P.A. Deift, S. Venakides and X. Zhou, New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Internat. Math. Res. Notices, 6, 285--299 (1997).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the steepest descent type method to the Riemann Hilbert problem (RHP) 5. 4) The steepest descent method for RHP s, the Deift Zhou method, was introduced by Deift and Zhou in [18] and is developed further in [19] 16] and finally placed in a systematic form by Deift, Venakides and Zhou in [17]. The steepest descent analysis of the RHP (5.4) was first conducted in [3] The analysis of [3] has many similarities with [13, 14, 15] where the asymptotics of orthogonal polynomials on the real line with respect to a general weight is obtained, leading to a proof of universality conjectures in ....

....branch is chosen such that log(z io) is analytic in C ( cx , 1] U i0: 7r q 0 and behaves like logz as z 6 cx . The basic properties of g(z) are summarized in Lemma 4. 2 of [3] In general, the role of the g function in RHP analysis, first introduced in [19] and then generalized in [17], is to replace exponentially growing terms in the jump matrix by oscillating or exponentially decaying terms. In [13] the authors introduced a g function of a form similar to (10.87) to analyze an RHP associated to orthogonal polynomials on the real line. The above g function (10.87) introduced ....

P. Delft, S. Venakides, and X. Zhou. New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. Internat. Math. Res. Notices, 6:285-299, 1997.

.... based on the characterization of the orthogonal polynomials by means of a Riemann Hilbert problem for 2 Theta 2 matrix valued functions due to Fokas, Its, and Kitaev [16] together with an application of the steepest descent method of Deift and Zhou, introduced in [13] and further developed in [4, 12, 14] and the papers cited before. See [8, 11] for an introduction, and [15, 19, 21, 27] for the latest developments. The orthogonal polynomials considered in [9, 10] are orthogonal with respect to an exponential weight e GammaQ(x) on R or with respect to varying weights e GammanV (x) on R. ....

P. Deift, S. Venakides, and X. Zhou, New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Internat. Math. Res. Notices

....modern approach to these semi classical limits combines the methods of Lax Levermore with 59 those of Riemann Hilbert problems with rapidly oscillating kernels. Here, the oscillations arise as the coefficient of dispersion vanishes, rather than as t 1. Recently, Deift, Venakides, and Zhou [51] have completed a Riemann Hilbert analysis of the small dispersion limit of the KdV equation. They are able to obtain a very detailed asymptotic description of the solution. It is remarkable to note that the maximization problem identified by Lax and Levermore appears as an essential component for ....

....to connect the initial value problem with the higher genus modulation equations of [70] So far such a Riemann Hilbert analysis has not been carried out for the case of the semi classical limit of the NLS equation, and there are some difficulties. One particularly interesting aspect is that in [51], the authors assume that there are no solitons present, and work exclusively with the reflection coefficient. The situation in which there are N solitons, and N 1 in the small dispersion limit (the easiest case for the Lax Levermore approach) seems somewhat more difficult in the ....

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P. Deift, S. Venakides, and X. Zhou. New results in small dispersion kdv by an extension of the steepest descent method for riemann--hilbert problems. International Mathematics Research Notices, pages 286--299, 1997.

....has not been established rigorously. However, experience from the interplay between whole line scattering theory and periodic spectral theory indicates that the existence of gaps implies the development of oscillations. For the small dispersion limit of the KdV equation, Deift, Venakides, and Zhou [7] have shown, under real analyticity assumptions on the spectral data, that the existence of gaps implies oscillations. Also, for the so called Toda shock problem, the connection between a gap and oscillations is well known [13, 28, 15] As already indicated, the formulation and analysis of a ....

....so called Toda shock problem, the connection between a gap and oscillations is well known [13, 28, 15] As already indicated, the formulation and analysis of a maximization problem like (1.12) 1. 13) lies at the heart of the Lax Levermore approach to the zero dispersion limit of the KdV equation [20, 27, 7, 10]. In a similar way, other singular limits of integrable systems have been treated recently. We mention the semiclassical limit of the defocing nonlinear Schrodinger equation [14] and the continuum limit of a discrete NLS chain [25] The long time asymptotic behavior was considered in [20] for the ....

P. Deift, S. Venakides, and X. Zhou, New results in small dispersion KdV by an extension of the steepest descent method for Riemann--Hilbert problems, Internat. Math. Research Notices 1997, no. 6, 286--299. 36

.... is based on the characterization of the orthogonal polynomials by means of a Riemann Hilbert problem for 2 2 matrix valued functions due to Fokas, Its, and Kitaev [16] together with an application of the steepest descent method of Deift and Zhou, introduced in [13] and further developed in [4, 12, 14] and the papers cited before. See [8, 11] for an introduction, and [15, 19, 21, 27] for the latest developments. The orthogonal polynomials considered in [9, 10] are orthogonal with respect to an exponential weight e Q(x) on R or with respect to varying weights e nV (x) on R. Inspired by these ....

P. Deift, S. Venakides, and X. Zhou, New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Internat. Math. Res. Notices

....the modern approach to these semi classical limits combines the methods of Lax Levermore with 59 those of Riemann Hilbert problems with rapidly oscillating kernels. Here, the oscillations arise as the coe cient of dispersion vanishes, rather than as t 1. Recently, Deift, Venakides, and Zhou [51] have completed a Riemann Hilbert analysis of the small dispersion limit of the KdV equation. They are able to obtain a very detailed asymptotic description of the solution. It is remarkable to note that the maximization problem identi ed by Lax and Levermore appears as an essential component for ....

....able to connect the initial value problem with the higher genus modulation equations of [70] So far such a Riemann Hilbert analysis has not been carried out for the case of the semi classical limit of the NLS equation, and there are some diculties. One particularly interesting aspect is that in [51], the authors assume that there are no solitons present, and work exclusively with the re ection coecient. The situation in which there are N solitons, and N 1 in the small dispersion limit (the easiest case for the Lax Levermore approach) seems somewhat more dicult in the Riemann Hilbert ....

[Article contains additional citation context not shown here]

P. Deift, S. Venakides, and X. Zhou. New results in small dispersion kdv by an extension of the steepest descent method for riemann{hilbert problems. International Mathematics Research Notices, pages 286-299, 1997.

No context found.

P. Deift, S. Venakides, and X. Zhou. New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. Internat. Math. Res. Notices, 6:285-299, 1997.

No context found.

P. Deift, S. Venakides, and X. Zhou. New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. Internat. Math. Res. Notices, 6:285-299, 1997.

....and hence prove the convergence of moments for 1 and 2 , using the steepest descent method for the Riemann Hilbert problem (RHP) naturally associated with T n and K n above. The steepest descent method for RHP was introduced in [14] and extended to include fully non linear oscillations in [13]. The asymptotic analysis in [3] 4] is closely related to the analysis in [11, 12] The main motivation for this paper was to nd a formula for the joint distribution of 1 ; k , which generalized (1.11) and to which the above Riemann Hilbert steepest descent methods could be ....

....contribution to the RHP comes from the part of the circle near z = 1. But in the later case, due to the existence of both factorization of the jump matrix, the RHP localizes in the limit just to two points on the circle. We refer the reader to [14] for an example of the second type, and to [15, 13, 12] for examples of the rst type. Remark. The di erent analysis for s = 1 and 0 s 1 gives us di erent estimates. Indeed, when s = 1, instead of (5.18) below, we have (see (6.42) of [3] log m 11 (0; k) k( 2t k log 2t k 1) 5.17) which imply (4.30) 4.31) Thus, in order to obtain ....

P. Deift, S. Venakides, and X. Zhou. New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. Internat. Math. Res. Notices, 6:285-299, 1997.

.... polynomials on the circle of the RHP introduced in [FIK] for orthogonal polynomials on the line) and then apply the steepest descent method for RHP s introduced by Deift and Zhou in [DZ1] further developed in [DZ2] DVZ1] and finally placed in a systematic form by Deift, Venakides and Zhou in [DVZ2], to compute the asymptotics as #, n # #. A general reference for RHP s is, for example, CG] The calculations in [BDJ] have many similarities to the calculations in [DKMVZ] The methods of [BDJ] apply here because the operator Kn in (16) is an example of a so called integrable operator, whose ....

P.A.Deift, S.Venakides and X.Zhou, New Results in Small Dispersion KdV by an Extension of the Steepest Descent Method for Riemann-Hilbert Problems, Internat. Math. Res. Notices, no. 6, 285-299, (1997).

....introduced a steepest descent type method to compute the asymptotic behavior of RHP s containing large oscillatory and or exponentially growing decaying factors as in (1. 26) This method was further extended in [DVZ1] and eventually placed in a very general form by Deift, Zhou and Venakides in [DVZ2], making possible the analysis of the limiting behavior of a large variety of asymptotic problems in pure and applied mathematics (see, e.g. DIZ] As we will see, the application of this method to (1.26) makes it possible to control the large k; behavior of 2 k ( The calculation in ....

....Painlev e III equation is presented in Section 3. Section 4 is the starting point for the analysis of the RHP (1.26) In this section, 1.26) is transformed into an equivalent RHP via a so called g function. The role of g function, rst introduced in [DZ2] and then analyzed in full generality in [DVZ2], is to replace exponentially growing terms in a RHP by oscillatory or exponentially decreasing terms. It turns out that in the case of (1.26) as in [DKMVZ1] the g function can be constructed in terms of an associated equilibrium measure d (s) as follows, g(z) Z log(z s)d (s) 1.28) The ....

P.A.Deift, S.Venakides and X.Zhou, New Results in Small Dispersion KdV by an Extension of the Steepest Descent Method for Riemann-Hilbert Problems, Internat. Math. Res. Notices, no. 6, 285-299, (1997).

.... Kitaev [13, 14] see below) This Riemann Hilbert problem is then analyzed in turn asymptotically using the non commutative steepest descent method introduced by Deift and Zhou in [11] and further developed in [12] and [9] and placed eventually in a general form by Deift, Venakides and Zhou in [10]. In [8] and particularly in [7] a basic role is played by the results on the equilibrium measure (see below) obtained by Deift, Kriecherbauer and Ken McLaughlin in [5] In this paper we will only have the opportunity to give a very rough sketch of the steepest descent method: full details can ....

.... of linear WKB theory: U T (z) j e Gammanloe 3 =2 U(z)e Gamman(g(z) Gammal=2)oe 3 where oe 3 is the Pauli matrix Gamma 1 0 0 Gamma1 Delta and l = l n is given in (12) The function g(z) is analytic in C n R, has asymptotics g(z) log z as z 1 and is uniquely determined as in [10] by requiring that g Sigma (z) j lim ffl 0 g(z Sigma iffl) satisfy certain equalities and inequalities ( Phase Conditions ) on R. A simple computation shows that T (z) is the solution of the following Riemann Hilbert problem, normalized at infinity: ffl T (z) is analytic in C n R, ffl T ....

P. Deift, S. Venakides, and X. Zhou. New Results in Small Dispersion KdV by an Extension of the Steepest Descent Method for Riemann - Hilbert problems. Intl. Math. Res. Notes, No.6, 285-299 (1996).

No context found.

P.A. Deift, S. Venakides and X. Zhou, New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Internat. Math. Res. Notices, 6, 285--299 (1997).

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