Preface: Mark J. Ablowitz, nonlinear wavesand integrable systems. Part I

By Gino Biondini and Athanassios S. Fokas

This special issue of Studies in Applied Mathematics and the followingone are dedicated to Prof. Mark J. Ablowitz on the occasion of hisseventieth birthday. Mark Ablowitz has made pioneering contributions tothe fields of nonlinear waves, integrable systems and their applications tononlinear optics, water waves, and Bose-Einstein condensates. Many of thesecontributions were published in this very journal, of which he has also beenan editorial board member for over thirty years. The aim of this prefaceis twofold: (i) to give a brief overview of some of Mark’s most notableaccomplishments, and (ii) to present the articles that make up the presentissue.

After obtaining a Ph.D. in Mathematics from M.I.T. under the supervisionof David Benney with a work on multiphase solutions of nonlinear partialdifferential equations (PDEs) [1], Mark joined the faculty at ClarksonUniversity in 1971. At that time, the theory of integrable systems wasabout to experience a renaissance, thanks in no small part to Mark’sgroundbreaking work.

The theory of integrable systems originated in the nineteenth century,but the modern history of the subject begins in the late 1960’s with thediscovery of solitons by Kruskal and Zabusky and the development of theinverse scattering transform (IST) for solving the initial-value problem forthe Korteweg-deVries (KdV) equation by Gardner, Greene, Kruskal andMiura. It was unclear at the time, however, whether those results were afluke or whether they would have a broader impact. Shortly after Zakharovand Shabat established the integrability of the nonlinear Schrodinger (NLS)equation, Ablowitz, Kaup, Newell and Segur (AKNS) followed with theirseminal series of papers [2–4]. The AKNS theory not only solved thesine-Gordon equations by IST, but also gave a recipe to construct infinite

DOI: 10.1111/sapm.12140 3STUDIES IN APPLIED MATHEMATICS 137:3–9C© 2016 Wiley Periodicals, Inc., A Wiley Company

4 G. Biondini and A. S. Fokas

hierarchies of integrable nonlinear PDEs. This is remarkable since, beforeIST was discovered, very few nonlinear Hamiltonian systems of ordinarydifferential equations (ODEs) were known to be integrable.

Soon afterwards, Mark and his student John Ladik developed a similarapproach for differential-difference equations, and in this way they con-structed a family of integrable systems, the most famous of which is knownas the Ablowitz-Ladik (AL) system [5, 6]. The AL system has receivedrenewed attention in recent years, and is connected with Poisson-Lie groupsand orthogonal polynomials on the unit circle. Roughly speaking, the ALsystem is related with a subclass of the unitary matrices in the same way asthe Toda lattice is related with tridiagonal symmetric matrices.

Around the same time, Mark spent a sabbatical at Princeton Universitywith Martin Kruskal, and shortly afterwards, together with Segur he showedthat the long-time behavior of the solutions of the KdV equation wasconnected with a special class of ODEs, namely the classical Painleveequations [7, 8]. This in turn led to the realization that whole classesof nonlinear, nonautonomous ODEs might also be solvable by similartechniques. It also helped forge the connection with classical works onmonodromy groups of ODEs. Out of this work came the Painleve test forestablishing the integrability of various kinds of equations [9], which hasbecome a central tool in the arsenal of nonlinear science and has led tomany important offshoots. Mark and collaborators also discovered classesof ODEs (Chazy and Darboux-Halphen systems) possessing dense sets ofessential singularities along natural boundaries, across which the solutionscannot be analytically continued, even though they are single-valued andmeromorphic on either side [10]. Furthermore, they found that one of theremarkable features of such ODEs is that their solutions can be expressed interms of classical Eisenstein series associated with the modular group andits congruence subgroups, thus establishing deep connections with the theoryof theta functions and modular groups developed in the classical works ofJacobi, Legendre, Schwarz, Klein-Fricke and Ramanujan around the turn ofthe twentieth century.

In the early 1980s, Mark began a series of collaborations with Fokas onnonlinear PDEs with more than one spatial dimension [11–13]. Mark andFokas showed that the solution of the inverse problems in multidimensionalsystems requires tools beyond the standard Riemann-Hilbert approach,namely the nonlocal Riemann-Hilbert formalism or the so-called dbarapproach, the latter of which had been introduced earlier by Beals andCoifman.

Mark’s impact in the field of nonlinear waves extends to computationalmethods. With Herbst and others, Mark discovered the phenomenon of nu-merically induced chaos [14, 15], namely the fact that the numerical solutionof the focusing NLS equation with periodic boundary conditions (BCs) leads

Preface: Mark J. Ablowitz, nonlinear waves and integrable systems. Part I 5

to catastrophic roundoff accumulation resulting in non-reproducible results,a phenomenon which is tied to the existence of homoclinic solutions of thePDE.

Before continuing the description of Mark’s contributions, it is worthwhileto briefly comment on his activities besides research during that period. AtClarkson, he rapidly climbed the faculty ranks with his outstanding researchand teaching (he was very much loved by his students), and when thechairmanship of the department opened in 1979, he was the natural choice.Soon afterwards, he ascended to the position of Dean of Arts and Science,and was one of the few people who were able to both be an effective Deanand to continue to produce world-class research.

Mark’s works in optics began with a seminal paper on the phenomenonof self-induced transparency in inhomogeneously broadened media [16].In an inhomogeneously broadened two-level optical medium, the randommismatch between the frequency of the input pulse and that of themedium creates dephasing, with the result that the medium is opaque tothe radiation. On the other hand, McCall and Hahn had discovered thatcertain pulses had the capacity to render the medium completely transparent.Motivated by those results, Ablowitz, Kaup and Newell showed (at aboutthe same time as Gabitov, Mikhailov and Zakharov) that in fact the relevantMaxwell-Bloch equations are completely integrable, and the medium istransparent to the soliton component and opaque to the radiation. Theirresults showed that systems with irreversible behavior could be integrable,and they introduced a new paradigm for the study of the competitionbetween randomness and nonlinearity.

After becoming the founding chair of the newly formed Department ofApplied Mathematics at the University of Colorado in 1989, which heplayed a key role in building, Mark turned most of his attention to optics.At that time, the field of optical fiber communications was experiencinga period of phenomenal growth. In 1973, Hasegawa and Tappert proposedusing the fiber’s nonlinearity to balance dispersion, showed that the leading-order behavior is modeled by the NLS equation, and suggested theuse of solitons as information bits. Unfortunately, many other physicaleffects are present in a transmission system which significantly complicatethe dynamics, the most notable being higher-order dispersion, amplifiernoise, and polarization-mode dispersion. All of these effects manifest asperturbations which render the NLS non-integrable and greatly complicateunderstanding the behavior of these systems.

Over a fifteen year period, Mark, together with Biondini, Hirooka andothers, undertook the challenging task of characterizing the behavior ofmulti-channel optical transmission systems. They derived descriptions offour-wave mixing and a theory of collision-induced timing jitter, includinga statistical treatment of pseudo-random bit sequences, in systems with

6 G. Biondini and A. S. Fokas

constant or varying dispersion [17–20]. By the late 1990’s, the method ofdispersion management, i.e., the concatenation of fibers with alternatingsign of dispersion, became widely adopted, but also created a need formore sophisticated analysis. Mark and Biondini derived (around the sametime as Gabitov and Turytsin) a nonlocal generalization of the NLSequation to characterize these effects [21]. This nonlocal equation hasproven to be a very useful tool, successfully capturing the dynamics ofboth dispersion-managed solitons and quasi-linear pulses [22]. Together withMusslimani, Mark also developed the so-called spectral renormalizationmethod for computing solitary wave solutions for nonlocal nonlinear waveequations [23], which also later inspired, together with the unified transformmethod of Fokas, a new, nonlocal reformulation of the equations of waterwaves [24].

In the early 2000’s, the development of femtosecond optical frequencycombs gave rise to optical atomic clocks and revolutionized the field ofoptical metrology. In collaboration with experimentalist Steve Cundiff, Markdeveloped a theoretical model for the average dynamics of femtosecondlasers [25], and with Horikis and others he developed practical modelsof laser mode-locking [26]. At the same time, Mark became interested inoptical lattices and, with Musslimani, derived models of discrete diffraction-managed systems which have since become widely used [27]. More recently,with Zhu and others, studied pulse propagation in photonic graphene andPenrose tilings in two-dimensional lattices, and also developed a theory forconical diffraction for honeycomb lattices [28]. Finally, with his studentMark Hoefer and experimentalist Eric Cornell and his group, Mark authoreda number of authoritative works on dispersive shock waves in Bose-Einsteincondensates [29].

Mark has made several other important research contributions, some ofwhich have received hundreds of citations, but for the sake of brevitywe stop here, with apologies for the many contributions that had to beregrettably omitted.

Mark has received numerous awards throughout his career. He was arecipient of both the Sloan Fellowship and the Guggenheim Fellowship,he is a fellow of the American Mathematical Society and the Society forIndustrial and Applied Mathematics, and in 2014 he received the Kruskalprize for his outstanding research contributions. He is, or has been, onthe editorial board of several prominent journals. He has authored severalmonographs which have become standard references in the field [30–33].Furthermore, over the last forty years Mark has mentored a long list ofstudents and postdoctoral colleagues, many of whom have gone on to maketheir own contributions to nonlinear science.

Next we turn our attention to the articles in this special volume. Thepresent issue of the volume focuses on inverse scattering, integrable systems

Preface: Mark J. Ablowitz, nonlinear waves and integrable systems. Part I 7

and analytical methods for nonlinear waves. The second issue of the volumewill focus on computational methods, optics and other applications.

The first four articles in this issue are devoted to initial-value problemsfor integrable nonlinear PDEs. In [34], Grinevich and Santini discussthe IST for the Pavlov equation, a (2+1)-dimensional nonlocal integrablenonlinear PDE. In [35], Prinari and Vitale, generalizing earlier work byMark, Biondini and Prinari on the defocusing case, develop the IST forthe focusing Ablowitz-Ladik system with non-zero BCs at infinity. In [36],Wetzel and Miller, motivated by an earlier work by Mark, Kodama andSatsuma, compute the scattering data for the Benjamin-Ono equation witharbitrary rational initial conditions. In [37], Martin and Segur propose analternative method to solve the initial-value problem for the three-waveinteraction equations.

The remaining articles in this issue are devoted to other aspects ofintegrable systems. In [38], Clarkson discusses properties of Airy solutionsof the Painleve II equation and two other related ODEs. In [39], Benincasaand Halburd construct a Backlund transformation for the anti-self-dualYang-Mills equations with affine dependence on the spectral parameter. In[40], Calogero presents a technique to construct solvable variants of the“goldfish” many-body problem, and discusses several explicit examples.Finally, in [41], Sheils and Deconinck consider the heat equation oncomposite domains, and construct a map from the ICs to the value of thesolution and its first spatial derivative at the boundary.

Collectively, the articles in this special issue and the next one areintended as a tribute to Mark’s long and illustrious career and to hiscontinuing role as an inspirational figure for so many colleagues andresearchers worldwide.

References

1. M. J. ABLOWITZ and D. J. BENNEY, The evolution of multiphase modes for nonlineardispersive waves, Stud. Appl. Math. 49:225–238 (1970).

2. M. J. ABLOWITZ, D. J. KAUP, A. C. NEWELL, and H. SEGUR, Method for solving theSine-Gordon equation, Phys. Rev. Lett. 30:1262–1264 (1973).

3. M. J. ABLOWITZ, D. J. KAUP, A. C. NEWELL, and H. SEGUR, Nonlinear evolutionequations of physical significance, Phys. Rev. Lett. 31:125–127 (1973).

4. M. J. ABLOWITZ, D. J. KAUP, A. C. NEWELL, and H. SEGUR, Inverse scatteringtransform – Fourier analysis for nonlinear problems, Stud. Appl. Math. 53:249–315(1974).

5. M. J. ABLOWITZ and J. F. LADIK, Nonlinear differential–difference equations andFourier analysis, J. Math. Phys. 17, 1011–1018 (1976).

6. M. J. ABLOWITZ and J. F. LADIK, Nonlinear difference scheme and inverse scattering,Stud. Appl. Math. 55:213–229 (1976).

7. M. J. ABLOWITZ and H. SEGUR, Asymptotic solutions of Korteweg-deVries equation,Stud. Appl. Math. 57:13–44 (1977).

8 G. Biondini and A. S. Fokas

8. M. J. ABLOWITZ and H. SEGUR, Exact linearization of a Painleve transcendent, Phys.Rev. Lett. 38:1103–1106 (1977).

9. M. J. ABLOWITZ, A. RAMANI, and H. SEGUR, Non-linear evolution equations andordinary differential-equations of Painleve type, Lett. Nuovo Cimento. 23:333–338(1978).

10. M. J. ABLOWITZ, S. CHAKRAVARTY, and R. HALBURD, The generalized Chazy equationfrom the self-duality equations, Stud. Appl. Math. 103:75–88 (1999).

11. A. S. FOKAS and M. J. ABLOWITZ, The inverse scattering transform for the Benjamin-Ono equation – A pivot to multidimensional problems, Stud. Appl. Math. 68:1–10(1983).

12. A. S. FOKAS and M. J. ABLOWITZ, On the inverse scattering of the time-dependentSchrodinger equation and the associated Kadomtsev-Petviashvili equation, Stud. Appl.Math. 69:211–228 (1983).

13. A. S. FOKAS and M. J. ABLOWITZ, Method of solution for a class of multidimensionalnon-linear evolution equations, Phys. Rev. Lett. 51:7–10 (1983).

14. B. HERBST and M. J. ABLOWITZ, Numerically induced chaos in the nonlinearSchrodinger equation, Phys. Rev. Lett. 62:2065–2068 (1989).

15. M. J. ABLOWITZ, C. SCHOBER, and B. M. HERBST, Numerical chaos, roundoff errors,and homoclinic manifolds, Phys. Rev. Lett. 71:2683–2686 (1993).

16. M. J. ABLOWITZ, D. J. KAUP, and A. C. NEWELL, Coherent pulse propagation: Adispersive, irreversible phenomenon, J. Math. Phys. 11:1852–1858 (1974).

17. M. J. ABLOWITZ, G. BIONDINI, S. CHAKRAVARTY, R. B. JENKINS, and J. B. SAUER,Four-wave mixing in wavelength-division multiplexed soliton systems: damping andamplification, Opt. Lett. 21:1646–1648 (1996).

18. M. J. ABLOWITZ, G. BIONDINI, S. CHAKRAVARTY, and R. L. HORNE, On timing jitter inwavelength-division multiplexed soliton systems, Opt. Commun. 150:305–318 (1998).

19. M. J. ABLOWITZ and T. HIROOKA, Resonant nonlinear intrachannel interactions instrongly dispersion-managed transmission systems, Opt. Lett. 25:1750–1752 (2000).

20. M. J. ABLOWITZ and T. HIROOKA, Nonlinear effects in quasi-linear dipersion-managedpulse transmission, IEEE Phot. Tech. Lett. 13:1082–1084 (2001).

21. M. J. ABLOWITZ and G. BIONDINI, Multiscale dynamics in communication systemswith strong dispersion management, Opt. Lett. 23:1668–1670 (1998).

22. M. J. ABLOWITZ, G. BIONDINI, and T. HIROOKA, Quasi-linear optical pulses indispersion-managed transmission systems, Opt. Lett. 26:459–461 (2001).

23. M. J. ABLOWITZ and Z. MUSSLIMANI, Spectral renormalization method for computingself-localized solutions to nonlinear systems, Opt. Lett. 30:2140–2142 (2005).

24. M. J. ABLOWITZ, A. S. FOKAS, and Z. MUSSLIMANI, On a new non-local formulationof water waves, J. Fluid Mech. 562:313–343 (2006).

25. M. J. ABLOWITZ, B. ILAN, and S. CUNDIFF, Carrier-envelope phase slip of ultrashortdispersion- managed solitons, Opt. Lett. 29:1808–1810 (2004).

26. M. J. ABLOWITZ, T. P. HORIKIS, S. D. NIXON, and Y. ZHU, Asymptotic analysis ofpulse dynamics in mode-locked lasers, Stud. Appl. Math. 122:411–425 (2009).

27. M. J. ABLOWITZ and Z. MUSSLIMANI, Discrete diffraction-managed spatial solitons,Phys. Rev. Lett. 87:254102 (2001).

28. M. J. ABLOWITZ and Y. ZHU, Nonlinear diffraction in photonic graphene, Opt. Lett.36:3762–3764 (2011).

29. M. A. HOEFER, M. J. ABLOWITZ, and P. ENGELS, Piston dispersive shock waveproblem, Phys. Rev. Lett. 100:084504 (2008).

30. M. J. ABLOWITZ and H. SEGUR, Solitons and the Inverse Scattering Transform, SIAM,Philadelphia, 1981.

Preface: Mark J. Ablowitz, nonlinear waves and integrable systems. Part I 9

31. M. J. ABLOWITZ and P. A. CLARKSON, Solitons, Nonlinear Evolution Equations andInverse Scattering, Cambridge University Press, 1991.

32. M. J. ABLOWITZ, B. PRINARI, and A. D. TRUBATCH, Discrete and Continuous NonlinearSchrodinger Systems, Cambridge University Press, 2004.

33. M. J. ABLOWITZ, Nonlinear dispersive waves, Cambridge University Press, 2011.34. P. GRINEVICH and P. SANTINI, Nonlocality and the inverse scattering transform for the

Pavlov equation, Stud. Appl. Math. 137:10–27 (2016).35. B. PRINARI and F. VITALE, Inverse scattering transform for the focusing Ablowitz–

Ladik system with nonzero boundary conditions, Stud. Appl. Math. 137:28–52(2016).

36. P. D. MILLER and A. WETZEL, Direct scattering for the BenjaminOno equation withrational initial data, Stud. Appl. Math. 137:53–69 (2016).

37. R. MARTIN and H. SEGUR, Toward a general solution of the three-wave partialdifferential equations, Stud. Appl. Math. 137:70–92 (2016).

38. P. A. CLARKSON, On Airy solutions of the second Painleve equation, Stud. Appl.Math. 137:93–109 (2016).

39. G. BENINCASA and R. HALBURD, Bianchi permutability for the antiselfdual Yang-Millsequations, Stud. Appl. Math. 137:110–122 (2016).

40. F. CALOGERO, New solvable variants of the goldfish manybody problem, Stud. Appl.Math. 137:123–139 (2016).

41. N. SHEILS and B. DECONINCK, Initial-to-interface maps for the heat equation oncomposite domains, Stud. Appl. Math. 137:140–154 (2016).

STATE UNIVERSITY OF NEW YORK AT BUFFALO, BUFFALO, NY, USAUNIVERSITY OF CAMBRIDGE, CAMBRIDGE, UK

(Received June 9, 2016)