Phase-driven interaction of widely separated nonlinear Schrödinger solitons
PPHASE-DRIVEN INTERACTION OF WIDELY SEPARATEDNONLINEAR SCHR ¨ODINGER SOLITONS
JUSTIN HOLMER AND QUANHUI LIN
Abstract.
We show that, for the 1d cubic NLS equation, widely separated equalamplitude in-phase solitons attract and opposite-phase solitons repel. Our resultgives an exact description of the evolution of the two solitons valid until the soli-tons have moved a distance comparable to the logarithm of the initial separation.Our method does not use the inverse scattering theory and should be applicableto nonintegrable equations with local nonlinearities that support solitons with ex-ponentially decaying tails. The result is presented as a special case of a generalframework which also addresses, for example, the dynamics of single solitons sub-ject to external forces as in [7, 8]. Introduction
We consider the 1d nonlinear Schr¨odinger equation (NLS)(1.1) i∂ t u + ∂ x u + | u | u = 0 . It has a single soliton solution u ( x, t ) = e it/ sech x . The invariances of (1.1) can beapplied to produce a whole family of solutions. To describe them, let (1.2) η ( x, µ, a, θ, v ) = e iθ e iµ − v ( x − a ) µ sech( µ ( x − a ))for parameters θ, a, v ∈ R , µ >
0. Then u ( x, t ) = η ( x, µ ( t ) , a ( t ) , θ ( t ) , v ( t )) solves (1.1)provided(1.3) µ ( t ) = µ a ( t ) = a + tv µ − θ ( t ) = θ + 12 t ( µ + µ − v ) v ( t ) = v In this paper, we study the evolution of initial data that is the sum of two widelyseparated solitons:(1.4) u ( x ) = η ( x, µ , a , θ , v ) + η ( x, µ , a , θ , v ) We order the parameters as ( µ, a, θ, v ) to mimic ( q , q , p , p ) as canonical coordinates for thefour dimensional symplectic space with symplectic form dp ∧ dq + dp ∧ dq . a r X i v : . [ m a t h . A P ] A ug JUSTIN HOLMER AND QUANHUI LIN where | a − a | (cid:29)
1. In particular, we focus on two illustrative cases. In both cases,we consider identical mass solitons with zero initial velocity. In Case 0, we take thesame initial phase, corresponding to an even superposition and in Case 1, we takeopposite initial phase corresponding to an odd superposition.(1.5) u ( x ) = (cid:40) η ( x, , − a , ,
0) + η ( x, , a , ,
0) Case σ = 0 η ( x, , − a , π,
0) + η ( x, , a , ,
0) Case σ = 1We find that in the same phase case (Case 0), the two solitons are drawn towardeach other and in the opposite phase case (Case 1) they are pushed apart– see Fig.1.1. In either case, the solution u to (1.1) is well-approximated by(1.6) u z ( x ) = η ( x, µ , a , θ , v ) + η ( x, µ , a , θ , v )where z represents coordinates (1.7) z = ( z , z , z , z , z , z , z , z ) = ( µ , a , µ , a , θ , v , θ , v )As parity is preserved by the flow (1.1), we have(1.8) µ def = µ = µ , a def = − a = a , v def = − v = v , and θ def = θ = θ in the same phase case (Case 0), while θ def = θ − π = θ in theopposite phase case (Case 1). Theorem 1.1.
Suppose that u ( t ) is the solution to (1.1) with initial data (1.5) . Let h = e − a (cid:28) (so a = log h − (cid:29) ). Let T ∼ (cid:40) h − Case σ = 0 h − log h − Case σ = 1 Let ( a ( t ) , v ( t )) solve (1.9) (cid:40) ˙ a = v ˙ v = − − σ e − a with initial data ( a , . Let µ solve (1.10) ˙ µ = ( − σ (8 a − ve − a , and then let θ solve (1.11) ˙ θ = 12 µ + 12 v µ − + 18( − σ e − a . Then on ≤ t ≤ T , we have (cid:107) u ( t ) − u z (cid:107) H x (cid:46) h − , Superscripts are used on z to conform with geometric summation conventions used later in thepaper. LS SOLITON INTERACTION 3 Figure 1.1.
The top plot is a depiction of Case 0 (same phase; evensolution), where the two solitons are pulled toward each other. Thebottom plot depicts Case 1 (opposite phase; odd solution), where theyrepel. In each case, the solution is modeled in Theorem 1.1 as u ≈ u z = η ( µ, − a, θ + σπ, − v ) + η ( µ, a, θ, v ) where ( µ, a, θ, v ) solve a specific ODEsystem. where (1.12) u z = η ( µ, − a, θ + σπ, − v ) + η ( µ, a, θ, v )Let us make some remarks on the ODE system (1.9). The energy associated tothis system is H eff = v − − σ e − a = − − σ e − a In the case σ = 0 (same phase), we have v ≤ a ≤ a and (cid:40) a ( t ) = a − log sec(2 ht ) v ( t ) = 2 h tan(2 ht )valid for 0 ≤ t (cid:46) e a = h − . In the case σ = 1 (opposite phase), we have v ≥ a ≥ a and (cid:40) a ( t ) = a + log cosh(2 ht ) v ( t ) = 2 h tanh(2 ht ) JUSTIN HOLMER AND QUANHUI LIN valid for 0 ≤ t (cid:46) a e a = h − log h − . In either case, µ evolving according to (1.10)satisfies | µ − | (cid:46) h and can thus be replaced by 1 in (1.12). However, the O ( h )behavior of µ is dynamically significant in that it yields O ( h ) effects in θ through(1.11). It is evident from the explicit forms for a ( t ) given above that, on the indicatedtime scale, the soliton has moved a distance comparable to log a .We remark that although (1.1) is completely integrable, we do not use the inversescattering theory of Zakharov-Shabat [20]. We expect that one could compute thescattering data associated to our initial condition and conduct an analysis using in-verse scattering theory that would describe the dynamics for all time. Our argument,however, has the merit of being relatively simple and should adapt to most nonin-tegrable nonlinearities that support stable solitons with exponentially decaying tails.An important example of such a nonintegrable equation is the 1d cubic-quintic NLS: i∂ t u + 12 ∂ x u + | u | u − (cid:15) | u | u = 0Furthermore, our goal was not just to obtain Theorem 1.1 but to present it in theconceptual (yet rigorous) framework of symplectic restriction that illustrates its con-nection to previous work of the first author, Holmer-Zworski [7, 8].We cite two papers from the physics literature as motivation for our problem.Stegeman-Segev [17] provide an overview of phase-driven two-soliton interaction inthe context of optics, beginning with an account of the 1d case (1.1) that we study(see their Fig. 4) and proceeding to a discussion of two-soliton interaction in twodimensions in which the attractive forces between in-phase solitons can lead to spi-raling structures – see their Fig. 6. The NLS equation also arises in a completelydifferent physical setting, Bose-Einstein condensation. Strecker et.al. [18] describe anexperiment producing multiple solitons, in which the model is (1.1) with a confiningpotential. A train of five solitons with successively opposite phases are produced andoscillate in a well. At the peak of the oscillations, the solitons bunch up but retainsome separation; [18] explains this in terms of their phase differences.We will now give an explanation of Theorem 1.1 and an overview of the proof.Consider L ( R ; C ) as a manifold with metric g u ( v , v ) = (cid:104) v , v (cid:105) def = Re (cid:90) v ¯ v for u ∈ L , v , v ∈ T u L (cid:39) L . Introduce J = − i , viewed as an operator T u L → T u L . The corresponding symplec-tic form is(1.13) ω u ( v , v ) = (cid:104) v , J − v (cid:105) LS SOLITON INTERACTION 5
Take as Hamiltonian the (densely defined, with domain D = H ) function H : L → R given by(1.14) H ( u ) = 14 (cid:90) | u x | − (cid:90) | u | Then H (cid:48) ( u ) ∈ T ∗ u L (cid:39) metric g T u L The corresponding flow is ∂ t u = J H (cid:48) ( u ) yielding (1.1).Recalling that η is given by (1.2), consider the manifold of solitons M = { η ( · , µ, a, θ, v ) | µ > , θ ∈ R , a ∈ R , v ∈ R } . Computations show that the restriction of the symplectic form ω to M is i ∗ ω = dθ ∧ dµ + dv ∧ da , while the restriction of the Hamiltonian H to M is H ( η ( · , µ, a, θ, v )) = 12 µ − v − µ , Note that the free single soliton flow (1.3) is just the solution to the Hamilton equa-tions of motion for H ( η ) with respect to i ∗ ω : ˙ µ = ∂ θ H ( η ) = 0˙ a = ∂ v H ( η ) = µ − v ˙ θ = − ∂ µ H ( η ) = 12 µ − v + 12 µ ˙ v = − ∂ a H ( η ) = 0Turning to the double soliton problem, recall that we model the u in terms of u z given by (1.6) where z = ( z , . . . , z ) is given by (1.7). We introduce the shorthandnotation η j def = η ( · , µ j , a j , θ j , v j ) , j = 1 , . Also recall that h = e − a (cid:28)
1, and the initial soliton separation is 2 a = 2 log h − (cid:29) H p ( u z ) = H p ( η ) + (cid:104) H (cid:48) p ( η ) , η (cid:105) + H p ( η ) + (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h )The last two terms are dominant near a (on the effective support of η ), so that thesecond soliton sees an “effective” Hamiltonian(1.16) H eff ( µ , a , θ , v ) = H ( η ) + (cid:104) H (cid:48) p ( η ) , η (cid:105) This domain is chosen so that JH (cid:48) ( u ) = − i ( − u xx − | u | u ) ∈ L . Although we restrict to u ∈ D = H here, we will prove estimates on the corresponding flow in H . This parallels thesituation in the theory of linear self-adjoint operators A , where a dense domain is specified but theflow associated to − iA extends to a unitary operator on all of L . JUSTIN HOLMER AND QUANHUI LIN and thus its expected equations of motion are(1.17) ˙ µ = ∂ θ H ( η ) + ∂ θ (cid:104) H (cid:48) p ( η ) , η (cid:105) ˙ a = ∂ v H ( η ) + ∂ v (cid:104) H (cid:48) p ( η ) , η (cid:105) ˙ θ = − ∂ µ H ( η ) − ∂ µ (cid:104) H (cid:48) p ( η ) , η (cid:105) ˙ v = − ∂ a H ( η ) − ∂ a (cid:104) H (cid:48) p ( η ) , η (cid:105) Likewise, the first two terms in (1.15) are dominant near a so the first soliton seesan effective Hamiltonian H eff ( µ , a , θ , v ) = H ( η ) + (cid:104) H (cid:48) p ( η ) , η (cid:105) and thus its expected equations of motion are(1.18) ˙ µ = ∂ θ H ( η ) + ∂ θ (cid:104) H (cid:48) p ( η ) , η (cid:105) ˙ a = ∂ v H ( η ) + ∂ v (cid:104) H (cid:48) p ( η ) , η (cid:105) ˙ θ = − ∂ µ H ( η ) − ∂ µ (cid:104) H (cid:48) p ( η ) , η (cid:105) ˙ v = − ∂ a H ( η ) − ∂ a (cid:104) H (cid:48) p ( η ) , η (cid:105) Pulling (1.17) and (1.18) together gives us a systems of eight equations in eightunknowns:(1.19) ˙ µ = ∂ θ H ( η ) + ∂ θ (cid:104) H (cid:48) p ( η ) , η (cid:105) ˙ a = ∂ v H ( η ) + ∂ v (cid:104) H (cid:48) p ( η ) , η (cid:105) ˙ θ = − ∂ µ H ( η ) − ∂ µ (cid:104) H (cid:48) p ( η ) , η (cid:105) ˙ v = − ∂ a H ( η ) − ∂ a (cid:104) H (cid:48) p ( η ) , η (cid:105) ˙ µ = ∂ θ H ( η ) + ∂ θ (cid:104) H (cid:48) p ( η ) , η (cid:105) ˙ a = ∂ v H ( η ) + ∂ v (cid:104) H (cid:48) p ( η ) , η (cid:105) ˙ θ = − ∂ µ H ( η ) − ∂ µ (cid:104) H (cid:48) p ( η ) , η (cid:105) ˙ v = − ∂ a H ( η ) − ∂ a (cid:104) H (cid:48) p ( η ) , η (cid:105) After the even/odd symmetry assumption is imposed, one can distill from (1.19) theequations appearing in the statement of Theorem 1.1. We find that the above argument yielding (1.19) is a little too vague to adapt to arigorous proof. We now consider a different perspective that informally produces the In fact, the above heuristic argument does not invoke the even/odd symmetry assumption andthus we might expect the equations (1.19) even without this assumption. However, the equations(1.19) are only expected to be accurate to order O ( h ). In the presence of the symmetry assumptionthe eight equations in (1.19) dramatically decouple as (1.9), (1.10), (1.11) which permits a directanalysis of these ODEs that shows that an O ( h ) unknown can only only have a limited O ( h ) effectthe solution. In the general case, the eight equations in (1.19) are more interdependent and we arenot certain as to the effect of O ( h ) perturbations. This is not the only obstacle to removing thesymmetry assumption; see comments below. LS SOLITON INTERACTION 7 same set of equations (1.19) but adapts to yield a proof of Theorem 1.1 and in factextends and unifies the results of [3, 6, 7, 8]. Recalling z defined in (1.7), considernow the eight-dimensional two-soliton manifold M = { u z = η + η = η ( · , µ , a , θ , v ) + η ( · , µ , a , θ , v ) } The symplectic form (1.13) restricted to M is(1.20) i ∗ ω = 12 (cid:88) (cid:96),m =1 a (cid:96)m ( z ) dz (cid:96) ∧ dz m where A ( z ) = ( a (cid:96)m ( z )) , a (cid:96)m ( z ) = (cid:104) ∂ z (cid:96) u z , J − ∂ z m u z (cid:105) Let H ( u z ) denote the restriction to M of the Hamiltonian (1.14). The expectedequations of motion for z m are Hamilton’s equations for H ( u z ) with respect to i ∗ ω .These equations are:(1.21) ˙ z m = − (cid:88) (cid:96) =1 ∂ z (cid:96) H ( u z ) a (cid:96)m ( z ) , m = 1 , . . . , a (cid:96)m denotes the components of the inverse of the matrix A = ( a (cid:96)m ).The matrix A contains O ( h ) terms that result from the pairing of directions paral-lel to the first soliton with directions parallel to the second soliton. Moreover, H ( u z )contains additional O ( h ) terms arising from the quadratic part of (1.14) not repre-sented in (1.19). It turns out that O ( h ) terms in a (cid:96)m and O ( h ) terms in H ( u z ) eachgive rise to terms which cancel in (1.21). This hinges upon the fact that(1.22) ∂ z (cid:96) H ( u z ) = − (cid:88) j =1 b j a j(cid:96) + ∂ z (cid:96) (cid:104) η , H (cid:48) p ( η ) (cid:105) + ∂ z (cid:96) (cid:104) η , H (cid:48) p ( η ) (cid:105) where b = ∂ v H ( η ) , b = ∂ v H ( η ) , b = − ∂ µ H ( η ) , b = − ∂ µ H ( η )and all other b j = 0. When this equation is substituted into (1.21), once can witnessthe simplification arising from the pairing of A and A − , and this shows that (1.21)is equivalent to (1.19). We elaborate upon this in Appendix A.The merit in this point of view is that the equations (1.21) readily follow from thesymplectic decomposition of the flow–that is, we select z (via the implicit functiontheorem) so that(1.23) u = u z + w where w ∈ T z M ⊥ (the symplectic orthogonal complement to T z M in T u z L ). In § JUSTIN HOLMER AND QUANHUI LIN with errors of size h + (cid:107) w (cid:107) H . This argument exploits the fact that (1.21), witherrors of size h + (cid:107) w (cid:107) H , is equivalent to(1.24) ∂ t u z = Π z J H (cid:48) ( u z ) + O ( h + (cid:107) w (cid:107) H )where Π z : T u z L → T z M is the symplectic orthogonal projection operator givenexplicitly by Π z f = (cid:88) (cid:96),m =1 (cid:104) f, J − ∂ z (cid:96) u z (cid:105) a (cid:96)m ( z ) ∂ z m u z The proof of Lemma 3.1 makes no reference to the specific meaning of H or u z , and asimilar result with nearly identical proof would yield the equations of motion in manyother problems, including those studied in [6, 7, 8]. The fact that the equations ofmotion follow automatically but rigorously from the symplectic decomposition (1.23)is one of the main advantages of this geometric approach to our problem, as opposedto a more ad hoc approach based on the discussion surrounding (1.16). It then remains to show that (cid:107) w ( t ) (cid:107) H x (cid:46) h on the time scale O ( h − ), which wewould like to prove using a suitable adaptation of the Lyapunov functional methodinitiated into the theory of orbital stability of single solitons by Weinstein [19]. Un-fortunately, the presence of the Π z projection in (1.24) corrupts this computationand only yields a bound (cid:107) w ( t ) (cid:107) H x (cid:46) h . To eliminate this problem, we construct afunction ν z = O ( h ), whose only time dependence is through the parameter z , suchthat the distorted double-soliton function ˜ u z = u z + ν z satisfies(1.25) ∂ t ˜ u z = J H (cid:48) (˜ u z ) + O ( h + (cid:107) w (cid:107) H )which is just (1.24) without the Π z projection. The construction of ν z is carried outin § v z to our soliton manifold M and consider the distortedmanifold ˜ M = { ˜ u z } . The solution u to (1.1) now has a decomposition u = ˜ u z + ˜ w where ˜ u z satisfies (1.25) and it suffices to prove that (cid:107) ˜ w ( t ) (cid:107) H x (cid:46) h . In other words,we would like to show that the exact solution to (1.1) is approximately equal tothe solution to the approximate equation (1.25). In §
5, a Lyapunov functional isemployed to obtain the needed control on ˜ w . The Lyapunov functional used is asuperposition of two copies–one for each soliton–of the classical functional, built fromenergy, momentum and mass, employed by Weinstein [19] to prove orbital stabilityof single solitons. This superposition was previously used by Martel-Merle-Tsai [10] The idea that the equations of motions should be Hamilton’s equations for the restricted Hamil-tonian with respect to the restricted symplectic form was introduced in [7, 8] and supported in-formally with an argument involving Darboux’s theorem. The equations of motion thus obtainedwere used as a guide in the analysis in [6, 7, 8] but the general rigorous connection between thesymplectic decomposition of the flow and the equations of motion, as obtained in our Lemma 3.1,was not obtained in [6, 7, 8].
LS SOLITON INTERACTION 9 in their study of the orbital stability of spreading multiple solitons. Our presentationof this component of the argument is a little different from [19] or [10] and more inline with the abstract orbital stability theory developed by Grillakis-Shatah-Strauss[4, 5]. Roughly, we prove that if W : L → R is a (densely defined) functional suchthat the derivative is of order O ( h ) on ˜ M , and if we define L to be the quadraticpart of W above ˜ M , then ∂ t L is essentially the quadratic part of the Poisson bracket { H, W } ( u ) above ˜ M , which we show is of order O ( h ).Let us note that h − δ losses occur in several estimates, which were not necessarilyindicated in the above introduction, owing to the fact that in the attractive case | v | can exceed h by a factor of log h − and a decreases below a , as well as the presence ofan x -multiplication factor in terms involving ∂ v j u z in both the attractive and repulsivecases. We indicate the presence of such losses by writing, for example, h − . Theselosses are more carefully quantified in the concluding summary of the proof in § § §
5, although stated only for the problemat hand, are fairly general and widely applicable to problems in orbital stability ofsingle [19, 4, 5] and multiple [11] solitons and the dynamics of solitons in slowlyvarying potentials [3, 6, 8, 2, 13], weak rough potentials [1, 7, 15], and the interactionof two soliton tails, as considered here. The portion of the analysis most specific to theproblem at hand appears in §
4, where the approximate solution is constructed. In thissection, we consider the two components of the double-soliton separately and exploitthe group structure of each individual soliton to pull-back to a nearly-stationaryproblem, which can be solved by operator inversion. This method was introduced byHolmer-Zworski [8] to produce an improvement of the result by Fr¨ohlich-Gustafson-Jonsson-Sigal [3] on the dynamics of single solitons in a slowly-varying potential,eliminating the uncontrollable errors in the ODEs appearing in [3].Let us point out some related papers. Marzuola-Weinstein [12] consider the dynam-ics of symmetric and antisymmetric states in a double well-potential. Krieger-Martel-Rapha¨el [9] construct two-soliton solutions with separating components asymptoti-cally as t → + ∞ for the nonlinear Hartree equation, where the long-range effects ofthe nonlinearity complicate the analysis but also lead to nonnegligible perturbationsof the asymptotic trajectories. Our analysis is similar in several ways to that of [9],although our priorties are different – we study the dynamics for a finite (but dynam-ically significant) time of an initial data that is close to a double-soliton, wheras theyprovide infinite time dynamics for an exact double-soliton solution. The problem ofstability of nonintegrable NLS multiple solitons, with components that separate as t → ∞ , has been considered by Perelman [14], Rodnianski-Soffer-Schlag [16], andMartel-Merle-Tsai [10].We now remark on where we rely upon the even/odd symmetry assumption onthe solution. While the arguments in § §
4, whenconstructing the solution ˜ u z to the approximate equation (1.25), we do make use of the symmetry assumption, although we have sketched an argument (not included in thispaper) showing how one can adapt the argument to the general case. The symmetryassumption also greatly simplifies the computations carried out in Appendix A whichultimately yield the ODEs (1.9), (1.10), (1.11) in Theorem 1.1. The integrals in thegeneral case appear very complicated, and we are less confident that we could controlthe propagation of O ( h ) errors, as previously remarked. However, the one placewhere the symmetry assumption is used critically is in obtaining the upper boundon the Lyapunov function used in § u and the solution ˜ u z of the approximate equation (1.25). Our guess is that to resolvethis issue, one would need to restructure the Martel-Merle-Tsai Lyapunov functionin a substantial way. The lower bound on the Lyapunov function, however, carriesthrough in general.1.1. Acknowledgements.
We thank Maciej Zworski and Galina Perelman for help-ful discussion related to this paper. J.H. was partially supported by NSF GrantDMS-0901582 and a Sloan Research Fellowship.2.
Background on solitons, Hamiltonian structure, and Lyapunovfunctionals
The NLS equation (1.1) can be put into Hamiltonian form as follows. Take asthe ambient symplectic manifold L = L ( R ; C ) with metric (cid:104) v , v (cid:105) u = Re (cid:82) v ¯ v for u ∈ L , v , v ∈ T u L = L . Let J = − i , viewed as an operator T u L → T u L .The corresponding symplectic form is ω u ( v , v ) = (cid:104) v , J − v (cid:105) u (we henceforth dropthe u -subscript). Define the (densely defined, with domain D = H ) Hamiltonian H : L → R as H ( u ) = 14 (cid:90) | u x | dx − (cid:90) | u | . Using the metric (cid:104)· , ·(cid:105) defined above, H (cid:48) ( u ) ∈ T ∗ u L is identified with an element of T u L . The free NLS equation (1.1) is(2.1) ∂ t u = J H (cid:48) ( u )Solutions to (1.1) also satisfy conservation of mass M ( u ) and momentum P ( u ), where M ( u ) = 12 (cid:90) | u | , P ( u ) = 12 Im (cid:90) ¯ u u x . Let φ ( x ) = sech x and η ( x, µ, a, θ, v ) = e iθ e iµ − v ( x − a ) µφ ( µ ( x − a ))Direct computation shows that M ( η ) = µ and P ( η ) = v . Consider the manifold ofsolitons M = { η ( · , µ, a, θ, v ) | µ > , θ ∈ R , a ∈ R , v ∈ R } . LS SOLITON INTERACTION 11
The tangent space at η = η ( · , µ, a, θ, v ) is T ( µ,a,θ,v ) M = span { ∂ µ η, ∂ θ η, ∂ a η, ∂ v η } . Note that
J H (cid:48) ( η ) ∈ T ( µ,a,θ,v ) M , and thus the flow associated to (1.1) will remainon M if it is initially on M . Specifically, direct computation shows(2.2) J H (cid:48) ( η ) = ( 12 µ − v + 12 µ ) ∂ θ η + µ − v∂ a η . To gain a better understanding of (1.3) and (2.2), we can restrict ω to M to obtain i ∗ ω = dθ ∧ dµ + dv ∧ da , where i : M → L denotes the inclusion and restrict H to M to obtain H ( η ) = 12 µ − v − µ , and then note that (1.3) is just the solution to the Hamilton equations of motion for H ( η ) with respect to i ∗ ω :(2.3) ˙ µ = ∂ θ H ( η ) = 0˙ a = ∂ v H ( η ) = µ − v ˙ θ = − ∂ µ H ( η ) = 12 µ − v + µ ˙ v = − ∂ a H ( η ) = 0Suppose we knew that J H (cid:48) ( η ) ∈ T ( µ,a,θ,v ) M and wanted to recover the coefficients asin (2.2). This could be achieved by noting that J H (cid:48) ( η ) = ∂ t η = ˙ µ∂ µ η + ˙ a∂ a η + ˙ θ∂ θ η + ˙ v∂ v η = ∂ v H ( η ) ∂ a η − ∂ µ H ( η ) ∂ θ η Moreover, the functionals M and P , considered as auxiliary Hamiltonians, have as-sociated Hamilton vector fields
J M (cid:48) ( η ) = − ∂ θ η J P (cid:48) ( η ) = ∂ a η . This enables us to write(2.4)
J H (cid:48) ( η ) = ∂ v H ( η ) J P (cid:48) ( η ) + ∂ µ H ( η ) J M (cid:48) ( η ) . From this, we learn that W (cid:48) ( µ,a,θ,v ) ( η ) = 0, where(2.5) W ( µ,a,θ,v ) ( u ) def = − ∂ µ H ( η ) M ( u ) − ∂ v H ( η ) P ( u ) + H ( u ) . The functional L ( µ,a,θ,v ) ( u ) = W ( µ,a,θ,v ) ( u ) − W ( µ,a,θ,v ) ( η ) is the Lyapunov functionalused in the classical orbital stability theory for (1.1) due to Weinstein [19]. Effective dynamics
Now we turn to the double soliton problem and begin the proof of Theorem 1.1.Consider the two-soliton submanifold M of L given by M = { u z def = η ( · , µ , a , θ , v ) + η ( · , µ , a , θ , v ) } . Note that M is just the linear superposition of two single solitons. We adopt thenotation z = ( z , z , z , z , z , z , z , z ) = ( µ , a , µ , a , θ , v , θ , v ) , for coordinates on this manifold M . Next, we give the form of the symplectic orthog-onal projection operator Π z : T u z L → T z M ,
Note that T u z L is naturally identified with L . A consequence of the requirementthat (cid:104) f − Π z f, J − ∂ z (cid:96) u z (cid:105) = 0, (cid:96) = 1 , . . . , z f = (cid:88) (cid:96),m =1 (cid:104) f, J − ∂ z (cid:96) u z (cid:105) a (cid:96)m ( z ) ∂ z m u z where A ( z ) = ( a (cid:96)m ( z )) is the 8 × a (cid:96)m ( z ) = (cid:104) ∂ z (cid:96) u z , J − ∂ z m u z (cid:105) and A ( z ) − = ( a (cid:96)m ( z )) is the inverse matrix.Let i : M → L denote the inclusion. It follows from the definition of A ( z ) thatthe restricted symplectic form i ∗ ( ω ) takes the form(3.2) i ∗ ( ω ) = 12 (cid:88) (cid:96),m =1 a (cid:96)m dz (cid:96) ∧ dz m . It also follows by substitution into (3.1) thatΠ z J H (cid:48) ( u z ) = − (cid:88) (cid:96),m =1 ∂ z (cid:96) H ( u z ) a (cid:96)m ( z ) ∂ z m u z Consequently, the equation ∂ t u z = Π z J H (cid:48) ( u z ) is equivalent to the system of equations˙ z m = − (cid:88) (cid:96) =1 ∂ z (cid:96) H ( u z ) a (cid:96)m ( z ) m = 1 , . . . , , which are precisely Hamiltonian’s equations of motion for the restricted (to M ) Hamil-tonian z (cid:55)→ H ( u z ) with respect to the restricted (to M ) symplectic form i ∗ ( ω ).We propose to model the solution u to (2.1) by(3.3) u = u z + w where u z ∈ M is chosen so that the symplectic orthogonality conditions(3.4) (cid:104) w, J − ∂ z (cid:96) u z (cid:105) = 0 , (cid:96) = 1 , . . . , . LS SOLITON INTERACTION 13 hold. The fact that such a z exists follows from the implicit function theorem andthe assumed smallness of w . Note that if we assume u ( t ) solves (2.1), this inducestime dependence on the parameters z ∈ R . Lemma 3.1 (effective dynamics) . Suppose that u evolves according to (2.1) and z , w are defined by (3.3) so that the orthogonality conditions (3.4) hold. Then (3.5) (cid:107) ∂ t u z − Π z J H (cid:48) ( u z ) (cid:107) T z M (cid:46) (cid:107) w (cid:107) H + max ≤ n ≤ (cid:107) J − ∂ z n Π ⊥ z J H (cid:48) ( u z ) (cid:107) H . Equivalently, considering M as an 8-dimensional symplectic manifold equipped withthe symplectic form i ∗ ( ω ) given in (3.2) , the Hamilton’s equations of motion for z induced by the restricted Hamiltonian z (cid:55)→ H ( u z ) approximately hold as follows: (3.6) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ z m + (cid:88) (cid:96) =1 ∂ z (cid:96) H ( u z ) a (cid:96)m ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) (cid:107) w (cid:107) H + max ≤ n ≤ (cid:107) J − ∂ z n Π ⊥ z J H (cid:48) ( u z ) (cid:107) H m = 1 , . . . , (cid:107) · (cid:107) T z M is the one induced by the metric (cid:104)· , ·(cid:105) u z . As T z M is finite-dimensional, we have the norm-equivalence to (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) (cid:96) =1 γ ( z (cid:96) ) ∂ z (cid:96) u z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T z M ∼ max ≤ (cid:96) ≤ | γ ( z (cid:96) ) | Proof.
Since u solves (2.1), we obtain from (3.3) the equation for w :(3.7) ∂ t w = − ( ∂ t u z − Π z J H (cid:48) ( u z )) + Π ⊥ z J H (cid:48) ( u z ) + J H (cid:48)(cid:48) ( u z ) w + O H ( (cid:107) w (cid:107) H )By applying ∂ t to (3.4), we obtain0 = (cid:104) ∂ t w, J − ∂ z n u z (cid:105) + (cid:104) w, J − ∂ z n ∂ t u z (cid:105) Here we have used that ∂ t ∂ z n = ∂ z n ∂ t , which holds provided we adopt the con-vention that ∂ z (cid:96) ˙ z m = 0 for all 1 ≤ (cid:96), m ≤
8. Substituting (3.7) and using that (cid:104) Π ⊥ z J H (cid:48) ( u z ) , J − ∂ z n u z (cid:105) = 0, we obtain(3.8) 0 = A + B + C + Dwhere A = −(cid:104) ∂ t u z − Π z J H (cid:48) ( u z ) , J − ∂ z n u z (cid:105) B = (cid:104)
J H (cid:48)(cid:48) ( u z ) w, J − ∂ z n u z (cid:105) C = (cid:104) w, J − ∂ z n ∂ t u z (cid:105) D = (cid:104) O ( w ) , J − ∂ z n u z (cid:105) Note that here w is properly understood as an element of T u z L and in (3.3) we mean that,starting at u z we take the flow-forward (by “time” 1) in the direction w . However, using the naturalidentification between T u z L and L , (3.3) makes sense as an equation involving functions in L . Since J ∗ J − = − H (cid:48)(cid:48) ( u z ) is self-adjoint,B = −(cid:104) w, H (cid:48)(cid:48) ( u z ) ∂ z n u z (cid:105) = −(cid:104) w, ∂ z n H (cid:48) ( u z ) (cid:105) = −(cid:104) w, J − ∂ z n J H (cid:48) ( u z ) (cid:105) Hence B + C = (cid:104) w, J − ∂ z n ( ∂ t u z − J H (cid:48) ( u z )) (cid:105) = (cid:104) w, J − ∂ z n ( ∂ t u z − Π z J H (cid:48) ( u z )) (cid:105) − (cid:104) w, J − ∂ z n Π ⊥ z J H (cid:48) ( u z ) (cid:105) Let R = ∂ t u z − Π z J H (cid:48) ( u z ) ∈ T z M , and expand with respect to the basis of T z M as R = (cid:88) (cid:96) =1 γ (cid:96) ( z ) ∂ z (cid:96) u z . It follows from (3.8) that(3.9) (cid:104)
R, J − ∂ z n u z (cid:105) = (cid:104) w, J − ∂ z n R (cid:105) − (cid:104) w, J − ∂ z n Π ⊥ z J H (cid:48) ( u z ) (cid:105) + O ( (cid:107) w (cid:107) H ) . We have ∂ z n R = (cid:88) (cid:96) =1 ∂ z n γ (cid:96) ( z ) ∂ z (cid:96) u z + (cid:88) (cid:96) =1 γ (cid:96) ( z ) ∂ z n ∂ z (cid:96) u z . Since w ∈ T z M ⊥ , (cid:104) w, J − ∂ z n R (cid:105) = (cid:88) (cid:96) =1 γ (cid:96) ( z ) (cid:104) w, J − ∂ z n ∂ z (cid:96) u z (cid:105) and hence(3.10) |(cid:104) w, J − ∂ z n R (cid:105)| (cid:46) (cid:107) w (cid:107) H (cid:107) R (cid:107) T z M . The lemma follows from (3.9), (cid:107) R (cid:107) T z M = max ≤ n ≤ |(cid:104) R, J − ∂ z n u z (cid:105)| , (3.10), andCauchy-Schwarz. (cid:3) In our case we shall have (cid:107) J − ∂ z n Π ⊥ z J H (cid:48) ( u z ) (cid:107) H (cid:46) h − . We carry out computations of (3.5) in Appendix A and show that (3.5) is equivalentto (1.19), with error terms O ( h − ), even without the even/odd assumption on thesolution. It is further shown in Appendix A that when the even/odd assumption isimposed and the integrals in (1.19) are explicitly computed, we obtain ˙ µ = ( − σ (8 a − ve − a + O ( h − )˙ a = µ − v + ( − σ ( − a + 23 π ) ve − a + O ( h − )˙ θ = 12 µ + 12 v µ − + 18( − σ e − a + O ( h − )˙ v = − − σ e − a + O ( h − ) LS SOLITON INTERACTION 15
The solution ( µ, a, θ, v ) is adequately approximated by the ODEs appearing in thestatement of Theorem 1.1. 4.
Approximate solution
By Lemma 3.1 and (3.7), the equation for w is (cid:40) ∂ t w = J H (cid:48)(cid:48) ( u z ) w + Π ⊥ z J H (cid:48) ( u z ) + O H ( (cid:107) w (cid:107) H + h − ) w (cid:12)(cid:12) t =0 = w The next step is to show that there exists a function ν z ( x ) such that (cid:107) ν z (cid:107) H (cid:46) h ,whose only time dependence occurs through the parameter z , such that(4.1) ∂ t ν z = J H (cid:48)(cid:48) ( u z ) ν z + Π ⊥ z J H (cid:48) ( u z ) + O H ( h − + (cid:107) w (cid:107) H )Here it is assumed that z ∈ R evolves according to Lemma 3.1, i.e.(4.2) ∂ t u z = Π z J H (cid:48) ( u z ) + O H ( h − + (cid:107) w (cid:107) H )The initial data ν z (cid:12)(cid:12) t =0 is not prescribed but our structural assumption on ν z is fairlyrigid. Note that given (4.2), the assertion that ν z solve (4.1) is equivalent to thestatement that ˜ u z def = u z + ν z solve(4.3) ∂ t ˜ u z = J H (cid:48) (˜ u z ) + O H ( h − + (cid:107) w (cid:107) H ) . This is an approximate solution to (2.1) (that does not, in general, satisfy the specifiedinitial data).Let g : L → L be the operator attached to the parameters ( µ, a, θ, v ) that actson a function ρ as follows:(4.4) ( gρ )( x ) = e iθ e iµ − v ( x − a ) µρ ( µ ( x − a )) . The inverse action is g − ρ ( x ) = e − iθ e − iµ − vx µ − ρ ( µ − x + a ) . The adjoint action g ∗ with respect to (cid:104)· , ·(cid:105) is g ∗ ρ ( x ) = e − iθ e − iµ − vx ρ ( µ − x + a ) = µg − ρ ( x ) . Denote φ ( x ) = sech x . Then u z = g φ + g φ . We look for a solution ν z to (4.1) in theform(4.5) ν z = (cid:88) j =1 α j g j ρ j , where α j = α j ( µ , a , θ , v , µ , a , θ , v ) and g j is the operator corresponding to( µ j , a j , θ j , v j ). That is, we assume ν z can be decomposed into two pieces, each ofwhich can be pulled back to a stationary equation and solved by operator inversion.The time dependence of ν z occurs only through z . The next step is to substitute (4.5) into (4.1). The resulting equation simplifiesprovided we assume that each ρ j satisfies | ρ j ( x ) | (cid:46) h e ( − | x | for | x | ≥ O H ( h − ). In this case, (4.1) will be satisfied provided for both j = 1 ,
2, we have ∂ t ( α j g j ρ j ) = J H (cid:48)(cid:48) ( g j φ )( α j g j ρ j )+Π ⊥ z J (( g j φ ) g − j φ +2 | g j φ | g − j φ )+ O H ( h − + (cid:107) w (cid:107) H )In the proof, we delete the j -subscripts, denote ˜ g = g − j and moreover assume that˙ α j = O ( h ). Then we aim to solve ∂ t ( gρ ) = J H (cid:48)(cid:48) ( gφ )( gρ ) + α − Π ⊥ J (( gφ ) ˜ gφ + 2 | gφ | ˜ gφ ) + O H ( h − + (cid:107) w (cid:107) H )The form of the operator Π ⊥ can be simplified, since we only need to keep the O (1)and O ( h ) parts. This equation takes the form(4.6) ∂ t ( gρ ) = J H (cid:48)(cid:48) ( gφ )( gρ ) + α − Π ⊥ J gf + O H ( h − + (cid:107) w (cid:107) H )where, in the case j = 1,(4.7) f = g − [( g φ ) g φ ] + 2 g − [ | g φ | g φ ]= e iω e iω x µ j µ − j φ ( µ − j µ − j x + ( a j − a − j )) φ ( x ) + 2 e iω e iω x µ j µ − j φ ( µ − j µ − j x + ( a j − a − j )) φ ( x ) with ω = θ − θ − µ − − j v − j ( a j − a − j ) ω = µ − j v j − µ − − j µ − j v − j ω = θ − θ + µ − − j v − j ( a j − a − j ) ω = µ − j v j + µ − − j µ − j v − j In the case j = 2,(4.8) f = g − [( g φ ) g φ ] + 2 g − [ | g φ | g φ ] . with a similar expansion. The only important feature of these expressions is that e (1 − ) (cid:104) x (cid:105) f = O ( h ) and e (1 − ) (cid:104) x (cid:105) ∂ t f = O ( h ) when θ − θ is a constant (as in the caseof the even/odd symmetry assumption in Theorem 1.1.Now we begin the task of pulling back (4.6) – applying g ∗ to (4.6), we obtain(4.9) g ∗ ∂ t gρ = g ∗ [ J H (cid:48)(cid:48) ( gφ )( gρ )] + α − g ∗ Π ⊥ J gf + O H ( h − + (cid:107) w (cid:107) H ) . First, we aim to simplify the term g ∗ [ J H (cid:48)(cid:48) ( gφ )( gρ )] in (4.9). Let K g ( φ ) = H ( gφ ). Itfollows that K (cid:48) g ( φ ) = g ∗ H (cid:48) ( gφ ) and K (cid:48)(cid:48) g ( φ ) = g ∗ [ H (cid:48)(cid:48) ( gφ )( g • )]. By direct substitution,we compute: K g ( u ) = µ H ( u ) + µvP ( u ) + 12 v µ − M ( u ) LS SOLITON INTERACTION 17
Since g ∗ J = J g ∗ , we have(4.10) g ∗ [ J H (cid:48)(cid:48) ( gφ )( g • )] = J K (cid:48)(cid:48) g ( φ )= µ J H (cid:48)(cid:48) ( φ ) + µvJ P (cid:48)(cid:48) ( φ ) + v µ − J M (cid:48)(cid:48) ( φ )Second, we seek to simplify the term g ∗ ∂ t gρ in (4.9). Define the operators˜ ∂ µ def = ∂ µ (cid:12)(cid:12) (1 , , , = ∂ x x ˜ ∂ a def = ∂ a (cid:12)(cid:12) (1 , , , = − ∂ x ˜ ∂ θ def = ∂ θ (cid:12)(cid:12) (1 , , , = i ˜ ∂ v def = ∂ v (cid:12)(cid:12) (1 , , , = ix Let(4.11) ¯ ∂ µ def = g ∗ ∂ µ g = ∂ x x − iµ − vx = ˜ ∂ µ − µ − v ˜ ∂ v ¯ ∂ a def = g ∗ ∂ a g = − µ ∂ x − iv = µ ˜ ∂ a − v ˜ ∂ θ ¯ ∂ θ def = g ∗ ∂ θ g = iµ = µ ˜ ∂ θ ¯ ∂ v def = g ∗ ∂ v g = iµ − x = µ − ˜ ∂ v It follows from the chain rule that g ∗ ∂ t g = ˙ µ ¯ ∂ µ + ˙ a ¯ ∂ a + ˙ θ ¯ ∂ θ + ˙ v ¯ ∂ v . Using (3.5),(4.12) g ∗ ∂ t g = vµ − ¯ ∂ a + µ ¯ ∂ θ + O H ( h )= vµ ˜ ∂ a + µ ˜ ∂ θ + O H ( h )Finally, we aim to simplify the term g ∗ Π ⊥ J gf in (4.9). We will show that(4.13) g ∗ Π ( µ,a,θ,v ) J gf = µ Π (1 , , , J f
Using that J ∗ J − = − g ∗ Π J gf = (cid:104) J gf, J − ∂ a gφ (cid:105) g ∗ ∂ v gφ − (cid:104) J gf, J − ∂ v gφ (cid:105) g ∗ ∂ a gφ + (cid:104) J gf, J − ∂ µ gφ (cid:105) g ∗ ∂ θ gφ − (cid:104) J gf, J − ∂ θ gφ (cid:105) g ∗ ∂ µ gφ = −(cid:104) f, ¯ ∂ a φ (cid:105) ¯ ∂ v φ + (cid:104) f, ¯ ∂ v φ (cid:105) ¯ ∂ a φ − (cid:104) f, ¯ ∂ µ φ (cid:105) ¯ ∂ θ φ + (cid:104) f, ¯ ∂ θ φ (cid:105) ¯ ∂ µ φ Substituting (4.11), after a few cancelations we obtain µ − g ∗ Π J gf = −(cid:104) f, ˜ ∂ a φ (cid:105) ˜ ∂ v φ + (cid:104) f, ˜ ∂ v φ (cid:105) ˜ ∂ a φ − (cid:104) f, ˜ ∂ µ φ (cid:105) ˜ ∂ θ φ + (cid:104) f, ˜ ∂ θ φ (cid:105) ˜ ∂ µ φ = + (cid:104) J f, J − ˜ ∂ a φ (cid:105) ˜ ∂ v φ − (cid:104) J f, J − ˜ ∂ v φ (cid:105) ˜ ∂ a φ + (cid:104) J f, J − ˜ ∂ µ φ (cid:105) ˜ ∂ θ φ − (cid:104) J f, J − ˜ ∂ θ φ (cid:105) ˜ ∂ µ φ which establishes (4.13). Note that it follows from (4.13) that g ∗ Π ⊥ ( µ,a,θ,v ) gJ f = µ Π ⊥ (1 , , , J f
Using the expressions (4.10), (4.12), and (4.13), the equation (4.9) converts to vµ ˜ ∂ a ρ + µ ˜ ∂ θ ρ + µ∂ t ρ = µ J H (cid:48)(cid:48) ( φ )( ρ ) + µvJ P (cid:48)(cid:48) ( φ )( ρ )+ α − µ Π ⊥ (1 , , , J f + O H ( h − + (cid:107) w (cid:107) H )Noting that J P (cid:48)(cid:48) ( φ ) = ˜ ∂ a and J M (cid:48)(cid:48) ( φ ) = − ˜ ∂ θ , the equation becomes µ J M (cid:48)(cid:48) ( φ ) ρ + µ J H (cid:48)(cid:48) ( φ ) ρ − µ∂ t ρ = − α − µ Π ⊥ (1 , , , J f + O H ( h − + (cid:107) w (cid:107) H )Hence we see we should take α = µ − so that the equation becomes J M (cid:48)(cid:48) ( φ ) ρ + J H (cid:48)(cid:48) ( φ ) ρ − µ − ∂ t ρ = − Π ⊥ (1 , , , J f + O H ( h − + (cid:107) w (cid:107) H )Now apply J − to obtain the equation(4.14) Sρ = J − µ − ∂ t ρ − J − Π ⊥ (1 , , , J f + O ( h − + (cid:107) w (cid:107) H )where the operator S ( ρ ) def = M (cid:48)(cid:48) ( φ )( ρ ) + H (cid:48)(cid:48) ( φ )( ρ ) = ρ − ∂ x ρ − | φ | ρ − φ ¯ ρ is self-adjoint with respect to the inner product (cid:104) u, v (cid:105) = Re (cid:82) u ¯ v . The kernel isspanned by ˜ ∂ a φ and ˜ ∂ θ φ . Lemma 4.1 (properties of S ) . (1) For any f ∈ H , let F = J − Π ⊥ (1 , , , J f . Then F satisfies the orthogonalityconditions (cid:104) F, ˜ ∂ θ φ (cid:105) = 0 , (cid:104) F, ˜ ∂ a φ (cid:105) = 0(4.15) (cid:104) F, ˜ ∂ µ φ (cid:105) = 0 , (cid:104) F, ˜ ∂ v φ (cid:105) = 0(4.16)(2) For any F satisfying (4.15) , S − F is defined and satisfies the boundednessproperties (cid:107) S − F (cid:107) H (cid:46) (cid:107) F (cid:107) L , (4.17) (cid:107) e σ (cid:104) x (cid:105) S − F (cid:107) H (cid:46) (cid:107) e σ (cid:104) x (cid:105) F (cid:107) L . (4.18) for ≤ σ < . (3) For any F satisfying (4.15) and (4.16) , S − F satisfies the orthogonality prop-erties (cid:104) S − F, ˜ ∂ θ φ (cid:105) = 0 , (cid:104) S − F, ˜ ∂ a φ (cid:105) = 0(4.19) (cid:104) J − S − F, ˜ ∂ θ φ (cid:105) = 0 , (cid:104) J − S − F, ˜ ∂ a φ (cid:105) = 0(4.20) LS SOLITON INTERACTION 19
Proof.
Item (1) is immediate from the definition of Π (1 , , , . For item (2), we recallthat ker S = span { ˜ ∂ a φ, ˜ ∂ θ φ } and moreover, 0 is an isolated point in the spectrum of S . Thus S − : (ker S ) ⊥ → (ker S ) ⊥ is bounded as an operator on L . The inequality(4.17) follows from this and elliptic regularity. To prove (4.18), it suffices to showthat for any G and any | σ | <
1, we have(4.21) (cid:107) G (cid:107) H (cid:46) (cid:107) e σx Se − σx G (cid:107) L + (cid:107) e − | x | G (cid:107) L Indeed, (4.18) follows by taking G = e σx S − F , appealing to (4.17), and separatelyconsidering σ > σ < | σ | <
1. To establish (4.21), we calculate(4.22) e σx Se − σx G = ( S + σ∂ x − σ ) G = ( 12 (1 − σ ) + σ∂ x − ∂ x ) G − φ G − φ ¯ G and hence ( 12 (1 − σ ) + σ∂ x − ∂ x ) G = e σx Se − σx G + 2 φ G + φ ¯ G On the left-hand side, we have an operator with symbol (1 − σ ) + σiξ + ξ , whichdominates (cid:104) ξ (cid:105) under our assumption on σ . From this and the fact that | φ ( x ) | ≤ e − | x | , we conclude (4.21).For item (3), (4.19) follows from the fact that S − : (ker S ) ⊥ → (ker S ) ⊥ . Toestablish (4.20), we note that by (4.16),0 = (cid:104) F, ˜ ∂ µ φ (cid:105) = (cid:104) S − F, S ˜ ∂ µ φ (cid:105) and similarly 0 = (cid:104) F, ˜ ∂ v φ (cid:105) = (cid:104) S − F, S ˜ ∂ v φ (cid:105) and thus it suffices to establish that S ( ˜ ∂ v φ ) = J − ˜ ∂ a φ and S ( ˜ ∂ µ φ ) = J − ˜ ∂ θ φ . Toprove these equalities, recall(4.23) 0 = W (cid:48) ( η ) = ( 12 µ + 12 µ − v ) M (cid:48) ( η ) − µ − vP (cid:48) ( η ) + H (cid:48) ( η )Taking ∂ v and evaluating at ( µ, a, θ, v ) = (1 , , ,
0) gives S ( ˜ ∂ v φ ) = P (cid:48) ( φ ) = J − ˜ ∂ a φ . Taking ∂ µ of (4.23) and evaluating at ( µ, a, θ, v ) = (1 , , ,
0) gives S ( ˜ ∂ µ φ ) = − M (cid:48) ( φ ) = J − ˜ ∂ θ µ (cid:3) Recall that the task is to solve (4.14) where f is either (4.7) or (4.8). At this point,we impose the even/odd solution assumption as in Theorem 1.1 which implies that θ − θ is constant. The other time dependent parameters in (4.7), (4.8) are all slowlyvarying so that ∂ t f j = O ( h ). Thus, we can solve (4.14) by iteration. Let (4.24) ρ = − S − J − Π ⊥ (1 , , , J f
By Lemma 4.1(1)(2), this is well-defined with ρ = O ( h − ) and satisfying all theneeded regularity properties. With ρ as yet undefined, we plug ρ + ρ into (4.14)to obtain Sρ = J − µ − ∂ t ρ + J − µ − ∂ t ρ + O ( h − + (cid:107) w (cid:107) H )As mentioned previously, ∂ t f = O ( h − ) and thus J − µ − ∂ t ρ = − J − µ − S − J − Π ⊥ (1 , , , J ∂ t f is also O ( h − ). By Lemma 4.1(3), in particular (4.20), with F = J − Π ⊥ (1 , , , J ∂ t f ,we have that ˜ F def = J − µ − ∂ t ρ = − J − µ − S − F satisfies the condition (4.15), and hence we can apply Lemma 4.1(2) with F replacedby ˜ F . That is, the function(4.25) ρ = S − ˜ F = − S − J − µ − S − J − Π ⊥ (1 , , , J ∂ t f satisfies all the needed regularity properties. Note further that ∂ t ρ = O ( h − ). Uponsubstituting ρ + ρ into (4.14) with ρ defined by (4.24) and ρ defined by (4.25), wefind that equality holds with O ( h − ) error.Thus we have successfully constructed a solution to the approximate equation (4.1).We summarize our conclusions in the next lemma. Lemma 4.2 (approximate solution) . Recall the operator g j associated to ( µ j , a j , θ j , v j ) defined in (4.4) and f j defined in (4.7) ( j = 1 ) or (4.8) ( j = 2 ). Let ρ j be given by (4.24) and then let ρ j be given by (4.25) . Then ρ kj for ≤ j, k ≤ satisfy (cid:107) e (1 − ) (cid:104) x (cid:105) ρ kj (cid:107) H (cid:46) h k − , Let ν z = µ g ( ρ + ρ ) + µ g ( ρ + ρ ) Suppose that the parameter z ∈ R evolves according to the ODEs obtained fromLemma 3.1 (in the same phase or opposite phase case). Then ν z ( x ) solves (4.1) . As indicated earlier, ρ can stand for either ρ or ρ . The superscript introduced here is differentand meant to indicate part of an asymptotic expansion for either function. In other words, we have ρ j = ρ j + ρ j for both j = 1 , LS SOLITON INTERACTION 21 Lyapunov functional
The final step is to show that the true solution u to (2.1) is approximately theapproximate solution ˜ u z = u z + ν z . For this purpose, we introduce a Lyapunovfunctional. First, some general considerations. We consider the “perturbed” 8-dimensional manifold ˜ M = { ˜ u z | z ∈ R } Introduce the notation ˜ w = u − ˜ u z (so that w = ˜ w + ν z ). Now it follows from (4.1)that(5.1) ∂ t ˜ u z = J H (cid:48) (˜ u z ) + F . where F = O H ( h − + (cid:107) ˜ w (cid:107) H ).Suppose that W z : L → R is a densely defined functional. We write ∂ z (cid:96) W z : L → R to indicate partial derivatives with respect to z and W (cid:48) z ( u ) ∈ T ∗ u L (cid:39) metric g T u L (cid:39) L to indicate partial derivatives with respect to u (ignoring the interdependence between z and u given by (3.3), (3.4)).Suppose that W z can be extended to a differentiable functional H → R ; thenfor each u ∈ H , we have a bounded linear map W (cid:48) z ( u ) : H → R which, underthe aforementioned identification, becomes a function belonging to H − . In fact, ourchoice of W z is differentiable at all orders as a map H → R , which is to say that W ( k ) z ( u ) : H × · · · × H (cid:124) (cid:123)(cid:122) (cid:125) k copies → R is a bounded k -multilinear map.We further assume that ∂ z (cid:96) W z ( u ) = 0 unless | ˙ z (cid:96) | (cid:46) h . Let(5.2) L z ( u ) = W z ( u ) − W z (˜ u z ) − (cid:104) W (cid:48) z (˜ u z ) , ˜ w (cid:105) . That is, L z ( u ) is the quadratic part of W z ( u ) above the base manifold ˜ M . Nowviewing u = u ( t ) and z = z ( t ) in accordance with (3.3), (3.4) (and thus reinstatingthe interdependence between z and u ), we have, for any functional G z : L → R , ∂ t G z ( u ) = (cid:104) G (cid:48) z ( u ) , ∂ t u (cid:105) + (cid:88) k =1 [ ∂ z k G ]( u ) ˙ z k . This leads to:
Lemma 5.1.
Suppose that u solves (2.1) and z evolves so that ˜ u z solves (5.1) , andthat L z ( u ) is given by (5.2) , the quadratic part of W z ( u ) above ˜ M . Then (5.3) ∂ t L z ( u ) = { H, W z } ( u ) − { H, W z } (˜ u z ) − (cid:104){ H, W z } (cid:48) (˜ u z ) , ˜ w (cid:105) − E + E , where E = (cid:104) W (cid:48)(cid:48) (˜ u z ) F, ˜ w (cid:105) + (cid:104) W (cid:48) (˜ u z ) , [ J H (cid:48) ( u ) − J H (cid:48) (˜ u z ) − J H (cid:48)(cid:48) (˜ u z ) ˜ w ] (cid:105) and E = (cid:88) k =1 ([ ∂ z k W z ]( u ) − [ ∂ z k W z ](˜ u z ) − (cid:104) [ ∂ z k W (cid:48) z ](˜ u z ) , ˜ w (cid:105) ) ˙ z k In other words, ∂ t L z ( u ) is, up to error E and E , the quadratic part of { H, W z } ( u ) above ˜ M . Note that E just involves the quadratic part of [ ∂ z k W ]( u ) above ˜ M . In the typical application of this lemma (as for our W z , defined below), we havebounded operators W (cid:48)(cid:48) z (˜ u z ) : H → H − and W (cid:48)(cid:48)(cid:48) z (˜ u z ) : H × H → L which impliesthe bound | E | (cid:46) (cid:107) F (cid:107) H (cid:107) ˜ w (cid:107) H + (cid:107) W (cid:48) (˜ u z ) (cid:107) H (cid:107) ˜ w (cid:107) H Thus, one just needs (cid:107) F (cid:107) H (cid:46) h and (cid:107) W (cid:48) (˜ u z ) (cid:107) H (cid:46) h ; in our case we in fact havethe stronger statements (cid:107) F (cid:107) H (cid:46) h − and (cid:107) W (cid:48) (˜ u z ) (cid:107) H (cid:46) h − . Moreover, in our casewe will have | E | (cid:46) h − (cid:107) ˜ w (cid:107) H since ˙ µ, ˙ v = O ( h − ). Proof.
By (5.2), ∂ t L z ( u ) = ∂ t W z ( u ) − ∂ t W z (˜ u z ) − ∂ t (cid:104) W (cid:48) z (˜ u z ) , ˜ w (cid:105) We compute each of the three terms on the right-hand side separately. ∂ t W z ( u ) = (cid:104) W (cid:48) z ( u ) , ∂ t u (cid:105) + (cid:88) k =1 [ ∂ z k W z ]( u ) ˙ z k = (cid:104) W (cid:48) z ( u ) , J H (cid:48) ( u ) (cid:105) + (cid:88) k =1 [ ∂ z k W z ]( u ) ˙ z k (5.4)where we invoked (2.1). Second, we compute ∂ t W z (˜ u z ) = (cid:104) W (cid:48) z (˜ u z ) , ∂ t ˜ u z (cid:105) + (cid:88) k =1 [ ∂ z k W z ](˜ u z ) ˙ z k = (cid:104) W (cid:48) z (˜ u z ) , J H (cid:48) (˜ u z ) (cid:105) + (cid:104) W (cid:48) z (˜ u z ) , F (cid:105) + (cid:88) k =1 [ ∂ z k W z ](˜ u z ) ˙ z k (5.5) LS SOLITON INTERACTION 23 where we invoked (5.1). Finally, we compute ∂ t (cid:104) W (cid:48) z (˜ u z ) , u − ˜ u z (cid:105) = (cid:104) W (cid:48)(cid:48) z (˜ u z ) ∂ t ˜ u z , u − ˜ u z (cid:105) + (cid:104) W (cid:48) z (˜ u z ) , ∂ t u − ∂ t ˜ u z (cid:105) + (cid:88) k =1 (cid:104) [ ∂ z k W ] (cid:48) (˜ u z ) , u − ˜ u z (cid:105) ˙ z k = (cid:104) W (cid:48)(cid:48) z (˜ u z ) J H (cid:48) (˜ u z ) , ˜ w (cid:105) + (cid:104) W (cid:48)(cid:48) (˜ u z ) F, ˜ w (cid:105) + (cid:104) W (cid:48) z (˜ u z ) , J H (cid:48) ( u ) − J H (cid:48) (˜ u z ) (cid:105)− (cid:104) W (cid:48) (˜ u z ) , F (cid:105) + (cid:104) (cid:88) k =1 [ ∂ z k W ] (cid:48) (˜ u z ) , ˜ w (cid:105) ˙ z k (5.6)Taking (5.4) minus (5.5) minus (5.6), noting the cancelation of + (cid:104) W (cid:48) z (˜ u z ) , F (cid:105) in (5.5)with −(cid:104) W (cid:48) (˜ u z ) , F (cid:105) in (5.6), we obtain (5.3). (cid:3) To produce W z ( u ), we use an idea of Martel-Merle-Tsai [10]. LetΨ( x ) = (cid:40) x ≥
10 if x ≤ − (cid:48) ( x )) (cid:46) min(Ψ( x ) , − Ψ( x )). Set δ = 4 / (log h − ) = 4 /a , so 0 < δ (cid:28) ψ ( x ) = Ψ( δx ) and ψ ( x ) = 1 − Ψ( δx ), and set M j ( u ) = M ( ψ / j u ) and P j ( u ) = P ( ψ / j u ). Define(5.7) W z ( u ) def = − (cid:88) j =1 ∂H ( η j ) ∂µ j M j ( u ) − (cid:88) j =1 ∂H ( η j ) ∂v j P j ( u ) + H ( u )= 12 (cid:88) j =1 ( µ j + µ − j v j ) M j ( u ) − (cid:88) j =1 µ − j v j P j ( u ) + H ( u )The Lyapunov functional L z ( u ) we use is then defined as in (5.2).Lemma 5.1 facilitates the computation of ∂ t L z ( u ), since W z ( u ) is built from “nearlyconserved” quantities. Indeed, we have the following Poisson brackets: { H, M j } ( u ) = 12 Im (cid:90) ψ (cid:48) j ¯ uu x { H, P j } ( u ) = (cid:90) ψ (cid:48) j ( 12 | u x | − | u | ) − (cid:90) ψ (cid:48)(cid:48)(cid:48) j | u | It thus follows from Lemma 5.1 that(5.8) ∂ t L z ( u ) = 12 (cid:88) j =1 ( µ j + µ − j v j ) (cid:104){ H, M j } (cid:48)(cid:48) (˜ u z ) ˜ w, ˜ w (cid:105) − (cid:88) j =1 µ − j v j (cid:104){ H, P j } (cid:48)(cid:48) (˜ u z ) ˜ w, ˜ w (cid:105) + O ( (cid:107) w (cid:107) H ) − E + E For our choice of W z ( u ), as remarked earlier, we have suitable bounds for E and E .Moreover, once one imposes the even/odd solution assumption of Theorem 1.1, wehave µ = µ and v = v , so the first term in (5.8) disappears. Hence(5.9) | ∂ t L z ( u ) | (cid:46) (( | v | + | v | ) δ + h ) (cid:107) ˜ w (cid:107) H + h (cid:107) ˜ w (cid:107) H + (cid:107) ˜ w (cid:107) H Since | v j | (cid:46) h − log h − and δ ∼ (log h − ) − , the term ( | v | + | v | ) δ (cid:46) h .Now we turn to the matter of obtaining a lower bound for L z ( u ). First note that (cid:104) W (cid:48)(cid:48) z (˜ u z ) ˜ w, ˜ w (cid:105) = L z ( u ) + O ( (cid:107) ˜ w (cid:107) H ) . Given that (cid:107) ˜ ν z (cid:107) H (cid:46) h − , we have(5.10) (cid:104) W (cid:48)(cid:48) z ( u z ) w, w (cid:105) = L z ( u ) + O ( h − ) + O ( h ) (cid:107) w (cid:107) H . The needed lower bound for the left-hand side will be established below in Lemma5.2.For the single-soliton case, we have coercivity for the classical functional fromWeinstein [19], which we now recall. Taking η = η ( · , µ, a, θ, v ) and R ( µ,a,θ,v ) ( u ) def = − ∂H ( η ) ∂µ M ( u ) − ∂H ( η ) ∂v P ( u ) + H ( u )= 12 ( µ + µ − v ) M ( u ) − µ − vP ( u ) + H ( u )then(5.11) (cid:107) w (cid:107) H (cid:46) (cid:104) R (cid:48)(cid:48) ( η ) w, w (cid:105) , provided we assume the orthogonality conditions (cid:104) w, J − ∂ µ η (cid:105) = 0 , (cid:104) w, J − ∂ a η (cid:105) = 0 , (cid:104) w, J − ∂ θ η (cid:105) = 0 , (cid:104) w, J − ∂ v η (cid:105) = 0 . A direct proof of (5.11) is possible; see [7, Prop. 4.1].We now prove a similar argument for the double-soliton functional W z ( u ) definedin (5.7). Before proceeding, we record the formulae(5.12) M (cid:48)(cid:48) j ( u ) = ψ j P (cid:48)(cid:48) j ( u ) = − iψ / j ∂ x ψ / j = − iψ (cid:48) j − iψ j ∂ x H (cid:48)(cid:48) ( u ) = − ∂ x − | u | − u C where C denotes the operator of complex conjugation. In fact, this is more easily seen by observing that once µ = µ and v = v , we have that thefirst term in (5.7) becomes M ( u ), whose Poisson bracket vanishes. We included the localization inthis term to illustrate the difficulty in treating the asymmetric case – one would not have that thefirst term in (5.8) is O ( h ). LS SOLITON INTERACTION 25
Lemma 5.2.
Suppose w satisfies the orthogonality conditions (3.4) . Then (5.13) (cid:107) w (cid:107) H (cid:46) (cid:104) W (cid:48)(cid:48) z ( u z ) w, w (cid:105) . Proof.
Denote w j = ψ / j w , j = 1 ,
2. Note that w + w (cid:54) = w , although 1 = ψ + ψ ≤ ψ / + ψ / ≤
2. Define functionals W j ( u ) = − ∂H ( η j ) ∂µ j M ( u ) − ∂H ( η j ) ∂v j P ( u ) + H ( u )= ( µ j + µ − j v j ) M ( u ) − µ − j v j P ( u ) + H ( u )We claim that(5.14) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) W (cid:48)(cid:48) ( u z ) w, w (cid:105) − (cid:88) j =1 (cid:104) W (cid:48)(cid:48) j ( η j ) w j , w j (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) δ (cid:107) w (cid:107) H and(5.15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:107) w (cid:107) H − (cid:88) j =1 (cid:107) w j (cid:107) H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) δ (cid:107) w (cid:107) H We now establish (5.14). Note that (see (5.12)) (cid:104) W (cid:48)(cid:48) ( u z ) w, w (cid:105) − (cid:88) j =1 (cid:104) W (cid:48)(cid:48) j ( η j ) w j , w j (cid:105) = (cid:104) ( H (cid:48)(cid:48) ( u z ) − ψ / H (cid:48)(cid:48) ( η ) ψ / − ψ / H (cid:48)(cid:48) ( η ) ψ / ) w, w (cid:105) The operator appearing on the right-hand side can be decomposed into A + A + A where A = −
12 ( ∂ x − ψ / ∂ x ψ / − ψ / ∂ x ψ / ) A = − | u z | − ψ | η | − ψ | η | ) A = − ( u z − ψ η − ψ η ) C We compute A explicitly: A = (cid:88) j =1 ( − ψ (cid:48) j ∂ x − ψ − j ( ψ (cid:48) j ) − ψ (cid:48)(cid:48) j ) = − (cid:88) j =1 ψ − j ( ψ (cid:48) j ) where we have used that ψ + ψ = 1 in the second equality. We have ( ψ (cid:48) j ) (cid:46) δ ψ j bythe corresponding property of Ψ and thus A is a multiplication operator with symbolbounded by δ . By the support properties of ψ , ψ , we obtain that the multiplicationoperators A , A have symbols bounded by h . This completes the proof of (5.14),and the proof of (5.15) is similar. By the orthogonality conditions (3.4), (cid:104) w , J − ∂ z j u z (cid:105) = −(cid:104) (1 − ψ / ) w, J − ∂ z j u z (cid:105) , and we have, for example (cid:104) w , J − ∂ µ u z (cid:105) = −(cid:104) (1 − ψ / ) w, J − ∂ µ u z (cid:105) = −(cid:104) (1 − ψ / ) w, J − ∂ µ η (cid:105) (cid:46) h / (cid:107) w (cid:107) L , due to the fact that (cid:107) (1 − ψ / ) J − ∂ µ η (cid:107) L (cid:46) h / . Hence, by the coercivity of theclassical Lyapunov functional (see discussion surrounding (5.11)), we have that (cid:88) j =1 (cid:104) W (cid:48)(cid:48) j ( η j ) w j , w j (cid:105) + h (cid:107) w (cid:107) L (cid:38) (cid:88) j =1 (cid:107) w j (cid:107) H , From this and (5.14), (5.15), we obtain (5.13). (cid:3) Conclusion of proof
In this section, we conclude the proof of Theorem 1.1.Recall that h = e − a which implies that a = log h − , and that we are in theeven/odd solution setting with (1.7), (1.8) in place.We introduce 0 < δ (cid:28)
1. The constant δ is absolute and is chosen sufficientlysmall in terms of the accumulation of numerous other absolute constants appearingin several estimates. In our argument, c will represent a large absolute constant thatmay change (typically enlarge) from one line to the next. At the conclusion of theargument, we can finally declare that δ should be taken small enough that cδ < .This does not constitute circular reasoning since one could tally up all of the absoluteconstants (the c ’s) in each estimate in advance of executing the argument and suitablydefine δ a priori but this is not a practical manner of exposition.Recall that we started by defining w ( t ) = u ( t ) − u z ( t ) where z was selected by the implicit function theorem so that orthogonality conditions(3.4) hold. By continuity of the flow in H , this is possible at least up to some smallpositive time. Let T be the supremum of all times 0 < T ≤ h − − δ for which (cid:107) w (cid:107) L ∞ [0 ,T ] H x ≤ h / (6.1) | v | ≤ h − δ (6.2) a ≥ a − δ (6.3)Note that the requirement (6.1) implies (cid:107) w (cid:107) H ≤ h (cid:107) w (cid:107) H + h (cid:107) w (cid:107) H , and enables us to discard cubic error terms in w in our estimates. LS SOLITON INTERACTION 27
In the course of the argument that follows, we work on the time interval [0 , T ]. Atthe conclusion of the argument, we are able to assert that either T = δh − log h − orthat (6.2) or (6.3) fail to hold at t = T .It follows from the decomposition (1.22) and the bootstrap assumptions (6.2), (6.3)above that (see Appendix A)sup ≤ n ≤ (cid:107) J − ∂ z n Π ⊥ z J H (cid:48) ( u z ) (cid:107) H x ≤ h − cδ Let (cid:15) def = h − cδ + (cid:107) w (cid:107) L ∞ [0 ,T ] H x By Lemma 3.1 and the computations in Appendix A, the ODEs ˙ µ = ( − σ (8 a − ve − a + O ( (cid:15) )˙ a = µ − v + ( − σ ( − a + 23 π ) ve − a + O ( (cid:15) )˙ θ = 12 µ + 12 v µ − + 18( − σ e − a + O ( (cid:15) )˙ v = − − σ e − a + O ( (cid:15) )hold on [0 , T ]. By the first of these equations and (6.1), (6.2), (6.3), we have | µ − | ≤ h − cδ . From the above ODEs and (6.1), (6.2), (6.3), we can deduce bounds on ˙ µ , ˙ a , ˙ θ ,and ˙ v that justify the estimates involved in the construction of ν z in § (cid:107) ν z (cid:107) H x (cid:46) h − cδ and (5.1) holds with (cid:107) F (cid:107) H x (cid:46) (cid:15) . By (5.9),(6.5) | ∂ t L z ( u ) | (cid:46) h (cid:107) ˜ w (cid:107) H + h (cid:107) ˜ w (cid:107) H + (cid:107) ˜ w (cid:107) H where we recall that ˜ w = w − ν z . By (6.4), (cid:107) ˜ w (cid:107) H (cid:46) (cid:107) w (cid:107) H + h − cδ , we obtain from(6.5) that(6.6) | ∂ t L z ( u ) | (cid:46) h (cid:107) w (cid:107) H + h − cδ . From (5.13) and (5.10), the bound(6.7) (cid:107) w (cid:107) H (cid:46) L z ( u ) + h − cδ holds. Combining (6.6) and (6.7), we obtain the bound | ∂ t L z ( u ) | (cid:46) hL z ( u ) + h − cδ By Gronwall’s inequality, it follows that L z ( u ) (cid:46) e cth h − cδ . Provided we restrict to t (cid:46) δh − log h − , this implies L z ( u ) (cid:46) h − cδ Reapplying (6.7), we obtain(6.8) (cid:107) w (cid:107) H (cid:46) h − cδ . At this point, we can declare that δ should have been taken sufficiently small so that cδ < , where c is as it appears in (6.8). It follows that (6.1), (6.2), (6.3) can onlybreak down provided T (cid:38) δh − log h − or if either (6.2) or (6.3) fails at t = T .We will see the from the following ODE analysis that (6.3) always holds; in the samephase (even solution, attractive) case, the assumption (6.2) first fails at T ∼ h − , andin the opposite phase (odd solution, repulsive) case, (6.2) remains valid and we canreach T ∼ h − log h − .Since we now restrict to t (cid:46) δh − log h − , we can assume that (6.8) holds and thus (cid:15) (cid:46) h − cδ .Let ˜ z = (˜ µ, ˜ a, ˜ θ, ˜ v ) solve ˙˜ µ = ( − σ (8˜ a − ve − a ˙˜ a = ˜ v ˙˜ θ = 12 ˜ µ + 12 ˜ v ˜ µ − + 18( − σ e − a ˙˜ v = − − σ e − a These tilde equations appear in the statement of Theorem 1.1 without tildes. Notethat the ˙˜ a and ˙˜ v equations can be solved separately as discussed in §
1. Let ¯ a = µa − ˜ a and ¯ v = v − ˜ v . Then we get the system (cid:40) ˙¯ a = ¯ v + O ( h − cδ )˙¯ v = − − σ e − a ¯ a + O ( h − cδ )Let γ = (¯ a ) + h − ¯ v . Then, substituting˙ γ (cid:46) h ¯ a ( h − ¯ v ) + ( h / ¯ a ) h − cδ + ( h − / ¯ v )( h − cδ )By the inequality αβ ≤ α + β , we obtain˙ γ (cid:46) hγ + h − cδ By Gronwall’s inequality, γ (cid:46) e cht ( γ + h − cδ )It follows that | ¯ a | (cid:46) h − δ , | ¯ v | (cid:46) h − cδ LS SOLITON INTERACTION 29
These errors only affect the ˙ µ equation at order h − cδ so µ is only affected at order h − cδ . Given this, the ˙ θ equation is only affected at order h − cδ . Thus, the impact on θ is of size h − cδ . In conclusion | ¯ θ | (cid:46) h − cδ | ¯ µ | (cid:46) h − cδ Thus (cid:107) u z − u ˜ z (cid:107) H (cid:46) h − cδ Since u z in Theorem 1.1 in fact means u ˜ z , this completes the proof of Theorem 1.1. Appendix A. Computations
We shall carry out the computations of the ODEs appearing in (3.5) in Lemma 3.1and show that they are equivalent to (1.19), with errors of size O ( h − ). This is carriedout without making the symmetry assumption on the solution. When the even/oddsymmetry assumption is imposed, we will carry out the integrals appearing in (1.19)and show that the ODEs claimed in the statement of Theorem 1.1 hold.Denote u z = η + η . Let L = { , , , } denote the indices that refer to the leftsoliton and R = { , , , } denote the indices that refer to the right soliton. Thecoefficient matrix of the symplectic form is( a (cid:96)m ) = A = (cid:20) − II (cid:21) + O ( h − )where the O ( h − ) contributions come from a (cid:96)m with (cid:96) ∈ L and m ∈ R (and vice-versa,but of course a (cid:96)m = − a m(cid:96) ). Fortunately, we do not need to compute these terms.Note that ( a (cid:96)m ) = A − = (cid:20) I − I (cid:21) + O ( h − )In fact, we can substantially reduce the complexity of computation in applyingLemma 3.1 by observing that J H (cid:48) ( u z ) decomposes into terms parallel to M plusother terms which are O ( h − ). To this end, we expand: H (cid:48) ( u z ) = H (cid:48) ( η ) + H (cid:48) ( η ) + H (cid:48)(cid:48) p ( η ) η + H (cid:48)(cid:48) p ( η ) η + O ( h ) , where H p ( u ) = − (cid:90) | u | . Moreover, we have
J H (cid:48) ( η ) = ∂ v H ( η ) ∂ a η − ∂ µ H ( η ) ∂ θ η . Hence,(A.1) H (cid:48) ( u z ) = (cid:88) j =1 b j J − ∂ z j u z + H (cid:48)(cid:48) p ( η ) η + H (cid:48)(cid:48) p ( η ) η where b = ∂ v H ( η ) , b = ∂ v H ( η ) , b = − ∂ µ H ( η ) , b = − ∂ µ H ( η )and all other b j = 0. Observe that (cid:104) H (cid:48)(cid:48) p ( η ) η , ∂ z (cid:96) u z (cid:105) = O ( h − ) for any (cid:96) ∈ R and (cid:104) H (cid:48)(cid:48) p ( η ) η , ∂ z (cid:96) u z (cid:105) = O ( h − ) for any (cid:96) ∈ L . Note further that for (cid:96) ∈ L (and hence ∂ z (cid:96) u z = ∂ z (cid:96) η ) we have(A.2) (cid:104) H (cid:48)(cid:48) p ( η ) η , ∂ z (cid:96) u z (cid:105) = (cid:104) η , H (cid:48)(cid:48) p ( η ) ∂ z (cid:96) η (cid:105) = ∂ z (cid:96) (cid:104) η , H (cid:48) p ( η ) (cid:105) Similarly, for (cid:96) ∈ R and (and hence ∂ z (cid:96) u z = ∂ z (cid:96) η ) we have(A.3) (cid:104) H (cid:48)(cid:48) p ( η ) η , ∂ z (cid:96) u z (cid:105) = (cid:104) η , H (cid:48)(cid:48) p ( η ) ∂ z (cid:96) η (cid:105) = ∂ z (cid:96) (cid:104) η , H (cid:48) p ( η ) (cid:105) From (A.1),(A.2), and (A.3), we obtain ∂ z (cid:96) H ( u z ) = − (cid:88) j =1 b j a j(cid:96) + ∂ z (cid:96) (cid:104) η , H (cid:48) p ( η ) (cid:105) + ∂ z (cid:96) (cid:104) η , H (cid:48) p ( η ) (cid:105) It follows that the equations (3.6) reduce to˙ z m = b m − (cid:88) (cid:96) ∈ L ∂ z (cid:96) (cid:104) H (cid:48) p ( η ) , η (cid:105) a (cid:96)m − (cid:88) (cid:96) ∈ R ∂ z (cid:96) (cid:104) H (cid:48) p ( η ) , η (cid:105) a (cid:96)m + O ( h − )It suffices in this sum to discard O ( h − ) terms in a (cid:96)m . Thus we obtain the equations ˙ z = b + ∂ z (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ z = b + ∂ z (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ z = b + ∂ z (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ z = b + ∂ z (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ z = b − ∂ z (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ z = b − ∂ z (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ z = b − ∂ z (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ z = b − ∂ z (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − ) LS SOLITON INTERACTION 31
In more direct language, these equations are˙ µ = + ∂ θ (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ a = + ∂ v H ( η ) + ∂ v (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ µ = + ∂ θ (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ a = + ∂ v H ( η ) + ∂ v (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ θ = − ∂ µ H ( η ) − ∂ µ (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ v = − ∂ a (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ θ = − ∂ µ H ( η ) − ∂ µ (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )˙ v = − ∂ a (cid:104) H (cid:48) p ( η ) , η (cid:105) + O ( h − )We note that these equations hold in general, without assuming that the solutionis even or odd.The next step is then to compute (cid:104) H (cid:48) p ( η ) , η (cid:105) and (cid:104) H (cid:48) p ( η ) , η (cid:105) . Let φ ( x ) = sech x .We have(A.4) (cid:104) H (cid:48) p ( η ) , η (cid:105) = − Re (cid:16) e i ( θ − θ ) e i ( µ − v a − µ − v a ) µ µ × (cid:90) e i ( µ − v − µ − v ) x φ ( µ ( x − a )) φ ( µ ( x − a ) dx (cid:17) At this point we will make the even/odd assumption. In the even case, we may set(A.5) ( µ, a, θ, v ) def = ( µ , − a , θ , − v ) = ( µ , a , θ , v )Then θ − θ = 0 . In the odd case, we may set(A.6) ( µ, a, θ, v ) def = ( µ , − a , θ − π, − v ) = ( µ , a , θ , v )Then θ − θ = π .In either the even or odd case, we find that ˙ µ = ˙ µ = ˙ µ = O ( h − ), from which itfollows that(A.7) µ = µ = µ = 1 + O ( h − )Take σ = 0 in the even case and σ = 1 in the odd case. We compute the equations for˙ µ , ˙ a , ˙ θ , ˙ v by carrying out the appropriate derivative of (A.4), and then evaluatingthe resulting expression using (A.7), (A.5), (A.6). By residue calculus computationsand asymptotic expansion, α ( ξ, a ) def = (cid:90) + ∞−∞ e − ixξ φ ( x − a ) φ ( x + a ) dx = e − a [4 + (2 − a ) iξ + ( − π a − a ) ξ + O ( ξ )] + O ( e − a ) and β ( ξ, a ) def = (cid:90) + ∞−∞ e − ixξ [ φ ] (cid:48) ( x − a ) φ ( x + a ) dx = e − a [4 + (6 − a ) iξ ] + O ( h − )We find that ˙ µ = ( − σ Re[ − iα ] + O ( h − )˙ a = µ − v + ( − σ Re[+ iaα + ∂ ξ α ] + O ( h − )˙ θ = 12 µ + 12 v µ − + ( − σ Re[( iva + 3) α + v ( ∂ ξ α )] + Re( aβ − i∂ ξ β ) + O ( h − )˙ v = ( − σ Re[ − ivα − β ] + O ( h − )where α , ∂ ξ α , and β are evaluated at ξ = − v .Substituting, we obtain ˙ µ = ( − σ (8 a − ve − a + O ( h − )˙ a = µ − v + ( − σ ( − a + 23 π ) ve − a + O ( h − )˙ θ = 12 µ + 12 v µ − + 18( − σ e − a + O ( h − )˙ v = − − σ e − a + O ( h − )The system ( µa, v ) can be solved with error O ( h − ); from which ( a, v ) can berecovered with error O ( h − ). At this accuracy the dynamics are comparable to (cid:40) ˙ a = v ˙ v = − − σ e − a Then µ can be solved with “explicit” order h term coming from the order h termin the equation for ˙ µ , and then ˙ θ can be obtained with error of size h . References [1] W.K. Abou Salem,
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