Unique continuation for discrete nonlinear wave equations

PROCEEDINGS OF THE AMERICAN MA THEMA TICAL SOCIETY V olume 140, Numb er 4, Pa ges 1321–1 330 S 0002-9939(0 6)08550-9 UNIQUE CONTINUA TION F OR DISCRETE NONLINEAR W A VE EQUA TIONS HELGE KR ¨ UGER AND GERALD TESCHL (Comm unicated by W alte r V an Assche) Abstract. W e establish unique con tin uation for v arious discrete nonlinear wa v e equations. F or example, we sho w that i f t wo solutions of the T o da lattice coincide f or one lattice point i n some arbitraril y small time interv al, then they coincide ev erywhere. Moreov er, w e establish analogous results for the T oda, Kac–v an M oerb eke, and Abl o witz–Ladik hierarchies. Although all these equa- tions are integrable, the proof does not use integrabilit y and can be adapted to other equations as well. 1. Introduction Unique con tin uation results for wa ve equations hav e a long tradition and s eem to originate in cont ro l theory . One o f the first res ults seems to b e the one by Zhang [20], where he pr ov es that if a sho rt-range so lutio n o f the K orteweg–de V ries (KdV) equa tion v anishes on an o p en subset in the x/t -pla ne, then it must v anish everywhere. Since then, this result has been extended in v a rious directions and for different e q uations (see for example [1], the introduction in [10] for the case of the nonlinear Sc hr¨ odinger equation, [5], [11], [12] for the ge ner alized K dV equation, [14] for the Camassa–Holm equation). How ever, all the results so far seem to only deal with w av e equations whic h are contin uous in the spatial direction and this clearly raises the question for such unique contin uation r esults for wa ve equations which a re discrete in the spatial v ariable. In particular, to the be st of our knowledge, there ar e no results for example for the T oda equa tion, one o f the mos t prominent discrete systems. While in principle the stra tegy fr o m Zhang [20] would b e applicable to the T o da la ttice, it is the pur p os e of this pap er to advocate a m uch simpler direct appro ach in the dis crete case. W e will sta rt with the T o da la ttice as our prototypical example and then show how the ent ire T o da hierarch y as well as the Ka c–v an Mo erb eke and Ablo witz– Ladik hierar chies can b e treated. It is impo rtant to stress that our approa ch does not use integrabilit y of these equations and hence can b e ada pted to more general systems. On the o ther hand, our appr oach is restr icted to o ne dimension in the spatial v ariable and th us do es not a pply to the discrete Schr¨ odinger equation on Z d . Receiv ed by the editors April 1, 2009 and, in revised form, December 30, 2010. 2000 Mathematics Subje ct Classific ation. Pr imary 35L05, 37K60; Secondary 37K15, 37K10. Key wor ds and phr ases. Unique con tinuat ion, T o da l attice, Kac–v an Mo erbeke lattice, Ablowitz –Ladik equations, discrete nonlinear Sc hr¨ odinger equation, Sch ur flow. Researc h supp orted by the Austrian Science F und (FWF) under Grant No. Y330 and the National Science F ound ation (NSF) under Grant No. DMS–0800100. c  2011 American Mathematical Soci et y 1321 1322 H. KR ¨ UGER AND G. TESCHL Due to the connectio ns with lo calizatio n for discrete Ander s on–Berno ulli mo dels, unique con tinuation f or this mo del is an impo rtant op en pro blem; see [2 ], [3]. 2. The Toda la ttice In this section we w ant to treat the T o da la ttice as the prototypical exa mple. T o this end, recall the T oda lattice [19] (in Flaschk a’s v ariables [7]) ˙ a ( n, t ) = a ( n, t )  b ( n + 1 , t ) − b ( n, t )  , ˙ b ( n, t ) = 2  a ( n, t ) 2 − a ( n − 1 , t ) 2  , n ∈ Z , (2.1) where the dot deno tes a deriv ativ e with re spe c t to t . It is a well-studied physical mo del and one of the prototypical discrete integrable w av e equations. W e refer to the mono graphs [6], [16], [1 9] or the r e v iew a rticles [1 3], [17] for further information. Theorem 2.1. Assume that a 0 ( n, t ) , b 0 ( n, t ) and a ( n, t ) , b ( n, t ) ar e c omplex-value d solutions of t he T o da lattic e (2.1) with a 0 ( n, t ) 6 = 0 for al l ( n, t ) ∈ Z × R such that ther e is one n 0 ∈ Z and t wo times t 0 < t 1 such t hat (2.2) a 0 ( n 0 , t ) 2 = a ( n 0 , t ) 2 , b 0 ( n 0 , t ) = b ( n 0 , t ) , for t ∈ ( t 0 , t 1 ) . Then (2.3) a 0 ( n, t ) 2 = a ( n, t ) 2 , b 0 ( n, t ) = b ( n, t ) for al l ( n, t ) ∈ Z × R . Pr o o f. It suffices to prov e that (2.2) for n 0 implies ( 2.2 ) for n 0 − 1 and n 0 + 1. W e start with N 0 − 1 and first observe that (2.1) implies that 0 = ˙ b ( n 0 , t ) − ˙ b 0 ( n 0 , t ) = 2  a ( n 0 , t ) 2 − a 0 ( n 0 , t ) 2 − a ( n 0 − 1 , t ) 2 + a 0 ( n 0 − 1 , t ) 2  = − 2  a ( n 0 − 1 , t ) 2 − a 0 ( n 0 − 1 , t ) 2  and th us a ( n 0 − 1 , t ) 2 = a 0 ( n 0 − 1 , t ) 2 . Using this w e compute 0 = ˙ a ( n 0 − 1 , t ) a ( n 0 − 1 , t ) − ˙ a 0 ( n 0 − 1 , t ) a 0 ( n 0 − 1 , t ) = b ( n 0 , t ) − b 0 ( n 0 , t ) − b ( n 0 − 1 , t ) + b 0 ( n 0 − 1 , t ) = − b ( n 0 − 1 , t ) + b 0 ( n 0 − 1 , t ) , so b ( n 0 − 1 , t ) = b 0 ( n 0 − 1 , t ). Now for n 0 + 1, w e be gin with 0 = ˙ a ( n 0 , t ) a ( n 0 , t ) − ˙ a 0 ( n 0 , t ) a 0 ( n 0 , t ) = b ( n 0 + 1 , t ) − b 0 ( n 0 + 1 , t ) − b ( n 0 , t ) + b 0 ( n 0 , t ) = b ( n 0 + 1 , t ) − b 0 ( n 0 + 1 , t ) , so b ( n 0 + 1 , t ) = b 0 ( n 0 + 1 , t ). Now, us e that 0 = ˙ b ( n 0 + 1 , t ) − ˙ b 0 ( n 0 + 1 , t ) = 2  a ( n 0 + 1 , t ) 2 − a 0 ( n 0 + 1 , t ) 2 − a ( n 0 , t ) 2 + a 0 ( n 0 , t ) 2  = 2  a ( n 0 + 1 , t ) 2 − a 0 ( n 0 + 1 , t ) 2  to conclude that a ( n 0 + 1 , t ) 2 = a 0 ( n 0 + 1 , t ) 2 . This finishes the proof.  UNIQUE CONTINUA T ION FOR DISCR ETE NONLINE AR W A VE EQUA TIONS 1323 It is worth while to note that the a ssumption a 0 ( n, t ) 6 = 0 is c rucial. In fact, if a 0 ( n 0 , t ) = 0 for one (and hence for all) t ∈ R , then the T oda lattice decouples into t wo indep endent parts to the left and right of n 0 , and the ab ov e result is clea r ly wrong. How ever, it r emains v alid o n ev ery consecutive num b er of points for whic h a 0 ( n, t ) 6 = 0 holds true. In particular , our result applies to the half-line T o da lattice or to the finite T oda lattice. As a simple consequence, this also proves that the pro pagation sp eed for the T o da lattice is finite. Corollary 2.2. L et a ( n, t ) 6 = 0 , b ( n, t ) b e a c omplex-value d solution of the T o da lattic e (2.1) for wh ich a ( n, t 0 ) − 1 2 , b ( n, t 0 ) is supp orte d on a fin ite nu mb er of p o ints n at some initial time t 0 . Then this do es not r emain t r u e for t ∈ ( t 0 , t 1 ) unless a ( n, t ) = 1 2 , b ( n, t ) = 0 for al l ( n, t ) ∈ Z × R . In fact, in the cas e of real-v alued solutions, one can even show the somewhat stronger result that a ( n, t 0 ) − 1 2 , b ( n, t 0 ) ca n b e compactly suppo rted for at most one time [1 8]. How ever, on the other hand, the T o da la ttice do es preser ve cer tain asymptotic prop erties of the initial conditions; see again [18]. 3. Extension to the Toda and Kac–v an Mo erbeke hierarchy In this section we show that our main r esult extends to the entire T o da hierar- ch y (which will cov er the K ac–v an Mo erb eke hiera rch y a s well). T o this end, we int ro duce the T o da hierarch y using the s tandard La x fo r malism following [4] (see also [9], [16]). Asso ciated with t wo s equences a ( t ) 2 6 = 0 , b ( t ) is a Jacobi operato r (3.1) H ( t ) = a ( t ) S + + a − ( t ) S − + b ( t ) acting on sequences ov er Z , where S ± f ( n ) = f ± ( n ) = f ( n ± 1) are the usual shift op erators . Mor e ov er, choo se constants c 0 = 1, c j , 1 ≤ j ≤ r , c r +1 = 0 , and set (3.2) P 2 r +2 ( t ) = r X j =0 c r − j ˜ P 2 j +2 ( t ) , ˜ P 2 j +2 ( t ) = [ H ( t ) j +1 ] + − [ H ( t ) j +1 ] − , where [ A ] ± denote the upp er a nd low er triangula r parts o f an op er ator with resp ect to the sta nda rd basis δ m ( n ) = δ m,n (with δ m,n the usual K roneck er delta). Then the T o da hierarch y is equiv alen t to the Lax equation (3.3) d dt H ( t ) − [ P 2 r +2 ( t ) , H ( t )] = 0 , t ∈ R , where [ A, B ] = AB − B A is the usual comm utator. Abbreviating g j ( n, t ) = j X ℓ =0 c j − ℓ ˜ g ℓ ( n, t ) , ˜ g ℓ ( n, t ) = h δ n , H ( t ) ℓ δ n i , h j ( n, t ) = j X ℓ =0 c j − ℓ ˜ h ℓ ( n, t ) + c j +1 , ˜ h ℓ ( n, t ) = 2 a ( n, t ) h δ n +1 , H ( t ) ℓ δ n i , (3.4) one explicitly obtains (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 , r ∈ N 0 , 1324 H. KR ¨ UGER AND G. TESCHL for the r -th equation TL r ( a, b ) = 0 in the T o da hierarch y (where N 0 = N ∪ { 0 } ). Our main point in this se c tion is the following generaliza tion of Theorem 2.1 to the en tire T o da hierar ch y: Theorem 3.1. Assu me that a 0 ( n, t ) 6 = 0 , b 0 ( n, t ) and a ( n, t ) , b ( n, t ) ar e c omplex- value d solutions of some e quation in the T o da hier ar chy TL r such that ther e is one n 0 ∈ Z and t wo times t 0 < t 1 such t hat (3.6) a 0 ( n 0 + j, t ) 2 = a ( n 0 + j, t ) 2 , b 0 ( n 0 + j, t ) = b ( n 0 + j, t ) , j = 0 , . . . , r , for t ∈ ( t 0 , t 1 ) . Then (3.7) a 0 ( n, t ) 2 = a ( n, t ) 2 , b 0 ( n, t ) = b ( n, t ) for al l ( n, t ) ∈ Z × R . Pr o o f. Let us drop the dependence on t for no ta tional simplicity dur ing this pro o f. The key observ ation is the following structure for the homogenous quantities ˜ g j , ˜ h j : ˜ g j ( n ) =                                   k − 1 Q ℓ =0 a ( n + ℓ ) 2  b ( n + k ) + R ( n + k − 1 , n − k + 1)+ +  k Q ℓ =1 a ( n − ℓ ) 2   b ( n − k ) + 2 k − 1 P ℓ =0 b ( n − ℓ )  , j = 2 k + 1 ,  k − 2 Q ℓ =0 a ( n + ℓ ) 2   a ( n + k − 1) 2 + b ( n + k − 1) 2 +2 b ( n + k − 1) k − 2 P ℓ =0 b ( n + ℓ )  + + R ( n + k − 2 , n − k + 1) + k Q ℓ =1 a ( n − ℓ ) 2 , j = 2 k , and ˜ h j ( n ) =                                  2  k − 1 Q ℓ =0 a ( n + ℓ ) 2   a ( n + k ) 2 + b ( n + k ) 2 +2 b ( n + k ) k − 1 P ℓ =0 b ( n + ℓ )  + + R ( n + k − 1 , n − k + 1) + 2 k Q ℓ =0 a ( n − ℓ ) 2 , j = 2 k + 1 , 2  k − 1 Q ℓ =0 a ( n + ℓ ) 2  b ( n + k ) + R ( n + k − 1 , n − k + 2) +2 k − 1 Q ℓ =0 a ( n − ℓ ) 2  b ( n + 1) + b ( n − k + 1) + 2 k − 2 P ℓ =0 b ( n − ℓ )  , j = 2 k , for j > 1 . Here R ( n, m ) denotes terms which in volv e only a ( ℓ ) and b ( ℓ ) with m ≤ ℓ ≤ n and we set R ( n, m ) = 0 if n < m . In fact, this can b e verified using ˜ g 0 = 1, ˜ h 0 = 0, together with the recursions ([16, Chap. 6]) ˜ g j +1 = ˜ h j + ˜ h − j 2 + b ˜ g j , (3.8) ˜ h j +1 = 2 a 2 j X ℓ =0 ˜ g j − ℓ ˜ g + ℓ − 1 2 j X ℓ =0 ˜ h j − ℓ ˜ h ℓ , j ∈ N 0 . (3.9) UNIQUE CONTINUA T ION FOR DISCR ETE NONLINE AR W A VE EQUA TIONS 1325 Now we are rea dy for the main part of the pro of. It suffices to show that (3 .7) holds for n = n 0 − 1 and n = n 0 + r + 1. W e first lo ok at the case r + 1 = 2 k + 1. Then 0 = ˙ a ( n 0 + k ) a ( n 0 + k ) − ˙ a 0 ( n 0 + k ) a 0 ( n 0 + k ) = g r +1 ( n 0 + k + 1) − g r +1 ( n 0 + k ) − g 0 ,r +1 ( n 0 + k + 1) + g 0 ,r +1 ( n 0 + k ) = r Y ℓ = k +1 a 0 ( n 0 + ℓ ) 2 !  b ( n 0 + r + 1) − b 0 ( n 0 + r + 1)  shows that b ( n 0 + r + 1) = b 0 ( n 0 + r + 1). Similarly , 0 = ˙ b ( n 0 + k ) − ˙ b 0 ( n 0 + k ) = h r +1 ( n 0 + k ) − h r +1 ( n 0 + k − 1) − h 0 ,r +1 ( n 0 + k ) + h 0 ,r +1 ( n 0 + k − 1) = 2 k − 1 Y ℓ =0 a 0 ( n 0 + ℓ ) 2 !  a ( n 0 − 1) 2 − a 0 ( n 0 − 1) 2  shows that a ( n 0 − 1) 2 = a 0 ( n 0 − 1) 2 . Pro ceeding like this and using the res ult found in the previous steps, 0 = ˙ b ( n 0 + k + 1) − ˙ b 0 ( n 0 + k + 1) = h r +1 ( n 0 + k + 1) − h r +1 ( n 0 + k ) − h 0 ,r +1 ( n 0 + k + 1) + h 0 ,r +1 ( n 0 + k ) = r Y ℓ = k +1 a 0 ( n 0 + ℓ ) 2 !  a ( n 0 + r + 1) 2 − a 0 ( n 0 + r + 1) 2  shows that a ( n 0 + r + 1) 2 = a 0 ( n 0 + r + 1) 2 , and 0 = ˙ a ( n 0 + k − 1) a ( n 0 + k − 1) − ˙ a 0 ( n 0 + k − 1) a 0 ( n 0 + k − 1) = g r +1 ( n 0 + k ) − g r +1 ( n 0 + k − 1) − g 0 ,r +1 ( n 0 + k ) + g 0 ,r +1 ( n 0 + k − 1) = k Y ℓ = − 1 a 0 ( n 0 + ℓ ) 2 !  b ( n 0 − 1) − b 0 ( n 0 − 1)  shows that b ( n 0 − 1) = b 0 ( n 0 − 1), whic h finishes the ca se r + 1 = 2 k + 1. The case r + 1 = 2 k is analo gous.  Finally , since the Kac–v an Mo erb eke hier arch y can be obtained by setting b = 0 in the odd equations of the T oda hierarch y , KM r ( a ) = TL 2 r +1 ( a, 0 ) (see [15]), this last result also co vers the Kac – v an Mo erb eke hierarch y . In particular , Corollary 3.2. Assume that ρ 0 ( n, t ) 6 = 0 and ρ ( n, t ) 6 = 0 ar e solutions of the Kac–van Mo erb eke e quation (3.10) ˙ ρ ( n, t ) = ρ ( n, t )  ρ ( n + 1 , t ) − ρ ( n − 1 , t )  such that ther e is one n 0 ∈ Z and two times t 0 < t 1 such t hat (3.11) ρ 0 ( n 0 , t ) = ρ ( n 0 , t ) , ρ 0 ( n 0 + 1 , t ) = ρ ( n 0 + 1 , t ) , for t ∈ ( t 0 , t 1 ) . Then (3.12) ρ 0 ( n, t ) = ρ ( n, t ) 1326 H. KR ¨ UGER AND G. TESCHL for al l ( n, t ) ∈ Z × R . 4. The Ablowitz–Ladik hierarchy In this sectio n we show that our ma in result extends to the Ablowit z–La dik (AL) hierarch y [9]. W e first s tate the result for the simplest case, whose pr o of f ollows as the one of Theorem 2.1. Theorem 4.1. L et C 0 , ± , c 1 ∈ C \ { 0 } . Assume that α 0 ( n, t ) , β 0 ( n, t ) , with ρ 0 ( n, t ) 6 = 0 , and α ( n, t ) , β ( n, t ) ar e solutions of the Ablowi tz–L adik e quation i ˙ α ( n, t ) = − ρ ( n, t ) 2  c 0 , − α ( n − 1 , t ) + c 0 , + α ( n + 1 , t )  − c 1 α ( n, t ) , i ˙ β ( n, t ) = ρ ( n, t ) 2  c 0 , + β ( n − 1 , t ) + c 0 , − β ( n + 1 , t )  + c 1 β ( n, t ) , (4.1) wher e (4.2) ρ ( n, t ) = (1 − α ( n, t ) β ( n, t )) 1 / 2 , such that ther e is one n 0 ∈ Z and two times t 0 < t 1 such t hat (4.3) α 0 ( n 0 + j, t ) = α ( n 0 + j, t ) , β 0 ( n 0 + j, t ) = β ( n 0 + j, t ) , j = 0 , 1 , for t ∈ ( t 0 , t 1 ) . Then (4.4) α 0 ( n, t ) = α ( n, t ) , β 0 ( n, t ) = β ( n, t ) for al l ( n, t ) ∈ Z × R . The sp ecial choices c 0 , ± = 1, c 1 = − 2 , and β = ± α yield the fo cusing, defo cus - ing discrete nonlinea r Schr¨ odinge r equations, r e spe c tiv ely . The alterna tive choice c 0 , ± = ± i, c 1 = 0, and β = α yield the Sc hur flo w. W e next turn to the AL hiera rch y following [8], [9]. Asso ciated with t wo se- quences α ( t ) , β ( t ) is a CMV op erator L ( t ) = ρ − ( t ) ρ ( t ) δ even S −− + ( β − ( t ) ρ ( t ) δ even − α + ( t ) ρ ( t ) δ od d ) S − − β ( t ) α + ( t ) + ( β ( t ) ρ + ( t ) δ even − α ++ ( t ) ρ + ( t ) δ od d ) S + + ρ + ( t ) ρ ++ ( t ) δ od d S ++ , (4.5) acting on sequences ov e r Z , where δ even and δ od d denote the characteris tic functions of the ev en, odd integers, (4.6) δ even = χ 2 Z , δ od d = 1 − δ even = χ 2 Z +1 , resp ectively . Next, consider P p ( t ) = i 2 p + X ℓ =1 c p + − ℓ, +  [ L ℓ ( t )] + − [ L ℓ ( t )] −  − i 2 p − X ℓ =1 c p − − ℓ, −  [ L − ℓ ( t )] + − [ L − ℓ ( t )] −  − i 2 c p Q d , p ∈ N 2 0 , (4.7) with Q d denoting the doubly infinite diagonal matrix (4.8) Q d =  ( − 1) k δ k,ℓ  k,ℓ ∈ Z . Then the AL hierarch y is equiv a lent to the Lax equation (4.9) d dt L ( t ) − [ P p ( t ) , L ( t )] = 0 , t ∈ R . UNIQUE CONTINUA TION F OR DISCRETE NONLINEAR W A VE EQUA TIONS 1327 T o find an explicit expression w e in tro duce f ℓ, ± ( t ) = ℓ X k =0 c ℓ − k, ± ˆ f k, ± ( t ) , g ℓ, ± ( t ) = ℓ X k =0 c ℓ − k, ± ˆ g k, ± ( t ) , h ℓ, ± ( t ) = ℓ X k =0 c ℓ − k, ± ˆ h k, ± ( t ) , (4.10) where ˆ f ℓ, + ( n, t ) = α ( n, t ) h δ n , L ℓ +1 δ n i + ρ ( n, t ) ( h δ n − 1 , L ℓ +1 ( t ) δ n i , n even, h δ n , L ℓ +1 ( t ) δ n − 1 i , n o dd, ˆ f ℓ, − ( n, t ) = α ( n, t )( δ n , L − ℓ δ n ) + ρ ( n, t ) ( h δ n − 1 , L − ℓ ( t ) δ n i , n even, h δ n , L − ℓ ( t ) δ n − 1 i , n o dd, ˆ g 0 , ± = 1 / 2 , ˆ g ℓ, ± ( n, t ) = h δ n , L ± ℓ ( t ) δ n i , (4.11) ˆ h ℓ, + ( n, t ) = β ( n, t ) h δ n , L ℓ ( t ) δ n i + ρ ( n, t ) ( h δ n , L ℓ ( t ) δ n − 1 i , n even, h δ n − 1 , L ℓ ( t ) δ n i , n o dd, ˆ h ℓ, − ( n, t ) = β ( n, t ) h δ n , L − ℓ − 1 δ n i + ρ ( n, t ) ( h δ n , L − ℓ − 1 ( t ) δ n − 1 i , n ev en, h δ n − 1 , L − ℓ − 1 ( t ) δ n i , n odd. Then the p th equation, p = ( p − , p + ) ∈ N 2 0 , in the AL hierarc hy is given b y AL p ( α, β ) = − i ˙ α ( t ) − α ( g p + , + ( t ) + g − p − , − ( t )) + f p + − 1 , + ( t ) − f − p − − 1 , − ( t ) − i ˙ β ( t ) + β ( g − p + , + ( t ) + g p − , − ( t )) − h p − − 1 , − ( t ) + h − p + − 1 , + ( t ) ! = 0 , p = ( p − , p + ) ∈ N 2 0 . (4.12) Theorem 4. 2. Fix some p = ( p + , p − ) ∈ N 2 0 such that p − = p + > 0 and s et p = p + + p − − 1 . Assu me that α 0 ( n, t ) , β 0 ( n, t ) , with ρ 0 ( n, t ) 6 = 0 , and α ( n, t ) , β ( n, t ) ar e s olut ions of s ome e quation in the T o da hier ar chy AL p such that ther e is one n 0 ∈ Z and t wo times t 0 < t 1 such t hat (4.13) α 0 ( n 0 + j, t ) = α ( n 0 + j, t ) , β 0 ( n 0 + j, t ) = β ( n 0 + j, t ) , 0 ≤ j ≤ p, for t ∈ ( t 0 , t 1 ) . Then (4.14) α 0 ( n, t ) = α ( n, t ) , β 0 ( n, t ) = β ( n, t ) for al l ( n, t ) ∈ Z × R . Pr o of. Aga in we dr o p the dependence on t for notational s implicit y during this pro of and use t he same conv entions as in the pro of of Theorem 3.1. 1328 H. KR ¨ UGER AND G. TESCHL The homog eneous quantit ies ˆ f ℓ, ± , ˆ g ℓ, ± , ˆ h ℓ, ± are uniquely defined by the following recursion relations [9, Lem. C.5]: ˆ g 0 , + = 1 2 , ˆ f 0 , + = − α + , ˆ h 0 , + = β , ˆ g l +1 , + = l X k =0 ˆ f l − k, + ˆ h k, + − l X k =1 ˆ g l +1 − k, + ˆ g k, + , ˆ f − l +1 , + = ˆ f l, + − α ( ˆ g l +1 , + + ˆ g − l +1 , + ) , ˆ h l +1 , + = ˆ h − l, + + β ( ˆ g l +1 , + + ˆ g − l +1 , + ) , and ˆ g 0 , − = 1 2 , ˆ f 0 , − = α, ˆ h 0 , − = − β + , ˆ g l +1 , − = l X k =0 ˆ f l − k, − ˆ h k, − − l X k =1 ˆ g l +1 − k, − ˆ g k, − , ˆ f l +1 , − = ˆ f − l, − + α ( ˆ g l +1 , − + ˆ g − l +1 , − ) , ˆ h − l +1 , − = ˆ h l, − − β ( ˆ g l +1 , − + ˆ g − l +1 , − ) . F rom them we obtain ˆ f j, + ( n ) = − j Y l =1 ρ ( n + l ) 2 ! α ( n + j + 1) + R ( n + j, n − j + 2) + j − 2 Y l =0 ρ ( n − l ) 2 ! α ( n + 1) 2 β ( n − j + 1) , (4.15) ˆ f j, − ( n ) = − j − 1 Y l =1 ρ ( n + l ) 2 ! α ( n ) 2 β ( n + j ) + R ( n + j − 1 , n − j + 1) + j − 1 Y l =0 ρ ( n − l ) 2 ! α ( n − j ) , (4.16) ˆ g j, + ( n ) = − j − 1 Y l =1 ρ ( n + l ) 2 ! β ( n ) α ( n + j ) + R ( n + j − 1 , n − j + 2) − j − 2 Y l =0 ρ ( n − l ) 2 ! α ( n + 1) β ( n − j + 1) , (4.17) ˆ g j, − ( n ) = − j − 1 Y l =1 ρ ( n + l ) 2 ! α ( n ) β ( n + j ) + R ( n + j − 1 , n − j + 2) − j − 2 Y l =0 ρ ( n − l ) 2 ! β ( n + 1) α ( n − j + 1) , (4.18) UNIQUE CONTINUA TION F OR DISCRETE NONLINEAR W A VE EQUA TIONS 1329 ˆ h j, + ( n ) = − j − 1 Y l =1 ρ ( n + l ) 2 ! β ( n ) 2 α ( n + j ) + R ( n + j − 1 , n − j + 1) + j − 1 Y l =0 ρ ( n − l ) 2 ! β ( n − j ) , (4.19) ˆ h j, − ( n ) = − j Y l =1 ρ ( n + l ) 2 ! β ( n + j + 1) + R ( n + j, n − j + 2) + j − 2 Y l =0 ρ ( n − l ) 2 ! β ( n + 1) 2 α ( n − j + 1) (4.20) for j ∈ N . Note that it suffices to verify the + cas e s inc e the − ca se follows from ˆ f j, ± ( α, β ) = ˆ h j, ± ( α, β ) a nd ˆ g j, + ( α, β ) = ˆ g j, ( α, β ) ([9, Lem. 3.7]). Now we can procee d as in the case of the T oda hiera rch y . F or exa mple, 0 = i  ˙ α ( n + p − ) − ˙ α 0 ( n + p − )  = − c 0 , +   p Y l = p − ρ ( n − l ) 2    α ( n + p + 1) − α 0 ( n + p + 1)  implies that α ( n + p + 1) = α 0 ( n + p + 1), etc.  Int eres tingly , the above appro ach do es not seem to work for p − 6 = p + in g eneral. In any cas e , the ab ov e r esult cov ers the discr ete nonlinear Schr¨ odinger and Sch ur hierarchies via the a bove-men tioned special choices c 0 , ± = 1 , β = ± α and c 0 , ± = ± i, β = α . Ac knowledgmen ts. W e thank F. Gesztesy and the ano nymous referee for po int ing out erro r s in a previous version of this ar ticle. References [1] J. Bourgain, On the c omp actness of the supp ort of solutions of disp e rsi v e e quations , Int er- nat. M ath. Res. 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