The KdV equation on the half-line: Time-periodicity and mass transport

THE KD V EQUA TION ON THE HALF-LINE: TIME-PERIODICITY AND MASS TRANSPOR T JERR Y L. BONA AND JONA T AN LENELLS Abstract. The work presented here emanates from questions arising from exp er- imen tal observ ations of the propagation of surface water wa v es. The exp erimen ts in question featured a perio dically moving w av emak er located at one end of a flume that generated unidirectional wa v es of relatively small amplitude and long wa v e- length when compared with the undisturbed depth. It w as observed that the wa ve profile at any p oin t down the channel v ery quickly became perio dic in time with the same p erio d as that of the wa v emak er. One of the questions dealt with here is whether or not suc h a prop erty holds for mo del equations for such wa ves. In the present discussion, this is examined in the context of the Korteweg-de V ries equation using the recently dev elop ed version of the inv erse scattering theory for b oundary v alue problems put forward by F ok as and his collaborators. It turns out that the Korteweg-de V ries equation does possess the properly that solutions at a fixed point down the channel hav e the property of asymptotic p eriodicity in time when forced perio dically at the b oundary . Ho wev er, a more subtle issue to do with conserv ation of mass fails to hold at the second order in a small parameter which is the t ypical wa v e amplitude divided b y the undisturb ed depth. AMS Subject Classifica tions (2010) : 35B30, 35C15, 35Q53, 37K10, 37K15, 76B15, 86-05. Keywords : W ater wa v es, initial-b oundary-v alue problem, w a ve tank experiments, Korteweg- de V ries equation. 1. Introduction The propagation of long-crested, unidirectional, small-amplitude, long w av elength distur- bances o ver a featureless, flat b ottom in shallow water can b e approximately describ ed by Kortew eg-de V ries–type equations. A one-parameter class of suc h equations takes the form u t + u x + α uu x + β  µu xxx − (1 − µ ) u xxt  = 0 . (1.1) Here, the indep endent v ariable x is prop ortional to distance in the direction of propagation while t is prop ortional to elapsed time. The dep enden t v ariable u ( x, t ) is prop ortional to to the deviation of the free surface from its rest p osition at the p oint corresp onding to x at time t . The real parameters α = a h and β = h 2 λ 2 are defined in terms of a typical amplitude a and w av elength λ as well as the undisturb ed depth h of the w ater. The v ariables are scaled so that u and its partial deriv ativ es are formally all of order one while α and β are assumed to b e small compared to one. The parameter µ is a mo deling parameter that in principle can tak e an y real v alue. Ho wev er, the initial-v alue problem for the mo del will not b e w ell p osed unless µ ≤ 1. In a flat-b ottomed, lab oratory channel, Zabusky and Galvin [31] ran experiments showing qualitativ e agreement b etw een measurements and the mo del’s predictions for the case µ = 1, the classical Kortew eg-de V ries equation (KdV equation henceforth). Later work by Hammack and Segur [27] contin ued this line of inv estigation and also found qualitative agreement. The detailed accuracy of such models in the case µ = 0, the BBM equation, was in v estigated in a series of w av e tank exp eriments rep orted in [11]. In these exp eriments, a paddle-type w av emak er mounted at one end of the tank was oscillated p erio dically and the resulting wa ve 1 2 JERR Y L. BONA AND JONA T AN LENELLS motion w as monitored at several points down the channel. More precisely , four measurements of the w av e motion were tak en at p oin ts x = 0 < x 1 < x 2 < x 3 . This pro duced four time series, u (0 , t ) , u ( x 1 , t ) , u ( x 2 , t ) and u ( x 3 , t ) , t ≥ 0. These measuremen ts suggested an initial-b oundary-v alue problem for (1.1) wherein the measuremen t u (0 , t ) = g 0 ( t ) w as taken as b oundary data for the equation and the initial data u ( x, 0) = u 0 ( x ) was identically zero, corresp onding to the w ater b eing initially at rest. The predictions of this initial-b oundary-v alue problem w ere then compared directly to the other three time series. As the equation (1.1) is an approximation for wa v es moving only to the right, the exp eriment ceases as so on as the wa v es reac h the end of the channel and reflection b ecomes relev an t. Hence, a natural b oundary condition at x = L , the end of the tank, is u ( L, t ) = 0 and, in the case of the KdV equation, the second b oundary condition u x ( L, t ) = 0 is also required. How ev er, since a lateral b oundary condition at the end of the c hannel a wa y from the wa v emak er is irrelev an t to the motion prior to the w a ve reaching it, it is mathematically easier to simply push the righ t-hand b oundary to infinit y . Rigorous justification of this pro cedure on the time scale where there is no motion at x = L can b e found in [7] and [8]. The outcome of the comparisons made in [11] is that the just-describ ed initial-b oundary-v alue problem works quantitativ ely quite well, ev en for rather large v alues of the Stokes’ num ber S = α/β . In the course of examining the results of the exp eriments just describ ed, tw o qualitative features of the wa v e motion emerged. The goal of the presen t essay is to address rigorously these tw o aspects in the context of the KdV equation, µ = 1. A. Time-p erio dicity . The exp erimental data indicate that a p erio dically moving w av emak er giv es rise to measurements u (0 , t ) and u ( x j , t ) , j = 1 , 2 , 3 , which are asymptotically time- p erio dic with the same p erio d as that of the w a vemak er. In other words, if the wa v e maker oscillates with p erio d t p , then the functions u (0 , t ) and u ( x j , t ) approach functions whic h are p erio dic in t with p erio d t p as t → ∞ . Indeed, in the exp eriments, this approac h is seen to b e very rapid. B. Mass transp ort. In the exp erimental set-up, the total mass of the water in the channel is evidently constant. As the wa v emak er oscillates p erio dically , the net amoun t of water added to the region beyond the first measuremen t p oin t x = 0 oscillates accordingly . Thus the function M ( t ) = R ∞ 0 u ( x, t ) dx is exp ected to settle do wn to p erio dic oscillations around zero for large v alues of t . The plan of the pap er is to introduce the principal theorems for b oth the linear problem in which the nonlinearity is dropp ed and the nonlinear problem in the next section. This will include a discussion of previous work on these problems. Theorem 2.1 will b e prov ed in Section 3 while the nonlinear Theorem 2.2 and the resulting Corollary 2.3 will b e dealt with in Section 4. The pro ofs are inspired by the developmen ts in [25] where the nonlinear Sc hr¨ odinger equation with asymptotically time-perio dic data was considered. As mentioned, atten tion will be giv en only to the Kortew eg-de V ries equation, the case µ = 1. This is b ecause the main tool used in the presen t in vestigation is in verse scattering theory . As far as we know, the model (1.1) does not ha v e an in v erse scattering theory if µ 6 = 1. A brief concluding section includes not only a summary of the results, but brief remarks on the implications for w av e tank exp eriments. 2. Main resul ts In this se ction, the t wo principal results of our study are stated and discussed. The pro ofs are presented in Sections 3 and 4. An elementary rescaling of the indep endent and dependent v ariables assures that w e ma y take α = 6 and β = 1 in (1.1). Ho wev er, it must b e remembered that the resulting b oundary data αg 0 ( β 1 2 t ) / 6 now dep ends up on the parameters α and β . The mathematical problem under consideration is then the equation u t + u x + 6 uu x + u xxx = 0 , x > 0 , t > 0 , (2.1) THE KDV EQUA TION ON THE HALF-LINE 3 together with the initial and b oundary conditions u ( x, 0) = u 0 ( x ) ≡ 0 , for x ≥ 0 and u (0 , t ) = g 0 ( t ) , for t ≥ 0 , (2.2) where the giv en Diric hlet b oundary v alue g 0 is tak en to be smo oth and compatible with the v anishing initial data at t = 0, which is to sa y g 0 (0) = 0. This initial-b oundary v alue problem has receiv ed considerable attention. It is known to be globally well p osed in a v ariet y of circumstances to do with restrictions on the initial and b oundary data (these developmen t started with [16] and [17]; see [15] and the references contained therein for a more up-to-date appraisal). As the present discussion derives directly from exp erimental results, we are not going to b e sp ecially concerned with sharp hypotheses on the data. The answers to the issues just mentioned are the fo cus. Also, while the Korteweg-de V ries equation is kno wn rigorously to approximate w ell solutions of the full, inviscid water w a v e problem (see [1], [10], [12], [21]), that fact dep ends up on smo othness of the auxiliary data. Without sufficien t smoothness, there is no appro ximation. 2.1. The linear limit. The first theorem answers the questions p osed in the introduction in the affirmative in the case of the linearized version of equation (2.1). Theorem 2.1. L et u ( x, t ) b e a sufficiently smo oth solution of the line arize d KdV e quation u t + u x + u xxx = 0 , x > 0 , t > 0 . (2.3) with vanishing initial data and c omp atible, p erio dic Dirichlet data u (0 , t ) = g 0 ( t ) of p erio d t p > 0 , i.e., g 0 (0) = 0 and g 0 ( t + t p ) − g 0 ( t ) = 0 for t > 0 . ( a ) F or any fixe d x ≥ 0 , u ( x, t ) is asymptotic al ly time-p erio dic with p erio d t p . Mor e pr e cisely, for e ach x ∈ R and as t → ∞ , u ( x, t + t p ) − u ( x, t ) = O ( t − 3 2 ) . (2.4) ( b ) F or any fixe d x ≥ 0 , the mass function M ( x, t ) = R ∞ x u ( x, t ) dx has the pr op erty that as t → ∞ , M ( x, t + t p ) − M ( x, t ) = Z t p 0 g 0 ( t 0 ) dt 0 + O ( t − 3 2 ) . (2.5) In p articular, if g 0 ( t ) has zer o aver age, then M ( x, t ) is asymptotic al ly time-p erio dic with p erio d t p . Asymptotic perio dicity is established for the linear problem for both the KdV and the BBM equation in [18]. These results are obtained b y classical energy estimates. The theory rep orted there is not as sharp as that obtained here using inv erse scattering techniques. In any case, the linear inv erse scattering theory is needed for our analysis of the nonlinear problem. 2.2. The nonlinear problem. In the second theorem, nonlinear corrections to Theorem 2.1 are kept and estimated. Thus, consider a p erturbative solution u ( x, t ) = u 1 ( x, t ) +  2 u 2 ( x, t ) + O (  3 ) of (2.1) with Diric hlet data g 0 ( t ) = g 01 ( t ) +  2 g 02 ( t ) + O (  3 ) , where  > 0 is a small parameter. The first and second Neumann b oundary v alues of the solution are g 1 ( t ) = u x (0 , t ) and g 2 ( t ) = u xx (0 , t ) . Their resp ective p erturbative expansions are written as g 1 ( t ) = g 11 ( t ) +  2 g 12 ( t ) + O (  3 ) , g 2 ( t ) = g 21 ( t ) +  2 g 22 ( t ) + O (  3 ) . (2.6) F or definiteness, the results are presen ted when the Dirichlet data comprise a p erio dic sine- w av e. Similar results can b e obtained for other p erio dic b oundary forcings. It is w orth men tioning that the measured boundary conditions in the experiments rep orted in [11] closely 4 JERR Y L. BONA AND JONA T AN LENELLS resem ble a sine w av e while the b oundary conditions used in the sediment transp ort study [3] w ere modeled exactly as sine wa v es with the field measured amplitudes and frequencies. Theorem 2.2. L et ω ∈ R b e a non-zer o c onstant such that | ω | 6 = 2 3 √ 3 and | ω | 6 = 1 3 √ 3 . (2.7) L et u ( x, t ) b e a solution of (2.1) with b oundary data g 0 ( t ) =  sin ω t. L et K, L ∈ C denote the unique solutions of the cubic e quations 4 K 3 − K + ω 2 = 0 , 4 L 3 − L + ω = 0 , such that K and L b elong to the b oundary ∂ D 3 of the domain D 3 ⊂ C define d by D 3 = { Im k < 0 } ∩ { Im(4 k 3 − k ) > 0 } (se e Figur e 1). Then, the first and se c ond Neumann b oundary values of u at x = 0 ar e as in (2.6), wher e, as t → ∞ , g 11 ( t ) = − K e iω t − ¯ K e − iω t + O  t − 3 2  , (2.8a) g 21 ( t ) = 2 iK 2 e iω t − 2 i ¯ K 2 e − iω t + O  t − 3 2  , (2.8b) g 12 ( t ) = 1 8 i  L K 2 − 2 K  e 2 iω t + 1 2 Im  1 K  − 1 8 i  ¯ L ¯ K 2 − 2 ¯ K  e − 2 iω t + O  t − 3 2  , (2.8c) g 22 ( t ) =  1 − L 2 4 K 2  e 2 iω t + Re  K ¯ K  − 1 +  1 − ¯ L 2 4 ¯ K 2  e − 2 iω t + O  t − 3 2  . (2.8d) Theorem 2.2 reveals that, at least to second order in perturbation theory , the time-perio dic Diric hlet profile  sin ω t gives rise to time-p erio dic Neumann conditions. This suggests that the time-p erio dic Dirichlet data also generates a time-p erio dic solution in the nonlinear case. Results of this nature for the nonlinear problem, but with damping incorp orated, are discussed in [13]. That theory relies up on rather delicate F ourier analysis and is not as sharp as what is brough t forth here. The situation for mass transport is more complicated. Corollary 2.3. Under the assumptions of The or em 2.2, the mass function M ( t ) = R ∞ 0 u ( x, t ) dx satisfies M ( t + t p ) − M ( t ) = m 1 ( t ) +  2 m 2 ( t ) + O (  3 ) , (2.9) wher e t p = 2 π /ω and, as t → ∞ , m 1 ( t ) = O  t − 3 2  , (2.10a) m 2 ( t ) = π ω  1 + 2 Re  K ¯ K  + O  t − 3 2  . (2.10b) Pr o of. Calculate as follows: M ( t + t p ) − M ( t ) = Z ∞ 0 Z t + t p t u t ( x, s ) dsdx = − Z t + t p t Z ∞ 0  u x + 6 uu x + u xxx  ( x, s ) dxds = Z t + t p t  g 0 ( s ) + 3 g 2 0 ( s ) + g 2 ( s )  ds. THE KDV EQUA TION ON THE HALF-LINE 5 - 2 - 1 0 1 2 - 2 - 1 0 1 2 D 0 1 D 2 D 00 1 D 00 4 D 3 D 0 4 Figure 1. The domains { D j } 4 1 in the c omplex k -plane with D 1 = D 0 1 ∪ D 00 1 and D 4 = D 0 4 ∪ D 00 4 . It thus transpires that m 1 ( t ) = Z t + t p t  g 01 ( s ) + g 21 ( s )  ds and m 2 ( t ) = Z t + t p t  3 g 2 01 ( s ) + g 22 ( s )  ds. Since g 01 ( t ) = sin ω t , the expressions for g 21 and g 22 obtained in Theorem 2.2 yield (2.10). 2 Equation (2.10b) implies that m 2 ( t ) do es not approach zero as t → ∞ . Indeed, lim t →∞ m 2 ( t ) = 0 if and only if arg K = ± π 3 + π n for some n ∈ Z , and this equation is never satisfies for K ∈ ∂ D 3 and K 6 = 0. This suggests that a perio dic Diric hlet b oundary condition do es not in general give rise to an asymptotically p erio dic mass function M ( t ) = R ∞ 0 u ( x, t ) dx , although the discrepancy lies at second order. 3. Proof of Theorem 2.1 The first step is to derive an integral represen tation for the solution u ( x, t ) of the b oundary v alue problem for the linear equation (2.3). Define the op en subsets { D j } 4 1 of the complex k -plane b y D 1 = { Im k > 0 } ∩ { Im(4 k 3 − k ) > 0 } , D 2 = { Im k > 0 } ∩ { Im(4 k 3 − k ) < 0 } , D 3 = { Im k < 0 } ∩ { Im(4 k 3 − k ) > 0 } , D 4 = { Im k < 0 } ∩ { Im(4 k 3 − k ) < 0 } . Let D 1 = D 0 1 ∪ D 00 1 where D 0 1 = D 1 ∩ { Re k > 0 } and D 00 1 = D 1 ∩ { Re k < 0 } . Similarly , let D 4 = D 0 4 ∪ D 00 4 with D 0 4 = D 4 ∩ { Re k > 0 } and D 00 4 = D 4 ∩ { Re k < 0 } ; see again Figure 1. F or each k ∈ ¯ D 1 ∪ ¯ D 3 , the cubic p olynomial 4 ν 3 − ν − (4 k 3 − k ) v anishes at exactly one point in each of the three sets ¯ D 0 1 , ¯ D 00 1 , and ¯ D 3 . Denote these p oin ts b y ν 1 ( k ), ν 2 ( k ), and ν 3 ( k ), respectively . 6 JERR Y L. BONA AND JONA T AN LENELLS Lemma 3.1. The solution u ( x, t ) for the initial-b oundary-value pr oblem (2.2) for the lin- e arize d KdV e quation (2.3) has the r epr esentation u ( x, t ) = 1 π Z R e − 2 ikx − f ( k ) t ˆ u 0 ( k ) dk + 1 2 iπ Z ∂ D 3 e − 2 ikx − f ( k ) t  f 0 ( k ) ˜ g 0 ( f ( k ) , t ) − 2 ik ˆ u 0 ( ν 2 ( k )) − ˆ u 0 ( ν 1 ( k )) ν 1 ( k ) − ν 2 ( k ) − 2 i ν 2 ( k ) ˆ u 0 ( ν 1 ( k )) − ν 1 ( k ) ˆ u 0 ( ν 2 ( k )) ν 1 ( k ) − ν 2 ( k )  dk , (3.1) in terms of the initial data u 0 ( x ) and the Dirichlet data g 0 ( t ) . Her e, f ( k ) = 2 i (4 k 3 − k ) and ˆ u 0 ( k ) = Z ∞ 0 e 2 ikx u 0 ( x ) dx, Im k ≥ 0 , ˜ g 0 ( κ, t ) = Z t 0 e κs g 0 ( s ) ds, κ ∈ C . Pr o of. Equation (2.3) is the compatibilit y condition of the Lax pair ( ϕ x + 2 ik ϕ = − i 2 k u, ϕ t + f ( k ) ϕ = − 2 ik u + u x + i 2 k ( u + u xx ) , (3.2) where k ∈ C is the sp ectral parameter and ϕ ( x, t, k ) is a scalar-v alued eigenfunction. W rite (3.2) in differential form as d  e 2 ikx + f ( k ) t ϕ  = W , where the closed one-form W ( x, t, k ) is defined b y W = e 2 ikx + f ( k ) t  − i 2 k u dx +  − 2 ik u + u x + i 2 k ( u + u xx )  dt  . Stok es’ Theorem implies that the integral of W around the b oundary of the domain (0 , ∞ ) × (0 , t ) in the ( x, t )-plane v anishes. This yields the global relation ˆ u 0 ( k ) − e f ( k ) t ˆ u ( k , t ) + ˜ g ( k , t ) = 0 , k ∈ ¯ D 1 ∪ ¯ D 2 , (3.3) where ˜ g ( k , t ) = (1 − 4 k 2 ) ˜ g 0 ( f ( k ) , t ) − 2 ik ˜ g 1 ( f ( k ) , t ) + ˜ g 2 ( f ( k ) , t ) , ˆ u 0 ( k ) = Z ∞ 0 e 2 ikx u 0 ( x ) dx, ˆ u ( k , t ) = Z ∞ 0 e 2 ikx u ( x, t ) dx, ˜ g j ( κ, t ) = Z t 0 e κs g j ( s ) ds, j = 0 , 1 , 2 . Multiplying equation (3.3) by 1 π e − 2 ikx − f ( k ) t and integrating the result along R with resp ect to k , it transpires that u ( x, t ) = 1 π Z R e − 2 ikx − f ( k ) t ˆ u 0 ( k ) dk − 1 π Z ∂ D 3 e − 2 ikx − f ( k ) t ˜ g ( k , t ) dk , (3.4) where Jordan’s lemma has b een used to deform the con tour from R to − ∂ D 3 in the second in tegral. The final step consists of using the global relation to eliminate the tw o unkno wn functions ˜ g 1 ( k , t ) and ˜ g 2 ( k , t ) from (3.4). Letting k → ν j ( k ), j = 1 , 2, in (3.3) giv es ˆ u 0 ( ν j ( k )) − e f ( k ) t ˆ u ( ν j ( k ) , t ) + (1 − 4 ν 2 j ( k )) ˜ g 0 ( f ( k ) , t ) − 2 iν j ( k ) ˜ g 1 ( f ( k ) , t ) + ˜ g 2 ( f ( k ) , t ) = 0 , k ∈ ¯ D 3 , j = 1 , 2 . (3.5) THE KDV EQUA TION ON THE HALF-LINE 7 Solving these tw o equations for ˜ g 1 ( f ( k ) , t ) and ˜ g 2 ( f ( k ) , t ) leads to ˜ g 1 ( f ( k ) , t ) = i 2  ν 1 ( k ) − ν 2 ( k )  n − ˆ u 0 ( ν 1 ( k )) + ˆ u 0 ( ν 2 ( k )) + e f ( k ) t [ ˆ u ( ν 1 ( k ) , t ) − ˆ u ( ν 2 ( k ) , t )] + 4 ˜ g 0 ( f ( k ) , t )( ν 2 1 ( k ) − ν 2 2 ( k )) o , ˜ g 2 ( f ( k ) , t ) = 1 ν 1 ( k ) − ν 2 ( k ) n − ν 1 ( k ) ˆ u 0 ( ν 2 ( k )) + ν 2 ( k ) ˆ u 0 ( ν 1 ( k )) + e f ( k ) t [ ν 1 ( k ) ˆ u ( ν 2 ( k ) , t ) − ν 2 ( k ) ˆ u ( ν 1 ( k ) , t )] o − ˜ g 0 ( f ( k ) , t )(1 + 4 ν 1 ( k ) ν 2 ( k )) . Substituting these expressions in to the solution formula (3.4) and observing that, Jordan’s lemma implies that the con tributions from the terms in volving { ˆ u ( ν j ( k ) , t ) } 2 1 v anish leads immediately to (3.1). 2 No w supp ose that u 0 ( x ) = 0 and that g 0 ( t ) is perio dic with p erio d t p . T o prov e ( a ), note that equation (3.1) implies u ( x, t + t p ) − u ( x, t ) = 1 2 iπ Z ∂ D 3 f 0 ( k ) e − 2 ikx − f ( k ) t Z t + t p 0 e f ( k )( s − t p ) g 0 ( s ) dsdk − 1 2 iπ Z ∂ D 3 f 0 ( k ) e − 2 ikx − f ( k ) t Z t 0 e f ( k ) s g 0 ( s ) dsdk . Making the change of v ariables s 7→ s + t p in the part of the first s -in tegral that runs along ( t p , t + t p ) and using the perio dicity of g 0 , there obtains u ( x, t + t p ) − u ( x, t ) = 1 2 iπ Z ∂ D 3 f 0 ( k ) e − 2 ikx − f ( k ) t Z t p 0 e f ( k )( s − t p ) g 0 ( s ) dsdk . Deforming the con tour of in tegration from ∂ D 3 to the steepest descent contour Γ, defined in App endix A, and see also Figure 4, a steep est descent argument yields (2.4) (see Prop osition A.1). T o prov e ( b ), define m ( x, t ) by m ( x, t ) := M ( x, t + t p ) − M ( x, t ) = Z t + t p t ∂ t M ( x, t 0 ) dt 0 = − Z t + t p t Z ∞ x  u x ( x, t 0 ) + u xxx ( x, t 0 )  dxdt 0 = Z t + t p t  u ( x, t 0 ) + u xx ( x, t 0 )  dt 0 . (3.6) Equation (3.1) implies that u ( x, t ) = 1 2 iπ Z ∂ ˆ D 3 f 0 ( k ) e − 2 ikx − f ( k ) t  e f ( k ) t f ( k ) g 0 ( t ) − Z t 0 e f ( k ) s f ( k ) ˙ g 0 ( s ) ds  dk , where ∂ ˆ D 3 denotes the contour ∂ D 3 deformed so that it passes to the right of the remo v able singularit y at k = 0. Since the contour has b een deformed to ∂ ˆ D 3 , the k -in tegral can b e split and the part in volving g 0 ( t ) can b e calculated using Cauch y’s theorem to reac h the formula u ( x, t ) = Res k =0 f 0 ( k ) e − 2 ikx f ( k ) g 0 ( t ) − 1 2 iπ Z ∂ ˆ D 3 f 0 ( k ) f ( k ) e − 2 ikx − f ( k ) t Z t 0 e f ( k ) s ˙ g 0 ( s ) dsdk = g 0 ( t ) − 1 2 iπ Z ∂ ˆ D 3 f 0 ( k ) f ( k ) e − 2 ikx − f ( k ) t Z t 0 e f ( k ) s ˙ g 0 ( s ) dsdk . 8 JERR Y L. BONA AND JONA T AN LENELLS ∂ D 3 D 3 Re f < 0 Re f > 0 Re f < 0 Re f > 0 Re f < 0 Re f > 0 Figure 2. The c ontour ∂ D 3 and the r e gions of definite sign of Re f ( k ) . Substituting this expression for u ( x, t ) into (3.6) gives m ( x, t ) = Z t + t p t g 0 ( t 0 ) dt 0 − 1 4 π Z ∂ ˆ D 3 f 0 ( k ) k e − 2 ikx  Z t + t p t e − f ( k ) t 0 Z t 0 0 e f ( k ) s ˙ g 0 ( s ) dsdt 0  dk . (3.7) In tegrating by parts with resp ect to t 0 and using the p erio dicity of g 0 , it is inferred that the curly brack et in (3.7) equals − e − f ( k )( t + t p ) f ( k ) Z t + t p 0 e f ( k ) s ˙ g 0 ( s ) ds + e − f ( k ) t f ( k ) Z t 0 e f ( k ) s ˙ g 0 ( s ) ds + Z t + t p t 1 f ( k ) ˙ g 0 ( t 0 ) dt 0 = − e − f ( k )( t + t p ) f ( k ) Z t p 0 e f ( k ) s ˙ g 0 ( s ) ds. Hence, for any x ≥ 0 and t ≥ 0, m ( x, t ) = Z t + t p t g 0 ( t 0 ) dt 0 + 1 4 π Z ∂ ˆ D 3 f 0 ( k ) k f ( k ) e − 2 ikx e − f ( k )( t + t p ) Z t p 0 e f ( k ) s ˙ g 0 ( s ) dsdk . Deforming the contour of integration in the k -in tegral from ∂ ˆ D 3 to Γ, a steep est descen t argumen t pro vides (2.5) (see again Appendix A and Figure 4). 4. Proof of Theorem 2.2 The pro of relies on the formulas g 11 ( t ) = 4 π Z ∂ D 3 k  ˆ χ 11 ( t, k ) + f 0 ( k ) f ( k ) g 01 ( t ) 4  dk , (4.1a) g 21 ( t ) = 8 π i Z ∂ D 3 k 2  ˆ χ 11 ( t, k ) + f 0 ( k ) f ( k ) g 01 ( t ) 4  dk , (4.1b) g 12 ( t ) = 4 π Z ∂ D 3 k  ˆ χ 12 ( t, k ) + f 0 ( k ) f ( k ) g 02 ( t ) 4  dk + 2 g 01 ( t ) π Z ∂ ˆ D 3 χ 21 ( t, k ) dk , (4.1c) g 22 ( t ) = 8 π i Z ∂ D 3 k 2  ˆ χ 12 ( t, k ) + f 0 ( k ) f ( k ) g 02 ( t ) 4  dk + 2 g 11 ( t ) π Z ∂ ˆ D 3 χ 21 ( t, k ) dk − 2 g 2 01 ( t ) , (4.1d) THE KDV EQUA TION ON THE HALF-LINE 9 where f ( k ) = 2 i (4 k 3 − k ) as b efore and ˆ χ 11 ( t, k ) = − f 0 ( k ) 4 Z t 0 e f ( k )( t 0 − t ) g 01 ( t 0 ) dt 0 , (4.2a) χ 21 ( t, k ) = f 0 ( k ) 2 if ( k ) Z t 0 ( g 01 ( t 0 ) + g 21 ( t 0 )) dt 0 , (4.2b) ˆ χ 12 ( t, k ) = 3 X m =1 ν m ( k ) ν 0 m ( k )Φ 12 ( t, ν m ( k )) , (4.2c) with Φ 12 ( t, k ) = Z t 0 e f ( k )( t 0 − t )  i 2 k ( g 01 + g 21 )Φ 11 − 2 ik g 02 + g 12 + i 2 k ( g 02 + 2 g 2 01 + g 22 ) +  − 2 ik g 01 + g 11 + i 2 k ( g 01 + g 21 )  Φ 21  dt 0 , Φ 11 ( t, k ) = Z t 0 e f ( k )( t 0 − t )  − 2 ik g 01 + g 11 + i 2 k ( g 01 + g 21 )  dt 0 , Φ 21 ( t, k ) = − i 2 k Z t 0 ( g 01 + g 21 ) dt 0 . These relations can b e extracted from the nonlinear in tegral equations c haracterizing the Diric hlet to Neumann map of (2.1) derived in [28]. The ro ots { ν j ( k ) } 3 1 satisfy the identities 3 X j =1 1 ν j ( k ) = − 2 i f ( k ) , 3 X j =1 ν 0 j ( k ) ν j ( k ) = f 0 ( k ) f ( k ) , 3 X j =1 ν 2 j ( k ) ν 0 j ( k ) = f 0 ( k ) 8 i , 3 X j =1 ν j ( k ) = 3 X j =1 ν j ( k ) ν 0 j ( k ) = 3 X j =1 ν 3 j ( k ) ν 0 j ( k ) = 0 , 3 X j =1 ν 4 j ( k ) ν 0 j ( k ) = f 0 ( k ) 32 i . In consequence, it transpires that ˆ χ 12 ( t, k ) = f 0 ( k ) 4 f ( k ) Z t 0 e f ( k )( t 0 − t ) ( g 01 + g 21 )( t 0 ) Z t 0 0 ( g 01 + g 21 )( t 00 ) dt 00 dt 0 − f 0 ( k ) 4 f ( k ) Z t 0 e f ( k )( t 0 − t ) ( g 01 + g 21 )( t 0 ) Z t 0 0 e f ( k )( t 00 − t 0 ) ( g 01 + g 21 )( t 00 ) dt 00 dt 0 − f 0 ( k ) 4 Z t 0 e f ( k )( t 0 − t ) g 02 ( t 0 ) dt 0 = f 0 ( k ) 4 f ( k ) Z t 0 e f ( k )( t 0 − t ) ( g 01 + g 21 )( t 0 )  Z t 0 0 − Z t t 0  ( g 01 + g 21 )( t 00 ) dt 00 dt 0 − f 0 ( k ) 4 Z t 0 e f ( k )( t 0 − t ) g 02 ( t 0 ) dt 0 . (4.3) F urthermore, since f 0 ( k ) f ( k ) = 3 k + O ( k − 3 ) , as k → ∞ , Cauc hy’s theorem implies Z ∂ ˆ D 3 χ 21 ( t, k ) dk =  − π 2 + π Res k =0 f 0 ( k ) f ( k )  Z t 0 ( g 01 ( t 0 ) + g 21 ( t 0 )) dt 0 = π 2 Z t 0 ( g 01 ( t 0 ) + g 21 ( t 0 )) dt 0 . (4.4) F or eac h integer n ≥ 1, the third-order p olynomial f ( k ) + inω has three zeros; one zero in eac h of the three sets ∂ D 0 1 , ∂ D 00 1 and ∂ D 3 . Denote the unique solutions of f ( k ) + inω = 0 in ∂ D 3 corresp onding to n = 1 and n = 2 by K and L , resp ectively . 10 JERR Y L. BONA AND JONA T AN LENELLS ∂ ˆ D 3 a − a K L − ¯ K − ¯ L 0 Figure 3. The c ontour ∂ ˆ D 3 in the c ase of ω > 2 3 √ 3 . Sev eral in tegrands will ha v e singularities at p oints in the set { 0 , K , − ¯ K , L, − ¯ L } . Let ∂ ˆ D 3 denote the con tour ∂ D 3 depicted in Figure 3 with indentations inserted so that ∂ ˆ D 3 passes to the righ t of the points 0 , K, − ¯ K , L , and − ¯ L , Assumption (2.7) implies that these inden tations can be chosen to lie in D 2 ∪ D 4 . Indeed, (2.7) implies that neither K nor L coincides with ± a ∈ ¯ D 1 ∩ ¯ D 3 where a = 1 2 √ 3 . In fact, for | ω | < 1 3 √ 3 , both K and L b elong to the interv al ( − a, a ). F or 1 3 √ 3 < | ω | < 2 3 √ 3 , K ∈ ( − a, a ) and Im L < 0. F or | ω | > 2 3 √ 3 , Im L < Im K < 0. In particular, K is a simple zero of f ( k ) + iω = 0 and L is a simple zero of f ( k ) + 2 iω = 0. Similarly , − ¯ K ∈ ∂ D 3 is a simple zero of f ( k ) − iω = 0 and − ¯ L ∈ ∂ D 3 is a simple zero of f ( k ) − 2 iω = 0. The discussion contin ues with the particular choice g 0 ( t ) =  sin( ω t ). Eac h of the expres- sions in (4.1) is considered in turn with this choice of boundary data. 4.1. Asymptotics of g 11 ( t ) . Supp ose g 0 ( t ) =  sin( ω t ). Equation (4.2a) becomes ˆ χ 11 ( t, k ) = − f 0 ( k ) 8 i Z t 0 e f ( k )( t 0 − t ) ( e iω t 0 − e − iω t 0 ) dt 0 = − f 0 ( k ) 8 i  e iω t f ( k ) + iω − e − iω t f ( k ) − iω  + A 1 ( t, k ) , (4.5) where A 1 ( t, k ) = f 0 ( k ) 8 i e − f ( k ) t  1 f ( k ) + iω − 1 f ( k ) − iω  . (4.6) Substituting this into (4.1a) and using Cauch y’s theorem gives the expression g 11 ( t ) = 4 π Z ∂ ˆ D 3 k 8 i  − f 0 ( k ) f ( k ) + iω + f 0 ( k ) f ( k )  dk e iω t + 4 π Z ∂ ˆ D 3 k 8 i  f 0 ( k ) f ( k ) − iω − f 0 ( k ) f ( k )  dk e − iω t + 4 π Z ∂ ˆ D 3 k A 1 ( t, k ) dk = − K e iω t − ¯ K e − iω t + 4 π Z ∂ ˆ D 3 k A 1 ( t, k ) dk , (4.7) where the formulas Res k = K k f 0 ( k ) f ( k ) + iω = K and Res k = − ¯ K k f 0 ( k ) f ( k ) − iω = − ¯ K THE KDV EQUA TION ON THE HALF-LINE 11 ha ve b een applied. Using Jordan’s lemma to deform the contour ∂ ˆ D 3 to the steep est descent con tour Γ depicted in Figure 4, Prop osition A.1 reveals that Z ∂ ˆ D 3 k A 1 ( t, k ) dk = O  t − 3 2  , t → ∞ , (4.8) thereb y establishing (2.8a). 4.2. Asymptotics of g 21 ( t ) . Substituting (4.5) into (4.1b) and using Cauch y’s theorem pro- vides the formula g 21 ( t ) = 8 π i Z ∂ ˆ D 3 k 2 8 i  − f 0 ( k ) f ( k ) + iω + f 0 ( k ) f ( k )  dk e iω t + 8 π i Z ∂ ˆ D 3 k 2 8 i  f 0 ( k ) f ( k ) − iω − f 0 ( k ) f ( k )  dk e − iω t + 8 π i Z ∂ ˆ D 3 k 2 A 1 ( t, k ) dk = 2 iK 2 e iω t − 2 i ¯ K 2 e − iω t + 8 π i Z ∂ ˆ D 3 k 2 A 1 ( t, k ) dk . (4.9) when g 0 ( t ) =  sin( ω t ). Here, we ha ve used that Res k = K k 2 f 0 ( k ) f ( k ) + iω = K 2 , Res k = − ¯ K k 2 f 0 ( k ) f ( k ) − iω = ¯ K 2 . Just as in the case of g 11 ( t ), a steep est descen t argumen t sho ws that Z ∂ ˆ D 3 k 2 A 1 ( t, k ) dk = O  t − 3 2  , t → ∞ , whic h pro ves (2.8b). 4.3. Asymptotics of g 12 ( t ) . The computation of g 12 ( t ) relies on (4.1c) and proceeds via a series of lemmas. Lemma 4.1. The inte gr al in the last term on the right-hand side of (4.1c) is given by Z ∂ ˆ D 3 χ 21 ( t, k ) dk = − π e iω t 8 K − π e − iω t 8 ¯ K + π 4 Re  1 K  − 4 i Z t 0 Z ∂ ˆ D 3 k 2 A 1 ( t 0 , k ) dk dt 0 . (4.10) Pr o of. Equation (4.9) together with the identities 1 − 4 K 2 = ω 2 K , and 1 − 4 ¯ K 2 = ω 2 ¯ K , imply that g 01 ( t ) + g 21 ( t ) = ω 4 iK e iω t − ω 4 i ¯ K e − iω t + 8 π i Z ∂ ˆ D 3 k 2 A 1 ( t, k ) dk . (4.11) After substituting this expression in to (4.4), formula (4.10) emerges. 2 Lemma 4.2. The function ˆ χ 12 ( t, k ) in (4.2c) has the r epr esentation ˆ χ 12 ( t, k ) = ω f 0 ( k ) 64 if ( k )  1 K 2  − 2 e 2 iω t f ( k ) + 2 iω + e iω t f ( k ) + iω + e 2 iω t f ( k ) + iω  + 1 | K | 2  e iω t f ( k ) + iω + 1 f ( k ) + iω − e − iω t f ( k ) − iω − 1 f ( k ) − iω  (4.12) + 1 ¯ K 2  2 e − 2 iωt f ( k ) − 2 iω − e − iω t f ( k ) − iω − e − 2 iω t f ( k ) − iω  + E ( t, k ) + A 2 ( t, k ) , 12 JERR Y L. BONA AND JONA T AN LENELLS wher e E ( t, k ) and A 2 ( t, k ) ar e given by E ( t, k ) = f 0 ( k ) 4 f ( k ) Z t 0 e f ( k )( t 0 − t )  8 π i Z ∂ ˆ D 3 k 2 A 1 ( t 0 , k ) dk  − 2 e iω t 0 − 1 − e iω t 4 K − 2 e − iω t 0 − 1 − e − iω t 4 ¯ K + 8 π i  Z t 0 0 − Z t t 0  Z ∂ ˆ D 3 k 2 A 1 ( t 00 , k ) dk dt 00  dt 0 + f 0 ( k ) 4 f ( k ) Z t 0 e f ( k )( t 0 − t )  ω 4 iK e iω t 0 − ω 4 i ¯ K e − iω t 0  ×  8 π i  Z t 0 0 − Z t t 0  Z ∂ ˆ D 3 k 2 A 1 ( t 00 , k ) dk dt 00  dt 0 (4.13) and A 2 ( t, k ) = ω f 0 ( k ) 64 if ( k )  1 K 2  2 e − f ( k ) t f ( k ) + 2 iω − e − f ( k ) t f ( k ) + iω − e iω t − f ( k ) t f ( k ) + iω  + 1 | K | 2  − e − f ( k ) t f ( k ) + iω − e − iω t − f ( k ) t f ( k ) + iω + e − f ( k ) t f ( k ) − iω + e iω t − f ( k ) t f ( k ) − iω  + 1 ¯ K 2  − 2 e − f ( k ) t f ( k ) − 2 iω + e − f ( k ) t f ( k ) − iω + e − iω t − f ( k ) t f ( k ) − iω  . (4.14) Pr o of. In view of (4.11), it transpires that  Z t 0 0 − Z t t 0  ( g 01 ( t 00 ) + g 21 ( t 00 )) dt 00 = − 2 e iω t 0 − 1 − e iω t 4 K − 2 e − iω t 0 − 1 − e − iω t 4 ¯ K + 8 π i  Z t 0 0 − Z t t 0  Z ∂ ˆ D 3 k 2 A 1 ( t 00 , k ) dk dt 00 . Inserting this into the expression (4.3) for ˆ χ 12 ( t, k ) leads to ˆ χ 12 ( t, k ) = ω f 0 ( k ) 64 if ( k ) Z t 0 e f ( k )( t 0 − t )  − 2 e 2 iω t 0 − e iω t 0 − e iω ( t + t 0 ) K 2 + e iω t 0 + e − iω ( t − t 0 ) − e − iω t 0 − e iω ( t − t 0 ) | K | 2 + 2 e − 2 iω t 0 − e − iω t 0 − e − iω ( t + t 0 ) ¯ K 2  dt 0 + E ( t, k ) , where E ( t, k ) is as in (4.13). Computing the in tegrals with resp ect to t 0 giv es (4.12). 2 Lemma 4.3. The first term on the right-hand side of (4.1c) is 4 π Z ∂ D 3 k ˆ χ 12 ( t, k ) dk = 1 8 i  L K 2 − 1 K  e 2 iω t + i 4 Re  1 K  e iω t + 1 4 Im 1 K − i 4 Re  1 K  e − iω t − 1 8 i  ¯ L ¯ K 2 − 1 ¯ K  e − 2 iωt + 4 π Z ∂ ˆ D 3 k [ E ( t, k ) + A 2 ( t, k )] dk . (4.15) Pr o of. This is a consequence of the expression (4.12) for ˆ χ 12 ( t, k ) and Cauc hy’s theorem. F or example, (4.12) implies that the co efficient of e 2 iω t is ω 16 iπ K 2 Z ∂ ˆ D 3 k f 0 ( k ) f ( k )  1 f ( k ) + iω − 2 f ( k ) + 2 iω  dk . Since the integrand has simple p oles at k = K and k = L , Cauch y’s theorem implies that the latter integral equals ω 8 K 2  Res k = K + Res k = L  k f 0 ( k ) f ( k )  1 f ( k ) + iω − 2 f ( k ) + 2 iω  = 1 8 i  L K 2 − 1 K  . THE KDV EQUA TION ON THE HALF-LINE 13 2 Using (4.10) and (4.15) in the expression (4.1c) for g 12 ( t ), one finds that g 12 ( t ) = 1 8 i  L K 2 − 2 K  e 2 iω t + 1 2 Im  1 K  − 1 8 i  ¯ L ¯ K 2 − 2 ¯ K  e − 2 iω t + F 1 ( t, k ) , (4.16) where F 1 ( t, k ) = − 4 e iω t − e − iω t π Z t 0 Z ∂ ˆ D 3 k 2 A 1 ( t 0 , k ) dk dt 0 + 4 π Z ∂ ˆ D 3 k [ E ( t, k ) + A 2 ( t, k )] dk . (4.17) T o complete the pro of of (2.8c), it is enough to show that F 1 ( t, k ) is O ( t − 3 / 2 ) as t → ∞ . The pro of of this fact is p ostp oned to Section 4.5. 4.4. Asymptotics of g 22 ( t ) . The computation of g 22 ( t ) relies on (4.1d). Lemma 4.4. The last two terms on the right-hand side of (4.1d) c an b e written as 2 g 11 ( t ) π Z ∂ ˆ D 3 χ 21 ( t, k ) dk − 2 g 2 01 ( t ) = 3 4 e 2 iω t − 1 4  1 + K ¯ K  e iω t + 1 2 Re  K ¯ K  − 1 − 1 4  1 + ¯ K K  e − iω t + 3 4 e − 2 iω t + 8 i π  K e iω t + ¯ K e − iω t  Z t 0 Z ∂ ˆ D 3 k 2 A 1 ( t 0 , k ) dk dt 0 + 1 π  2 Re  1 K  − 1 K e iω t − 1 ¯ K e − iω t  Z ∂ ˆ D 3 k A 1 ( t, k ) dk − 32 i π 2  Z ∂ ˆ D 3 k A 1 ( t, k ) dk  Z t 0 Z ∂ ˆ D 3 k 2 A 1 ( t 0 , k ) dk dt 0  . (4.18) Pr o of. This follo ws from the expression (4.7) for g 11 ( t ), the expression (4.10) for R ∂ ˆ D 3 χ 21 ( t, k ) dk , and the fact that g 01 ( t ) = sin ω t . 2 Lemma 4.5. The first term on the right-hand side of (4.1d) is given by 8 π i Z ∂ D 3 k 2 ˆ χ 12 ( t, k ) dk = 1 4  1 − L 2 K 2  e 2 iω t + 1 4  1 + K ¯ K  e iω t + 1 2 Re  K ¯ K  + 1 4  1 + ¯ K K  e − iω t + 1 4  1 − ¯ L 2 ¯ K 2  e − 2 iω t + 8 π i Z ∂ ˆ D 3 k 2 [ E ( t, k ) + A 2 ( t, k )] dk . (4.19) Pr o of. As in the pro of of Lemma 4.3, this is a consequence of the expression (4.12) for ˆ χ 12 ( t, k ) and Cauch y’s theorem. 2 According to (4.1d), g 22 ( t ) is the sum of the expressions in (4.18) and (4.19). Th us, g 22 ( t ) =  1 − L 2 4 K 2  e 2 iω t + Re  K ¯ K  − 1 +  1 − ¯ L 2 4 ¯ K 2  e − 2 iωt + F 2 ( t, k ) , (4.20) 14 JERR Y L. BONA AND JONA T AN LENELLS where F 2 ( t, k ) = 8 i π  K e iω t + ¯ K e − iω t  Z t 0 Z ∂ ˆ D 3 k 2 A 1 ( t 0 , k ) dk dt 0 + 1 π  2 Re  1 K  − 1 K e iω t − 1 ¯ K e − iω t  Z ∂ ˆ D 3 k A 1 ( t, k ) dk − 32 i π 2  Z ∂ ˆ D 3 k A 1 ( t, k ) dk  Z t 0 Z ∂ ˆ D 3 k 2 A 1 ( t 0 , k ) dk dt 0  + 8 π i Z ∂ ˆ D 3 k 2 [ E ( t, k ) + A 2 ( t, k )] dk . (4.21) The proof of (2.8d) is completed b y establishing that F 2 ( t, k ) is O ( t − 3 2 ) in the next subsection. 4.5. Asymptotics of F 1 ( t, k ) and F 2 ( t, k ) . In this subsection, the pro of of Theorem 2.2 is completed by showing that F j ( t, k ) is of order O ( t − 3 / 2 ) as t → ∞ , j = 1 , 2. Lemma 4.6. The functions A 1 ( t, k ) and A 2 ( t, k ) define d in (4.6) and (4.14) satisfy Z t 0 Z ∂ ˆ D 3 k 2 A 1 ( t 0 , k ) dk dt 0 = π i 2 ω Re( K 2 ) + O  t − 3 2  , t → ∞ , (4.22) and Z ∂ ˆ D 3 k j A 2 ( t, k ) dk = O  t − 3 2  , t → ∞ , j = 1 , 2 . (4.23) Pr o of. A computation using (4.6), Proposition A.1, and Cauch y’s theorem shows that the left-hand side of (4.22) equals Z ∂ ˆ D 3 k 2 f 0 ( k ) 8 i  1 f ( k ) + iω − 1 f ( k ) − iω  Z t 0 e − f ( k ) t 0 dt 0 dk = − 1 8 i Z ∂ ˆ D 3 k 2 f 0 ( k ) f ( k )  1 f ( k ) + iω − 1 f ( k ) − iω  e − f ( k ) t dk + 1 8 i Z ∂ ˆ D 3 k 2 f 0 ( k ) f ( k )  1 f ( k ) + iω − 1 f ( k ) − iω  dk = O  t − 3 2  + π 4  Res k = K k 2 f 0 ( k ) f ( k ) 1 f ( k ) + iω − Res k = − ¯ K k 2 f 0 ( k ) f ( k ) 1 f ( k ) − iω  = π i 4 ω ( K 2 + ¯ K 2 ) + O  t − 3 2  . (4.24) This prov es (4.22). Recalling the definition (4.14) of A 2 and deforming the con tour to Γ, equation (4.23) follows immediately from Prop osition A.1.  Lemma 4.7. The function E ( t, k ) define d in (4.13) satisfies 4 π Z ∂ ˆ D 3 k E ( t, k ) dk = 2 i ω Re( K 2 )( e iω t − e − iω t ) + O  t − 3 2  , t → ∞ , (4.25a) 8 π i Z ∂ ˆ D 3 k 2 E ( t, k ) dk = 4 ω Re( K 2 )( K e iω t + ¯ K e − iω t ) + O  t − 3 2  , t → ∞ . (4.25b) Pr o of. First note that the definition (4.6) of A 1 and Cauch y’s theorem yield  Z t 0 0 − Z t t 0  Z ∂ ˆ D 3 k 2 A 1 ( t 00 , k ) dk dt 00 = π i 4 ω ( K 2 + ¯ K 2 ) − Z ∂ ˆ D 3 k 2 f 0 ( k ) 8 i 2 e − f ( k ) t 0 − e − f ( k ) t f ( k )  1 f ( k ) + iω − 1 f ( k ) − iω  dk . THE KDV EQUA TION ON THE HALF-LINE 15 Substituting this and the expression (4.6) for A 1 in to the definition (4.13) of E ( t, k ) and p erforming the integrals with resp ect to t 0 pro vides the formula E ( t, k ) = − f 0 ( k ) 4 π f ( k ) Z ∂ ˆ D 3 k 0 2 f 0 ( k 0 )  1 f ( k 0 ) + iω − 1 f ( k 0 ) − iω  ×  − e iω t − f ( k 0 ) t − e − f ( k ) t 2 K ( f ( k ) − f ( k 0 ) + iω ) + e − f ( k 0 ) t − e − f ( k ) t 4 K ( f ( k ) − f ( k 0 )) + e iω t − f ( k 0 ) t − e iω t − f ( k ) t 4 K ( f ( k ) − f ( k 0 )) − e − iω t − f ( k 0 ) t − e − f ( k ) t 2 ¯ K ( f ( k ) − f ( k 0 ) − iω ) + e − f ( k 0 ) t − e − f ( k ) t 4 ¯ K ( f ( k ) − f ( k 0 )) + e − iω t − f ( k 0 ) t − e − iω t − f ( k ) t 4 ¯ K ( f ( k ) − f ( k 0 )) + 4 e − f ( k 0 ) t − e − f ( k ) t ω ( f ( k ) − f ( k 0 )) Re( K 2 )  dk 0 − f 0 ( k ) 4 f ( k ) 1 π 2 Z ∂ ˆ D 3 k 0 2 f 0 ( k 0 ) ×  1 f ( k 0 ) + iω − 1 f ( k 0 ) − iω  Z ∂ ˆ D 3 k 00 2 f 0 ( k 00 )  1 f ( k 00 ) + iω − 1 f ( k 00 ) − iω  ×  2 e − f ( k 00 ) t − f ( k 0 ) t − e − f ( k ) t f ( k 00 )( f ( k ) − f ( k 0 ) − f ( k 00 )) − e − f ( k 00 ) t − f ( k 0 ) t − e − f ( k 00 ) t − f ( k ) t f ( k 00 )( f ( k ) − f ( k 0 ))  dk 00 dk 0 + ω f 0 ( k ) 4 π f ( k ) Z ∂ ˆ D 3 k 0 2 f 0 ( k 0 ) f ( k 0 )  1 f ( k 0 ) + iω − 1 f ( k 0 ) − iω  ×  e − f ( k 0 ) t + iω t − e − f ( k ) t 2 iK ( f ( k ) − f ( k 0 ) + iω ) − e − f ( k 0 ) t + iω t − e − f ( k 0 ) t − f ( k ) t 4 iK ( f ( k ) + iω ) − e − f ( k 0 ) t − iω t − e − f ( k ) t 2 i ¯ K ( f ( k ) − f ( k 0 ) − iω ) + e − f ( k 0 ) t − iω t − e − f ( k 0 ) t − f ( k ) t 4 i ¯ K ( f ( k ) − iω )  dk 0 + Re( K 2 ) f 0 ( k ) f ( k )  e iω t − e − f ( k ) t 4 iK ( f ( k ) + iω ) − e − iω t − e − f ( k ) t 4 i ¯ K ( f ( k ) − iω )  . (4.26) T o prov e (4.25), m ultiply (4.26) b y k j , j = 1 , 2, and integrate o v er ∂ ˆ D 3 . W e claim that all the terms on the right-hand side of (4.26) which in volv e a factor of e − f ( k ) t , e − f ( k 0 ) t , or e − f ( k 00 ) t are of order O ( t − 3 / 2 ). Assuming for the moment that this is v alid, it is inferred that the only O (1) contribution to R ∂ ˆ D 3 k j E ( t, k ) dk , j = 1 , 2, derives from the last line of (4.26), which is to say , Z ∂ ˆ D 3 k j E ( t, k ) dk = Re( K 2 ) Z ∂ ˆ D 3 k j f 0 ( k ) f ( k )  e iω t 4 iK ( f ( k ) + iω ) − e − iω t 4 i ¯ K ( f ( k ) − iω )  dk + O  t − 3 2  , t → ∞ , j = 1 , 2 . Computing the integral using the residue theorem, the asymptotic relations (4.25) emerge. It remains to show that any term in (4.26) inv olving e − f ( k ) t , e − f ( k 0 ) t , or e − f ( k 00 ) t yields a con tribution of order O  t − 3 2  . This follo ws from steepest descen t considerations. In Appendix B, the details of this calculation are pro vided for the case of the triple in tegral Q ( t ) := Z ∂ ˆ D 3 dk k 2 f 0 ( k ) f ( k ) Z ∂ ˆ D 3 dk 0 k 0 2 f 0 ( k 0 ) f ( k 0 ) + iω × Z ∂ ˆ D 3 dk 00 k 00 2 f 0 ( k 00 ) f ( k 00 )( f ( k 00 ) + iω ) e − f ( k 00 ) t − f ( k 0 ) t − e − f ( k ) t f ( k ) − f ( k 0 ) − f ( k 00 ) (4.27) and the double in tegral R ( t ) := Z ∂ ˆ D 3 dk k 2 f 0 ( k ) f ( k ) Z ∂ ˆ D 3 dk 0 k 0 2 f 0 ( k 0 ) f ( k 0 ) + iω e iω t − f ( k 0 ) t − e − f ( k ) t f ( k ) − f ( k 0 ) + iω . (4.28) The other terms can be treated similarly . The pro of of Lemma 4.7 is complete. 2 16 JERR Y L. BONA AND JONA T AN LENELLS Using equations (4.8), (4.22), (4.23), and (4.25) in the definitions (4.17) and (4.21) of F 1 ( t, k ) and F 2 ( t, k ), resp ectiv ely , there app ears F j ( t, k ) = O  t − 3 2  , t → ∞ , j = 1 , 2 , (4.29) whic h completes the pro of of Theorem 2.2. 2 5. Conclusion Tw o questions ab out solutions of a natural initial-b oundary-v alue problem for the Korteweg- de V ries equation ha ve b een addressed here. Both these questions arise naturally from ob- serv ed experimental data obtained in water tank exp eriments. Overall, these results, which are asymptotic, but exact in their large-time structure, raise a cautionary note. While the p ositiv e result of asymptotic perio dicity corresp onds well to what is seen in exp erimen ts, the lack of asymptotically conserv ed mass is troubling. Admittedly , this does not o ccur at first order, O (  ) in the notation in force here, but rather at the second order O (  2 ). As the Kortew eg-de V ries mo del is only an accurate approximation on the so-called Boussinesq time scale of order O (1 / ), the fact that mass is not conserved at the higher order does not mak e the initial-boundary-v alue problem considered here necessarily susp ect. Ho w ever, it do es re- inforce the view that the mo del should not be pushed beyond the Boussinesq time scale. If a unidirectional initial-b oundary-v alue problem v alid on a longer time scale is needed, a higher- order correct mo del should b e employ ed. A recent example of such a mo del is provided in [5] (and see the references therein to other, related models) and a relev an t initial-boundary-v alue problem was put forw ard and analysed in [20]. How ev er, this latter problem features a piece of b oundary data that migh t b e hard to obtain in a lab oratory setting. This issue deserv es further study . Appendix A. Steepest descent lemma In this app endix, the method of steepest descen t is used to determine the large t b eha vior of certain integrals inv olving the exp onen tial exp( − f ( k ) t ). Let a = 1 / (2 √ 3) as b efore. The function f 0 ( k ) = 2 i (12 k 2 − 1) v anishes at the t wo critical p oin ts ± a . Let Γ b e the steep est descent con tour sho wn in Figure 4. The contour Γ is c haracterized b y the condition that Im f ( k ) = Im f ( a ) on the part of Γ passing through k = a , while Im f ( k ) = Im f ( − a ) on the part of Γ passing through k = − a . F or j = 1 , . . . , 4, let Γ j denote the part of Γ that lies in the j ’th quadran t. Then Re f ( k ) is strictly increasing from 0 to + ∞ as k mov es aw a y from ± a to wards ∞ along any of the Γ j ’s. Prop osition A.1. L et q ( k ) b e a function which is analytic in a neighb orho o d of Γ . Supp ose that q ( k ) gr ows at most algebr aic al ly as k → ∞ . It fol lows that Z Γ q ( k ) f 0 ( k ) e − f ( k ) t dk = O  t − 3 2  , t → ∞ . (A.1) Pr o of. W rite the left-hand side of (A.1) as the sum P 4 j =1 I j ( t ), where I j ( t ) denotes the con- tribution from Γ j , viz. I j ( t ) := e − f ( a ) t Z Γ j q ( k ) f 0 ( k ) e − [ f ( k ) − f ( a )] t dk , j = 1 , . . . , 4 . Consider I 1 ( t ). F or k ∈ Γ 1 , let l = f ( k ) − f ( a ). The definition of the steep est descent con tour implies that the mapping l 7→ k ( l ) is a diffeomorphism from [0 , ∞ ) on to Γ 1 . Th us, a c hange of v ariables yields I 1 ( t ) = e − f ( a ) t Z ∞ 0 q ( k ( l )) e − lt dl. (A.2) F or an y  > 0, the assumption that q ( k ) gro ws at most algebraically as k → ∞ implies that the integral     Z ∞  q ( k ( l )) e − lt dl     ≤ e − t 2 Z ∞  | q ( k ( l )) | e − lt 2 dl ≤ C e − t 2 THE KDV EQUA TION ON THE HALF-LINE 17 Γ a − a Figure 4. The ste ep est desc ent c ontour Γ p assing thr ough the critic al p oints ± a . is exp onentially small. On the other hand, for k near a w e ha v e k ( l ) = a + e 3 iπ 4 2 · 3 1 4 √ l + O ( l ) , l → 0 . Th us, if q has the expansion q ( k ) = ∞ X n =0 q n ( k − a ) n , k → a, then q ( k ( l )) = q 0 + e 3 iπ 4 q 1 2 · 3 1 4 √ l + O ( l ) , l → 0 . Substituting this into (A.2) and ev aluating the integrals with resp ect to l leads to I 1 ( t ) = e − f ( a ) t  q 0 t + e 3 iπ 4 q 1 2 · 3 1 4 √ π 2 t 3 2 + O  t − 2   , t → ∞ . (A.3) The last step can b e made rigorous using standard argumen ts from the steep est descent metho d (see e.g. [29]). A similar argumen t applied to Γ 4 pro vides the asymptotic relation I 4 ( t ) = e − f ( a ) t  − q 0 t + e 3 iπ 4 q 1 2 · 3 1 4 √ π 2 t 3 2 + O  t − 2   , t → ∞ . (A.4) Equations (A.3) and (A.4) imply that I 1 ( t ) + I 4 ( t ) = O ( t − 3 / 2 ). Analogous computations give I 2 ( t ) + I 3 ( t ) = O ( t − 3 / 2 ), thereb y establishing Prop osition (A.1). 2 Remark A.2. The pro of of Prop osition A.1 can b e extended to give the expansion of the in tegral in formula (A.1) to all orders in t . In principle, a tedious computation can then pro vide the asymptotic expansions in (2.8) to all orders in t . Appendix B. Asymptotics of Q ( t ) and R ( t ) Here, a proof is offered that Q ( t ) and R ( t ) defined in (4.27)-(4.28) are also of order O ( t − 3 / 2 ) as t → ∞ . Asymptotics of Q ( t ) T o prov e that Q ( t ) = O ( t − 3 / 2 ), note that the integrand in (4.27) has remo v able singularities at the p oints where f ( k ) − f ( k 0 ) − f ( k 00 ) = 0. Moreov er, the 18 JERR Y L. BONA AND JONA T AN LENELLS ∂ ˇ D 3 Figure 5. The c ontour ∂ ˇ D 3 . in tegrand is non-singular for k in the set { 0 , K , − ¯ K , L, − ¯ L } , so the contour ∂ ˆ D 3 in the k - in tegral may b e replaced with ∂ D 3 . Deforming the contour of in tegration in the k 00 -in tegral from ∂ ˆ D 3 to Γ and in terchanging the order of the k 0 and k 00 in tegrals, it is found that Q ( t ) = Z ∂ D 3 dk k 2 f 0 ( k ) f ( k ) Z Γ dk 00 k 00 2 f 0 ( k 00 ) f ( k 00 )( f ( k 00 ) + iω ) Z ∂ ˆ D 3 dk 0 k 0 2 f 0 ( k 0 ) f ( k 0 ) + iω × e − f ( k 00 ) t − f ( k 0 ) t − e − f ( k ) t f ( k ) − f ( k 0 ) − f ( k 00 ) . Next, deform the contour in the k -integral so that it passes to the left (i.e. the indentation lies in D 3 ) of k = 0 as well as to the left of the solutions in ∂ D 3 of the equations f ( k ) = 2 f ( a ) and f ( k ) = − 2 f ( a ) (see Figure 5). Denote this deformed contour b y ∂ ˇ D 3 . F or each pair ( k , k 00 ) ∈ ∂ ˇ D 3 × Γ, one observes that Re( f ( k ) − f ( k 00 )) ≤ 0. In consequence, there exists a unique p oint K r ( k , k 00 ) ∈ ¯ D 3 suc h that f ( k ) − f ( k 00 ) = f ( K r ( k , k 00 )) . (B.1) Let ∂ ˆ D 3 ( k , k 00 ) denote the con tour ∂ ˆ D 3 with a small inden tation added so that it passes to the right of K r ( k , k 00 ) whenever K r ( k , k 00 ) ∈ ∂ D 3 . The following claim implies that this inden tation can b e chosen in suc h a wa y that ∂ ˆ D 3 ( k , k 00 ) ⊂ ¯ D 2 ∪ ¯ D 4 for all k , k 00 . Lemma B.1. Ther e exists an  > 0 such that K r ( k , k 00 ) stays an  -distanc e away fr om the critic al p oints ± a for ( k , k 00 ) ∈ ∂ ˇ D 3 × Γ . Pr o of. Let Γ + = Γ 1 ∪ Γ 4 denote the p ortion of Γ that passes through a . It will b e sho wn that K r ( k , k 00 ) stays well a wa y from ± a for k ∈ ∂ ˇ D 3 and k 00 ∈ Γ + . The case when k 00 ∈ Γ 2 ∪ Γ 3 can b e handled in a similar wa y . In view of (B.1), K r ( k , k 00 ) can approach ± a only if f ( k ) − f ( k 00 ) is close to f ( ± a ) = ∓ 2 i/ (3 √ 3). Let  > 0 b e such that dist( f ( k ) , {± 2 f ( a ) , 0 } ) > 2  for k ∈ ∂ ˇ D 3 . There exists a δ > 0 such that | f ( k 00 ) − f ( a ) | <  whenev er k 00 ∈ Γ + satisfies | k 00 − a | < δ . Thus, if k 00 ∈ Γ + is such that | k 00 − a | < δ and k ∈ ∂ ˇ D 3 , then by the triangle inequality , | f ( k ) − f ( k 00 ) − f ( ± a ) | ≥ | f ( k ) − f ( a ) − f ( ± a ) | − | f ( k 00 ) − f ( a ) | > 2  −  = , so that K r ( k , k 00 ) stays aw a y from ± a in this case. On the other hand, since Re f ( k 00 ) ≥ 0 for k 00 ∈ Γ + and Re f ( k 00 ) is small only for k 00 near a , there exists a c > 0 such that Re f ( k 00 ) > c whenev er | k 00 − a | ≥ δ . Also, Re f ( k ) ≤ 0 for k ∈ ∂ ˇ D 3 . Hence, if k 00 ∈ Γ + is suc h that THE KDV EQUA TION ON THE HALF-LINE 19 | k 00 − a | ≥ δ and k ∈ ∂ ˇ D 3 , then | f ( k ) − f ( k 00 ) − f ( ± a ) | ≥ | Re( f ( k ) − f ( k 00 ) − f ( ± a )) | = Re f ( k 00 ) − Re f ( k ) > c, sho wing that K r ( k , k 00 ) also stays aw a y from ± a in this case. 2 Since the contour ∂ ˆ D 3 ( k , k 00 ) av oids the point K r ( k , k 00 ), it is the case that f ( k ) − f ( k 0 ) − f ( k 00 ) 6 = 0 for all ( k , k 00 ) ∈ ∂ ˇ D 3 × Γ and k 0 ∈ ∂ ˆ D 3 ( k , k 00 ). Hence, the integral Q may b e split as follows: Q ( t ) = Z ∂ ˇ D 3 dk k 2 f 0 ( k ) f ( k ) Z Γ dk 00 k 00 2 f 0 ( k 00 ) f ( k 00 )( f ( k 00 ) + iω ) Z ∂ ˆ D 3 ( k,k 00 ) dk 0 k 0 2 f 0 ( k 0 ) f ( k 0 ) + iω × e − f ( k 00 ) t − f ( k 0 ) t f ( k ) − f ( k 0 ) − f ( k 00 ) − Z ∂ ˇ D 3 dk k 2 f 0 ( k ) f ( k ) Z Γ dk 00 k 00 2 f 0 ( k 00 ) f ( k 00 )( f ( k 00 ) + iω ) Z ∂ ˆ D 3 ( k,k 00 ) dk 0 k 0 2 f 0 ( k 0 ) f ( k 0 ) + iω × e − f ( k ) t f ( k ) − f ( k 0 ) − f ( k 00 ) =: Q 1 ( t ) − Q 2 ( t ) . T o compute Q 1 ( t ), first use Jordan’s lemma to deform the con tour ∂ ˆ D 3 ( k , k 00 ) to Γ and then deform the con tour ∂ ˇ D 3 in the k -in tegral to infinity . Since k 0 , k 00 ∈ Γ implies f ( k 0 ) , f ( k 00 ) ∈ ± 2 i 3 √ 3 + R ≥ 0 , it follows that | f ( k ) − f ( k 0 ) − f ( k 00 ) | ≥ dist  f ( k ) ,  ± 4 i 3 √ 3 , 0  , k ∈ ¯ D 3 , k 0 , k 00 ∈ Γ . This implies that there exists a c > 0 suc h that | f ( k ) − f ( k 0 ) − f ( k 00 ) | ≥ c | k | 3 for all large k ∈ ¯ D 3 . In particular, the integrand is O ( k − 2 ) as k → ∞ in ¯ D 3 . Thus, deforming the contour in the k -in tegral to infinit y , observing that f ( k ) − f ( k 0 ) − f ( k 00 ) has strictly negative real part and hence is nonzero throughout this deformation, it follows that Q 1 ( t ) = 0. T o compute Q 2 ( t ), remark that by deforming the contour ∂ ˆ D 3 ( k , k 00 ) to infinit y and using Cauc hy’s theorem, the formula Q 2 ( t ) = 2 π i Z ∂ ˇ D 3 dk k 2 f 0 ( k ) e − f ( k ) t f ( k ) Z Γ dk 00 k 00 2 f 0 ( k 00 ) f ( k 00 )( f ( k 00 ) + iω ) ×  Res k 0 = K + Res k 0 = K r ( k,k 00 )  k 0 2 f 0 ( k 0 ) f ( k 0 ) + iω 1 f ( k ) − f ( k 0 ) − f ( k 00 ) = 2 π i Z ∂ ˇ D 3 dk k 2 f 0 ( k ) e − f ( k ) t f ( k ) Z Γ dk 00 k 00 2 f 0 ( k 00 ) f ( k 00 )( f ( k 00 ) + iω ) K 2 − K 2 r ( k , k 00 ) f ( k ) − f ( k 00 ) + iω emerges. The function f ( k ) − f ( k 00 ) + iω = 0 v anishes at k 00 = K r ( k , K ), but this singularity is remov able since K r ( k , K r ( k , K )) = K . In consequence, deforming the contour Γ to infinity and using Cauch y’s theorem yields Q 2 ( t ) = (2 π i ) 2 Z ∂ ˇ D 3 dk k 2 f 0 ( k ) e − f ( k ) t f ( k ) Res k 00 = K  k 00 2 f 0 ( k 00 ) f ( k 00 )( f ( k 00 ) + iω ) K 2 − K 2 r ( k , k 00 ) f ( k ) − f ( k 00 ) + iω  = (2 π i ) 2 Z ∂ ˇ D 3 dk k 2 f 0 ( k ) e − f ( k ) t f ( k ) K 2 − iω K 2 − K 2 r ( k , K ) f ( k ) + 2 iω . Since K r ( L, K ) = K , the singularit y at k = L is remo v able. Deforming the con tour to Γ and applying Prop osition A.1 yields Q 2 ( t ) = O ( t − 3 / 2 ), thereby showing that Q ( t ) = O ( t − 3 / 2 ). Asymptotics of R ( t ) T o show that R ( t ) = O ( t − 3 / 2 ), notice that the in tegrand in (4.28) has remo v able singularities at the p oints where f ( k ) − f ( k 0 ) + iω = 0. T o pro ceed, first deform the contour in the k 0 -in tegral from ∂ ˆ D 3 to Γ and then deform the contour in the k -integral from ∂ ˆ D 3 to ∂ ˇ D 3 . How ev er, con trary to the earlier notation, the contour ∂ ˇ D 3 is no w obtained from ∂ D 3 b y inserting indentations so that it passes to the left of the origin and of the p oints 20 JERR Y L. BONA AND JONA T AN LENELLS K r ( K, ± a ) ∈ ∂ D 3 at which f ( k ) ± f ( a ) + iω = 0 (see again Figure 5). Since Re f ( k ) ≤ 0 for k ∈ ∂ ˇ D 3 and Re f ( k 0 ) ≥ 0 for k 0 ∈ Γ with equality only at k 0 = ± a , f ( k ) − f ( k 0 ) + iω can v anish only when k 0 = ± a and k = K r ( K, ∓ a ). In particular, f ( k ) − f ( k 0 ) + iω 6 = 0 for k ∈ ∂ ˇ D 3 and k 0 ∈ Γ. Thus the integral can be split, viz. R ( t ) = e iω t Z ∂ ˇ D 3 dk k 2 f 0 ( k ) f ( k ) Z Γ dk 0 k 0 2 f 0 ( k 0 ) f ( k 0 ) + iω e − f ( k 0 ) t f ( k ) − f ( k 0 ) + iω − Z ∂ ˇ D 3 dk k 2 f 0 ( k ) f ( k ) Z Γ dk 0 k 0 2 f 0 ( k 0 ) f ( k 0 ) + iω e − f ( k ) t f ( k ) − f ( k 0 ) + iω =: R 1 ( t ) − R 2 ( t ) . T o compute R 1 ( t ), observ e that k 0 ∈ Γ implies f ( k 0 ) ∈ ± 2 i 3 √ 3 + R ≥ 0 . It thus transpire that | f ( k ) − f ( k 0 ) + iω | ≥ dist  f ( k ) , iω ± 2 i 3 √ 3  , k ∈ ¯ D 3 , k 0 ∈ Γ . Th us, the in tegrand is O ( k − 2 ) as k → ∞ in ¯ D 3 . Deforming the contour in the k -integral to infinit y in D 3 , it follows that R 1 ( t ) = 0. F or the computation of R 2 ( t ), remark that R 2 ( t ) = 2 π i Z ∂ ˇ D 3 dk k 2 f 0 ( k ) e − f ( k ) t f ( k )  Res k 0 = K + Res k 0 = K r ( k,K )  k 0 2 f 0 ( k 0 ) f ( k 0 ) + iω 1 f ( k ) − f ( k 0 ) + iω = 2 π i Z ∂ ˇ D 3 dk k 2 f 0 ( k ) e − f ( k ) t f ( k ) K 2 − K 2 r ( k , K ) f ( k ) + 2 iω . Since K r ( L, K ) = K , the singularit y at k = L is remo v able. Deforming the con tour to Γ and applying Prop osition A.1, it is concluded that R 2 ( t ) = O ( t − 3 / 2 ), thereby establishing that R ( t ) = O ( t − 3 / 2 ). Appendix C. An al terna tive per turba tive approa ch In Theorem 2.2 it was shown that if g 0 ( t ) =  sin ω t , then g 1 and g 2 are asymptotically p erio dic as t → ∞ , at least to second order in p erturbation theory . W e also computed the large t asymptotics and ga v e rigorous error estimates for g 1 and g 2 to the same order. If one assumes that g 1 and g 2 are asymptotically p erio dic as t → ∞ with p erio d 2 π /ω , and if one does not w orry about precise error estimates, the coefficients in (2.8) can b e determined directly using an alternative p erturbative approach. This idea was first implemented for the nonlinear Schr¨ odinger equation in [26]. The KdV equation (2.1) admits the Lax pair ( ϕ x + ik σ 3 ϕ = V 1 ϕ, ϕ t + i (4 k 3 − k ) σ 3 ϕ = V 2 ϕ, (C.1) where ϕ ( x, t, k ) is a v ector-v alued eigenfunction, k ∈ C is the spectral parameter, { V j ( x, t, k ) } 2 1 are defined by V 1 = u 2 k ( σ 2 − iσ 3 ) , V 2 = 2 k uσ 2 + u x σ 1 + 2 u 2 + u + u xx 2 k ( iσ 3 − σ 2 ) , and { σ j } 3 1 denote the standard Pauli matrices. Letting ϕ 1 and ϕ 2 denote the first and second en tries of ϕ resp ectively , the t -part of (C.1) can b e written as ϕ 1 t + i 2 k (8 k 4 − 2 k 2 − u − 2 u 2 − u xx ) ϕ 1 + h 2 ik u − u x − i 2 k ( u + 2 u 2 + u xx ) i ϕ 2 = 0 , ϕ 2 t − i 2 k (8 k 4 − 2 k 2 − u − 2 u 2 − u xx ) ϕ 2 − h 2 ik u + u x − i 2 k ( u + 2 u 2 + u xx ) i ϕ 1 = 0 . This in turn implies that the quotient p = ϕ 1 /ϕ 2 satisfies the Ricatti equation ip t + 1 2 k ((1 − 4 k 2 ) u + 2 u 2 + 2 ik u x + u xx ) p 2 + 1 k (2 k ( k − 4 k 3 ) + u + 2 u 2 + u xx ) p + 1 2 k ((1 − 4 k 2 ) u + 2 u 2 − 2 ik u x + u xx ) = 0 . (C.2) THE KDV EQUA TION ON THE HALF-LINE 21 Assume that the functions { g j ( t ) } 2 0 and p ( t, k ) are asymptotically time-p erio dic of perio d t p = 2 π /ω as t → ∞ . The functions to whic h they are asymptotic then hav e F ourier series represen tations g 0 ( t ) ∼ ∞ X n = −∞ a n e inω t , g 1 ( t ) ∼ ∞ X n = −∞ b n e inω t , g 2 ( t ) ∼ ∞ X n = −∞ c n e inω t , p ( t, k ) ∼ ∞ X n = −∞ d n ( k ) e inω t , t → ∞ . (C.3) Substituting these representations into equation (C.2) ev aluated at x = 0 leads directly to the equation ∞ X n = −∞  − nω d n ( k ) + 1 − 4 k 2 2 k ∞ X l,m = −∞ a l d m ( k ) d n − l − m ( k ) + 1 k ∞ X j,l,m = −∞ a j a l d m ( k ) d n − j − l − m ( k ) + i ∞ X l,m = −∞ b l d m ( k ) d n − l − m ( k ) + 1 2 k ∞ X l,m = −∞ c l d m ( k ) d n − l − m ( k ) + 2( k − 4 k 3 ) d n ( k ) + 1 k ∞ X m = −∞ a m d n − m ( k ) + 2 k ∞ X l,m = −∞ a l a m d n − l − m ( k ) + 1 k ∞ X m = −∞ c m d n − m ( k ) + 1 − 4 k 2 2 k a n + 1 k ∞ X m = −∞ a m a n − m − ib n + 1 2 k c n  e inω t = 0 . This in turn yields the infinite hierarch y of equations d n ( k ) = i f ( k ) + inω  1 − 4 k 2 2 k ∞ X l,m = −∞ a l d m ( k ) d n − l − m ( k ) + 1 k ∞ X j,l,m = −∞ a j a l d m ( k ) d n − j − l − m ( k ) + i ∞ X l,m = −∞ b l d m ( k ) d n − l − m ( k ) + 1 2 k ∞ X l,m = −∞ c l d m ( k ) d n − l − m ( k ) + 1 k ∞ X m = −∞ a m d n − m ( k ) + 2 k ∞ X l,m = −∞ a l a m d n − l − m ( k ) + 1 k ∞ X m = −∞ c m d n − m ( k ) + 1 − 4 k 2 2 k a n + 1 k ∞ X m = −∞ a m a n − m − ib n + 1 2 k c n  , n ∈ Z . (C.4) C.1. Perturbativ e solution of the algebraic system. The algebraic system (C.4) may b e solved p erturbatively if the F ourier co efficients a n asso ciated with the Dirichlet data are kno wn. Indeed, a p erturbative analysis of (C.4) yields expressions for the co efficien ts d n ( k ) in terms of the a n , b n , and c n . The condition that d n ( k ) be non-singular in ¯ D 1 then yields expressions for the F ourier co efficien ts b n and c n asso ciated with the Neumann v alues in terms of the a n . This pro vides a straightforw ard, constructiv e approach to the Dirichlet to Neumann map. 22 JERR Y L. BONA AND JONA T AN LENELLS In more detail, first substitute the expansions a n = a 1 ,n +  2 a 2 ,n + O (  3 ) ,  → 0 , n ∈ Z , b n = b 1 ,n +  2 b 2 ,n + O (  3 ) ,  → 0 , n ∈ Z , c n = c 1 ,n +  2 c 2 ,n + O (  3 ) ,  → 0 , n ∈ Z , d n ( k ) = d 1 ,n ( k ) +  2 d 2 ,n ( k ) + O (  3 ) ,  → 0 , n ∈ Z , in to (C.4). The terms of O (  ) lead to the formulas d 1 ,n ( k ) = i f ( k ) + inω  1 − 4 k 2 2 k a 1 ,n − ib 1 ,n + 1 2 k c 1 ,n  , n ∈ Z . (C.5) F or n ∈ Z , let k 1 ( n ), k 2 ( n ), and k 3 ( n ) denote the three roots of f ( k ) + inω = 0, ordered so that k 1 ( n ) ∈ ∂ D 0 1 , k 2 ( n ) ∈ ∂ D 00 1 , k 3 ( n ) ∈ ∂ D 3 . These ro ots satisfy the iden tities k 1 ( n ) + k 2 ( n ) + k 3 ( n ) = 0 , k 1 ( n ) k 2 ( n ) + k 1 ( n ) k 3 ( n ) + k 2 ( n ) k 3 ( n ) = − 1 4 , k 1 ( n ) k 2 ( n ) k 3 ( n ) = − nω 8 , k 3 ( − n ) = − k 3 ( n ) , n ∈ Z . (C.6) F or d 1 ,n ( k ) to b e nonsingular at k 1 ( n ) and k 2 ( n ), it is required that 1 − 4 k 2 j ( n ) 2 k j ( n ) a 1 ,n − ib 1 ,n + 1 2 k j ( n ) c 1 ,n = 0 , n ∈ Z , j = 1 , 2 . Solving these equations for { b 1 ,n , c 1 ,n } and using (C.6), there app ears the form ulas b 1 ,n = − 2 ik 3 ( n ) a 1 ,n , c 1 ,n = − 4 k 2 3 ( n ) a 1 ,n , n ∈ Z . (C.7) The equations (C.5) th us lead to d 1 ,n ( k ) = ia 1 ,n f ( k ) + inω  1 − 4 k 2 2 k − 2 k 3 ( n ) − 2 k k 2 3 ( n )  , n ∈ Z . (C.8) Similarly , the terms of O (  2 ) give d 2 ,n ( k ) = i f ( k ) + inω  1 k ∞ X m = −∞ a 1 ,m d 1 ,n − m ( k ) + 1 k ∞ X m = −∞ c 1 ,m d 1 ,n − m ( k ) + 1 − 4 k 2 2 k a 2 ,n + 1 k ∞ X m = −∞ a 1 ,m a 1 ,n − m − ib 2 ,n + 1 2 k c 2 ,n  , n ∈ Z . (C.9) The condition that d 2 ,n ( k ) should not hav e singularities at k 1 ( n ) and k 2 ( n ) determines the co efficien ts b 2 ,n and c 2 ,n . This pro cess can b e contin ued indefinitely . Indeed, the terms of order O (  m ) yield an equation of the form d m,n ( k ) = i f ( k ) + inω  F mn ( k ) + 1 − 4 k 2 2 k a m,n − ib m,n + 1 2 k c m,n  , n ∈ Z , where the function F mn ( k ) is given in terms of (known) low er order terms. The condition that d m,n ( k ) should not hav e singularities at k 1 ( n ) and k 2 ( n ) then implies that b m,n = 2 ia m,n ( k 1 ( n ) + k 2 ( n )) − i k 1 ( n ) F mn ( k 1 ( n )) − k 2 ( n ) F mn ( k 2 ( n )) k 1 ( n ) − k 2 ( n ) , c m,n = − a m,n (1 + 4 k 1 ( n ) k 2 ( n )) + 2 k 1 ( n ) k 2 ( n )( F mn ( k 1 ( n )) − F mn ( k 2 ( n ))) k 1 ( n ) − k 2 ( n ) . Once { b m,n } ∞ n = −∞ , { c m,n } ∞ n = −∞ , { d m,n ( k ) } ∞ n = −∞ are determined, one ma y pro ceed to the next order. THE KDV EQUA TION ON THE HALF-LINE 23 Example C.1. Consider the example wherein g 0 ( t ) =  sin ω t with ω 6 = 0 satisfying (2.7). Then a m,n = 0 for all m ≥ 2 and a 1 ,n =      1 2 i , n = 1 , − 1 2 i , n = − 1 , 0 , otherwise . As in Theorem 2.2, write K and L for k 3 (1) and k 3 (2) resp ectively . Equations (C.7) and (C.8) provide the formulas b 1 , 1 = ¯ b 1 , − 1 = − K, c 1 , 1 = ¯ c 1 , − 1 = 2 iK 2 , d 1 , 1 ( k ) = 1 2( f ( k ) + iω )  1 − 4 k 2 2 k − 2 K − 2 K 2 k  , d 1 , − 1 ( k ) = − 1 2( f ( k ) − iω )  1 − 4 k 2 2 k + 2 ¯ K − 2 ¯ K 2 k  , b 1 ,n = c 1 ,n = d 1 ,n = 0 , n 6 = ± 1 , (C.10) whence g 11 ( t ) ∼ − K e iω t − ¯ K e − iω t and g 21 ( t ) ∼ 2 iK 2 e iω t − 2 i ¯ K 2 e − iω t , t → ∞ . Th us, apart from the error terms, the expressions in (2.8a) and (2.8b) hav e b een recov ered. Similarly , equation (C.9) yields d 2 , 2 ( k ) = i f ( k ) + 2 iω  1 k a 1 , 1 d 1 , 1 ( k ) + 1 k c 1 , 1 d 1 , 1 ( k ) + 1 k a 2 1 , 1 − ib 2 , 2 + 1 2 k c 2 , 2  , d 2 , 1 ( k ) = i f ( k ) + iω  − ib 2 , 1 + 1 2 k c 2 , 1  , d 2 , 0 ( k ) = i k f ( k )  ( a 1 , 1 + c 1 , 1 ) d 1 , − 1 ( k ) + ( a 1 , − 1 + c 1 , − 1 ) d 1 , 1 ( k ) + 2 a 1 , 1 a 1 , − 1 − ik b 2 , 0 + 1 2 c 2 , 0  , d 2 , − 1 ( k ) = i f ( k ) − iω  − ib 2 , − 1 + 1 2 k c 2 , − 1  , d 2 , − 2 ( k ) = i f ( k ) − 2 iω  1 k a 1 , − 1 d 1 , − 1 ( k ) + 1 k c 1 , − 1 d 1 , − 1 ( k ) + 1 k a 2 1 , − 1 − ib 2 , − 2 + 1 2 k c 2 , − 2  . F or d 2 , 2 ( k ) to not hav e singularities at k 1 (2) and k 2 (2), it must b e the case that ( ( a 1 , 1 + c 1 , 1 ) d 1 , 1 ( k 1 (2)) + a 2 1 , 1 − ik 1 (2) b 2 , 2 + c 2 , 2 2 = 0 , ( a 1 , 1 + c 1 , 1 ) d 1 , 1 ( k 2 (2)) + a 2 1 , 1 − ik 2 (2) b 2 , 2 + c 2 , 2 2 = 0 . Solving for b 2 , 2 and c 2 , 2 leads to b 2 , 2 = − i ( a 1 , 1 + c 1 , 1 ) d 1 , 1 ( k 1 (2)) − d 1 , 1 ( k 2 (2)) k 1 (2) − k 2 (2) , c 2 , 2 = − 2 a 2 1 , 1 − 2( a 1 , 1 + c 1 , 1 ) k 1 (2) d 1 , 1 ( k 2 (2)) − k 2 (2) d 1 , 1 ( k 1 (2)) k 1 (2) − k 2 (2) . Substituting in (C.10) and using the identities k 1 (2) k 2 (2) = − ω 4 L , k 1 (2) + k 2 (2) = − L, 4 L 2 − 4 K 2 = − ω L + ω 2 K , 24 JERR Y L. BONA AND JONA T AN LENELLS w e find after some algebraic manipulations that b 2 , 2 = 1 8 i  L K 2 − 2 K  , c 2 , 2 = 1 − L 2 4 K 2 . F or d 2 , 1 ( k ) to not hav e singularities at k 1 (1) and k 2 (1), it is required that ( − ib 2 , 1 + c 2 , 1 2 k 1 (1) = 0 , − ib 2 , 1 + c 2 , 1 2 k 2 (1) = 0 , whic h implies that b 2 , 1 = c 2 , 1 = 0. Similar considerations yield b 2 , 0 = 1 2 Im  1 K  , c 2 , 0 = Re  K ¯ K  − 1 , and b 2 , − 1 = c 2 , − 1 = 0 , b 2 , − 2 = ¯ b 2 , 2 , c 2 , − 2 = ¯ c 2 , 2 . Apart from the error terms, this recov ers the expansions in (2.8c) and (2.8d). Ac kno wledgemen t The gestation of this pr oje ct to ok plac e when the authors wer e b oth p articip ating in a c onfer enc e held at the Schr¨ odinger Institute in Vienna. The authors ar e esp e cial ly gr ateful to the Americ an Institute of Mathematics for pr oviding an exc el lent en- vir onment for discussions during a we eklong National Scienc e F oundation USA supp orte d workshop on b oundary-value pr oblems for nonline ar, disp ersive e quations. JL thanks the Uni- versity of Il linois at Chic ago for supp ort during a visit ther e and also acknow le dges supp ort fr om the Eur op e an R ese ar ch Council, Gr ant A gr e ement No. 682537, the Swe dish R ese ar ch Council, Gr ant No. 2015-05430, the G¨ or an Gustafsson F oundation, and the EPSR C, UK. JB is gr ateful to Southern University of Scienc e and T e chnolo gy in Shenzhen for pr oviding supp ort for a visit ther e in the c ourse of which the manuscript was finalize d. References [1] A. A. Alazman, J. P . Alb ert, J. 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