of stability of the solitonsolution does not affect its continuability, i.e. the solitonexists on either side of V = V c . In this case we have dP/dV = 0 at the point where the stability changes [32].The continuation may be obstructed only when an-other (the second) eigenvalue of the operator H crossesthrough zero at V = V c : H ~ Φ = 0. If the correspondingeigenvector ~ Φ is not orthogonal to the vector-function
J ~ Ψ ξ in the right-hand side of equation (13), its solution ~z = ~ Ψ V will not be bounded. This implies a saddle-nodebifurcation; the soliton solution ψ s cannot be continuedbeyond V = V c . Note that although ~ Φ is an eigenvectorof the symplectic operator J − H , the algebraic multiplic-ity of the symplectic eigenvalue remains equal to 2 in thiscase. Assume now that the eigenvector ~ Φ is orthogonal to J ~ Ψ ξ . This may happen if the soliton solution ψ s of equa-tion (9) with V = V c is a member of a two -parameterfamily of solutions ψ s = ψ s ( ξ − θ ; χ ), with χ equal tosome χ . Here we assume that each member of the fam-ily ψ s ( ξ − θ ; χ ) is a solution of Eq.(9) — with the same V = V c . Then ~ Φ is given by ~ Ψ χ ≡ ∂ ~ Ψ /∂χ (cid:12)(cid:12)(cid:12) χ = χ . If χ isa root of the equation F ( χ ) = 0 , (14a)where F ( χ ) ≡ Z ( ~ Ψ χ , J ~ Ψ ξ ) dξ, (14b)the vectors ~ Ψ χ and J ~ Ψ ξ will be orthogonal which, in turn,will imply that a bounded solution ~ Ψ V of the equation(13) exists. [In Eq.(14b) ( , ) stands for the R scalarproduct: ( ~a,~b ) ≡ a b + a b .] In this case the value V is not a turning point; the soliton solution ψ s exists onboth sides of V = V . The algebraic multiplicity of thezero symplectic eigenvalue increases at the point V = V c .In fact from the hamiltonian property it follows that itincreases up to 4 (rather than 3).Recalling the definition of the momentum integral (10)and writing it in terms of the real and imaginary part of ψ s , equation (14) becomes simply ∂P∂χ (cid:12)(cid:12)(cid:12)(cid:12) χ = χ = 0 . This condition ensures that a two-parameter family ofsolutions ψ s ( x − θ ; χ ), existing at the velocity V = V ,has a one-parameter subfamily ψ s ( x − θ ; χ ) continuableto V = V c [32]. IV. NON-PROPAGATING SOLITONSA. Simple solitons
The ordinary differential equation (9) with V = 0, ψ xx + 2 | ψ | ψ − ψ = − h, (15)has two real-valued localised solutions, ψ + and ψ − .These are given by explicit formulas [22]: ψ ± ( x ) = ψ (cid:20) β ± cosh β cosh( Ax ) (cid:21) , (16)where the parameter β (0 ≤ β < ∞ ) is in one-to-onecorrespondence with the driving strength h : h = √ β (1 + 2 cosh β ) / . −20 −10 0 10 20−1−0.500.51 xh=0.2 V=0 ψ + ψ − FIG. 1. Stationary ψ + and ψ − solitons As h increases from 0 to p / ≈ . β decreasesfrom infinity to zero. (Hence 0 ≤ h ≤ . ψ and inverse width A are also expressible through β : ψ = 1 √ p β , A = √ β p β . (Note that the asymptotic value ψ corresponds to thestable background, denoted ψ in Sec.II.)The stationary soliton ψ + has a positive eigenvalue inthe spectrum of the linearised operator (11); hence the ψ + is unstable for all h for which it exists [22]. Thespectrum of the stationary soliton ψ − with small h in-cludes two discrete eigenvalues λ , = iω , , ω , > h growsto 0.07749, λ and λ approach each other, collide andacquire real parts of the opposite sign. This is a hamil-tonian Hopf bifurcation. For h > . ψ − is prone to the oscillatory instability [22].When a damping term is added to the equation, thetwo stationary solitons ψ + and ψ − persist and can forma variety of multisoliton bound states, or complexes[16, 23–25]. In the next subsection, we show that un-damped directly driven solitons can also form stationarycomplexes. Some of these complexes are bound so tightlythat the solution represents a single entity. To distin-guish these objects from the solitons ψ + and ψ − , we willbe referring to the ψ + and ψ − as the simple solitons. B. The twist solitons
In addition to the two simple solitons expressible inelementary functions, the stationary equation (15) hastwo localised solutions that cannot be constructed an-alytically. Unless h is extremely small, each of these two solutions has the form of a single entity [Fig.2(a,b)]— a soliton whose phase does not stay constant butgrows, monotonically, as x changes from large negativeto large positive values. When visualised in the three-dimensional ( x, Re ψ, Im ψ )-space, it looks like a twistedribbon (twisted by 360 ◦ ); hence we will be calling thesetwo solutions simply “twists”. For the reason that willbecome obvious in the paragraph following the next one,we denote the two solutions ψ T and ψ T , respectively.The twist solitons were previously encountered in theparametrically driven (undamped) nonlinear Schr¨odingerequation [32]. For each h , the parametrically driven twistis a member of a two-parameter family of stationary two -soliton solutions. The first parameter is the overall trans-lation of the complex; the second one is the separationdistance between the two bound solitons. The twist cor-responds to a very small separation, where the two sim-ple solitons bind to form a single entity. (The resultingobject does not have even a slightest reminiscence of atwo-soliton state; without knowing the whole family, therelation would hardly be possible to guess.)The two simple solitons, ψ + and ψ − , detach from the U (1)-symmetric family of solitons of the unperturbednonlinear Schr¨odinger at h = 0 [21]. The two twist so-lutions of (15) also hail from the solitons of the unper-turbed equation; however this time the relation is morecomplicated. Reducing h , the two solutions transforminto complexes of well-separated solitons [Fig.2(c,d)].Namely, one of the two twist solutions becomes a complexof two solitons: ψ T → e iπ/ sech( x + x ) + e − iπ/ sech( x − x ) , where x → ∞ as h →
0. The other twist continues to acomplex of three unperturbed solitons: ψ T → i sech( x + x ) − sech x − i sech( x − x ) , and again, the separation x grows without bound as h →
0. The “full names” of the two twists, ψ T and ψ T ,were coined to reflect this multisoliton ancestry.Despite being quiescent, nonpropagating objects, thetwists carry nonzero momentum. Since equation (15)is invariant under the space inversion, the twist solitonwith momentum P has a partner with momentum − P which is obtained by changing x → − x . This trans-formation leaves the absolute value of ψ ( x ) intact butchanges the sign of the phase derivative, ( d/dx )arg ψ ( x ).By analogy with the right-hand rule of circular motion,the twist whose phase decreases as x grows from −∞ to + ∞ [that is, the trajectory on the (Re ψ, Im ψ ) phaseplane is traced clockwise], will be called right-handed.The twist with the increasing phase (i.e. with a trajec-tory traced counter-clockwise) will be called left-handed.One can readily verify that the left-handed twist has apositive momentum, whereas the right-handedness im-plies P < −20 −10 0 10 20−1−0.500.51 xh=0.2 V=0P=1.74(a) Re ψ T2 Im ψ T2 −20 −10 0 10 20−1−0.500.51 xh=0.2 V=0P=3.34(b) Re ψ T3 Im ψ T3 −20 −10 0 10 20−1−0.500.51 x h=0.0000215 V=0P=0.0005746 (c) Re ψ T2 Im ψ T2 x h=0.0000215 V=0P=0.0005746 (c) Re ψ T2 Im ψ T2 −20 −10 0 10 20−1−0.500.51 x h=0.0000215V=0P=0.0015781 (d) Re ψ T3 Im ψ T3 FIG. 2. (a,b): The two nonpropagating twist solutions for h ∼
1. (Here h = 0 . h . (Here h = 2 . × − ). All twist solutions shown in these figures are left-handed. Consider some particular value of the driving strength, h = h . Unlike the twist solution in the parametrically driven NLS, the directly driven twist with h = h is amember of a one-parameter (rather than two-parameter)family of solutions. (The only free parameter is the trans-lation, −∞ < θ < ∞ , whereas the intersoliton separation χ is fixed by h .) This can be concluded from the factthat the corresponding operator H has only one, trans-lational, zero eigenvalue. Had the twist been a memberof a family of solutions parametrised by two continuousparameters, say θ and χ , the operator H would have hadan additional zero eigenvalue with the eigenvector ~ Ψ χ .Letting ψ = x + ix , the stationary equation (15) canbe written as a classical mechanical system on the plane,with the Lagrangian L = 12 ( ˙ x + ˙ x ) −
12 ( x + x ) + 12 ( x + x ) − hx . The existence of a one-parameter family of homoclinicorbits ~x = ~x χ ( t ), where ~x ≡ ( x , x ), would imply thatthe above system has the second integral of motion, inaddition to the energy. However, equation (15) is knownnot to have any additional conserved quantities [34].Finally, we need to comment on the stability of the twotwist solutions. When h is equal to 0 and the two solu-tions represent a doublet and a triplet of infinitely sepa-rated solitons of the unperturbed nonlinear Schr¨odinger,the symplectic spectrum includes 8 and 12 zero eigen-values, respectively. When h is small nonzero, only twoeigenvalues remain at the origin in each case. In addi-tion, the spectrum of the ψ T twist includes a complexquadruplet ± λ, ± λ ∗ and a pair of opposite pure imagi-nary eigenvalues. As h is increased, the imaginary paircollides with another imaginary pair emerging from thecontinuum, producing the second complex quadruplet.The spectrum of the ψ T twist includes two complex −20 −10 0 10 20−0.4−0.200.20.4 xh=0.01 V=1.97P=0.339Re ψ + Im ψ + FIG. 3. As V → c , the ψ + solitons (for all h ) and ψ − soli-tons (for small h ) approach linear waves with slowly decayingenvelopes. Shown is the ψ + solution with V close to c . (Inthis plot, h = 0 .
01; the corresponding c = 1 . ψ − solutions with V close to c have a similar shape. quadruplets and a pair of pure imaginary eigenvalues;this arrangement remains in place for all h , from verysmall to h = p /
27. The bottom line is that both twistsolutions are unstable for all h ; the instability is alwaysof the oscillatory type. V. NUMERICAL CONTINUATION OF SIMPLESOLITONSA. The travelling ψ + soliton Travelling solitons are sought as solutions of the ordi-nary differential equation (9) under the boundary condi-tions ψ ξ → | ξ | → ∞ .We begin with the continuation of the quiescent soliton ψ + . For a sequence of h sampling the interval (0 , p / ψ + was path followed all the wayto V = c , where c is given by Eq.(8). As V increases,the amplitude of the solution decreases while the widthgrows. A typical solution with V close to c is shown inFig.3. As V → c , the momentum P tends to zero.The resulting P ( V ) diagram is shown in Fig.4(a). Foreach h , the unstable stationary ψ + soliton remains un-stable when travelling sufficiently slow. The instability isdue to a real eigenvalue λ > V grows, the unstable eigenvalue moves towards theorigin along the real axis. Eventually, as the momentum P reaches its maximum, the positive eigenvalue λ collideswith its opposite partner λ ′ = − λ , after which both realeigenvalues move onto the imaginary axis and the solitonacquires stability. The soliton remains stable all the wayfrom the point V c , where the momentum is maximum, tothe value V = c where P = 0 and the soliton ceases toexist.The resulting P ( V ) dependence shows a remarkablesimilarity to the P ( V ) diagram [32] for the parametrically driven nonlinear Schr¨odinger, iψ t − iV ψ ξ + ψ ξξ + 2 | ψ | ψ − ψ = hψ ∗ . (17)The “parametrically driven” diagram is reproduced inFig.4(b) for the sake of comparison. One should keep inmind here that the notation used for the parametricallydriven solitons is opposite to the notation employed inthe externally driven situation. Thus, the parametricallydriven stationary ( V = 0) soliton with a positive sym-plectic eigenvalue in its spectrum is denoted ψ − (andnot ψ + as its externally driven counterpart). On theother hand, the parametrically driven stationary solitondenoted ψ + is stable for sufficiently small h (like the ex-ternally driven soliton ψ − ). For this reason, the objectsfeaturing P ( V ) diagrams similar to those of our exter-nally driven solitons ψ + , are the parametrically drivensolitons ψ − . B. The travelling ψ − soliton; h < . In the case of the ψ − solitons, there are two charac-teristic scenarios. When h lies between 0 and 0 .
06, thesoliton ψ − exists for all V between 0 and c . As V is in-creased from zero, the momentum P grows from P = 0and reaches its maximum at some point V c , 0 < V c < c .As V is changed from V c to c , the momentum decays tozero [see Fig.4(a)]. On the other hand, when h equals0 .
06 or lies above this value, the curve P ( V ) does notexhibit a point of maximum.Consider, first, the case h < .
06. The transformationscenario here is similar to the case of the soliton ψ + ; seeFig.4. What makes the bifurcation curves for the ψ + and ψ − solitons different, is the stability properties ofthe two solutions. Unlike the ψ + solution, the stationary ψ − soliton with h ≤ . V grows to the value V c where the momentum reachesits maximum, two opposite pure imaginary eigenvaluescollide at the origin on the (Re λ, Im λ ) plane and crossto the positive and negative real axis, respectively. Forthe driving strengths h ≤ . . ≤ h ≤ .
06, here the instabil-ity sets in earlier, as V reaches some V = V (where V FIG. 4. (a) The momentum of the ψ + , ψ − solitons continued to positive velocities. Decimal fractions attached to brancheslabel the corresponding values of h , with the superscripts + and − indicating the ψ + and ψ − solitons. (For example, 0 . + marks the branch emanating from the stationary ψ + soliton with h = 0 . ψ + and ψ − branches with h = 0 . 01; the curve just above h = 0 is 0 . − and the curve just below the h = 0 branch is0 . + . Solid curves mark stable and dashed ones unstable branches. (b) The corresponding P ( V ) diagram for the travellingparametrically driven solitons from [32]. eigenvalues ± λ, ± λ ∗ . (Here λ has a small real and fi-nite imaginary part.) This is a point of the hamiltonianHopf bifurcation, associated with the oscillatory instabil-ity [31, 32]. As V is increased to V (where V < V < V c ),two pairs of complex-conjugate λ converge on the realaxis, becoming two positive ( λ = λ > 0) and two neg-ative ( − λ = − λ ) eigenvalues. Finally, when V crossesthrough V c , the eigenvalues λ and − λ move on to theimaginary axis. The soliton does not restabilise at thispoint though; the real pair ± λ persists for all V ≥ V c .The bifurcation values V and V are, naturally, func-tions of h . The value V decreases (and V increases) as h is increased from 0 . h reaches0 . V reaches zero. It is interesting to note thatthere is a gap between V and V c for all h . Thereforethe oscillatory and nonoscillatory instability coexist forno V ; for smaller V ( V < V < V ) the instability isoscillatory whereas for larger V ( V > V ) the instabilityhas a monotonic growth.Finally, it is appropriate to mention here that the bi-furcation curve for the ψ − solitons with small h < . P ( V ) dependence for the small- h parametrically driven solitons (more specifically, para-metrically driven ψ - plus solitons) — see Fig.4(b). C. The travelling ψ − soliton; h ≥ . The P ( V ) graphs for h ≥ . 06 are qualitatively differ-ent from the small- h bifurcation curves. For these larger h , the bifurcation curve emanating from the origin on the ( V, P )-plane turns back at some V = V max , withthe derivative ∂P/∂V remaining strictly positive for all V ≤ V max .For h in the interval 0 . ≤ h < . 25, the P ( V ) curvecrosses the P -axis [Fig.4(a), Fig.5]. The solution arisingat the point V = 0 is nothing but the ψ T twist soliton,shown in Fig.2(a).As we continue this branch to the V < ψ − solitons. The P ( V ) curve makes one more turn and even-tually returns to the origin on the ( V, P )-plane (Fig.5).As V and P approach zero, the distance between thesolitons in the complex tends to infinity.An interesting scenario arises when h is greater orequal than 0 . 25. Here, as V grows from zero, the soli-ton ψ − gradually transforms into a three-soliton complex ψ (+ − +) . The branch turns back towards V = 0 but doesnot cross the P -axis. Instead of continuing to negative V ,the branch reapproaches the origin in the ( V, P ) plane,remaining in the positive ( V, P ) quadrant at all times.The ingoing path is almost coincident with the outgoingtrajectory; as a result, the branch forms a lasso-lookingloop [Fig.4(a)].Turning to the stability properties of solutions alongthe branch continued from ψ − , we start with a short in-terval 0 . ≤ h ≤ . V, P ) plane, are similar to the in-terval 0 . < h < . 06 that we discussed in the previousparagraph. The stationary ψ − soliton is stable and sta-bility persists for small V . As V reaches a certain V > −1 −0.5 0 0.5 1 1.5−3−2−10123 h=0.2P V ψ − ψ T2 ψ T3 ψ (−−) ψ (−−−) ψ (+−+) ψ (++) ψ (++++) FIG. 5. The full P ( V ) bifurcation diagram for the ψ − soli-ton with h = 0 . 2. Also shown is the continuation of the T ψ (++) branch. More solution branches can beobtained by the reflection V → − V , P → − P . All branchesshown in this figure correspond to unstable solutions. a quadruplet of complex eigenvalues is born and oscil-latory instability sets in. Subsequently two pairs of thecomplex eigenvalues converge on the real axis, dissociate,then recombine and diverge to the complex plane again;a pair of opposite pure imaginary eigenvalues moves tothe real axis and back — however, despite all this activityon the complex plane, the soliton solution never regainsits stability.For larger h , h > . ψ − solitonis unstable, with a complex quadruplet in its spectrum.As we continue in V , two pairs of opposite pure imagi-nary eigenvalues move on to the real axis, one after an-other. For h ≥ . 25, the resulting arrangement (two pairsof opposite real eigenvalues and a complex quadruplet)persists until the branch reaches the origin on the ( V, P )plane. On the other hand, when h lies in the interval0 . < h < . 25, the four real eigenvalues collide,pairwise, producing the second complex quadruplet atsome point on the curve before it crosses the P axis inFigs.4(a) and 5. Two complex quadruplets persist in thespectrum as we continue the curve further. Thus the un-stable stationary soliton ψ − with h > . V . VI. NUMERICAL CONTINUATION OF THETWIST SOLITON When 0 . ≤ h < . 25, the branch resulting from thecontinuation of the stationary ψ − soliton turns back andcrosses the P -axis; the point of crossing corresponds to the T h liesoutside the (0 . , . 25) interval, the stationary T ψ − soliton and can beused as a starting point for a new, independent, branch.Another new branch is seeded by the T A. Travelling twist T ( h < . ) We start with the situation of small h : h < . 06, andconsider the T ψ T twist is path followed to pos-itive V , it transforms into a ψ (++) complex. At somepoint, the P ( V ) curve makes a U-turn [Fig. 6(a)] andconnects to the origin on the ( V, P ) plane. The entirepositive- V branch is unstable. The stationary twist hasa complex quadruplet in its spectrum; as the curve iscontinued beyond the turning point, the complex eigen-values converge, pairwise, on the positive and negativereal axis. In addition, a pair of opposite pure imaginaryeigenvalues moves onto the real axis as V passes throughthe point of maximum of the momentum in Fig.6(a).As the curve approaches the origin, the distance be-tween the two solitons in the complex increases and be-comes infinite when V = P = 0. The spectrum becomesthe spectrum of two infinitely separated ψ + solitons, i.e.it includes two positive eigenvalues λ ≈ λ ; their nega-tive counterparts − λ ≈ − λ ; and four eigenvalues nearthe origin.Continuing the T V direction, ittransforms into a complex of two ψ − solitons. At somepoint along the curve, a quadruplet of complex eigen-values converges on the imaginary axis and the complexstabilises. (For the value h = 0 . 05 which was used toproduce Fig.6(a), the stabilisation occurs at the point V = − . V , thebranch turns back; shortly after that (at V = − . h = 0 . 05) the momentum reaches its minimum. Twoopposite imaginary eigenvalues collide at this point andmove onto the real axis; the solution loses its stability.When continued beyond the turning point and thepoint of minimum of momentum, the curve connects tothe origin on the ( V, P ) plane (Fig. 6(a)). As V, P → ψ − solitons grows withoutbound. The two opposite real eigenvalues decay in ab-solute value but remain in the spectrum all the way to V = 0.It is interesting to note a similarity between the bi-furcation diagram resulting from the continuation of thesmall- h T −0.5 0 0.5 1 1.5−1012 P V ψ (−−) ψ (−−−) ψ (+−+) ψ T3 ψ (++) ψ (++) ψ T2 ψ (−−) (a) h=0.05 −0.6 −0.3 0 0.3 0.6−1−0.500.51 VP ψ (++) ψ (−−) ψ T (b) h=0.05 FIG. 6. (a) Continuation of the T T h . Also shown is the two-soliton branch whichconnects the origin to itself without intersection the vertical axis. (b) The continuation of the twist soliton in the case of theparametrically driven NLS equation (adapted from [32]). More solution branches can be obtained by the reflection V → − V , P → − P both in (a) and (b). B. Travelling twist T , h < . Figs.6(a) and 5 also show the continuation of the T h < . 06 and 0 . ≤ h < . 25 are qualitatively similar.Continuing the stationary T ψ (+ − +) complex. If we, in-stead, continue to negative velocities, the twist trans-forms into a triplet of ψ − solitons. Both V > V < V, P ) plane. As V and P approach the origin oneither side, the distance between the three solitons boundin the complex grows without limit.The stationary T h , both or one of these convergeon the real axis as we continue it to V > V < λ = 0 at theextrema of P ( V ). Finally, as V and P approach theorigin, the spectrum transforms into the union of spectraof three separate solitons. C. Travelling twists T and T , h ≥ . Another parameter region where the continuation ofthe ψ − does not cross the P -axis, is h ≥ . 25. The resultof the continuation of the two twist solutions is shownin Fig.7(a). The continuation of T h = 0 . h = 0 . 05: the twist transforms into a complex of twosolitons ψ − . At some negative V the curve turns backand connects to the origin on the ( V, P ) plane, with thedistance between the two solitons bound in the complexincreasing without bound. The eigenvalues evolve ac-cordingly: two complex quadruplets in the spectrum ofthe stationary T V < 0, supplemented bya pair of real eigenvalues which arrive from the imaginaryaxis at the point of minimum of P ( V ). As V, P → 0, thediscrete spectrum becomes the union of the eigenvaluesof two simple solitons.The continuation of T V produces a lessexpected outcome. Instead of turning clockwise and con-necting to the origin as in Fig.6(a), the curve turns coun-terclockwise and crosses through the P -axis once again.The solution arising at the point V = 0 is nothing butthe twist T 3. Two complex quadruplets in the spectrumof T T V, P )-plane. Thecorresponding solution is a complex of three ψ − soli-tons, shown in Fig.7(b). The third complex quadrupletemerges at some V before the turning point, and a pair ofopposite real eigenvalues arrives from the imaginary axisat the point of minimum of the momentum. As V, P → ψ − . D. Other branches It is appropriate to note that there are branches whichdo not originate on any of the four stationary solutions listed above ( ψ ± , ψ T or ψ T ). The simplest of these1 −1 −0.5 0−202 h=0.25(a)P V ψ (−−) ψ (−−−) ψ T2 ψ T3 −20 −10 0 10 20−1−0.500.51 x h=0.25 V=−0.85P=−3.19(b) Re ψ (−−−) Im ψ (−−−) FIG. 7. (a) The P ( V ) curve resulting from the continuation of the twist for h = 0 . 25. The starting point of the continuationis marked by an open circle. All branches shown in this figure are unstable. (b) A ψ ( −−− ) solution on the lower branch in (a).Here V = − . P = − . 2. In (b), the solid lines show the real and dashed imaginary part. emerge from the origin on the ( V, P ) plane as boundstates of simple solitons with large separation. Onebranch of this sort arises for h ≥ . 06 (Fig. 5). It emergesfrom the origin as the ψ (++) and returns as the ψ (++++) complex. The entire branch is unstable.Next, unlike in the parametrically driven NLS, thesame pair of externally driven travelling solitons maybind at various distances. In particular, when h is smallerthan 0.06, there is more than one bound state of two ψ + solitons and more than one complex of two ψ -minuses.Fig.6(a) shows a branch ψ ( −− ) that emerges from theorigin in the first quadrant of the ( V, P ) plane, describesa loop and re-enters the origin — this time as a ψ (++) branch. Note that for small V and P , the re-entering ψ (++) branch is indistinguishable from the other ψ (++) branch — the one that continues from the twist solution.(In a similar way, the V → − V , P → − P reflection ofthe ψ ( −− ) branch overlaps with the small- V, P section ofthe ψ ( −− ) branch arriving from the twist.) All solutionsconstituting this branch are unstable. VII. CONCLUDING REMARKS In this paper, we studied stationary and moving soli-tons of the externally driven nonlinear Schr¨odinger equa-tion, iψ t + ψ xx + 2 | ψ | ψ − ψ = − h. (18)Our continuation results are summarised in Fig.8(a)which shows ranges of stable velocities for each value ofthe driving strength h .The notation ψ + and ψ − in this figure is used for thetravelling waves obtained by the continuation of the sta- tionary ψ + and ψ − solitons, respectively. The travellingsoliton preserves some similarity with its stationary an-cestor; this justifies the use of the same notation.The uppermost curve in this figure is given by V = c ( h )where c is the maximum velocity of the soliton propaga-tion, Eq.(8). This curve serves as the upper bound of thetravelling ψ + soliton existence domain. The dotted curvedemarcates the existence domain of the travelling ψ − soli-ton. For h between 0 and 0.06 it coincides with the V = c ;for 0 . ≤ h ≤ . V = V max ( h ) where V max is the position of the turning point in Fig.4(a).The area shaded in blue (light grey) gives the stabilityregion of the soliton ψ + and the area shaded by purple(dark grey) is the ψ − stability domain. Note that theblue and purple regions partially overlap: for small h ,there is a range of “stable” velocities accessible to solitonsof both families. The light (yellow) strip inside the purple(dark grey) region represents the stability domain of thebound state of two ψ − solitons.As we cross the right-hand “vertical” boundary of thepurple (dark grey) region, the ψ − soliton loses its stabi-ilty to an oscillatory mode. If we had damping in thesystem, the onset of instability would correspond to theHopf bifurcation giving rise to a time-periodic solution.In the absence of damping, the oscillatory instability pro-duces an oscillatory structure with long but finite lifetime[31]. These solitons with oscillating amplitude and width,travelling with oscillatory velocities, were observed in[30]. These are expected to exist to the right of the purple(dark grey) region.Where possible, we tried to emphasise the similar-ity of the arising bifurcation diagrams with the corre-sponding diagrams for the parametrically driven nonlin-2 V h stable ψ + stable ψ − stable ψ (−−) (a)V=c hV stable ψ + stable ψ − (b) FIG. 8. (a) The chart of the stable one-soliton solutions of the externally driven nonlinear Schr¨odinger equation (18). Here h varies from 0 to p / ≈ . ear Schr¨odinger equation: iψ t + ψ xx + 2 | ψ | ψ − ψ = hψ ∗ . (19)Fig.8(b) reproduces the soliton attractor chart forEq.(19) [32]. The structure of the stability regions in thetwo figures is remarkably similar. 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