A $\dbar$-steepest descent method for oscillatory Riemann-Hilbert problems
aa r X i v : . [ n li n . S I] N ov A ¯ ∂ -steepest descent method for oscillatoryRiemann-Hilbert problems Fudong Wang , Wen-Xiu Ma a Department of Mathematics and StatisticsUniversity of South Florida
Abstract
We study the asymptotic behavior of Riemann-Hilbert problems (RHP) aris-ing in the AKNS hierarchy of integrable equations. Our analysis is based onthe ¯ ∂ -steepest descent method. We consider RHPs arising from the inversescattering transform of the AKNS hierarchy with H , ( R ) initial data. Theanalysis will be divided into three regions: fast decay region, oscillating re-gion and self-similarity region (the Painlev´e region). The resulting formulascan be directly applied to study the long-time asymptotic of the solutions ofintegrable equations such as NLS, mKdV and their higher order generaliza-tions.
1. Introduction
It is well known that matrix Riemann-Hilbert problem (RHP) plays anfundamental role in studying certain 1 + 1 dimensional integrable systems.In particular, one can study the long-time asymptotics for some nonlinearintegrable PDEs via RHPs. A countless number of papers have devoted tostudy the asymptotic behavior of a certain type of oscillatory 2 × × ∂ -steepestdescent method[1, 2]. Email addresses: [email protected] (Fudong Wang), [email protected] (Wen-XiuMa)
Preprint submitted to TBD December 1, 2020 n the study of the Cauchy problem of integrable systems by means ofinverse scattering, the following RHP appears: given a matrix-valued func-tion v θ ∈ GL ( R , m ∈ Hol ( C \ R ) such that m + = m − v θ ( z ) , z ∈ R , (1)where m ± ( z ) = lim ǫ → + m ( z ± iǫ ) , z ∈ R , and m ( z ) = I + o (1) , z → ∞ .The function v θ is called jump matrix and is generated by performingdirect scattering to the AKNS hierarchy. In this paper, we consider thefollowing defocusing type AKNS hierarchy: ψ x ( x, t ; z ) = (cid:18) izσ + (cid:18) q ( x, t ) q ( x, t ) 0 (cid:19)(cid:19) ψ ( x, t ; z ) , (2) ψ t ( x, t ; z ) = N X k =0 Q k ( x, t ) z k ! ψ ( x, t ; z ) , (3)where q ( x, t ) is the potential which solves certain 1+1 dimensional integrableequation, Q k is determined by the recursion relation (for details see [3]).In most of the applications, Q N ( x, t ) is constant with respect to x, t . Thecorresponding nonlinear integrable PDEs are carried out by the the zerocurvature condition which is also equivalent to ψ xt = ψ tx . In this paper, wewill study the Cauchy problem of the PDEs generated by the AKNS hierarchyprovided the initial data belongs to H , ( R ) = { f ∈ L | f ′ ∈ L , xf ∈ L } .Due to Xin Zhou’s result [4], after direction scattering, the coefficient R ( z )also belongs to H , ( R ). Then perform the time evolution, we arrive at thejump matrix v θ , which reads v θ ( z ) = (cid:18) − | R ( z ) | − ¯ R ( z ) e − itθ ( z ) R ( z ) e itθ ( z ) (cid:19) , (4)where the phase function θ (or weight function) shares the following proper-ties:(1). θ is a real polynomial of degree N .(2). θ ′ ( z j ) = 0 , θ ′′ ( z j ) = 0 for j = 1 · · · l , l is the number of stationary phasepoints. 2 .1. Main result Theorem 1.
In the region I, the long-time asymptotics for the potentials q ( x, t ) , associated with a generic oscillatory RHP whose phase function is θ ( z ; x, t ) and initial data q ( x, ∈ H , , reads q ( x, t ) = q as ( x, t ) + O ( t − / ) , t → ∞ (5) where q as ( x, t ) = − i l X j =1 | η ( z j ) | / p tθ ′′ ( z j ) e iϕ ( t ) , (6) and ϕ ( t ) = π − arg Γ( − iη ( z j )) − tθ ( z j ) − η ( z j )2 ln | tθ ′′ ( z j ) | + 2 arg( δ j ) + arg( R j ) , (7) and { z j } lj =1 are the stationary phase points, the definitions of η, R j and δ j are given in the main content. Corollary 2.
For the AKNS system, the phase function θ ( z ) = xz + ctz n and only have two real stationary phase points: z ± = ± (cid:12)(cid:12) − cnct (cid:12)(cid:12) n − , then thelong-time asymptotics for the potentials in the AKNS hierarchy is a specialcase of the theorem (1) . Theorem 3.
In the region III, the long-time asymptotics for the potentialsreads q ( x, t ) = O ( t − ) , t → ∞ . (8) Theorem 4.
In the region II, the long-time asymptotics for the potentialsreads q ( x, t ) = ( nt ) − n u n ( x ( nt ) − n ) + O ( t − n ) , t → ∞ , (9) where u n solves the n th member of the Painlev´e II hierarchy. t I : Oscillating III : Fast decaying II : Painlev´e region Figure 1: Three main asymptotic regions
In the following sections, we first review the inverse scattering for theAKNS hierarchy, then we will execute all steps which have been commonlyshown in many literatures. The first step is called conjugation, which coin-cides with previous chapters. After conjugation, we are able to to factorizethe jump matrix into lower/upper or upper/lower matrices whose diagonalsare all one and the exponential terms will decay due to the signature ofRe( itθ ). The next step is so called ”lenses opening”. In each interval where θ is monotonic, we can deform those intervals into new contours off the realline such that those exponential terms will decay as t goes to infinity. Alsoin this step, we will use a ¯ ∂ -RH problem. In the spirit of method of steepestdescent, the asymptotics are dominated near those saddle points. Hence thenext step is to study how can one separate the contributions from each saddlepoint. Then in each saddle point, one can study the RHP part of the ¯ ∂ − RHPthrough so called small norm technique. The last step is the estimate theerrors from the pure ¯ ∂ problems which dominate the errors generated fromsmall norm techniques. Then we conclude this chapter by represent a generalformula for our setting of the phase function.4 . Inverse scattering transform and Riemann-Hilbert problem in L In this section, we will review the inverse scattering transform for theAKNS hierarchy in some weighted L spaces. For more details, we directreaders to Zhou’s paper[4].The AKNS hierarchy is the integrable hierarchy associated with the fol-lowing spectral problem: ψ x ( x, t ; z ) = ( izσ + Q ( x, t )) ψ ( x, t ; z ) , (10)where σ = (cid:18) − (cid:19) and Q ( x, t ) = (cid:18) q ( x, t ) r ( x, t ) 0 (cid:19) .In current paper, we only consider the defocusing type reduction: r ( x, t ) = q ( x, t ) ∈ R . (11)Moreover, we assume q ( x, t = 0) ∈ H , ( R , dx ) and q decays to 0 as | x | − ∞ .For t = 0, we are looking for solutions (so-called Jost solutions) of equation(10) in L ( R , dx ), which satisfy the following boundary conditions at infinity: ψ ± = e ixzσ + o (1) , x → ±∞ . (12)The scattering matrix S ( z ) is defined as S ( z ) := ψ − − ψ + . (13)It is well known that S enjoys the following properties: for z ∈ R , S ( z ) = (cid:18) a ( z ) ¯ b ( z ) b ( z ) ¯ a ( z ) (cid:19) , (14)where a, b can be represented in terms of potentials q and eigenfunctions ψ .To find such representations, we consider µ ( ± ) = ψ ± e − izxσ . (15)Then the spetral problem (10) becomes:( µ ( ± ) ) x = iz [ , σ , µ ( ± ) ] + Qµ ( ± ) . (16)5hen the representations of a, b read a ( z ) = µ (+)11 ( x → −∞ ) = 1 − Z R q ( y ) µ (+)21 ( y, z ) dy, (17) b ( z ) = µ (+)12 ( x → −∞ ) = − Z R e iyz q ( y ) µ (+)22 ( y, z ) dy. (18)From the above representations, it is straightforward to show that a ( z ) =1 + O (1 /z ) and b ( z ) = O (1 /z ) as z → ∞ , and a ( z ) can be analyticallyextended to upper half plane.Now, setting m + ( x, z ) = ( µ (+)1 ( x, z ) /a ( z ) , µ ( − )2 ( x, z )) , Im z ≥ m − ( x, z ) = ( µ ( − )1 ( x, z ) , µ (+)2 ( x, z ) / ˘ a ( z )) , Im z ≤ , (20)we can then define the jump matrix on the real line by v ( z ) = e − izx ad σ [ m − − m + ] . (21)A direct comuptation shows v ( z ) := (cid:18) − | R ( z ) | − ¯ R ( z ) R ( z ) 1 (cid:19) , (22)where R ( z ) = b ( z ) a ( z ) . (23)Now if we deform the spectral problem (10) with respect to t governedby the following equation: ψ t ( x, t ; z ) = N X k =0 Q k ( x, t ) z k ! ψ ( x, t ; z ) , (24)such that the spectral is perserved, or equivalently speaking, the flow isisospectral. Such Q k ’s are uniquely determined if we assume the integrationconstants to be zeros. One can systematically determines the Q k ’s via asso-ciated Lie algebra techeneques, see for example[5]. Through the Lie algebra,one can show the AKNS hierarchy is integrable, i.e, there are infinite many6onvervation laws. Moreover, using the powerful trace identity[6], one caneasily show the bi-hamiltonian structure of the AKNS hierarchy. Moreover,under the same framework, one can show that any linear combinations of thetime-evolution problem are also integrable.The compatibility condition of (10) and (24) generates integrable PDEs,including the defocusing nonlinear Schrodinger equation, the modified KdVequation, the fifth order modified KdV equation. Due to the decaying of thepotential Q , it is easy to show the time evolution of the jump matrix v istrivial. Finnaly, we formulate the direct scattering problems as a Riemann-Hilbert problem as follows: Looking for a 2 by 2 matrix-valued functions m ( z ; x, t ) such that(1). m ( z ) is analytic in C \ R ,(2). m + = m − v θ ,(3). m = I + m ( x, t ) /z + O (1 /z ), z → ∞ ,where v θ is mentioned in the introduction, see equation (4) and m ± =lim ǫ ↓ m ( z ± iǫ ).From the equation (16), and the definition of the jump matrix v , we canrevocery the potentials by q ( x, t ) = lim z →∞ { z ( m − I ) } , (25)= − i ( m ( x, t )) . (26)In the following sections, we will perform the ¯ ∂ − steepest descent methodand study the asymptotic behavior for t is sufficient large.
3. Conjugation
As we have already knew that that our jump matrix can be factorizedinto upper/lower triangular matrices in the interval where θ is increasing andlower/diagonal/upper matrices in where the phase is decreasing. So let usdenote D ± = { z ∈ R : ± θ ′ ( z ) > } . To eliminate the diagonal matrix inthe second factorization, we introduction a new scalar RHP, IRHP −| R | ) χ D − + χ D + , I, R ), denoting the solution by δ as usual. Then we conjugate v θ by δ and arrive at a new RHP, IRHP δ σ − v θ δ σ + , I, R ), and denote thesolution by m [1] ( z ) ∈ I + ∂C ( L ). 7ere δ shares all the properties previously showed in the last chapter. Inthis section, we will study IRHP − | R | ) χ D − + χ D + , I, R ) in more detail.This scalar RHP can be easily solved by Plemelj formula, one obtainsln ( δ ( z )) = ( C D − (ln(1 − | R | )))( z ) , z ∈ C \ D − , (27)where the Cauchy operator C D − f = πi R D − f ( s ) s − z ds . Now if we assume R ∈ H , ( R ), then it is obvious that ln(1 − | R | ) is in H , , then by Sobolevembedding, we know it is also H¨older continuous with index 1 /
2. Thenthe Plemelj-Privalov theorem, which says that Cauchy operator perseveresH¨older continuity with index less than 1, tell us ln( δ ( z )) is H¨older continuouswith index 1 / η ( z ) = − π ln(1 − | R ( z ) | ) , z ∈ R . (28)We will prove the following proposition.First we define a function supported on the interval [ − ǫ, ǫ ], s ǫ ( z ) = ( , | z | ≥ ǫ ± ǫ z + 1 , < ± z < ǫ. (29) Proposition 5.
For each ǫ > , and ǫ ≤ min j = k | z j − z k | , there existsinterval I = I ( ǫ ) , such that the identity ln( δ ( z )) = i Z D − \ I η ( s ) s − z ds + i l X j =1 [ η ( z j )(1 + ln( z − z j ))] ε j (30)+ i l X j =1 Z I ∩ D − η ( s ) − η j ( s ) s − z ds (31)+ i l X j =1 ǫ η ( z j )[( z − z j ) ln( z − z j ) − ( z − z j + ε j ǫ ) ln( z − z j + ε j ǫ )](32) is true, where ε j = sgn ( θ ′′ ( z j )) and η j ( z ) = η ( z j ) s ǫ ( z − z j ) and for thelogarithm function, the branch is chosen such that arg ∈ ( − π, π ) . roof. Let I = ∪ lj =1 ( I j + ∪ I j − ), where I j ± = { z : 0 < ± ( z − z j ) < ǫ } . Nowwe have ln( δ ( z )) = i Z D − \ I η ( s ) s − z ds + i l X j =1 ( Z I j − ∩ D − + Z I j + ∩ D − η ( s ) s − z ds ) . For each j , we have Z I j − η ( s ) s − z ds = Z I j − η ( s ) − η j ( s ) s − z ds + Z I j − η j ( s ) s − z ds. The first integral on the right hand side non-tangential limit as z → z j andthe second one generates a logarithm singularity near z j . In fact, directcomputation shows Z I j − η j ( s ) s − z ds = η ( z j ) + 1 ǫ [( z − z j ) ln( z − z j ) − ( z − z j + ǫ ) ln( z − z j + ǫ )] η ( z j )+ η ( z j ) ln( z − z j ) . Similarly, for I j + , Z I j + η j ( s ) s − z ds = − η ( z j ) + 1 ǫ [( z − z j ) ln( z − z j ) − ( z − z j − ǫ ) ln( z − z j − ǫ )] − η ( z j ) ln( z − z j ) . And note that only one of the I j ± ∩ D − is nonempty, which depends on thesign of the second derivative of the phase function θ . Assembling all together,the proof is done. Remark 6.
The above proposition tells us how the function δ ( z ) behaviornear the saddle points. In fact, near those saddle points, there are mildsingularities ( z − z j ) iη ( z j ) . Fortunately, those singularities are bounded alongany ray off R hence in some sense they do not affect asymptotics much. Alsoit is worth mention that this general treatment has been taken in [7] and [8]before via different analysis tools. 9et us denote v [1] ( z ) = δ σ − v θ δ σ + . Now the conjugated jump matrix v [1] enjoys the following factorization: v [1] = − ¯ R ( z ) δ ( z ) e − itθ ( z ) ! R ( z ) δ − ( z ) e itθ ( z ) ! , z ∈ D + R ( z ) δ − − ( z ) e itθ ( z ) −| R ( z ) | ! − ¯ R ( z ) δ ( z ) e − itθ ( z ) −| R ( z ) | ! , z ∈ D − .
4. Lenses opening
The purpose of opening lenses is to deform the real line to some newcontours where the exponential terms will decay as t → ∞ . We begin withstudy the signature of Im θ near the saddle point z j . Since there are only θ ′ = 0 z j I j + I j − α Σ j, Σ j, Σ j, Σ j, Ω j, Ω j, Ω j, Ω j, Ω j, Ω j, Figure 2: Notations for studying signatures of Im( θ ( z )) near z j finite number of saddle points, we choose α sufficiently small such that Im( θ )share the same sign in the shaded region. Let I j + = [ z j , z j + z j +1 ] and I j − =[ z j + z j − , z j ] Two cases need to be discussed. The first case, if θ ′′ ( z j ) > I j ± ⊂ D ± . Recall the factorization of the conjugated jumpmatrix, to deform I j + to Σ j, and keep the exponential term ( e iyθ ( z ) ) decay,we need to discuss Im θ on Σ j, . Consider Taylor expansion of θ ( z ) at z j ,we have θ ( z ) = θ ( z j ) + ε j A j ( z − z j ) + O (( z − z j ) ), where A j = (cid:12)(cid:12)(cid:12) θ ′′ ( z j )2 (cid:12)(cid:12)(cid:12) Let z − z j = u + iv = ρe iφ , then Im( θ ( z )) = ε j A j ρ sin(2 φ ) + O ( ρ ), where φ ∈ (0 , α ] is fixed. Since α is sufficiently small,say less then π/
2, then 2 φ willless than π/
2. And we have ρ sin(2 φ ) = 2 uv. ρ k sin( kφ ) ≥ cρ k sin( φ ) cos k − ( φ ) = cu k − v. Put all together, we conclude | e itθ ( z ) | is dominated by e − t | A j | uv . Similarly,one can show in all other contours Σ j,k , k = 2 , ,
4, the exponential terms willdecay as t → ∞ . And now we are in the position to open lenses.First we introduce a bounded smooth function K defined on [0 , α ] suchthat K (0) = 1 and K ( α ) = 0. Consider ε j = 1 first. And ¯ ∂ extension func-tions as follows. Let z − z j = u + iv = ρe iφ , For the case ε j = −
1, oneonly needs to switch the index 1 with 3 and 4 with 6 due to the difference offactorization based on the local monotonicity of phase function. The bound-ary value of those E j,k , k = 1 , , , R are just the original conjugatedjump matrices. And the boundary values on the new contours are just somefunction with a mild singularity compare to exponential decay. Now usingthose E j,k , we construct the lens-opening matrix O ( z ) as follows: O ( z ) = O n ( z ) = − n E j,n e itθ ( z ) ! , z ∈ Ω j,n , n = 1 , ,O m ( z ) = − m E j,m e − itθ ( z ) ! , z ∈ Ω j,m , m = 3 , ,O k ( z ) = I, z ∈ Ω j,k , k = 2 , . (33)Now let m [2] ( z ) = m [1] ( z ) O ( z ) , z ∈ C \ R , due to the lacking of analyticity of O ( z ), we arrived at a mixed ¯ ∂ − Riemann-Hilbert problem( ¯ ∂ -RHP):1. The RHP(1.a). m [2] ( z ) = m [2] ( u, v ) ∈ C ( R \ Σ) and m [2] ( z ) = I + O ( z − ) , z → ∞ .(1.b). On the new contour Σ j,k , j = 1 · · · l, k = 1 , , , v [2] ( z ) = O j ( z ) , z ∈ Σ j,k .
2. The ¯ ∂ problemFor z ∈ C , we have ¯ ∂m [2] ( z ) = m [2] ( z ) ¯ ∂O ( z ) . (34)11 emark 7. By multiple O ( z ) to m [1] ( z ), we actually remove the jumpson the real line, where the exponential factor e ± itθ ( z ) is oscillating. Theregularity of m [2] is determined by the regularity of O ( z ) which inheritedfrom the construction of E j,k , the fact R ( z j + u ) is no longer analytic but is C in the weak sense. Moreover, due to the boundedness of E j,k ( z ) along anynon-real ray, and the fact the exponential factors are all exponential decayingas z → ∞ , we will have O ( z ) = I + o (1) , z → ∞ , i.e. uniformly in t . In thefollowing sections, we will see the error is eventually dominated by a pure ¯ ∂ problem.To closed the section, we state a bound estimate for ¯ ∂E j,k , which will beused in later sections. Lemma 8.
For j = 1 · · · l, k = 1 , , , , and z ∈ Ω j,k , u = Re( z − z j ) , | ¯ ∂E j,k ( z ) | ≤ c ( | z − z j | − / + | R ′ ( u + z j ) | ) . (35) Proof.
In polar coordinates, ¯ ∂ = e iφ ( ∂ ρ + iρ − ∂ φ ), then for z in any ray (notparallel to R ) and away from z j , we have¯ ∂E j, ( z ) = ie iφ K ′ ( φ )2 ρ [ R ( u + z j ) δ − ( z ) − R ( z j ) δ − j ( z − z j ) − iη ( z j ) ]+ K ( φ ) R ′ ( u + z j ) δ − ( z ) , where δ j = lim z = z j + ρe iφ ,ρ → ,φ ∈ (0 ,π/ δ ( z )( z − z j ) iη ( z j ) . From the Proposition (5), one can easily see that | δ ( z ) − δ j ( z − z j ) iη ( z j ) | ≤ c | z − z j | / . In fact, | δ ( z ) − δ j ( z − z j ) iη ( z j ) | ≤ | ln( δ ( z )) − ln( δ j ) − iη ( z j ) ln( z − z j ) |≤ | Z D − \ I η ( s ) s − z ds + l X k = j Z I ∩ D − η ( s ) − η k ( s ) s − z ds + l X k = j ǫ η ( z k )[( z − z k ) ln( z − z k ) − ( z − z k + ε k ǫ ) ln( z − z k + ε j ǫ )] − ln( δ j ) | . R ∈ H , , from standard Sobolev embedding, we know η is H¨older con-tinuous with index 1 / / R D − \ I η ( s ) s − z ds . Similarly onecan show R I ∩ D − η ( s ) − η k ( s ) s − z ds is also H¨older continuous with index 1 /
2. Nowlet us denote g ( z ) = ( z − z k ) ln( z − z k ) − ( z − z k + ε k ǫ ) ln( z − z k + ε j ǫ ) . Direct computation (using the fact that ln( z ) has a mild singularity at 0which is integrable.) shows that g ′ ∈ L along any rays that are not parallelto R . And again by Sobolev embedding, this term is also H¨older continuouswith index 1 /
2. All those three together eventually approach ln( δ j ) and rateof convergence is controlled by | z − z j | / .Thus | ¯ ∂E j, ( z ) | ≤ cρ − | z − z j | / + c | R ′ ( u + z j ) | (36) ≤ c ( | z − z j | − / + | R ′ ( u + z j ) | ) . (37)Note that ρ = | z − z j | and δ ( z ) and K ( φ ) are bounded along any rays thatis not parallel to R .Note also that sup | R | <
1, we have R −| R | ≤ R − sup | R | , and thus by Dom-inated converge theorem, all estimations for E j, can be smoothly moved to E j,k , k = 3 , ,
5. Separate Contributions and phase reduction
Since there are multiple saddle points on the real line, we have to separatecontributions from each saddle point. And in each, saddle point, we mayapproximate the RHP by a model RHP locally which will be discussed inthe next section. Also as we assumed all saddle points are of order 1, so thephase function can be approximated by θ ( z j ) + θ ′′ ( z j )2 ( z − z j ) . Thus we needto estimate the error generated by reduce the order of the phase function.Since our phase function is polynomial, we can always choose sufficient small α , such that the small triangular region along two saddle points shares thesame signature of Im( θ ( z )). And a difference between multiple saddle pointssituation and single saddle point situation is that there are jump in thatsmall triangular region. So in this section, we will show in fact, those jumpscan be ignored with sufficiently fast decaying error, say faster than the errorgenerated from the pure ¯ ∂ − problem.13et us consider two saddle points z j , z j +1 , and discuss ε j = 1 = − ε j +1 forexample. z j z j +1 z j + Σ j + Ω j, Ω j +1 , Ω j, Ω j +1 , Σ j, Σ j +1 , Figure 3: Jumps in a small triangular region.
Recall the constructions of E j, and E j +1 , , the boundary value of m [2] ( z )on Σ j + from Ω j, is m [1] ( z j +1 / + iv ) O j, ( z j +1 / + iv ) , while from Ω j +1 , is m [1] ( z j +1 / + iv ) O j +1 , ( z j +1 / + iv ) . Both are corresponding to locally increasing part of phase function, thus cor-responding to a upper/lower factorization. So the jump on the new contourΣ j +1 / is O j +1 , O − j, , where the nontrivial entry is(1 − K ( φ ))[ R ( z j ) δ − j ( z j +1 / − z j + iv ) − iη ( z j ) − R ( z j +1 ) δ − j +1 ( z j +1 / − z j +1 + iv ) − iη ( z j +1 ) ] e itθ ( z j +1 / + iv ) , where v ∈ (0 , ( z j +1 / − z j ) tan( α )).Note that | ( z j +1 / − z j + iv ) − iη ( z j ) | = e η ( z j ) φ ≤ e η ( z j ) α . and | e itθ ( z j +1 / + iv ) | ≤ ce − tdv , d = ( z j +1 + z j ) / O j +1 , O − j, = I + O ( e − t ) , t → ∞ . (38)14ince m [1] is analytic in a neighborhood of Σ j +1 / , O ( z j +1 / + iv ) is at least C with respect to v , then we havelim z →∞ | z ( m [2] ( z ) − I ) | ≤ π Z d tan( α )0 | m [1] − ( z j +1 / + s ) | e − tds ds by integration by parts since O is C near Σ j +1 / = O ( t − ) . This limit is in fact from the construction of potential, and this estimationshow that if we drop the contour Σ j +1 / , the potential induced by the newRHP will bring an error term O ( t − ), which is dominated by the error gen-erated by the ¯ ∂ − problem. Actually, we will later show that the ¯ ∂ − problemwill generate error O ( t − / ).For the triangular region below and for the cases when ε j = −
1, one cando slight modification to carry out the errors are still O ( t − ). Thus we willremove the jumps on Σ j +1 / for the RHP of m [2] .Now for the ¯ ∂O j, and ¯ ∂O j +1 , ,we have, for z − z j = ρe iφ , ρ = d/ tan( φ ), | ¯ ∂Q j, | ≤ c | ie iφ cos( φ ) d K ′ ( φ ) |≤ c. The implicit constant comes from the fact that both R ( z ) and δ ( z ) arebounded on the contour Σ j +1 / . Comparing to the estimation on Lemma(8),we see that the ¯ ∂ − estimations on Σ j +1 / are also dominated by | z − z j | − / + | R ′ (Re( z )) | .Even more, one can actually drop segments away from the stationaryphase points. It is well-known[9, 8] that the | E j, e itθ | ≤ ce − t tan( α ) u , wherelet u ≥ u >
0, then the jump matrix will go to I with decaying rateat O ( e − t ). Together with the analysis of RHP on Σ j +1 / , and a priori es-timate that pure ¯ ∂ − problem will generate error O ( t − / ), we can in facttruncate our contours to new one by dropping Σ j +1 / , j = 1 , · · · , l . Andwe arrive at a new ¯ ∂ − RHP by simply drop those contours which contributeless than the ¯ ∂ − problem. See Figure4 about the new contours, withoutgenerating any confusing, we will continuous to denote new contours byΣ j,k , j = 1 , c . . . , l, k = 1 , , , j z j +1 Figure 4: New contours, dashed line segments are those deleted parts. for multiple stationary phase points cases, which are based on the analysisof Beals-Coifman operators, have already shown the following lemma. Forconvenience, we will reprove it based on our settings.
Lemma 9. As t → ∞ , Z Σ ((1 − C w ) − I ) w = l X j =1 Z Σ j ((1 − C w j ) − I ) w j + O ( t − ) , (39) where w j is the factorization data supported on Σ j = ∪ k =1 Σ j,k , w = P lj =1 w j and Σ = ∪ lj Σ j .Proof. First, recall the following observation by Varzugin[7],(1 − C w )(1 + X j C w j (1 − C w j ) − ) = 1 − X j = k C w j C w k (1 − C w k ) − . With the hints from this observation, we need to estimate the norms of C w j C w k from L ∞ to L and from L to L . Also from next section (with asmall norm argument), we know (1 − C w j ) − are uniformly bounded in L sense. Now let us focus on the contour Σ j, , and ε = 1. Then the nontrivialentry of the factorization data is E j, ( z ) e − itθ ( z ) , z ∈ Σ j, , thus we have | w j ↾ Σ j, | ≤ ce − t tan( α ) u , k w j ↾ Σ j, k L = O ( t − / ) and k w j ↾ Σ j, k L = O ( t − / ).Then follow exactly the same steps in the proof of [9], Lemma 3.5, we havefor j = k k C w j C w k k L (Σ) = O ( t − / ) , k C w j C w k k L ∞ → L (Σ) = O ( t − / ) . Then use the resolvent identities and Cauchy-Schwartz inequality,((1 − C w ) − I ) = I + l X j =1 C w j (1 − C w j ) − I + [1 + l X j =1 C w j (1 − C w j ) − ](1 − X j = k C w j C w k (1 − C w k ) − ) − ( X j = k C w j C w k (1 − C w k ) − ) I = I + l X j =1 C w j (1 − C w j ) − I + ABDI. thus | Z Σ ABCIw | ≤ k A k L k B k L k D k L ∞ → L k w k L ≤ ct − / t − / = O t − . In the proof over the L boundedness, we actually use the triangularity of w j ’s, which gives us mild orthogonality[8]. The we apply the restrictionlemma([9],Lemma 2.56), and also by the mild orthogonality, one obtains Z Σ ( I + C w j (1 − C w j ) − I ) w ↾ Σ j = Z Σ j ( I + C w j (1 − C w j ) − I ) w = Z Σ j ((1 − C w j ) − I ) w j All together, the proof is done. 17 . Model RH problem
In this section, we will discuss a model RHP which can be solved explicitlyby solving a parabolic-cylinder equation. Consider the following RHP:1. P + ( ξ ; R ) = P − ( ξ ; R ) J ( R ) , ξ ∈ R , where J ( ξ ) = (cid:18) − | R | − ¯ RR (cid:19) is a constant matrix with respect to ξ .2. P ( ξ ; R ) = ξ iησ e − i ξ σ ( I + P ξ − + O ( ξ − )) , ξ → ∞ , where P = (cid:18) β ¯ β (cid:19) Then by Liouville’s argument, P ′ P − is analytic and thus P ′ ( ξ ) = ( − iξ σ − i σ , P ]) P ( ξ ) , (40)which can be solved in terms parabolic-cylinder equation, and apply theasymptotics formulas we can eventually determine that β = √ πe iπ/ e − πη/ R Γ( − a ) , (41)where a = iη. (42)The above result has been presented in considerable literatures in many ways.Here we follows the representations in [9]. Next, we will connect this modelRHP to the original RHP. Recall, near stationary phase point z j , we need toestimate integral R Σ j ((1 − C w j ) − I )( w j + + w j − ), which is equivalent to solvethe following RHP: 18. m [3 ,j ]+ ( z ) = m [3 ,j ] − ( z ) v [3 ,j ] ( z ) , z ∈ Σ j . The jump matrix ( ε j >
0) is v [3 ,j ] ( z ) = R ♯j ( z − z j ) − iη ( z j ) e itθ ′′ ( z j )( z − z j ) ! , z ∈ Σ j, − ¯ R ♯j −| R ♯j | ( z − z j ) iη ( z j ) e − itθ ′′ ( z j )( z − z j ) , z ∈ Σ j, R ♯j −| R ♯j | ( z − z j ) iη ( z j ) e itθ ′′ ( z j )( z − z j ) , z ∈ Σ j, − ¯ R ♯j ( z − z j ) − iη ( z j ) e − itθ ′′ ( z j )( z − z j ) ! , z ∈ Σ j, , (43)where R ♯j = R j δ − j e itθ ( z j ) .2. m [3 ,j ] = I + O ( z − ) , z → ∞ . Set ξ = (2 tθ ′′ ( z j )) / ( z − z j ) and by closing lenses, we arrive at an equiv-alent RHP on the real line:1. m [4 ,j ] ( ξ ) + = m [4] − v [4] ( ξ ) , ξ ∈ R . The new jump is v [4 ,j ] ( ξ ) = (2 θ ′′ ( z j ) t ) − iη ( zj )2 ad σ ξ iη ( z j ) ad σ e − iξ ad σ (cid:18) − | R ♯j | − ¯ R ♯j R ♯j (cid:19) (44)2. m [4 ,j ] = I + O ( ξ − ) , ξ → ∞ . Comparing with the model RHP, we observe that m [4 ,j ] ( ξ )(2 θ ′′ ( z j ) t ) − iη ( zj )2 σ ξ iη ( z j ) σ e − iξ σ solves the model RHP, whichleads to m [4]1 , = √ πe iπ/ e − πη ( z j ) / R ♯j Γ( − iη ( z j )) (45) m [4]1 , = −√ πe − iπ/ e − πη ( z j ) / ¯ R ♯j Γ( iη ( z j )) (46)19hange the variable ξ back to z , we have m [3]1 , ( t ) = l X j =1 (2 tθ ′′ ( z j )) − − iη ( zj )2 √ πe iπ/ e − πη ( z j ) / R ♯j Γ( − iη ( z j )) , (47) m [3]1 , ( t ) = − l X j =1 (2 tθ ′′ ( z j )) − + iη ( zj )2 √ πe − iπ/ e − πη ( z j ) / ¯ R ♯j Γ( iη ( z j )) . (48)Note that R ♯j = R j δ − j e itθ ( z j ) , one can rewrite in a neat way: m [3]1 , ( t ) = l X j =1 | η ( z j ) | / p tθ ′′ ( z j ) e iϕ ( t ) , (49) m [3]1 , ( t ) = l X j =1 | η ( z j ) | / p tθ ′′ ( z j ) e − iϕ ( t ) . (50)where the phase is ϕ ( t ) = π − arg Γ( − iη ( z j )) − tθ ( z j ) − η ( z j )2 ln | tθ ′′ ( z j ) | + 2 arg( δ j ) + arg( R j ) . (51)Here we have used the fact that | β | = η . From the relation connecting RHPand potential, we have q as ( x, t ) = − i l X j =1 | η ( z j ) | / p tθ ′′ ( z j ) e iϕ ( t ) , (52)The variable x is implicitly contained in z j ’s.
7. Errors from the pure ¯ ∂ − problem In this section, we will discuss the error generated from the pure¯ ∂ − problem of m [2] . Let us denote E ( z ) = m [2] ( m [3] ) − . (53)Since m [ k ] = I + m [ k ]1 z − + O ( z − ) , k = 2 ,
3, we have E ( z ) = 1 + ( m [2]1 − m [3]1 ) z − + O ( z − ) , (54)20hich can be regarded as the error of replacing m [2] by m [3] . Moreover, fromthis constructing, there is no jump on the contours Σ j,k , k = 1 , , , ∂ − problem is left due to the non-analyticity, which reads¯ ∂E = EW, (55)where W ( z ) = m [3] ¯ ∂O ( z )( m [3] ) − . (56)From the normalization condition of m [3] , we see it is uniformly bounded by c − sup R . And to estimate the errors of recovering the potential, one actuallyneeds to estimate lim z →∞ z ( E − I ), where the limit can be chosen along anyrays that are not parallel to R . For simplicity, we will take the imaginary axis.The ¯ ∂ − problem is equivalent to the following Fredholm integral equation bya simple application of generalized Cauchy integral formula: E ( z ) = I − π Z C E ( s ) W ( s ) s − z dA ( s ) . (57)In the following, we will show for each fixed z ∈ C , K ( W )( z ) := R C E ( s ) W ( s ) s − z dA ( s ) is bounded and then by the dominated convergence the-orem, we will show lim z →∞ z ( E − I ) = O ( t − / ). First of all, since m [3] isuniformly bounded, set z = z j + u + iv ,we have k W k . ( | ¯ ∂E j,k | e − tθ ′′ ( z j ) uv , z ∈ Ω j,k , k = 1 , | ¯ ∂E j,k | e tθ ′′ ( z j ) uv , z ∈ Ω j,k , k = 3 , , (58)where k · k stands for matrix norm and 0 ≤ a . b means there exists C > a ≤ Cb . Then we have K ( W ) ≤ k E k Z C k W ( s ) k| s − z | dA ( s ) . (59)We claim the following lemma: Lemma 10.
Let
Ω = { s : s = ρe φ , ρ ≥ , φ ∈ [0 , π/ } , and z ∈ Ω ,then Z Ω | u + v | − / e − tuv | u + iv − z | dudv = O ( t − / ) . (60)21 roof. Since there are two singularities of the integrand at z and (0 , z = 0, and let d = dist ( z, ∪ Ω ∪ Ω , where Ω = { s : | s | < d/ } ∩ Ω , Ω = { s : | s − z | < d/ } ∩ Ωand Ω = Ω \ (Ω ∪ Ω ).In the region Ω , | s − z | ≥ d/
3, thus | Z Ω | u + v | − / e − tuv | u + iv − z | dudv | ≤ d Z ∞ Z u e − tuv ( u + v ) / dvdu substituted v = wu ≤ d Z ∞ Z e − tu w (1 + w ) / u / dwdu ≤ d Z ∞ Z e − tu w u / dwdu = 32 d Z ∞ − e − tu tu / du = 32 d t − / Z ∞ − e − u u / du = 3 d Γ(3 / t − / . In the region Ω , | s | − / ≤ (2 d/ − / , | Z Ω | u + v | − / e − tuv | u + iv − z | dudv | ≤ r d Z Ω e − tuv (( u − x ) + ( v − y ) ) / dvdu ≤ r d Z d/ Z π e − t ( x + ρ cos( θ ))( y + ρ sin( θ )) dθdρ ≤ π r d e − txy . While in the region Ω , | Z Ω | u + v | − / e − tuv | u + iv − z | dudv | ≤ Z ∞ Z u e − tuv dvdu = O ( t − ) . z = 0. We have | Z Ω e − tuv ( u + v ) / dA ( u, v ) | = Z ∞ Z u e − tuv ( u + v ) / dvdu = Z ∞ Z e − tu w (1 + w ) / u / dwdu ≤ Z ∞ Z e − tu w u / dwdu = Z ∞ − e − tu tu / du = Z ∞ − e − u tt − / u / t − / u − du = 12 t − / Z ∞ − e − u u / du = 38 t − / Γ(1 / . Assembling all together, the proof is done.
Remark 11.
The essential fact that makes the above true is the rapid decayof the exponential factor in the region. And the lemma also tells us thatthose mild singularities, which have rational order grow, can be absorbedby the exponential factor. Back to our situation, after some elementarytransformations (translation and rotation), the estimation of R C k W ( s ) k| s − z | dA ( s )will eventually reduce to similar situation discussed in the above lemma.Based on the Lemma(10), we know when t is sufficiently large, kKk < t , k E − I k = k (1 − K ) − K I k ≤ ct − / − ct − / ≤ ct − / , t > t (61)Now since for each z ∈ Ω j,k , we have | ¯ ∂E j,k ( z ) | ≤ c ( | z − z j | − / + | R ′ ( u + z j ) | ) , and apply the Dominated Convergence theorem, we havelim z →∞ | z ( E − I ) | ≤ π l X j =1 4 X k =1 k E k L ∞ Z Ω j,k k W k ds, z ∈ Ω ofLemma (10), and we eventually have: E = lim z →∞ | z ( E − I ) | = O ( t − / ) . (62)
8. Asymptotics representation
First, we summary all the steps as following:(1). Initial RHP m [0] .(2). Conjugate initial RHP to obtain m [1] = m [0] δ − σ .(3). Open lens to obtain m [2] = m [1] O ( z ), where O ( z ) = I + o ( z ) , z → ∞ inall sectors.(4). Preparing for separating contributions and phase reduction by remov-ing some contours, which generates error O ( e − t ).(5). Separating contributions and phase reduction generate error O ( t − ).(6). Connect each RHP( m [3 ,j ] ) near stationary phase point to a ModelRHP( m [4 ,j ] ).(7). Comparing m [2] and m [3] and computing the error by analysis a pure¯ ∂ − problem. The error term is O ( t − / ).Combining all previous results, and undo all steps, we arrive at: m [0] ( z ) = E ( z ) m [3] ( z ) O − ( z ) δ σ . Since O ( z ) uniformly converges to I as z → ∞ , and δ σ is diagonal matrix,they do not affect the recovering the potential. Then we obtain q ( x, t ) = − i ( m [3]1 , + E , ) (63)= q as ( x, t ) + O ( t − / ) . (64) Remark 12. q as is O ( t − / ) as t → ∞ in the region x > x/t = O (1).24 . Fast decay region: x < , t → ∞ , − x/t = O (1) Observe that the contour Im( z k +1 + z ) = 0 has no intersection with realaxis for k ∈ N , which corresponding to the MKdV hierarchy, i.e., the oddparts of the AKNS hierarchy. For those phase function θ ( z ) = z n + z, n =2 k + 1 , k = 1 , , ... , we have the following properties:1. There exits ǫ = ǫ ( n ) > ± Im( θ ) > { z : ± Im( z ) ∈ (0 , ǫ ) } .2. There exits M ∈ (0 , /ǫ ) such that Im( θ ) ≥ nvu n − for | u | ≥ M ǫ andIm( θ ) ≥ v (1 − ( M ǫ ) ) for | u | ≤ M ǫ . Here z = u + iv .Now, we will formulate a general RHP model as follows: Given R ( z ) ∈ H , ( R ), find a piecewise holomorphic matrix-value function m such that1. m + = m − e − itθ ( z ) ad σ v ( z ) , z ∈ R , where the jump matrix is given by v ( z ) = (cid:18) − | R | − ¯ RR (cid:19) = (cid:18) − ¯ R (cid:19) (cid:18) R (cid:19) (65)2. m = I + O ( z − ) , z → ∞ . Theorem 13.
For the above RHP, the solution m enjoys the followingasymptotics as t → ∞ : m ( t ) = O ( t − ) . (66) where m = I + m ( t ) /z + O ( z − ) , z → ∞ . Now we will prove the theorem again using the idea of ¯ ∂ -steepest descentmethod. Proof.
We will only prove for the z ∈ { z : Im z ∈ (0 , ǫ ) } , for the other half,the same analysis works just by a slight modification. First we open the lens( R ) by multiple m by a smooth( R ) matrix-valued function O ( z ), where O ( z )is given by e − itθ ( z ) − R (Re z )1+(Im z ) ! . (67)25et us denote Σ = { z : Im z = ǫ } and˜ m = ( m, z ∈ Ω mO, z ∈ Ω where Ω = { z : Im z ∈ (0 , ǫ ) } and Ω = { z : Im z > ǫ } .Now as usual, we obtain a ¯ ∂ − RHP, and based on a traditional small normargument, the ˜ m = I + o (1) [10]. Denote the solution to the pure RHP by m ♯ , and consider E = ˜ m ( m ♯ ) − . (68)Then E doesn’t have jump on σ anymore and it satisfies a pure ¯ ∂ − problem:¯ ∂E = EW, (69)where W = − m ♯ e − tθ ( z ) ¯ ∂ ( R (Re z )1+(Im z ) )( m ♯ ) − , here ¯ ∂ = ( ∂ Re z + i∂ Im z ).Since R ∈ H , , ¯ ∂ ( R (Re z )1+(Im z ) ) is uniformly bounded by a non-negative L function f (Re z ). Note that m ♯ is uniformly close to I , set z = u + iv , thenwe have k W k ≤ f ( u ) e − t Im θ ( u,v ) , u ∈ R , v ∈ (0 , ǫ ) . By the same procure as previous sections, the error of approximating m bythe identity matrix is given by the following integral:∆ := Z ǫ Z R f ( u ) e − t Im θ dudv. (70)Split the u into two region: (1) | u | ≤ M ǫ , (2) | u | ≥ M ǫ . And denote by ∆ ,∆ respectively. Then ∆ = ∆ + ∆ . And∆ ≤ Z ǫ Z Mǫ − Mǫ f ( u ) e − tv (1 − M ǫ ) dudv by Cauchy-Schwartz ≤ k f k L ( R ) (2 M ǫ ) / − e − tǫ (1 − M ǫ ) t (1 − M ǫ )= O ( t − ) .
26n the other hand,∆ ≤ Z ǫ Z | u |≥ Mǫ f ( u ) e − ntvu n − dudv = Z | u |≥ Mǫ f ( u ) Z ǫ e − ntvu n − dvdu ≤ t − k f k L ( Z | u |≥ Mǫ ( 1 − e − ntvu n − nu n − ) du ) / ≤ t − k f k L nn − M ǫ ) − ( n − = O ( t − ) . Thus the error term is O ( t − ) and this completes the proof.
10. Painlev´e Region
It is well-known that one can generate Painlev´e II hierarchy from simi-larity reduction of MKdV hierarchy[11]. In this section, we will provide analgorithm based on the Riemann-Hilbert problems to generate the Painlev´eII hierarchy. Let’s denote Θ( x, z ) = xz + cn z n , suppose m solves the followingRHP: m + = m − e i Θ σ v e − i Θ σ , z ∈ Σ n ,m = I + O ( z − ) , z → ∞ . where the contour Σ n consists of all stokes lines which are determined by Θ,and v is a constant 2 by 2 matrix that is independent of x, z .Now let ˜ m = me i Θ σ , we arrive at a new RHP:˜ m + = ˜ m − v , z ∈ Σ n , ˜ m = ( I + O ( z − )) e i Θ σ , z → ∞ . Since v is constant, it is easily to check, by Louisville’s argument, that both ∂ z ˜ m ˜ m − and ∂ x ˜ m ˜ m − are polynomial of z . Hence we obtain the followingtwo differential equations: ∂ x ˜ m ˜ m − = A ( x, z ) , (71) ∂ z ˜ m ˜ m − = B ( x, z ) . (72)27f we assume m = I + n − X j =1 m j ( x ) z − j + O ( z − ( n − ) , z → ∞ , (73) m = m − = I + n − X j =1 m j ( x ) z − j + O ( z − ( n − ) , z → ∞ , (74)then direct computation shows A = i [ m , σ ] + izσ , (75) B = ixσ + icz n − σ + iz n − [ m , σ ] + n − X k =2 icz n − − k ( m k σ + σ m k + k − X j =1 m k − j σ m j ) . (76)Since m x,z = m z,x , we have A z − B x + [ A, B ] = 0 . (77)Let the coefficients of z all vanish, and set m j = (cid:18) u j ( x ) u j ( x ) 0 (cid:19) , (78)we can solve the equations recursively from the high degree of z to low degree,and eventually, we will arrive at a nonlinear ODEs of u , which are in fact ahierarchy of Painlev´e II equations. We list the first few of them: n = 3 : − cu + cu xx − ux = 0 , (79) n = 4 : 12 icu u x − icu xxx − ux = 0 , (80) n = 5 : − cu + 10 cu u xx + 10 cuu x − c u xxxx − ux = 0 . (81)In this dissertation, we are interested on the odd members. In particular, n = 3 corresponds to the MKdV equation, n = 5 corresponds to the 5thorder MKdV, and so on. In the following section, we will show how can weconnect the long-time asymptotics MKdV hierarchy with solutions to thePainlev´e II hierarchy. 28 Recall the phase functions of the AKNS hierarchy of MKdV type equa-tions are θ ( z ; x, t ) = xz + ctz n , n is odd . (82)The Painlev´e region is the region of x = O ( t n ), by rescaling z → ( nt ) − n ξ ,and let s = x ( nt ) − n , we haveΘ( ξ ) = sξ + cn ξ n . (83)Now the modular of the stationary phase points of (82) is | z | = (cid:12)(cid:12)(cid:12) − xct (cid:12)(cid:12)(cid:12) n − = O ( t − n ) , however, after scaling, the modular of the stationary phase points of Θ( ξ ) is | ξ | = z t n , (84)which is fixed as t → ∞ . A direct computation shows for any odd n , thesignature of Re ( iθ ) is just similar to Fig5. Since the original RHP only hasa jump on the real line, all the stokes lines except those cross real line canbe ignored.Note that e − iθ ( z ) ad σ v ( z ) = e − i Θ( ξ ) ad σ v ( ξ )= (cid:18) − | R | − ¯ Re − i Θ Re i Θ (cid:19) = (cid:18) − ¯ Re − i Θ (cid:19) (cid:18) Re i Θ (cid:19) = e − i Θ( ξ ) ad σ v − − v + We can deform the the contour { z ∈ R : | z | > | ξ |} as before and getthe deformed contour as follows: As before, set the original RHP as m [1] with jump e − iθ ( z ) ad σ v ( z ). After re-scaling and ¯ ∂ − lens opening, we set29 ξ |−| ξ | Figure 5: Signature of Re( iθ ). The gray region indicates Re( iθ ) > m [2] ( ξ ) = m [1] O ( γ ), where the lens-opening matrix is O ( γ ) = − E + e i Θ( γ ) ! , γ ∈ Ω ∪ Ω , − E − e − i Θ( γ ) ! , γ ∈ Ω ∪ Ω ,I, γ ∈ Ω ∪ Ω , (85)where E + ( γ ) = K ( φ ) R (cid:16) ( nt ) − n ξ (cid:17) + (1 − K ( φ )) R ( ξ ( nt ) − n ) (86) E − ( γ ) = E + ( γ ) , γ = ξ + ρe iφ , ξ = Re( γ ) . (87)Now we arrive at the following ¯ ∂ − RHP:30 ξ |−| ξ | Σ Σ Σ Σ Σ Ω Ω Ω Ω Ω Ω Figure 6: Contour for ¯ ∂ − RHP. [1] The RHP(1.a). m [2] ( γ ) ∈ C ( R \ Σ) and m [2] ( z ) = I + O ( γ − ) , γ → ∞ .(1.b). The jumps on Σ and Σ are e − i Θ( ξ ) ad σ v + , the jumps on Σ and Σ are e − i Θ( ξ ) ad σ v − , and the jump on Σ is e − i Θ ad σ v (( nt ) − n ξ ) . [2] The ¯ ∂ problemFor z ∈ C , we have ¯ ∂m [2] ( ξ ) = m [2] ( ξ ) ¯ ∂O ( ξ ) . (88)Again, we will need the following lemma in order to estimate the errors fromthe ¯ ∂ − problem. Lemma 14.
For γ ∈ Ω , , , , ξ = Re γ , | ¯ ∂E ± ( γ ) | ≤ ( nt ) − n | ( nt ) − n ( ξ − ξ ) | − k R k H , + ( nt ) − n | R ′ (( nt ) − n ξ ) | . (89) Proof.
For brevity, we only proof for the region Ω . Using the polar coordi-nates, we have | ¯ ∂E + ( γ ) | = (cid:12)(cid:12)(cid:12)(cid:12) ie iφ ρ K ′ ( φ ) h R (cid:16) ( nt ) − n ξ (cid:17) − R ( ξ ( nt ) − n ) i + K ( φ ) R ′ (cid:16) ( nt ) − n ξ (cid:17) ( nt ) − n (cid:12)(cid:12)(cid:12)(cid:12) by Cauchy-Schwartz inequality ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k R k H , | ( nt ) − n ξ − ξ ( nt ) − n | / γ − ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ( nt ) − n (cid:12)(cid:12)(cid:12) R ′ (cid:16) ( nt ) − n ξ (cid:17)(cid:12)(cid:12)(cid:12) ≤ ( nt ) − n | ( nt ) − n ( ξ − ξ ) | − k R k H , + ( nt ) − n | R ′ (( nt ) − n ξ ) | . Similarly, we can prove for other regions.31ext, consider a pure RHP m [3] which satisfies exactly the RHP part of¯ ∂ − RHP( m [2] ). Moreover, m [3] can be approximated by the RHP correspond-ing to a special solution of the Painlev´e II hierarchy. Since for γ ∈ Ω , (cid:12)(cid:12)(cid:12)(cid:16) R ( ξ ( nt ) − n ) − R (0) (cid:17) e i Θ( γ ) (cid:12)(cid:12)(cid:12) ≤ | ξ ( nt ) − n | k R k H , e i Θ( γ ) ≤ ( nt ) − n | Re γ | k R k H , e i Θ( γ ) , it is evident that k Re i Θ − R (0) e i Θ k L ∞ ∩ L ∩ L ≤ c ( nt ) − n . (90)Let m [4] be solve to the RHP by change the jumps of m [3] to R (0) and¯ R (0), then, via small norm techniques, the errors between the correspondingpotential is given by error , = lim γ →∞ | γ ( m [4]12 − m [3]12 ) | (91) ≤ c Z Σ | ( R (Re( s )( nt ) − n ) − R (0)) e i Θ( s ) | ds (92) ≤ c ( nt ) − n . (93)Then since now the jumps are all analytic, we can perform the analyticdeformation and arrive at the green contours as show in Fig7. Let’s denotethe new RHP by m [5] ( γ ), which is exactly equivalent to m [4] ( γ ). | ξ |−| ξ | Σ Σ [4]1 Σ Σ [4]2 Σ Σ [4]3 Σ Σ [4]4 Figure 7: Contour for m [4] (Green part). m [5] reads: e − i Θ( γ ) v [4] (0) = R (0) e i Θ( γ ) ! , γ ∈ Σ [4]1 , R (0) e − i Θ( γ ) ! , γ ∈ Σ [4]3 , . (94)Then according to the previous section, the (1 ,
2) entry of the solution m [5] ,so is the solution m [4] , is the solution to the Painlev´e II hierarchy, i.e., m [4]12 ( γ ) = m [5]12 ( γ ) , (95)Recall P IIn ( s ) = lim γ →∞ γm [5]12 where P IIn solves the n th equation in Painlev´eII hierarchy.Now let’s consider the error generated from the ¯ ∂ -extension. Recall theerror E satisfies a pure ¯ ∂ problem:¯ ∂E = EW,W = m [3] ¯ ∂O ( m [3] ) − . As before, the ¯ ∂ equation is equivalent to an integral equation which reads E ( z ) = I + 1 π Z C E ( s ) W ( s ) z − s ds = I + K ( E ) . As before, we can show that the resolvent is always exist for large t . So weonly need to estimate the true error which is:lim z →∞ z ( E − I ). In fact, wehave lim z →∞ | z ( E − I ) | = | Z C EW ds |≤ c k E k ∞ Z Ω | ¯ ∂O | ds. For the sake of simplicity, we only estimate the integral on the right handside in the region of the top right corner. Note there is only one entry isnonzero in ¯ ∂O , which is one of the E ± and we split the integral into two33arts in the obvious way, i.e., Z Ω | ¯ ∂O | ds ≤ I + I = Z Ω ( nt ) − n | Re s − ξ |k R k H , e i Θ( s ) ds + Z Ω ( nt ) − nt | R ′ (( nt ) − n s ) | e i Θ( s ) ds. As we know from previous sections, e Re 2 i Θ( s ) ≤ ce − | Θ ′′ ( ξ ) | uv in the region { z = u + iv : u > ξ , < v < αu } for some small α , where s = u + iv + ξ .Then we have I ≤ ( nt ) − n Z Ω | Re s − ξ | − / e − cuv dudv ≤ ( nt ) − n Z ∞ Z αu u − / e − cuv dudv ≤ C ( nt ) − n Z ∞ − e − α | Θ ′′ ( ξ ) | u / du = O (cid:16) ( nt ) − n (cid:17) . And I ≤ ( nt ) − n Z | R ′ (( nt ) − n Re s ) | e − cuv dudv by Cauchy-Schwartz inequality ≤ ( nt ) − n k R k H , Z ∞ ( Z ∞ αv e − cuv du ) / dv ≤ ( nt ) − n k R k H , Z ∞ e − cαv √ αcv dc = O (( nt ) − n ) . Thus, we arrive at ¯ ∂ Error = O (( nt ) − n ) . (96)34nd we undo all the deformations, we obtain m [1] (( nt ) − n γ ) = m [2] ( γ ) O − ( γ )= (1 + O t n γ ) m [3] ( γ ) O − ( γ )= (1 + O t n γ )(1 + O t n γ ) m [4] ( γ ) O − ( γ ) , and can be rewrite in terms variable z : m [1] ( z ) = (cid:18) O ( t − / (2 n ) ) z ( nt ) /n (cid:19) m [5] (( nt ) /n z ) . (97)Since m [
5] is corresponding to the RHP for the Painlev´e II hierarchy, we have m [5] ( γ ) = I + m [5]1 ( s ) γ + O ( γ − ) , (98)where γ = z ( nt ) /n . Thus, m [1] ( z ) = O ( t − n ) z ( nt ) /n ! m [5]1 ( s ) z ( nt ) /n + O ( z − ) ! (99)= I + m [5]1 ( s ) z ( nt ) /n + O ( t − n ) z ( nt ) /n + O ( z − ) , (100)since m [5]1 ( s ) is connected to solutions of the Painlev´e II hierarchy, we con-clude that q ( x, t ) = lim z →∞ z ( m [1] − I ) (101)= ( nt ) − n u n ( x ( nt ) − n ) + O ( t − n ) , (102)where u n solves the n th member of the Painlev´e II hierarchy. For the case ofMKdV type defocusing reduction, n only takes odd number.
11. Conclusion
In the current paper, we have established a general large-parameterasymptotic result for an oscillatory Riemann-Hilbert problem with polyno-mial phase function. The result can be directly applied to study the long-time35ehavior of nonlinear integrable dispersive PDEs which admit a Lax pair rep-resentation from the AKNS hierarchy. The main tool we used for analysis theasymptotic of the RHP is so-called ¯ ∂ -steepest descent method[10], which canbe regarded as a generalization of the original Deift-Zhou’s method. Thoughthe ¯ ∂ -steepest descent method simplifies a lot deep harmonic analysis, thereare still some work to be resolved in order to fully replace the Deift-Zhou’smethod. On the one hand, it seems Deift-Zhou’s method can generate a bet-ter leading term estimation provided a more regular initial data[12]. However,the ¯ ∂ -steepest descent method only requires the initial data to be H , theclassical Sobolev space. To get a better estimation, the key is to constructa better interpolation function E in the lens-opening step. On the otherhand, it is well known that for the Schwartz initial data, the error term ofthe asymptotic solution is O ( log( t ) t ). It is natural to guess the error term willdecay faster provided more regular initial data. How to build the connectionbetween the regularity of the initial data with the decaying rate of the errorterm? This question remains open. References [1] M. Dieng, K. D. T. R. McLaughlin, Long-time asymptotics for the nlsequation via dbar methods, 2008.[2] K. T. R. McLaughlin, P. D. Miller, The dbar steepest descent methodand the asymptotic behavior of polynomials orthogonal on the unit circlewith fixed and exponentially varying nonanalytic weights, 2004.[3] M. A. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equationsand Inverse Scattering, London Mathematical Society Lecture Note Se-ries, Cambridge University Press, 1991.[4] X. Zhou, L2-sobolev space bijectivity of the scattering and inverse scat-tering transforms, Communications on Pure and Applied Mathematics51 (1998) 697–731.[5] W.-X. Ma, A soliton hierarchy associated with so(3,r), Applied Mathe-matics and Computation 220 (2013) 117 – 122.[6] G. Tu, The trace identity, a powerful tool for constructing the hamilto-nian structure of integrable systems, Journal of Mathematical Physics30 (1989) 330–338. 367] G. Varzugin, Asymptotics of oscillatory riemann–hilbert problems,Journal of Mathematical Physics 37 (1996) 5869–5892.[8] Y. Do, A nonlinear stationary phase method for oscillatory riemann-hilbert problems, 2009.[9] P. Deift, X. Zhou, A steepest descent method for oscillatory riemann–hilbert problems. asymptotics for the mkdv equation, Annals of Math-ematics 137 (1993) 295–368.[10] M. Dieng, K. D. T. R. McLaughlin, P. D. Miller, Dispersive asymptoticsfor linear and integrable equations by the ∂∂