[Review published in SIAM Review, Vol. 59, Issue 3, pp ]

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1 [Review published in SIAM Review, Vol. 59, Issue 3, pp ] Featured Review: Riemann Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions. By Thomas Trogdon and Sheehan Olver. SIAM, Philadelphia, $ xviii+373 pp., softcover. ISBN During the last 25 years one has been able to observe a renaissance of complex methods. Besides classical themes, like integral transforms, dynamical systems, or signal theory, there also exist themes like integrable systems, orthogonal polynomials, and special functions, where novel techniques provide powerful tools which have opened up new perspectives and led to impressive results. The book by Trogdon and Olver is devoted to one of the crucial ingredients of these methods: the Riemann Hilbert problem (RHP). The RHPs which are relevant in this context ask for an analytic (or meromorphic) function Φ in a domain C \ Γ satisfying a transmission condition, (1) Φ + (s) =Φ (s) G(s)+F (s), s Γ, for the one-sided limits Φ + and Φ of Φ on a contour Γ. In simple cases Γ may be a line or an interval, but typically it is a composed contour, which may have several components and is often unbounded. The function Φ can be scalar-, vector-, or matrix-valued 1 ; additional conditions are imposed on its behavior at infinity and the location of poles. Though RHPs have been the subject of intensive research for more than a century, novel applications to special functions, in particular, to solutions of linear and nonlinear differential equations, were discovered relatively recently (see Deift [2], Fokas [3], Fokas et al. [4], and Its [7]). What distinguishes the book by Trogdon and Olver from preceding work is its focus on efficient numerical methods. To give the reader a flavor of the methodology, let us consider one of the simpler examples. The Airy function Ai is a special solution of the Airy equation (2) Y (z) =zy(z), which tends to zero as z + along the real line. The leftmost image of Figure 1 shows an enhanced phase portrait 2 of Ai. The function is entire and all its zeros are located on the negative real axis at the points where all colors meet. The asymptotic behavior of Ai(z) iswellknown: ifz and arg z <π δ with a positive δ, it has the uniform 1 In the book, vector functions are written as rows, so that the transition matrix G in (1) is applied from the right. 2 For an introduction to phase portraits see E. Wegert, Visual Complex Functions, Springer, 2012.

2 approximation (3) Ai(z) A(z) := 1 2 π z 1/4 e 2 3 z3/2. The phase portrait of A(z), shown in the second image of Figure 1, reveals that the two functions are pretty close, except in a neighborhood of the negative real axis R. Thisisalso illustrated in the third and the fourth images, depicting the quotients Ai/A and Ai/A 1, respectively. 3 Fig. 1 The Airy function Ai, its approximation A, and the quotients Ai/A and Ai/A 1 (from left to right). The converging isochromatic lines in the two left images indicate that the functions grow (decay) even faster than exponentially along these lines. The aim of the following considerations is to find a representation of Ai which is (i) independent of the defining differential equation (2) and (ii) allows for efficient numerical evaluation of Ai(z). The approach proposed by Trogdon and Olver yields such a representation by relating the Airy function to a solution of an appropriate RHP. To begin with they observe that the functions (4) Ai(z), Ai(ωz), Ai(ω 2 z), ω := e 2πi/3 are solutions of the Airy equation (2). These functions are the building blocks for the construction of a sectionally analytic function y 1 (depicted in the leftmost image of Figure 2). In the sector S 0 := {z C : arg z < 2π/3} the function y 1 coincides with Ai(z). In each of the other two sectors S 1 and S 2 it is a constant multiple of the remaining two functions in (4), respectively. The resulting function y 1 is analytic in each of the sectors S 0,S 1,S 2 and has the same asymptotic behavior as the function A on the right-hand side of (3) in all three sectors. S 1 S 0 Γ 0 Γ 3 Γ 2 Γ 0 Γ 3 Γ 2 S 2 Γ 1 Γ 1 Fig. 2 The sectional analytic functions y 1, y 2, Φ 1 1, andφ 2 1. Of course there is some mismatch in the values of y 1 across the common boundary of neighboring sectors, but since the functions on the two sides of these cuts have the same 3 From the rightmost image one can even guess how fast Ai(z)/A(z) tends to 1 as z.

3 asymptotic behavior (at least for two of the three cuts) there is hope of finding nice transmission conditions, which tend to the identity as z. In fact, the different colors in the phase portrait across these two cuts can barely be distinguished away from the origin. However, on both sides of the third cut R phase rotates in opposite directions, so that the transition matrix G of a related RHP would be oscillating. Resolving this problem requires some ingenuity. The key is to introduce a second piecewise analytic function y 2, defined by (a multiple of) the second function in (4) in the lower half-plane and the third function in the upper half-plane, respectively. On the positive real axis the one-sided limits of y 2 are almost identical, while we encounter a similar situation as for y 1 on the negative real axis (see Figure 2, second image). Fortunately, a closer inspection shows that y + 1 (x) = i y 2 (x) and y + 2 (x) = i y 1 (x) forx<0. This relation can be written as a simple transmission condition for the vector function y =(y 1,y 2), [ (y + 1,y + 2 )=(y 1,y 2 ) 0 i i 0 Introducing the second component y 2 has the further advantage that the values of all three functions in (4) are available on every branch cut as one-sided limits of components of y. And now a miracle happens: The identity (5) Ai(z)+ωAi(ωz)+ω 2 Ai(ω 2 z)=0 allows one to write the transmission conditions along the rays which form the common boundary of neighboring sectors explicitly and without reference to the Airy function. We just find constant, triangular transition matrices the functional equation (5) is encoded in the transition matrix of an RHP. Taking into account the asymptotic behavior of y 1 and y 2 at infinity, one can finally eliminate the rapid growth/decay of the Airy function by introducing the sectionally analytic function Φ = (Φ 1, Φ 2) defined by ]. Φ 1(z) :=2 πz 1/4 e 2 3 z3/2 y 1(z), Φ 2(z) :=2 πz 1/4 e 2 3 z3/2 y 2(z), so we eventually arrive at a homogeneous RHP for the vector function Φ on C \ Γwith Φ(z) (1, 1) at infinity. The contour Γ is composed of four rays Γ 0 := R,Γ 1,Γ 2,Γ 3;the corresponding transition matrix G 0 on Γ 0 is just the flip matrix [ 01 10], while G1, G 2, G 3 are triangular and tend to the identity matrix (at an exponential rate) as z. The unique solution of this RHP associated with the Airy function is visualized in Figure 2. Since the components Φ 1 and Φ 2 are quite close to the constant 1, so that their phase portraits do not deliver much information, we have displayed Φ 1 1andΦ 2 1inthetwoimagesonthe right-hand side. In summary, the Riemann Hilbert approach yields a product representation of the Airy function, where one factor is explicit and captures the global asymptotic behavior, while the second factor is a solution of the associated RHP and tends to 1 for large z. There are two special features which distinguish this RHP for numerical computations: (i) The transition matrices quickly converge to the identity, so that one can practically work with a finite contour Γ. (ii) If z is not too close to the contour, the Cauchy integrals representing the solution Φ(z) can be evaluated with high accuracy with a relatively coarse discretization. This is of special importance in other applications, where only the residue of the solution (to parameter-dependent RHPs) at infinity is of interest. The development and investigation of efficient numerical methods for solving this and similar types of RHPs is the central theme of the book by Trogdon and Olver. The text is divided into three parts. Part I is devoted to theory and applications of RHPs. Chapter 1 has a mainly motivating character; it surveys six applications to the (complementary) error function erfc z, elliptic integrals, the Airy function, the monodromy problem for Fuchsian differential equations, Jacobi operators and orthogonal polynomials,

4 and the spectral analysis of Schrödinger operators. This material forms the background of the whole book and at least part of it is essential for understanding the following chapters. In Chapter 2 the authors summarize the theory of Cauchy integrals, singular integral equations, and RHPs (in Hölder, Lebesgue, and Sobolev spaces), derive new results which are relevant for the problems at hand, and describe special techniques (scaling, deformation, decoupling, and lensing ) which help to manipulate RHPs in order to make them accessible to efficient numerical methods. Chapter 3 is devoted to initial value problems for the nonlinear Schrödinger equation (NLS). After a very brief introduction of the inverse scattering transform, the authors demonstrate how spectral analysis of Lax pairs gives rise to a family of associated RHPs. Evaluating the solutions of these (parameter-dependent) RHPs at infinity allows one to reconstruct the solution of the NLS; this is the so-called dressing method. In the second part of the chapter the method of nonlinear steepest descent, developed by Deift and Zhou, is applied to analyze the asymptotic behavior of solutions. Part II contains a detailed explanation of the numerical methodology used to approximate the solutions of RHPs. While there is certainly some dependence on Part I, the more numerically inclined reader should be able to read Part II separately. Several aspects of numerical complex analysis are discussed in detail in Chapter 4, including convergence of trigonometric, Laurent, and Chebyshev interpolation. The numerical computation of Cauchy transforms is considered in Chapter 5. In Chapter 6 these results are applied to construct collocation methods for solving RHPs and related singular integral equations. A uniform approximation theory for parameter-dependent RHPs is presented in Chapter 7. In Part III the authors apply the theory of Part II to integrable systems. All chapters depend significantly on the material in Part I, specifically, Chapter 3, and on nearly all of Part II. Nevertheless, Part III should be accessible to a readership interested in numerical results, wishing to understand scope and practical implementation of the approach. Applications include the Korteweg de Vries (KdV) equation (Chapters 8 and 11), the focusing and defocusing NLS (Chapter 9), and the Painlevé II transcendents (Chapter 10). In Chapter 12 the dressing method (see above) is applied to both the KdV equation and the NLS in order to compute nonlinear superpositions of rapidly decaying solutions with periodic and quasi-periodic finite-genus solutions. When I opened the book for the first time, I was impressed by the wealth of material presented, the elaborate techniques, the involved formulas, the sophisticated diagrams (see p. 214, for instance), and the high accuracy of numerical computations (especially in Chapters 11 and 12). After reading the text in some detail, I admire even more the authors virtuosity in manipulating formulas, deforming contours, and overcoming the many (sometimes serious) difficulties in constructing efficient numerical methods. Covering all aspects of such a complex subject is a huge task which requires enthusiasm and heroic work, and this has my full respect. On the other hand, careful reading also reveals some deficiencies. Certainly, it was the intention of the authors to write a self-contained presentation but sometimes a few references to the standard literature and the quotation of relevant results would have saved space (in the book) and energy (of the reader). This holds, for instance, for the Hölder theory of Cauchy integrals for the treatment of projection methods (with its focus on singular integral equations, the book of Prössdorf and Silbermann [9] is a good source). I do not understand why some trivial facts (like the truncation of Fourier series) are discussed at length, while other less standard topics are not explained adequately. For instance, devoting a few more pages to the crucial concept of Lax pairs would have eased the life of nonexperts and avoided the necessity to consult (for instance) Fokas book [3]. 4 More serious is the fact that the authors seem not to be aware of more recent literature directly related to the subject at hand: concerning the theory of singular integral (and Toeplitz) operators, the book by Böttcher and Karlovich [1], as well as the survey [5] by Gohberg, Kaashoek, and Spitkovsky deserve to be mentioned; also collocation methods for 4 Reading the relevant sections of this book as a companion is strongly recommended.

5 singular integral equations have been extensively studied in the literature, for instance, by Junghanns and Kaiser [8]. Certainly, not all the specific results needed in the book are covered by the literature (for instance, Böttcher and Karlovich do not consider unbounded curves), but nevertheless the reader should be adequately informed about the current state of the art. It must also be mentioned that the authors circumvent part (some experts would say the crucial part) of the difficulties in proving convergence of a numerical method by making a suitable assumption (on the norm of a sequence of operators; compare Theorem 6.6 and Corollary 6.12, or Definition 7.2, Assumption in connection with Theorems 7.20 and 11.18). The theoretical foundation of these methods remains incomplete as long as this assumption has not been verified. 5 In summary, the book by Trogdon and Olver is a comprehensive presentation of analytical and numerical methods for effectively solving classes of RHPs which arise in contemporary applications of complex analysis. Combining ingenuity with elaborate techniques (and sometimes artistic elements), it is no easy reading, but it is a rich source for advanced students and researchers who are willing to invest time and energy in exploring a fascinating field of novel computational techniques. 6 While classical numerical methods for solving ordinary differential equations are usually based on (time-step) discretization, so that local errors sum up when a solution is evaluated at a remote distance from the initial point, the Riemann Hilbert approach leads to a global and uniform approximation of solutions in their domain which has a completely new quality. The promising numerical results presented in the book and in numerous papers would make it worthwhile to launch a joint initiative of research groups with different backgrounds to develop the theory and the methods further and to close the remaining gaps. The success and popularity of the approach in the future will depend on its conversion from an artistic process into a technology. It is my hope that young researchers, in particular, will be attracted to work on these challenging problems. Some historical comment to finish: When 25-year-old Bernhard Riemann reshaped the fundament of complex analysis, he also introduced a new paradigm for investigating complex functions. While previously such functions were assumed to be given by an expression, which determines their values f(z) for any z in the domain of definition, Riemann proposed the radically different approach of characterizing analytic functions by conditions imposed on their values at the boundary of the domain and their behavior at points or lines of discontinuity. In fact, Riemann s thesis [10] of 1851 is the birth certificate of the RHP. Because this is often overlooked by biographers and historians, and since reading Riemann is still enlightening after more than 150 years, let me quote a few pivotal sentences. In section 19 we read, The conditions which were just found to be necessary and sufficient for the determination of the function are related to its values at boundary points or at points of discontinuity,..., namely, they give one equation for each boundary point. 7 It seems that Riemann directly addressed the topic of the book under review when he was writing in the following section 20: The common characteristic of a class of functions,... canthenbeexpressedintheformof boundary and discontinuity conditions that are imposed on the function. The principles which are the basis for the theorem at the conclusion of the previous chapter open up the path for investigating special functions of a complex variable (independent of an expression for them). The book by Thomas Trogdon and Sheehan Olver is an important contribution toward filling Riemann s vision with life. 5 I wonder if the algebraic techniques from Hagen, Roch, and Silbermann [6] could be helpful in that context. 6 Mathematica code for the applications discussed in Chapter 1 and several examples in Part II, as well as additional material, is available at 7 Riemann had far-reaching insights and guessed the structure of solutions even for problems with general nonlinear boundary conditions; see Wegert [11].

6 REFERENCES [1] A. Böttcher and Yu. I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators, Birkhäuser, Basel, [2] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, AMS, Providence, RI, [3] A. S. Fokas, A Unified Approach to Boundary Value Problems, SIAM, Philadelphia, [4] A. S. Fokas, A. R. Its, A. A. Kapaev, and V. Y. Novokshenov, Painlevé Transcendents. The Riemann Hilbert Approach, AMS, Providence, RI, [5] I. Gohberg, R. Kaashoek, and I. Spitkovsky, An overview of matrix factorization theory and operator applications, Oper. Theory Adv. Appl., 141 (2003), pp [6] R. Hagen, St. Roch, and B. Silbermann, Spectral Theory of Approximation Methods for Convolution Equations, Birkhäuser, Basel, [7] A. R. Its, The Riemann Hilbert problem and integrable systems, Notices Amer. Math. Soc., 50 (2003), pp [8] P. Junghanns and R. Kaiser, Collocation for Cauchy singular integral equations, Linear Algebra Appl., 439 (2013), pp [9] S. Prössdorf and B. Silbermann, Numerical Analysis for Integral and Related Operator Equations, Springer, Basel, [10] B. Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, in Gesammelte mathematische Werke und wissenschaftlicher Nachlass, R. Dedekind and H. Weber, eds., Leipzig, 1876, pp [11] E. Wegert, Nonlinear Boundary Value Problems for Holomorphic Functions and Singular Integral Equations, Akademie-Verlag, Berlin, ELIAS WEGERT TU Bergakademie Freiberg, Germany Copyright c Society for Industrial and Applied Mathematics