Time evolution of the scattering data for a fourth-order linear differential operator

TIME EV OLUTION OF THE SCA TTERING D A T A F OR A F OUR TH-ORDER LINEAR DIFFERENTIAL OPERA TOR T unca y Aktosun Departmen t of Mathematics Univ ersity of T exas at Arl ington Arlington, TX 76019-0408, USA V assilis G. Papanicolaou Departmen t of Mathematics National T ec hnical Universit y of Athens Zografou Campus, 157 80, A thens, Greece Abstract : The time ev olution of the scattering and sp ectral dat a is obtained for the differen tial op erat or d 4 dx 4 + d dx u ( x, t ) d dx + v ( x, t ) , where u ( x , t ) and v ( x, t ) are real-v alued p oten tials deca ying exp onen tially as x → ±∞ at each fixed t. The result is rel ev an t in a crucial step of the in v erse scattering transform metho d that is used in solving the initial - v alue problem for a pair o f coupled nonlinear partial differential equatio ns satisfied by u ( x, t ) and v ( x, t ) . Mathemat ics Su b je ct Classifi c a tion (2000): 37K15 35Q51 Keyw ords: Time evolution of scattering data for a fourth-order ODE, i n v erse scattering transform for a fourth-order ODE, Gelfand-Dic k ey hierarc h y Short title: Time evolution for a fourth-order ODE 1 1. INTR ODUCTION Consider the fourth-order ordinary differen tial equati on ϕ ′′′′ + ( uϕ ′ ) ′ + v ϕ = k 4 ϕ, x ∈ R , (1.1) where the p rime denotes the deriv ativ e with respect to the indep enden t v ariable x, and the p oten tials u and v are real v alued i n suc h a wa y that u, u ′ , v are integrable and they deca y exp onentially or faster [8, 9] as x → ±∞ . Let us writ e (1. 1 ) as L ϕ = k 4 ϕ, where L is the linear op erator defined as L := D 4 + D uD + v , D := d dx . (1.2) W e wi ll consider ( 1.1) in the sector Ω in the compl ex k -plane where arg k ∈ [0 , π / 4] with the understanding that w e excl ude the p oin t k = 0 . A complete analysis at k = 0 for (1.1) do es not exist and t his deserv es a careful future study . Some partial a nal ysis at k = 0 is a v ailable in [8] . W e will omit the analysis at k = 0 in this pap er. The differential equati on app earing in (1.1) is the canonical equiv al en t [ 4 ,6] of the Euler-Bernoulli equation, and the former is obtai ned from the latter vi a a transforma- tion of b o th indep enden t and dep enden t v ariables [4]. Suc h fourth-order eq uations arise in analyzing vi brat ions of b eams, whereas second-order equations a re used in describing vibrations of strings. W e refer the reader to [8,9] and [7,10] for t w o complemen tary studies of (1.1). U nder appropriate restrictions on t he p otentials u and v , Iw asaki [8,9] studied (1.1) b y analyzing it in the sector Ω under the a ssumption that there a r e no b ound states and no spectral or nonsp ectral singularities. He obtained [8] v arious prop erties of solutions t o (1.1) and form ulated [9] the in ve rse problem of recov ery of u and v from some appropriate scatter- ing data. Iw asaki p o sed [9] the in v erse scatt ering problem for ( 1.1) as a b o undary-v alue problem where the scattering data consisted of a reflection co efficien t and a connection co efficien t sp ecified on the b oundary of Ω , and he pro vided the pro o f of uniqueness for 2 the sol ution to that in ve rse problem. A sp ecial case of (1.1) w as studied i n [ 7,10] under the very restrictiv e assumption that the reflection and connection co efficien ts a re all zero. The corresp onding in v erse problem was formulated [10] a s a Riemann-Hilb ert problem on the whole complex k -plane, where the set of scattering data is sp ecified on the rays arg k = ( l − 1) π / 4 for l = 1 , 2 , . . . , 8 . It seems to b e the case that the authors of [7,10] ha v e not b een aw ar e of [8 , 9]. Some examples of expli cit soluti ons to (1.1) w ere pro v ided in [7,10] under t he restricti on that the reflection and connection co efficients are all zero. W e should indicate that the terminology used in [7 ,10] differs from that used in [8, 9]. F o r the b enefit of the reader t he rela t ionships b et w een the q uan tities in [ 7,10] and those in [8,9] are indicated in Section 5 of our pap er. It i s already kno wn [5,10,1 2 ] that (1.1) is rel a ted to the coupled system of nonlinear partial differen tial equat i ons given i n (4.2), which is solv able b y the inv erse scattering transform metho d and is related to the Gelfand-Dick ey hierarch y [ 5 ]. The time evolution of the scattering and b ound-state data forms a crucial step in that in v erse scattering transform. One of our goals in this pap er is to analyze the b ound states for (1.1) by analyzing it in the sector Ω of the complex k -plane, whic h was also t he domain used i n [8,9] , and to obtain the time ev olution of the corresp onding b ound-state data; namely , t he time ev olution of the dep endency and norming constan ts. The t ime evolution o f the b o und-state norming constan ts is give n in [ 10] b y analyzing (1 .1) in the entire complex k -plane. Another goal of ours is to obtain the time ev olutions of the scattering and connection co efficients as w ell as other co efficien ts related to scattering, which is absen t in [7,10]. In our pap er w e also clarify ho w the rel ev an t quan tities in [7, 10] a re related to t hose of [8, 9], and w e pro vide some illustrati v e examples of explicit solutions to (1.1) helping to understand the corresp onding scatt ering and b ound sta tes b etter. Our pap er is organized as follows. In Section 2 we pro vide the preliminaries and in tro duce the Jost solutio ns ψ + and ψ − as w ell as the exp onen tial solutions φ + and φ − 3 to (1 . 1), and we presen t the scattering data in terms of the spatial a sy mptotics of t he Jost and exp onen tial solutions. In Section 3 w e analyze the b ound states asso ciated with (1.1) and intro duce the dep endency constan ts and norming constants for eac h b ound state. In Section 4 w e consider the system of integrable nonlinear partial differen tial equations satisfied b y the time-ev o lv ed po ten tials u ( x, t ) and v ( x, t ) , and w e obtain the t ime evolution of t he corresp onding scattering and b ound-state data. In Section 5 we explain ho w the quan tities asso ciated with ( 1.1) and used in [7,10] are related to those used i n [8,9]. Finally , in Section 6 we pro vide some ill ustrative ex pl i cit solutions to (1. 1 ). 2. PRELIMINARIES Consider the case t hat the po t en tials u and v in ( 1 .1) are functions of x alone and do not dep end on the parameter t. In Section 4 w e will consider the case w here u and v dep end on b oth x and t. As in [8] let us i n tro duce the Jost solutions ψ + and ψ − to (1.1) in the sector arg k ∈ (0 , π / 4) with the asy mptotics ψ ± ( k , x ) = e ± ikx [1 + o (1)] , x → ±∞ , (2.1) in suc h a wa y that e ∓ ikx ψ ± ( k , x ) remains b ounded for al l x ∈ R . Similarly , let us in- tro duce the exp onen tial solutions φ + and φ − in the sector arg k ∈ (0 , π / 4) satisfying the asymptotics φ ± ( k , x ) = e ∓ kx [1 + o (1)] , x → ±∞ . (2.2) It follows that t hese solutions satisfy the resp ective in tegral relations given b y φ + ( k , x ) = e − kx + 1 4 k 4 4 X j =1 Z ∞ x dy [ k 3 j u ( y ) − k 2 j u ′ ( y ) + k j v ( y )] e k j ( x − y ) φ + ( k , y ) , ψ + ( k , x ) = e ikx + 1 4 k 4 X j =1 , 2 , 4 Z ∞ x dy [ k 3 j u ( y ) − k 2 j u ′ ( y ) + k j v ( y )] e k j ( x − y ) ψ + ( k , y ) − 1 4 k 4 Z x −∞ dy [ k 3 3 u ( y ) − k 2 3 u ′ ( y ) + k 3 v ( y )] e k 3 ( x − y ) ψ + ( k , y ) , 4 φ − ( k , x ) = e kx − 1 4 k 4 4 X j =1 Z x −∞ dy [ k 3 j u ( y ) − k 2 j u ′ ( y ) + k j v ( y )] e k j ( x − y ) φ − ( k , y ) , ψ − ( k , x ) = e − ikx + 1 4 k 4 Z ∞ x dy [ k 3 1 u ( y ) − k 2 1 u ′ ( y ) + k 1 v ( y )] e k 1 ( x − y ) ψ − ( k , y ) − 1 4 k 4 X j =2 , 3 , 4 Z x −∞ dy [ k 3 j u ( y ) − k 2 j u ′ ( y ) + k j v ( y )] e k j ( x − y ) ψ − ( k , y ) , where we ha v e defined k 1 := k , k 2 := ik , k 3 := − k , k 4 := − ik . (2.3) The spatial asym pto tics of the Jost solutio ns and the exp onen tial sol utions can b e obtained w i th the help of the four in tegral relations gi v en ab o v e. Belo w, we list those asymptotics, t hrough which we intro duce the scattering and connection co efficients for (1.1). F or arg k ∈ (0 , π / 4) w e hav e                    ψ ± ( k , x ) = e ± ikx [1 + o (1)] , x → ±∞ , ψ ± ( k , x ) = e ± ikx  1 T ( k ) + o ( 1 )  , x → ∓∞ , φ ± ( k , x ) = e ∓ kx [1 + o (1)] , x → ±∞ , φ ± ( k , x ) = e ∓ kx [ A ( k ) + o (1 )] , x → ∓∞ . (2.4) F or arg k = 0 we ha v e                    ψ ± ( k , x ) = e ± ikx [1 + o (1)] , x → ±∞ , ψ ± ( k , x ) = e ± ikx  1 T ( k ) + R ± ( k ) T ( k ) e ∓ 2 ikx + o ( 1 )  , x → ∓∞ , φ ± ( k , x ) = e ∓ kx [1 + o (1)] , x → ±∞ , φ ± ( k , x ) = e ∓ kx [ A ( k ) + o (1 )] , x → ∓∞ . (2.5) F or arg k = π / 4 w e ha v e                  ψ ± ( k , x ) = e ± ikx [1 + C ± ( k ) e ∓ kx ∓ ik x + o (1)] , x → ±∞ , ψ ± ( k , x ) = e ± ikx  1 T ( k ) + o (1)  , x → ∓∞ , φ ± ( k , x ) = e ∓ kx [1 + o (1)] , x → ±∞ , φ ± ( k , x ) = e ∓ kx  A ( k ) + B ± ( k ) e ± kx ± ik x + o ( 1 )  , x → ∓∞ . (2.6) 5 W e emphasize that A ( k ) and T ( k ) are defined for arg k ∈ [0 , π / 4] , R + ( k ) and R − ( k ) are defined for arg k = 0 , and the four co efficients B + ( k ) , B + ( k ) , C + ( k ) , and C − ( k ) are defined only for arg k = π / 4 . The co efficien t T i s known as the transmission co efficien t, R + and R − are t he left and right reflection coefficien ts, resp ectiv ely , C + and C − are k no wn as the connection co efficien ts, and B + and B − are some coefficients that can be expressed in terms of A, C + , and C − , as w e wil l see. It is either kno wn [8] or can easily b e sho wn that for arg k = 0 we ha v e 1 + | R ± ( k ) | 2 | T ( k ) | 2 = 1 | T ( k ) | 2 , R ± ( k ) T ( k ) = − R ∓ ( k ) ∗ T ( k ) ∗ , A ( k ) = A ( k ) ∗ , (2.7) and for arg k = π / 4 we ha ve                            B ∓ ( k ) = − iB ± ( k ) ∗ , C − ( k ) = iT ( k ) ∗ C + ( k ) ∗ T ( k ) , | B − ( k ) | = | B + ( k ) | , | C − ( k ) | = | C + ( k ) | , 1 T ( k ) = A ( k ) ∗ + B ± ( k ) C ± ( k ) , B ± ( k ) ∗ + A ( k ) C ± ( k ) = 0 , A ( k ) T ( k ) = | A ( k ) | 2 − | B ± ( k ) | 2 = | A ( k ) | 2  1 − | C ± ( k ) | 2  , (2.8) where the asterisk denotes complex conjugation. 3. BOUND ST A TES Since the co efficien t of the third deriv ati v e in ( 1.1) is zero, it follo ws from the general theory of ordinary differen tial equati ons that the W ronskian of any four solutions to ( 1 .1) is indep enden t of x, and that W ronskian is zero i f and only if those four soluti ons are linearly dependen t. Recall t hat a W ronskian is defined with the help of a determinan t. F or example, the W ronskian i nv olving the Jost and exp onen tial solutions is given b y W 4 [ ψ + , ψ − , φ + , φ − ] :=           ψ + ψ − φ + φ − ψ ′ + ψ ′ − φ ′ + φ ′ − ψ ′′ + ψ ′′ − φ ′′ + φ ′′ − ψ ′′′ + ψ ′′′ − φ ′′′ + φ ′′′ −           . 6 Using (2. 4)-(2.6) we obtain W 4 [ ψ + ( k , x ) , ψ − ( k , x ) , φ + ( k , x ) , φ − ( k , x )] = − 16 ik 6 A ( k ) T ( k ) , arg k ∈ [0 , π / 4] . (3.1) The linear indep endence and b oundedness properties o f v arious solutions to (1. 1) help to iden tify b ound-state soluti ons. Recall that ei g enfunctions of L corresp ond to square- in tegrable soluti o ns to (1.1 ) , which are also k no wn as b ound-state solutions. It is easy to verify that L is selfadjoin t and hence i ts eigen v alues can o ccur only for real v alues o f k 4 , i .e. when k l ies on the b oundary of the regio n Ω in tro duced in Section 1. Thus, an y p ositive eig env al ue o f L can o ccur o nly on the ray arg k = 0 a nd an y negative eigen v alue can o ccur only on the ra y a rg k = π / 4 . If A ( k ) = 0 at some k -v alue on t he boundary of the region Ω and a square-in tegrable solution to (1.1) at that k -v alue do es not exist, then w e call that k -v alue a sp ectral singula ri t y of (1. 1). If A ( k ) = 0 at some k -v alue in the in terior of Ω , then w e call that k -v alue a nonspectral singularity of (1.1). By a singula rit y w e refer to either a spectral or nonspectral singularity . It is already kno wn that at a singular p oin t the t w o integral relations given in Section 2 for ψ + ( k , x ) and ψ − ( k , x ) , resp ectively , are not solv able [8]. Spectral and nonsp ectral singulari ties for (1.1) ma y exist, and some explicit examples are illustrat ed in Section 6. Note that a b ound state in the region Ω can o ccur only when A ( k ) /T ( k ) = 0 some- where on the ray arg k = 0 or arg k = π / 4 . Otherwise, as seen from (3.1) , the four solutions ψ + , ψ − , φ + , and φ − are linearly independen t, and the asymptotics of those four solutions give n in (2.5) and (2.6) indicate that no l inear comb ination of them can decay sim ultane- ously b o t h as x → + ∞ and x → −∞ . If t here is a b o und state a t k = κ on the ra y a rg k = 0 , then we mu st ha v e A ( κ ) = 0. This follows from the first iden tit y in (2. 7) implying t hat 1 /T ( κ ) 6 = 0 a nd the fact that A ( κ ) /T ( κ ) = 0 at the b o und state. Since four l inearly independen t solutio ns to (1.1) m ust hav e resp ective asymptoti cs prop ortional to e κx [1 + o (1)] , e − κx [1 + o (1)] , e iκx [1 + o (1)] , e − iκx [1 + o (1)] as x → + ∞ , and appropriately simila r asymptotics as x → −∞ , it foll o ws that a b ound state at k = κ must 7 deca y exp onen tially as b o t h as x → + ∞ and x → −∞ . I n t hi s case, w e see from (2.5) that ψ + ( κ, x ) and ψ − ( κ, x ) are tw o linearly indep enden t solutions to (1.1) and they do not v anish sim ultaneously b ot h as x → + ∞ and x → −∞ . Th us, from (2.5) we conclude that a b ound-state eigenfunction ϕ ( κ, x ) m ust b e i n the form ϕ ( κ, x ) = d 1 φ + ( κ, x ) = d 2 φ − ( κ, x ) , (3.2) for some nonzero constan ts d 1 and d 2 . Since an y constan t mu ltiple o f an eigenfunction is still an eigenfunc tion of L , only the rati o d 2 /d 1 is relev ant and w e can call i t a de p endency constan t at k = κ, i.e. η ( κ ) := φ + ( κ, x ) φ − ( κ, x ) , (3.3) where η ( κ ) is the dep endency constan t at k = κ on the ra y arg k = 0 . Defining the bound- state norming constan ts d + and d − as d ± ( κ ) :=  Z ∞ −∞ dx φ ± ( κ, x ) φ ± ( κ, x ) ∗  − 1 / 2 , (3.4) w e see that d − = η d + and that d ± ( κ ) φ ± ( κ, x ) i s a normalized b ound-state eigenfunction of the op erator L . Ha ving clari fied the sta t us of b ound sta tes on the ra y arg k = 0 , let us now consider the b ound states on the ray arg k = π / 4 . If there i s a b ound state a t k = κ o n the ray arg k = π / 4 , then w e hav e three p ossibiliti es: (i) A ( κ ) = 0 and 1 /T ( κ ) 6 = 0 . In this case, an ar g umen t simil ar to the case given on the ra y arg k = 0 sho ws t hat φ + ( κ, x ) and φ − ( κ, x ) ar e linearly dep enden t and the bound state is simple and has t he form gi v en in (3. 2). Recall that a b ound state o ccurring at k = κ is simple i f there is o nl y one l i nearly i ndep enden t square-in tegrable solution to (1 . 1) when k = κ. (ii) A ( κ ) 6 = 0 and 1 /T ( κ ) = 0 . In this case, a si mi lar argumen t indicates that a b ound-state eigenfunction m ust hav e the form ϕ ( κ, x ) = c 1 ψ + ( κ, x ) = c 2 ψ − ( κ, x ) , 8 for some nonzero constan ts c 1 and c 2 . Since a b ound-state eigenfun ction is defined up to a constant multiple, only the ra tio c 2 /c 1 is relev an t and w e can call t hat ratio a dep endency constan t at k = κ, i.e. γ ( κ ) := ψ + ( κ, x ) ψ − ( κ, x ) , (3.5) where γ ( κ ) is the dep endency constan t at k = κ on the ray arg k = π / 4 . Defining the b ound-state norming constants c + and c − as c ± ( κ ) :=  Z ∞ −∞ dx ψ ± ( κ, x ) ψ ± ( κ, x ) ∗  − 1 / 2 , (3.6) w e see that c − = γ c + and that c ± ( κ ) ψ ± ( κ, x ) i s a normal ized b ound-state eigenfunc- tion of the operator L . In this case there is only one linearly independen t b ound-state eigenfunction at k = κ, and hence the b ound sta t e is simple. (iii) A ( κ ) = 0 and 1 /T ( κ ) = 0 . In this case, ψ + ( κ, x ) and ψ − ( κ, x ) are linearly dep enden t solutions deca ying exp onen tially as x → + ∞ as well as x → −∞ , and φ + ( κ, x ) and φ − ( κ, x ) are al so t w o l inearly dep enden t solutions deca ying ex p onen tially as x → + ∞ as well as x → −∞ . On the ot her hand, ψ + ( κ, x ) and φ + ( κ, x ) are l inearly indep en- den t, and hence they form a basis for the tw o-dimensional eig enspace consisting of b ound-state sol utions at k = κ. In other w ords, the multiplicit y of the b ound state in this case is t w o. The corresp o nding dep endency constants η ( κ ) and γ ( κ ) and the norming constan ts c + ( κ ) , c − ( κ ) , d + ( κ ) , d − ( κ ) are defined as in ( 3.3)-(3.6) . 4. TIME EVOLUTION OF S CA TTERING AND OTHER COEFFICIENTS In the previo us sections we hav e considered the Jost and exp o nen tial solutio ns, t he scattering and connection co efficien ts, a nd the b ound st a tes asso ciated wit h ( 1.1) in the case where the p oten tials u and v ar e functions of x alone. L et us now assume that the p oten tials u and v app earing in (1.1) also dep end on the extra parameter t. In that case all the rel ev an t quan tities asso ciated with (1. 1) ma y a lso dep end on t as w ell. In t his sectio n w e anal yze suc h a dep endence on t b y interpreting t a s the time v ariable. 9 No w consider the time evolution of the p oten tials u ( x, t ) and v ( x, t ) from their ini - tial v alues u ( x, 0) and v ( x, 0) , resp ectively , so that the time-evolv ed linear o p erator L corresp onding t o (1. 2 ) is given b y L := ∂ 4 x + ∂ x u ( x, t ) ∂ x + v ( x, t ) , (4.1) where ∂ x = ∂ /∂ x, and l et [5, 1 0,12] A : = − 8 ∂ 3 x − 6 u ( x, t ) ∂ x − 3 u x ( x, t ) , so that L a nd A form a Lax pair. As easily ve rified, the differen tial op erat o r L t + LA − AL reduces to a scala r m ultiplicati on op erator, and in fact we get L t + LA − AL = 0 , whic h is equiv alent to the system of nonlinear ev olution equat i ons ( u t = 10 u xxx + 6 uu x − 2 4 v x , v t = 3 u xxxxx + 3 uu xxx + 3 u x u xx − 6 uv x − 8 v xxx . (4.2) Note that w e use subscripts to denote the a ppropriat e partial deriv atives. Since u and v v anish as x → ±∞ at eac h fixed t, we ha ve A → − 8 ∂ 3 x as x → ±∞ . F rom (4.1) we also see t hat L t = u t ∂ 2 x + u xt ∂ x + v t . In order to solv e the initial-v a l ue problem relat ed to (4.2), i.e. to determine u ( x, t ) and v ( x, t ) that solve (4. 2) when u ( x, 0) and v ( x , 0) are sp ecified, we are i n terested in analyzing the t ime evolutions of the scattering a nd other co efficien ts asso ciated wit h (1.1) . T ow ards our goal, w e first anal y ze the time ev olutions of t he Jost solutions ψ + ( k , x, t ) and ψ − ( k , x, t ) and the exp onen tial solutions φ + ( k , x, t ) and φ − ( k , x, t ) , from whic h the time evolutions of other relev ant co efficien ts are easily extracted. Theorem 4.1 In the r e gion a r g k ∈ [0 , π / 4] the ti me evolutions of ψ + ( k , x, t ) , ψ − ( k , x, t ) , φ + ( k , x, t ) , and φ − ( k , x, t ) ar e giv en by [ ∂ t − A ] ψ ± = ∓ 8 ik 3 ψ ± , [ ∂ t − A ] φ ± = ∓ 8 k 3 φ ± . (4.3) 10 Mor e over, the time evolutions of various c o efficients app e aring in (2.4 ) -(2.6) ar e given by T ( k , t ) = T ( k, 0) , A ( k , t ) = A ( k , 0) , arg k ∈ [0 , π / 4] , R ± ( k , t ) = R ± ( k , 0) e ∓ 16 ik 3 t , arg k = 0 , B ± ( k , t ) = B ± ( k , 0) e ± (8 ik 3 − 8 k 3 ) t , C ± ( k , t ) = C ± ( k , 0) e ∓ (8 ik 3 − 8 k 3 ) t , arg k = π / 4 . PR OOF: The pro ofs in (4. 3) can all b e given as in the case of the time ev olution of ψ + for arg k = π / 4 , whic h is outl ined below. It is kno wn [1-3,11] that [ ∂ t − A ] ψ + m ust satisfy L ϕ = k 4 ϕ, where L is the op erator in (4.1). Th us, w e hav e [ ∂ t − A ] ψ + = c 1 ( k , t ) ψ + + c 2 ( k , t ) ψ − + c 3 ( k , t ) φ + + c 4 ( k , t ) φ − , (4.4) for some co efficien ts c j ( k , t ) to b e determined. By ev aluating (4.4 ) as x → − ∞ and x → + ∞ , with the help o f (2 . 6) w e obtai n  e ikx T  t − 8 ik 3 e ikx T = c 1 e ikx T + c 2 [ e − ikx + C − e kx ] + c 3 [ Ae − kx + B + e ikx ] + c 4 e kx , (4.5) ( C + ) t e − kx − 8 ik 3 e ikx − 8 k 3 C + e − kx = c 1 [ e ikx + C + e − kx ] + c 2 e − ikx T + c 3 e − kx + c 4 [ Ae kx + B − e − ikx ] . (4.6) By matching the corresp onding co efficien ts of the exp onen tial t erms in (4.5) and ( 4.6) we get c 1 = − 8 ik 3 , c 2 = c 3 = c 4 = 0 , T t = 0 , ( C + ) t = ( 8 k 3 − 8 ik 3 ) C + , and hence the first equation in (4.3) for ψ + is confirmed when arg k = π / 4 , and we also get the time evolutions of T and C + when arg k = π / 4 , as stated. The remaining parts o f the pro of are obtained i n a simil ar wa y . The implication of Theorem 4. 1 that T ( k , t ) and A ( k , t ) do not c hange in t is signifi- can t. As w e ha v e seen in Sectio n 3, at a b ound stat e k = κ w e m ust ha v e A ( κ, t ) /T ( κ, t ) = 0 , and at a singularity k = κ w e m ust ha v e A ( κ, t ) = 0 . Hence, the k -v alues corresponding to b o und states or singularities of t he op erator L of (4.1) also remain unc hanged in time. 11 Theorem 4.2 Ass ume that k = κ c orr esp onds to a b ound state of (1. 1). The time evolution of the b ound-state dep endency c onstants γ ( κ, t ) and η ( κ, t ) and the e v o lution of the norming c onstants c ± ( κ, t ) and d ± ( κ, t ) ar e gi ven by c ± ( κ, t ) = c ± ( κ, 0) e ± 4(1+ i ) κ 3 t , d ± ( κ, t ) = d ± ( κ, 0) e ± 4(1+ i ) κ 3 t , (4.7) γ ( κ, t ) = γ ( κ, 0) e − 8(1+ i ) κ 3 t , η ( κ, t ) = η ( κ, 0) e − 8(1+ i ) κ 3 t . (4.8) PR OOF: Let us assume that there i s a b ound state at k = κ somewhere on arg k = π / 4 with 1 /T ( κ, 0) = 0 . Then, ψ + ( κ, x, t ) is a b ound-state solution and the norming constan t c + ( κ, t ) can b e defined as in (3.6) via c + ( κ, t ) :=  Z ∞ −∞ dx ψ + ( κ, x, t ) ψ + ( κ, x, t ) ∗  − 1 / 2 , (4.9) so t hat c + ( κ, t ) ψ + ( κ, x, t ) is normalized, i.e. its L 2 -norm i s equal t o one. Let us now find the t ime evolution of c + ( κ, t ) . F rom (4.3) and its complex conjugate w e obtain [ ∂ t + 8 ∂ 3 x + 6 ∂ x u ( x, t ) + 3 u x ( x, t )] ψ + ( κ, x, t ) = − 8 iκ 3 ψ + ( κ, x, t ) , (4.10) [ ∂ t + 8 ∂ 3 x + 3 ∂ x u ( x, t ) + 3 u x ( x, t )] ψ + ( κ, x, t ) ∗ = 8 i ( κ ∗ ) 3 ψ + ( κ, x, t ) ∗ , (4.11) where w e recall that the p oten tials u and v are assumed t o be real v alued. Multiply ing (4.10) b y ψ + ( κ, x, t ) ∗ and (4.11) b y ψ + ( κ, x, t ) , and a dding the resulting equations we obtain ∂ t | ψ + | 2 + ∂ x  8 ψ ∗ + ( ∂ 2 x ψ + ) + 8 ψ + ( ∂ 2 x ψ ∗ + ) − 8( ∂ x ψ + )( ∂ x ψ ∗ + ) + 6 u | ψ + | 2  = − 8(1 + i ) κ 3 | ψ + | 2 , where w e ha v e used the fact that k ∗ = − ik on the ray arg k = π / 4 . In tegrating ov er t he real axis and using the v anishing of ψ + , ∂ x ψ + , ∂ 2 x ψ + , and u as x → + ∞ and x → −∞ , w e obtain d dt Z ∞ −∞ dx | ψ + ( κ, x, t ) | 2 = − 8(1 + i ) κ 3 Z ∞ −∞ dx | ψ + ( κ, x, t ) | 2 . (4.12) Using (4. 9) w e can write (4.12) as d dt  1 c + ( κ, t ) 2  = − 8(1 + i ) κ 3 c + ( κ, t ) 2 , 12 or equiv alen tly we obtain dc + ( κ, t ) dt = 4( 1 + i ) κ 3 c + ( κ, t ) , whic h yi elds c + ( κ, t ) = c + ( κ, 0) e 4(1+ i ) κ 3 t . The time evolution o f the norming constants c − ( κ, t ) , d + ( κ, t ) , and d − ( κ, t ) app earing in the anal ogs of (3.4) and (3. 6 ) can b e o bt a i ned in a simi lar wa y . With the help of (4.3) w e obtain (4.7) , and hence the dep endency constan ts γ ( κ, t ) and η ( κ, t ) app eari ng in the analogs of (3.3) and (3.5), resp ecti v ely , ev olv e according to (4. 8). 5. A COM P ARISON OF REFERENCES [8] AND [10] As seen from (1 . 1), if f ( k , x ) is a soluti on to (1 .1), so are f ( − k , x ) , f ( ik , x ) , and f ( − ik , x ) . Th us, a solutio n to (1.1) k no wn in the region Ω in the complex k -plane can b e extended to the three regions obtai ned by rotati ng Ω b y π / 2 , π , and 3 π / 2 , resp ectively , around the origi n of the complex k -plane. Moreov er, since the p oten tials u and v are real v alued, f ( k ∗ , x ) ∗ is also a solution and hence a soluti on kno wn in a region in the complex k -plane can b e extended to the symmetric region wi th respect to the real axis. Th us, solutions known in Ω can b e extended to the en tire complex k -plane. In [10], some four solutions Ψ j ( k , x ) to ( 1 .1) for j = 1 , 2 , 3 , 4 are presen ted on the whole complex k -plane with spatial asymptoti cs Ψ j ( k , x ) = ( e k j x [1 + o (1)] , x → + ∞ , a j ( k ) e k j x [1 + o (1)] , x → −∞ , where the a j ( k ) are certain co efficien ts and t he k j are as in (2 .3). Comparing the asy mp- totics as x → ±∞ , w e see that those four soluti ons are related to the Jost and exp onen tial solutions app eari ng in (2.1) and (2.2) as follows: Ψ 1 ( k , x ) = φ − ( k , x ) A ( k ) , Ψ 2 ( k , x ) = ψ + ( k , x ) , 13 Ψ 3 ( k , x ) = φ + ( k , x ) , Ψ 4 ( k , x ) = T ( k ) ψ − ( k , x ) , where A and T ar e the co efficien ts app earing i n some o f (2.4)-(2.6 ). Then, as k mov es to the b oundary of arg k ∈ (0 , π / 4) from the in terior, we see that the reflection co efficien ts r 0 ( k ) , r 1 ( k ) , and r 2 ( k ) defined in [ 1 0] are relat ed as fol lo ws to the co efficien ts used in [8] and i n our pap er: r 0 ( k ) = R − ( k ) , r 1 ( k ) = C + ( k ) ∗ , r 2 ( k ) = B − ( k ) ∗ A ( k ) ∗ . Moreo v er, the quantities a j ( k ) app earing in [10] are rela t ed to the quan tities used in [8] and i n our pap er as a 1 ( k ) = 1 A ( k ) , a 2 ( k ) = 1 T ( k ) , a 3 ( k ) = A ( k ) , a 4 ( k ) = T ( k ) . Th us, the in v erse problem has b een analyzed in [10] in the sp ecial case R ± ( k ) = C ± ( k ) = B ± ( k ) = 0 . In that case, A ( k ) and T ( k ) simply b ecome rational functions of k wi th asymptotics 1 + O (1 /k ) as k → ∞ and with appropriate jump conditions on the rays arg k = ( l − 1) π / 4 for l = 1 , 2 , . . . , 8 . One can then form ulate the inv erse problem o n t he en tire complex k -plane as a Riemann-Hilb ert problem and solve it explicit ly . 6. EXAMPLES In t hi s section we presen t some explicit examples of solutions and relev ant quan tities asso ciated with (1. 1). Such examples should help to understand b et t er the scattering and b ound-state data for (1.1). It is already k no wn [5,8,10] that if f ( k , x ) is a solution to the Sc hr¨ odinger equation − f ′′ ( k , x ) + q ( x ) f ( k , x ) = k 2 f ( k , x ) , then f ( k , x ) is al so a soluti o n to (1.1) when u ( x ) = − 2 q ( x ) , v ( x ) = q ( x ) 2 − q ′′ ( x ) , b ecause in that case we ha v e D 4 + D uD + v = ( − D 2 + q ) 2 . (6.1) 14 Note that (6 .1) holds in our first and fourth exa m pl es b elow, but it do es not hold for our second a nd third examples. Example 6.1 Let u ( x ) = v ( x ) = 0 for x < 0 and u ( x ) = 4 e x (1 + e x ) 2 , v ( x ) = 2 e x (1 − e x ) 2 (1 + e x ) 4 , x > 0 . By using the con tin uit y of ψ + , ψ ′ + , ψ ′′ + at x = 0 and the jump condition ψ ′′′ + ( k , 0 + ) − ψ ′′′ + ( k , 0 − ) = − u (0 + ) ψ ′ + ( k , 0) , w e can determine all the q uan tities relev a n t t o (1.1) . In t erms of f ( k , x ) := e ikx  1 − 2 i (2 k + i ) ( 1 + e x )  , w e hav e the Jost and ex p onen tial sol ut i ons ψ + ( k , x ) =      f ( k , x ) + C + ( k ) f ( ik , x ) , x ≥ 0 , 1 T ( k ) e ikx + R + ( k ) T ( k ) e − ikx + c 1 ( k ) e kx , x ≤ 0 , φ + ( k , x ) = ( f ( ik , x ) , x ≥ 0 , A ( k ) e − kx + B + ( k ) e ikx + c 2 ( k ) e kx + c 3 ( k ) e − ikx , x ≤ 0 , where A ( k ) = (8 k 2 + 1)(2 k − 1) 16 k 3 , B + ( k ) = (4 − 4 i ) k 2 + i 16 k 3 (2 k + 1 ) , c 1 ( k ) = (2 i − 2) k + ( 1 + i ) 2 k (8 k 2 + 1 ) , c 2 ( k ) = − 2 k + 1 16 k 3 , c 3 ( k ) = (4 + 4 i ) k 2 − i 16 k 3 (2 k + 1) , R + ( k ) = 16 k 4 + ( 2 + 4 i ) k 2 − 1 (8 k 2 − i )(16 k 4 + 2 ik 2 − 1 ) , C + ( k ) = (4 + 4 i ) k 2 + i (8 k 2 + 1 )(2 k + i )(2 k − 1 ) , T ( k ) = 2 k (2 k − 1 )(2 k + 1)(2 k + i )(8 k 2 + 1 ) (8 k 2 − i )( 16 k 4 + 2 ik 2 − 1) . The remaining co efficien ts R − , B − , and C − can easi l y b e ev al uat ed by using (2.7) and (2.8) . In t hi s example, there is exactl y one simple b o und st a te a t k = κ wit h κ := (1 + i ) / 4 , where T ( k ) has a simple p ol e. A corresp onding b ound-state eigenfunction is a constan t m ultiple of ψ + ( κ, x ) , and it can b e c hosen a s ϕ ( κ, x ) =      e − x/ 4 1 + e x { cos( x/ 4) + sin( x/ 4) + e x [cos( x/ 4) − 3 sin( x/ 4)] } , x ≥ 0 , e x/ 4 [cos( x/ 4) − 3 sin( x/ 4)] , x ≤ 0 . 15 Ev en though A and T each ha ve a zero at k = 1 / 2 on the ray ar g k = 0 , their ra tio A/T is nonzero at k = 1 / 2 , which corresponds to a sp ectral singularity and not to a bo und state. Example 6.2 Let u ( x ) = 0 and v ( x ) = − ǫδ ( x ) , wit h δ ( x ) denoting the Di rac delta distribution and ǫ b ei ng a real, nonzero parameter. W e w an t ψ , ψ ′ , ψ ′′ to b e con tin uous at x = 0 , and ψ ′′′ ( k , 0 − ) = − ǫ + ψ ′′′ ( k , 0 + ) . W e find ψ + ( k , x ) =      e ikx − ǫ 4 k 3 + ǫ e − kx , x ≥ 0 , 4 k 3 + (1 − i ) ǫ 4 k 3 + ǫ e ikx + iǫ 4 k 3 + ǫ e − ikx − ǫ 4 k 3 + ǫ e kx , x ≤ 0 , φ + ( k , x ) =    e − kx , x ≥ 0 , − iǫ 4 k 3 e ikx + iǫ 4 k 3 e − ikx − ǫ 4 k 3 e kx + 4 k 3 4 k 3 + ǫ e − kx x ≤ 0 . The co efficien ts relat ed to t he corresp onding scattering problem are given b y 1 T ( k ) = 4 k 3 + (1 − i ) ǫ 4 k 3 + ǫ , R ± ( k ) = iǫ 4 k 3 + ( 1 − i ) ǫ , A ( k ) = 4 k 3 + ǫ 4 k 3 , B ± ( k ) = − iǫ 4 k 3 , C ± ( k ) = − ǫ 4 k 3 + ǫ . Note that A ( k ) /T ( k ) v a ni shes on the rays arg k = 0 and arg k = π / 4 only when 4 k 3 + ( 1 − i ) ǫ = 0 . Since we assume ǫ 6 = 0 , w e find t hat there are no such k v alues if ǫ < 0 , and there exists exactly one k v al ue lying on the ra y a r g k = π / 4 when ǫ > 0 . Denoting that k -v al ue b y κ, w e obta i n a b ound state of mu ltipli ci t y one at k = κ, where κ := 1 + i 2 3 √ ǫ. Th us, a b ound-state eigenfunction i s obtained as ψ + ( κ, x ) = ( e iκx + ie − κx , x ≥ 0 , e − iκx + ie κx , x ≤ 0 . Since R ∞ −∞ dx | ψ + ( κ, x ) | 2 = 4 / 3 √ ǫ, a normalized b ound-state eig enfunction is given b y ϕ ( κ, x ) = ( 6 √ ǫe − 3 √ ǫ x/ 2 [cos( 3 √ ǫ x/ 2) + si n( 3 √ ǫ x/ 2)] , x ≥ 0 , 6 √ ǫe 3 √ ǫ x/ 2 [cos( 3 √ ǫ x/ 2) − sin( 3 √ ǫ x/ 2)] , x ≤ 0 . 16 Example 6.3 F or any p osit ive constant c, consider u ( x ) = 16 c 2 [1 + √ 2 cosh(2 cx )] [ √ 2 + cosh(2 cx )] 2 , v ( x ) = 4 c 4 [ √ 2 cosh(6 cx ) − 12 cosh(4 cx ) − 5 √ 2 cosh(2 cx ) + 4 ] [ √ 2 + cosh(2 cx )] 4 . W e then obtai n A ( k ) = [ k − (1 + i ) c ][ k − (1 − i ) c ] [ k + (1 + i ) c ][ k + (1 − i ) c ] , T ( k ) = [ k + (1 + i ) c ][ k − (1 − i ) c ] [ k − (1 + i ) c ][ k + (1 − i ) c ] , R ± ( k ) = 0 , C ± ( k ) = 0 , B ± ( k ) = 0 , ψ + ( k , x ) = e ikx  1 + iα ( x ) k + (1 + i ) c + iα ( x ) ∗ k − (1 − i ) c  , φ + ( k , x ) = ψ + ( ik , x ) , where we ha v e defined α ( x ) := − √ 2 c − (1 + i ) c e − 2 cx √ 2 + cosh(2 cx ) . This example w as presen ted in [10] in differen t terminology . Note that A ( k ) /T ( k ) has a double zero at k = κ with κ : = ( 1 + i ) c, whic h correspo nds t o a bound state of multiplicit y t w o. Two linearly independen t eigenfunctions are given b y ψ + ( κ, x ) and φ + ( κ, x ) , or they can b e c hosen as real v alued, e.g. as ϕ 1 ( κ, x ) = ( √ 2 e − cx + 2 e cx ) cos( cx ) + √ 2 e − cx sin( cx ) √ 2 + cosh(2 cx ) , ϕ 2 ( κ, x ) = ( √ 2 e − cx + 2 e cx ) sin( cx ) − √ 2 e − cx cos( cx ) √ 2 + cosh(2 cx ) . Example 6.4 Consider the p otentials u ( x ) = − 4 ( | x | + 1) 2 , v ( x ) = − 8 ( | x | + 1) 4 . Using (2. 4)-(2.6) a nd the contin uity of the solutions to (1.1) a nd the con t in uit y of their first, second, and t hi rd x -deriv at iv es, we get T ( k ) = k ( k 4 + k 3 + k 2 + 2 k + 2) ( k + i )[ k 4 + ( 1 + i ) k 3 + ik 2 + (1 − i ) k + 3] , 17 A ( k ) = ( k + 1) ( k 4 + k 3 + k 2 + 2 k + 2) k 5 , C + ( k ) = − 2( k + i ) k 4 + k 3 + k 2 + 2 k + 2 , B + ( k ) = − 2 i ( k + 1)( k + i ) k 5 , R + ( k ) = i ( k 2 + k + 3) ( k + i )[ k 4 + (1 + i ) k 3 + ik 2 + ( 1 − i ) k + 3] , ψ + ( k , x ) =      f ( k , x ) + C + ( k ) f ( ik , x ) , x ≥ 0 1 T ( k ) f ( − k , − x ) + R + ( k ) T ( k ) f ( k , − x ) + C + ( k ) f ( ik , − x ) , x ≤ 0 , and for x ≥ 0 w e ha v e φ + ( k , x ) = f ( ik , x ) , while for x ≤ 0 we ha v e φ + ( k , x ) = A ( k ) f ( − ik , − x ) + B + ( k ) f ( − k , − x ) + B + ( k ) ∗ f ( k , − x ) − k 2 − 2 k 5 f ( ik , − x ) , where we ha v e defined f ( k , x ) := e ikx  1 + i k ( x + 1)  . In this example there exists exactly o ne b ound state at k = κ wit h κ := 0 . 77 8(1 + i ) corresp onding to the simple p ole of T ( k ) on the ray arg k = π / 4 . Note that w e ha v e used an ov erline to denote the round-off on the di g it. An eigenfunction for that si mpl e b ound state is a constant m ultiple of ψ + ( κ, x ) . A t k = 0 . 47 6 + 1 . 183 i in the in terior of the sector Ω , b oth A and T ha v e simple zeros wi thout A/T v anishing there; thus, that k -v al ue do es not corresp ond to a b ound state and it corresp o nds to a nonsp ectral singulari t y . Ac kno wledgment . The researc h leading to this article was supp orted in part b y the Na- tional Science F oundation under grant DMS-0610494 and a National T ec hnical Universit y PEBE gran t. The first author is grateful to the colleagues in the Departmen t of Mathe- matics at Nati onal T ech nical Universit y of A thens for t heir hospitality during his recen t visit. REFERENCES [1] M. J. Abl owitz and P . A. 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