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APPEARED IN BULLETIN OF THE

arXiv:math/9201261v1 [math.AP] 1 Jan 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 1, Jan 1992, Pages 119-124A STEEPEST DESCENT METHODFOR OSCILLATORY RIEMANN-HILBERT PROBLEMSP. Deift and X. ZhouIn this announcement we present a general and new approach to analyzing theasymptotics of oscillatory Riemann-Hilbert problems.

Such problems arise, in par-ticular, in evaluating the long-time behavior of nonlinear wave equations solvableby the inverse scattering method. We will restrict ourselves here exclusively to themodified Korteweg de Vries (MKdV) equation,yt −6y2yx + yxxx = 0,−∞< x < ∞, t ≥0,y(x, t = 0) = y0(x),but it will be clear immediately to the reader with some experience in the field,that the method extends naturally and easily to the general class of wave equationssolvable by the inverse scattering method, such as the KdV, nonlinear Schr¨odinger(NLS), and Boussinesq equations, etc., and also to “integrable” ordinary differentialequations such as the Painlev´e transcendents.As described, for example, in [IN] or [BC], the inverse scattering method forthe MKdV equation leads to a Riemann-Hilbert factorization problem for a 2 × 2matrix valued function m = m(·; x, t) analytic in C\R,(1)m+(z) = m−(z)vx,t,z ∈R,m(z) →Ias z →∞,wherem±(z) = limε↓0 m(z ± iε; x, t),vx,t(z) ≡e−i(4tz3+xz)σ3v(z)ei(4tz3+xz)σ3,σ3 =100−1,andv(z) =1 −|r(z)|2−r(z)r(z)1=1−r01 10r1≡b−1−b+.Received by the editors February 13, 1991 and, in revised form, March 6, 19911980 Mathematics Subject Classification (1985 Revision).

Primary 35Q20; Secondary 35B40The work of the authors was supported in part by NSF Grants DMS-9001857 and DMS-9196033, respectivelyc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2P. DEIFT AND X. ZHOUIf y0(x) is in Schwartz space, then so is r(z) andr(z) = −r(−z),supz∈R|r(z)| < 1.From the inverse point of view, given v(z), one considers a singular integral equation(see [BC]) for the associated quantity µ(z; x, t)=m+(z; x, t)(b−1+ )x,t = m−(z; x, t)(b−1−)x,tand the solution of the inverse problem is then given by(2)y(x, t) =σ3,ZRµ(z; x, t)wx,t(z) dz2πi21wherewx,t = (w+)x,t + (w−)x,t,w± = ±(b± −I).Significant work on the long-time behavior of nonlinear wave equations solvableby the inverse scattering method, was first carried out by Manakov [M] and byAblowitz and Newell [AN] in 1973.The decisive step was taken in 1976 whenZakharov and Manakov [ZM] were able to write down precise formulae, depend-ing explicitly on the initial data, for the leading asymptotics for the KdV, NLS,and sine-Gordon equations, in the physically interesting region x = O(t).

A com-plete description of the leading asymptotics of the solution of the Cauchy problem,with connection formulae between different asymptotic regions, was presented byAblowitz and Segur [AS], but without precise information on the phase. The as-ymptotic formulae of Zakharov and Manakov were rigorously justified and extendedto all orders by Buslaev and Sukhanov [BS 1–2] in the case of the KdV equation,and by Novokshenov [N] in the case of NLS.The method of Zakharov and Manakov, pursued rigorously in [BS] and in [N],involves an ansatz for the asymptotic form of the solution and utilizes techniquesthat are somewhat removed from the classical framework of Riemann-Hilbert prob-lems.

In 1981, Its [I] returned to a method first proposed in 1973 by Manakov in[M], which was tied more closely to standard methods for the inverse problem. In[I] the Riemann-Hilbert problem was conjugated, up to small errors which decayas t →∞, by an appropriate parametrix, to a simpler Riemann-Hilbert problem,which in turn was solved explicitly by techniques from the theory of isomonodromicdeformations.

This technique provides a viable, and in principle, rigorous approachto the question of long-time asymptotics for a wide class of nonlinear wave equa-tions (see [IN]). Finally we note that in [B], Buslaev derived asymptotic formulaefor the KdV equation from an exact determinant formula for the solution of theinverse problem.In our approach we consider the Riemann-Hilbert problem (1) directly, and bydeforming contours in the spirit of the classical method of steepest descent, we showhow to extract the leading asymptotics of the MKdV equation.

In particular forx < 0, let ±z0 = ±p|x|/12t be the stationary phase points for i(4tz3 + xz). Thenthe first step in our method is to show that (1) can be deformed to a Riemann-Hilbert problem on a contour Σ of shape (see Figure 1), in such a way that thejump matrices vx,t on R ⊂Σ and on the compact partof Σ\R away from ±z0, converge rapidly to the identity as t →∞.

Thus we areleft with a Riemann-Hilbert problem on a pair of crosses ΣA ∪ΣB (see Figure 2).As t →∞, the interaction between ΣA and ΣB goes to zero to higher order and

A STEEPEST DESCENT METHOD3Figure 1Figure 2Figure 3the contribution to y(x, t) through (2) is simply the sum of the contributions fromΣA and ΣB separately. Under the scalings z →z(48tz0)−1/2 ∓z0, the problems onΣA and ΣB are reduced to problems on a fixed cross, with jump matrices whichare independent of time, and which can be solved explicitly in terms of paraboliccylinder functions, as in [I].

Substitution in (2) yields, finally, the asymptotics fory(x, t).Our result is the following: letφ(z0) = arg Γ(iν) −π4 −arg r(z0) + 1πZ z0−z0log |s −z0|d(log(1 −|r(s)|2)(here Γ is the standard gamma function) and letya = ν3tz01/2cos(16tz30 −ν log(192tz30) + φ(z0)),where ν = −(2π)−1 log(1 −|r(z0)|2) > 0. Set τ = tz30 = (|x|/12t1/3)3/2.Theorem.

Let y0(x) lie in Schwartz space with reflection coefficient r(z).Ast →∞, the solution y(x, t) of MKdV with initial data y0(x), has uniform lead-ing asymptotics conveniently described at fixed t ≫1, in the six regions shown inFigure 3.In region I, for any j,y(x, t) = ya + O((−x)−j + (−x)−3/4Cj(−x/t))where Cj(·) is rapidly decreasing. In region II,y(x, t) = ya + (tz0)−1/2O(τ −1/4).

4P. DEIFT AND X. ZHOUIn region III,y(x, t) = (3t)−1/3p(x/(3t)1/3) + O(τ 2/3/t2/3),where p is a Painlev´e function of type II.In region IV,y(x, t) = (3t)−1/3p(x/(3t)1/3) + O(t−2/3).In region V, for any j,y(x, t) = (3t)−1/3p(x/(3t)1/3) + O(t−j + t−2/3e−12ητ 2/3)for some η > 0.Finally, in region VI, for any j,y(x, t) = O((x + t)−j).□Remark 1.

The reader may check that in the overlap regions the asymptotic formsdo indeed match. Also the reader may check that in regions II and IV, the formulaefor the leading asymptotics agree with those in [IN].Remark 2.

The above error estimates are not the best possible and in region II inparticular, the τ −1/4 decay can certainly be improved.Remark 3. There is no obstacle in the method to obtaining an asymptotic expansionfor y(x, t) to all orders.Remark 4.

As noted at the beginning of this announcement, the method we havepresented extends naturally to the general class of nonlinear wave equations solv-able by the inverse scattering method. Also, there is no difficulty in incorporatingsolutions with solitons.References[AN]M. J. Ablowitz and A. C. Newell, The decay of the continuous spectrum for solutions ofthe Korteweg de Vries equation, J.

Math. Phys.

14 (1973), 1277–1284.[AS]M. J. Ablowitz and H. Segur, Asymptotic solutions of the Korteweg de Vries equation,Stud.

Appl. Math.

57 (1977), 13–14.[BC]R. Beals and R. Coifman, Scattering and inverse scattering for first order systems, Comm.Pure Appl.

Math. 37 (1984), 39–90.[B]V.

S. Buslaev, Use of the determinant representation of solutions of the Korteweg de Vriesequation for the investigation of their asymptotic behavior for large times, Uspekhi Mat.Nauk 34 (1981), 217–218. [BS1] V. S. Buslaev and V. V. Sukhanov, Asymptotic behavior of solutions of the Korteweg deVries equation, Proc.

Sci. Seminar LOMI 120 (1982), 32–50.

(Russian); transl. in J. SovietMath.

34 (1986), 1905–1920. [BS2], On the asymptotic behavior as t →∞of the solutions of the equation ψxx +u(x, t)ψ + (λ/4)ψ = 0 with potential u satisfying the Korteweg de Vries equation, I, Prob.Math.

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Seminar LOMI 138 (1984), 8–32. (Russian); transl.

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(Russian).[I]A. R. Its, Asymptotics of solutions of the nonlinear Schr¨odinger equation and isomon-odromic deformations of systems of linear differential equations, Soviet Math.

Dokl. 24(1981), 452–456.[IN]A.

R. Its and V. Yu. Novokshenov, The isomonodromic deformation method in the the-ory of Painlev´e equations, Lecture Notes in Math., vol.

1191, Springer-Verlag, Berlin andHeidelberg, 1986.

A STEEPEST DESCENT METHOD5[M]S. V. Manakov, Nonlinear Fraunhofer diffraction, Zh. `Eksper.

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65 (1973), 1392–1398. (Russian); transl.

in Soviet Phys.-JETP, 38 (1974), 693–696.[N]V. Yu.

Novokshenov, Asymptotics as t →∞of the solution of the Cauchy problem for thenonlinear Schr¨odinger equation, Soviet Math. Dokl.

21 (1980), 529–533. [ZM] V. E. Zakharov and S. V. Manakov, Asymptotic behavior of nonlinear wave systems in-tegrated by the inverse method, Zh.

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Phys.-JETP 44 (1976), 106–112.Department of Mathematics, New York University–Courant Institute, New York,New York 10012Department of Mathematics, Yale University, New Haven, Connecticut 06520

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