On the Spatial Asymptotics of Solutions of the Toda Lattice

ON THE SP A TIAL ASYMPTOTICS OF SOLUTIONS OF THE TOD A LA TTICE GERALD TESCHL Abstract. W e inv estigate the spatial asymptotics of decaying solutions of the T oda lattice and s ho w that the asymptotic b eha vior is preserve d by the time ev olution. In particular, we sho w that the leading asymptotic term is time independent . M oreo ver, w e establish infinite propagation sp eed for the T o da lattice. All r esults are extended to the en tire T o da as we ll as the Kac–v an Mo erbeke hierarch y . 1. Introduction Since the seminal work of Ga rdner et al. [9 ] in 1 967 it is k nown that c o mpletely int egra ble wa ve equations ca n b e solved b y vir tue of the inv erse sca ttering tra ns - form. In pa rticular, this implies that short-rang e perturba tions of the free solution remain short-rang e during the tim e evolution. This raise s the q uestion to what extend spatial as ymptotical prop erties are pr eserved during time ev olution. In [1], [2] (see also [1 3]) Bondar ev a and Shubin considered the initial v alue problem for the Korteweg–de V ries (KdV) equation in the c lass of initial conditions which ha ve a prescrib ed asymptotic expansion in terms o f pow ers of the spatial v ariable. As part of their a nalysis they o btained that the leading term o f this asymptotic expansion is time independent. Inspired by this in triguing fact, the a im of the present paper is to prov e a general res ult for the T o da equation which co ntains the ana log of this result plus the known res ults for short-r ange p er turbation alluded to b efor e a s a sp ecial case. More sp ecifically , recall the T o da lattice [2 3] (in Flaschk a’s v ar iables [8]) d dt a ( n, t ) = a ( n, t )  b ( n + 1 , t ) − b ( n, t )  , d dt b ( n, t ) = 2  a ( n, t ) 2 − a ( n − 1 , t ) 2  , n ∈ Z . (1.1) It is a well studied physical mo del and the prototypical discrete in tegra ble wa ve equation. W e refer to the monog raphs [7], [10], [21], [23] o r the revie w articles [14], [22] for further informa tion. Then our main re s ult, Theorem 2.5 b elow, implies for example tha t (1.2) a ( n, t ) = 1 2 + α n δ + O ( 1 n δ + ε ) , b ( n, t ) = β n δ + O ( 1 n δ + ε ) , n → ∞ , 2000 Mathematics Subje ct Classific ation. Primary 37K40, 37K15; Secondary 35Q53, 37K10. Key wor ds and phr ases. T o da lattice, spatial asymptotics, T o da hierarc hy , Kac–v an Mo erb ek e hierarch y . W or k supported by the Austrian Science F und (FWF) under Gran t No. Y330. Discrete Con tin. Dyn. Syst. 2 7:3 , 1233–12 39 (201 0). 1 2 G. TESCHL for all t ∈ R provided this holds for the initial condition t = 0. Here α, β ∈ R and δ ≥ 0, 0 < ε ≤ 1. A few re ma rks a re in order : First of all, it is imp o rtant to point out that the error terms will in genera l grow with t (see the discussion after Theorem 2.5 for a rough time dependent b ound on the error). An analogous result holds for n → −∞ . Moreov er, there is nothing sp ecial a b o ut the p ow ers n − δ , which can b e replaced by any bounded sequence whic h, roughly sp eaking , do es decay at most expo nent ially and whose difference is asymptotica lly o f lower o rder. Finally , similar r esults hold for the Ablowit z–La dik equation. How ever, s inc e the Ablowitz–Ladik system doe s not have the s ame difference structur e s ome mo difications are neccessa r y and will be given in Michor [17]. 2. The Ca uchy problem for the Toda la ttice T o se t the stage let us re c all some basic facts for the T o da lattice. W e will o nly consider b o unded solutions a nd hence require Hyp othesi s H. 2.1. Supp ose a ( t ) , b ( t ) satisfy a ( t ) ∈ ℓ ∞ ( Z , R ) , b ( t ) ∈ ℓ ∞ ( Z , R ) , a ( n, t ) 6 = 0 , ( n, t ) ∈ Z × R , and let t 7→ ( a ( t ) , b ( t )) b e differ entiable in ℓ ∞ ( Z ) ⊕ ℓ ∞ ( Z ) . First of all, to see co mplete integrability it suffices to find a s o -called Lax pair [16], that is, tw o op erato rs H ( t ) , P ( t ) in ℓ 2 ( Z ) such that the Lax equation (2.1) d dt H ( t ) = P ( t ) H ( t ) − H ( t ) P ( t ) is eq uiv alent to (1.1). Here ℓ 2 ( Z ) deno tes the Hilber t space of squa re summable (complex-v alued) sequences over Z . One can easily convince oneself that the right choice is H ( t ) = a ( t ) S + + a − ( t ) S − + b ( t ) , P ( t ) = a ( t ) S + − a − ( t ) S − , (2.2) where ( S ± f )( n ) = f ± ( n ) = f ( n ± 1) are the usual shift op er ators. Now the Lax equation (2.1) implies that the op erator s H ( t ) for different t ∈ R are unitarily equiv a lent (cf. [21, Thm. 12.4]): Theorem 2 .2. L et P ( t ) b e a family of b ounde d skew-adjoint op er ators, such that t 7→ P ( t ) is differ entiable. Th en t her e exists a family of u nitary pr op agators U ( t, s ) for P ( t ) , that is, (2.3) d dt U ( t, s ) = P ( t ) U ( t, s ) , U ( s, s ) = 1 l . Mor e over, the L ax e quation (2.1) implies (2.4) H ( t ) = U ( t, s ) H ( s ) U ( t, s ) − 1 . As p ointed out in [19], this r esult immediately implies globa l existence of b ounded solutions of the T o da lattice as follows: Considering the Banach space of a ll b ounded real-v alued co efficients ( a ( n ) , b ( n )) (with the sup norm), lo cal exis tence is a conse- quence o f sta nda rd res ults for differential equations in Banach spaces. Moreov er, ON THE SP A TIAL ASYMPTOTICS OF SOLUTIONS OF THE TOD A LA TTICE 3 Theorem 2.2 implies that the norm k H ( t ) k is constant, which in turn pro vides a uniform bo und on the co efficie n ts o f H ( t ), (2.5) k a ( t ) k ∞ + k b ( t ) k ∞ ≤ 2 k H ( t ) k = 2 k H (0) k . Hence s olutions o f the T o da la ttice ca nnot blow up and are globa l in time (see [21, Sect. 12.2 ] for details): Theorem 2.3. Supp ose ( a 0 , b 0 ) ∈ M = ℓ ∞ ( Z , R ) ⊕ ℓ ∞ ( Z , R ) . Then ther e exists a unique int e gr al curve t 7→ ( a ( t ) , b ( t )) in C ∞ ( R , M ) of the T o da lattic e (1.1) s u ch that ( a (0) , b (0 )) = ( a 0 , b 0 ) . How ev er, more can b e shown. In fact, when consider ing the inv erse sca ttering transform for the T o da lattice it is desirable to e s tablish existence of solutions within the Marchenko class, that is, solutions satisfying (2.6) X n ∈ Z (1 + | n | )  | a ( n, t ) − 1 2 | + | b ( n, t ) |  < ∞ for all t ∈ R . That this is indeed true was firs t established in [20] and rediscovered in [11] using a different metho d. F urthermo re, the w eight 1 + | n | ca n b e replaced by an (almost) arbitrary w eight function w ( n ). Lemma 2 .4. Su pp ose a ( n, t ) , b ( n, t ) is some b ounde d solution of the T o da lattic e (1.1) satisfyi ng (2.7) for one t 0 ∈ R . Then (2.7) X n ∈ Z w ( n )  | a ( n, t ) − 1 2 | + | b ( n, t ) |  < ∞ , holds for al l t ∈ R , wher e w ( n ) ≥ 1 is some weight with sup n ( | w ( n +1) w ( n ) | + | w ( n ) w ( n +1) | ) < ∞ . Moreov er, as was demonstr ated in [5 ] (see also [6]), o ne can ev en replace | a ( n, t ) − 1 2 | + | b ( n, t ) | by | a ( n, t ) − ¯ a ( n, t ) | + | b ( n, t ) − ¯ b ( n, t ) | , where ¯ a ( n, t ), ¯ b ( n, t ) is some other b ounded so lutio n of the T o da lattice. See also [12], where similar res ults a re shown. This result shows that the asy mptotic b ehavior as n → ± ∞ is preserved to leading o rder by the T o da lattice. The purp ose of this pap er is to show that even the leading term is preserved (i.e., time indep endent) by the time evolution. Set (2.8) k ( a, b ) k w, p =         P n ∈ Z w ( n )  | a ( n ) | p + | b ( n ) | p   1 /p , 1 ≤ p < ∞ sup n ∈ Z w ( n )  | a ( n ) | + | b ( n ) |  , p = ∞ . Then one has the fo llowing result: Theorem 2.5. Le t w ( n ) ≥ 1 b e some weight with sup n ( | w ( n +1) w ( n ) | + | w ( n ) w ( n +1) | ) < ∞ and fix some 1 ≤ p ≤ ∞ . Supp ose a 0 , b 0 and ˜ a 0 , ˜ b 0 ar e b oun de d se quenc es such that (2.9) k ( a + 0 − a 0 , b + 0 − b 0 ) k w, p < ∞ and k ( ˜ a 0 , ˜ b 0 ) k w, p < ∞ . Supp ose a ( t ) , b ( t ) is the unique solution of the T o da lattic e (1.1) c orr esp onding t o the initial c onditions (2.10) a (0) = a 0 + ˜ a 0 6 = 0 , b (0) = b 0 + ˜ b 0 . 4 G. TESCHL Then this solution is of the form (2.11) a ( t ) = a 0 + ˜ a ( t ) , b ( t ) = b 0 + ˜ b ( t ) , wher e k (˜ a ( t ) , ˜ b ( t )) k w, p < ∞ for al l t ∈ R . Pr o of. The T o da e q uation (1.1) implies the differen tial equation d dt ˜ a ( n, t ) = a ( n, t )  ˜ b ( n + 1 , t ) − ˜ b ( n, t ) + b 0 ( n + 1 ) − b 0 ( n )  , d dt ˜ b ( n, t ) =2   a ( n, t ) + a 0 ( n )  ˜ a ( n, t ) −  a ( n − 1 , t ) + a 0 ( n − 1)  ˜ a ( n − 1 , t ) + ( a 0 ( n ) + a 0 ( n − 1))( a 0 ( n ) − a 0 ( n − 1))  , n ∈ Z (2.12) for ( ˜ a, ˜ b ). Since our requirement for w ( n ) implies that the shift ope r ators a re con- tin uous with resp ect to the norm k . k w, p and the same is true for the multiplication op erator with a b ounded seq uence, this is a n inhomogeneous linear differential equation in o ur Banach spa ce which has a unique global solution in this Banach space (e.g., [4, Sect. 1.4]). Moreover, since w ( n ) ≥ 1 this solution is b ounded and the corresp onding co efficients ( a, b ) coincide with the solution of the T o da equation from Theorem 2.3.  Note that using Gr onw all’s inequalit y one can ea sily obtain an explic it bound (2.13) k (˜ a ( t ) , ˜ b ( t )) k w, p ≤ k (˜ a 0 ( t ) , ˜ b 0 ( t )) k w, p e C t + k ( a + 0 − a 0 , b + 0 − b 0 ) k w, p 1 C (e C t − 1) , where C = 4( k H k + k a 0 k ∞ ) (since k a ( t ) k ∞ ≤ k H k by (2.5)). T o se e the claim (1.2) from the in tro duction, let (2.14) a 0 ( n ) = 1 2 + α n δ , b 0 ( n ) = β n δ , α, β ∈ R , δ > 0 , for n > 0 and a 0 ( n ) = b 0 ( n ) = 0 for n ≤ 0. Now cho ose p = ∞ with (2.15) w ( n ) = ( (1 + | n | ) δ + ε , n > 0 , 1 , n ≤ 0 . and a pply the previo us theo rem. T o see Lemma 2.4, just c ho ose a 0 ( n ) = 1 2 , b 0 ( n ) = 0 and p = 1. Finally , let us remark tha t the requirement that w ( n ) do es not grow faster than exp onentially is imp o rtant. If it w ere no t present, our r esult would imply that a compact p erturbation of the free s olution a ( n, t ) = 1 2 , b ( n, t ) = 0 remains compact for all time if and only if it is equal to the free solution. This is well-known for the KdV equation [24], but we are not aware of a refer e nc e for the T o da equa tion. Theorem 2 .6. L et a ( n, t ) , b ( n, t ) b e a b ounde d solution of the T o da lattic e (1 .1 ) . If the se quenc es a ( n, t ) − 1 2 , b ( n, t ) ar e zer o for al l exc ept for a finite numb er of n ∈ Z for two differ ent times t 0 6 = t 1 , then t hey vanish identic al ly. Pr o of. Without loss we can choose t 0 = 0 and supp ose that the s equences a ( n, 0) − 1 2 , b ( n, 0) are zer o for a ll except for a finite num b er o f n . Then the ass o ciated reflection co efficients R ± ( k , 0) (see [21] Chapter 10) are rational functions with resp ect to k and by the in verse sc a ttering tra nsform ([21] Theorem 13.8 ) we hav e R ± ( k , t ) = R ± ( k , 0) exp( ± ( k − k − 1 ) t ), which is not ra tional for any t 6 = 0 unles s R ± ( k , t ) ≡ 0 . Hence it must be a pure N so liton solution, which has compa ct suppo rt if and o nly if it is trivial, N = 0.  ON THE SP A TIAL ASYMPTOTICS OF SOLUTIONS OF THE TOD A LA TTICE 5 F or related unique contin uation results for the T o da eq uation s ee Kr¨ ug e r and T eschl [15]. 3. Extension to the Toda and Kac–v an Moerbeke hierarchy In this section w e show that our main res ult extends to the en tire T o da hierar- ch y (which w ill co ver the Kac–v an Mo erb eke hier arch y as well). T o this end, we int ro duce the T o da hierar ch y using the standar d Lax forma lis m following [3] (see also [10], [21]). Cho ose constants c 0 = 1 , c j , 1 ≤ j ≤ r , c r +1 = 0, and set g j ( n, t ) = j X ℓ =0 c j − ℓ h δ n , H ( t ) ℓ δ n i , h j ( n, t ) = 2 a ( n, t ) j X ℓ =0 c j − ℓ h δ n +1 , H ( t ) ℓ δ n i + c j +1 . (3.1) The sequences g j , h j satisfy the recur sion relations g 0 = 1 , h 0 = c 1 , 2 g j +1 − h j − h − j − 2 bg j = 0 , 0 ≤ j ≤ r , h j +1 − h − j +1 − 2( a 2 g + j − ( a − ) 2 g − j ) − b ( h j − h − j ) = 0 , 0 ≤ j < r . (3.2) Int ro ducing (3.3) P 2 r + 2 ( t ) = − H ( t ) r +1 + r X j =0 (2 a ( t ) g j ( t ) S + − h j ( t )) H ( t ) r − j + g r +1 ( t ) , a straightforward computation shows that the Lax equation (3.4) d dt H ( t ) − [ P 2 r + 2 ( t ) , H ( t )] = 0 , t ∈ R , is equiv ale nt to (3.5) TL r ( a ( t ) , b ( t )) =   ˙ a ( t ) − a ( t )  g + r +1 ( t ) − g r +1 ( t )  ˙ b ( t ) −  h r +1 ( t ) − h − r +1 ( t )    = 0 , where the dot denotes a deriv a tive with respect to t . V ary ing r ∈ N 0 yields the T o da hier arch y TL r ( a, b ) = 0 . All res ults men tioned in the pr evious section, Theorem 2.2, Theo rem 2.3, and Lemma 2 .4 rema in v alid for the en tire T o da hierarch y (see [21]) a nd so do es o ur main result: Theorem 3.1. Le t w ( n ) ≥ 1 b e some weight with sup n ( | w ( n +1) w ( n ) | + | w ( n ) w ( n +1) | ) < ∞ and fix some 1 ≤ p ≤ ∞ . Supp ose a 0 , b 0 and ˜ a 0 , ˜ b 0 ar e b oun de d se quenc es such that (3.6) k ( a + 0 − a 0 , b + 0 − b 0 ) k w, p < ∞ and k ( ˜ a 0 , ˜ b 0 ) k w, p < ∞ . Supp ose a ( t ) , b ( t ) is the unique solution of some e quation of the T o da hi er ar chy, TL r ( a, b ) = 0 , c orr esp onding to the initial c onditions (3.7) a (0) = a 0 + ˜ a 0 > 0 , b (0) = b 0 + ˜ b 0 . Then this solution is of the form (3.8) a ( t ) = a 0 + ˜ a ( t ) , b ( t ) = b 0 + ˜ b ( t ) , wher e k (˜ a ( t ) , ˜ b ( t )) k w, p < ∞ 6 G. TESCHL for al l t ∈ R . Pr o of. The pro of is almost identical to the one of Theorem 2.5. F rom TL r ( a, b ) = 0 one obtains an inhomog eneous differential equa tion for (˜ a, ˜ b ). The homo genous part is a finite sum ov er shifts of (˜ a, ˜ b ) a nd the inhomogeneo us part is  a 0 ( g + 0 ,r +1 − g 0 ,r +1 ( t )) , h 0 ,r +1 − h − 0 ,r +1  , where g 0 ,r +1 , h 0 ,r +1 are formed from ( a 0 , b 0 ). Finally , it is stra ightf or ward to show that the k . k w, p norm of the inhomogeneo us pa rt is finite by induction using the recursive definition of g r +1 ( t ) and h r +1 ( t ).  Similarly we also obtain Theorem 3.2. L et a ( n, t ) , b ( n, t ) b e a b ounde d solution of the of some e quation of the T o da hier ar chy, TL r ( a, b ) = 0 . If t he se qu enc es a ( n, t ) − 1 2 , b ( n, t ) ar e zer o for al l exc ept for a finite numb er of n ∈ Z fo r two differ ent times t 0 6 = t 1 , then they vanish identic al ly. Finally since the Kac – v an Mo erb eke hier arch y can b e obtained by setting b = 0 in the o dd equations of the T o da hierarchy , KM r ( a ) = TL 2 r + 1 ( a, 0) (see [18]), this last result also coveres the Kac –v a n Mo erb eke hierarch y . Ackno wledgments I w ant to thank Ira Ego rov a, F ritz Gesztesy , Thomas Kapp eler, and Helge Kr ¨ uger for discussions on this topic. References [1] I. N. B ondarev a, The Kortewe g–de V ries e quation in classes of incr e asing functions with pr escribe d asymptotic b ehavior as | x | → ∞ , M at. USSR Sb. 5 0:1 , 125–13 5 (1985). [2] I. Bondarev a and M. Sh ubin, Incr e asing asymptotic solutions o f the Kortewe g–de V ries e quation and i t s higher analo gues , So v. Math. Dokl. 2 6:3 , 716–71 9 (1982 ). [3] W. Bulla, F. Geszt esy , H. Holden, and G. T eschl, Algebr o-Ge ometric Quasi-Perio dic Finite- Gap Solutions o f the T o da and Kac-van Mo erb eke Hier ar chies , Mem. Amer. Math. So c. 135-641 , (1998 ). [4] K. Deimling, Or dinary Differ ential Equations on Banach Sp ac es , Lecture Notes in Mathe- matics 596 , Springer, Ber lin, 1977 . [5] I. Egorov a, J. Michor, and G. T eschl, Inverse sc attering tr ansform for the T o da hier ar chy with quasi-p erio dic b ackgro und , Pro c. Amer. Math. Soc. 135 , 1817–182 7 (2007 ). [6] I. Egorov a, J. Michor, and G. T eschl, Inverse sc attering tr ansform for the T o da hier ar chy with ste plike finite-gap b ackgr ounds , J. Math. Ph ys. 50 , 103521 (20 09). [7] L. F addeev and L. T akh ta j an, Hamiltonian Metho ds in the The ory of Solitons , Spri nger, Berlin, 1987. [8] H. Fl asc hk a, The T o da lattic e. I. Existence of inte gr als , Ph ys. Rev. B 9 , 1924–1 925 (1974). [9] C. S. Gardner, J. M. Green, M. D. Krusk al, and R. M . Mi ura, A metho d for solving the Kortewe g-de V ries e quation , Ph ys. Rev. Letters 19 , 1095– 1097 (1967). [10] F. Gesztesy , H. Holden, J. Michor, and G. T eschl, Soliton Equations and Their Algebr o- Ge ometri c Solutions. V olume II: (1 + 1) -Dimensional Discr ete Mo dels , Cambridge Studies in Adv anced Mathematics 114 , Cam bridge Universit y Press, Cam bridge, 2008. [11] A. Kh. Khanmamedo v, O n the r apid ly de c r ea sing solution of the Cauchy pr oblem for the T o da chain , Theoret. and Math. Ph ys. 142:1 , 1–7 (2005 ). [12] A. Kh. Khanmamedo v, The solution of Cauchys pr oblem for the T o da lattic e with limit p erio dic initial data , Sb. Math. 199:3 , 449–458 (2008). [13] T. Kappeler, P . Perry , M . Shubin and P . T opalo v, Solutions of mKdV i n classes of functions unb ounde d at infinity , J. Geom. Anal. 18 , 443–477 (2008) . [14] H. Kr ¨ uger and G. T eschl, L ong-time asymptotics for t he T o da lattic e for de c aying initial data r evisite d , Rev. Math. Ph ys. 21:1 , 61–109 (2009). ON THE SP A TIAL ASYMPTOTICS OF SOLUTIONS OF THE TOD A LA TTICE 7 [15] H. Kr ¨ uger and G. T esc hl, Uni q ue c ontinuation for discr ete nonline ar wave e quations , [16] P . D. Lax Inte gr als of nonline ar e quations of evolution and solitary waves , Comm. Pure and Appl. Math. 2 1 , 467–490 (1 968). [17] J. Michor, O n the sp atial asymptotics of solutions of the A blowitz–L adik hier ar chy , [18] J. Mic hor and G. T esc hl, On the e quivalenc e of differ ent L ax p airs for the K ac-van Mo erb eke hier ar chy , in Mo dern Analysis and Applications, V. Adam ya n (ed.) et al., 445–453, Op er. Theory Adv. A ppl. 191 , Birkh¨ auser, Basel, 2009. [19] G. T esc hl, On the T o da and Kac–van Mo erb eke hier ar chies , Math. Z. 231 , 325–344 (1999). [20] G. T eschl, Inverse sc attering tr ansform for the T o da hier ar chy , Math. Nac h. 202 , 163–171 (1999). [21] G. T eschl, Jac obi Op er ators and Completely Inte gr able Nonline ar L attices , M ath. Surv. and Mon. 72 , Amer. Math. So c., Rhode Isl and, 2000. [22] G. T eschl, A lmost everything you always wante d to know ab out the T o da equa tion , Jahresb er. Deutsc h. Math.-V erein. 103 , no. 4, 149–162 (2001). [23] M. T o da, The ory of Nonline ar L attic es , 2 nd enl. edition, Springer, Berlin, 1989. [24] B. Zhang, Unique c ontinuation for the Korteweg– de V ries e quation , SIAM J. Math. A nal. 23 , 55–7 1 (1992). F acul ty of M a thema tics, University of Vienna, Nordber gstrasse 15, 1090 Wien, Aus - tria, and Interna tional Er win Schr ¨ odinger Institute for Ma thema tical Physics, Bol tz- manngasse 9, 1090 Wien, Austria E-mail addr ess : Gerald.Tesch l@univie.ac.at URL : http://w ww.mat.univi e.ac.at/ ~ gerald/