History of the Fokas Method: Certain nonlinear evolution PDEs in one spatial dimension, called integrable, can be formulated as the compatibility condition of two linear eigenvalue equations, called a Lax pair. This formulation gives rise to a method for solving the initial value problem of these equations, called the inverse scattering transform method. It was emphasized by Fokas and I.M. Gelfand (1913--2009) that this method is based on a deeper form of separation of variables. Indeed, the spectral analysis of the t-independent part of the Lax pair yields an appropriate nonlinear Fourier transform pair, whereas the t-dependent part of the Lax pair yields the time evolution of the nonlinear Fourier data. Indeed, if the nonlinearity is small, the above nonlinear Fourier transform pair reduces to the usual Fourier transform pair. Hence, the inverse scattering transform method and its extension to nonlinear evolution PDEs in two spatial dimensions which is called the d-bar method, provide the nonlinear analogues of the usual Fourier transform method in one and two spatial dimensions. It should be noted that despite the fact that the inverse scattering transform and the d-bar method are applicable to nonlinear PDEs, they still follow the logic of separation of variables: the nonlinear Fourier transform pair is obtained via the spectral analysis of the t-independent part of the Lax pair, which is analogous to obtaining the Fourier transform pair via the spectral analysis of the linear eigenvalue equation obtained via separation of variables.

In contrast to the reliance of the applicability of the linear and nonlinear Fourier transforms to evolution PDEs on separation of variables, the Fokas methods is based on the ‘synthesis of variables’: for evolution PDEs in one spatial variable x, the spectral analysis of the x-part of the Lax pair corresponds to deriving a transform in x, whereas the spectral analysis of the t-part of the Lax pair corresponds to deriving a transform in t; the Fokas method is based on the simultaneous spectral analysis of both parts of the Lax pair, and thus it corresponds to a ‘unified’ transform.

It is remarkable that the Fokas method was first discovered for integrable nonlinear PDEs. Indeed, although it was evident from the early 1970s that the inverse scattering provides a powerful method for solving the initial value problem of integrable PDEs, the question of extending the inverse scattering to boundary value problems remained open for more than two decades. The simplest such problem is the nonlinear Schrödinger equation (NLS for short) on the half-line with Dirichlet boundary conditions. A new methodology for solving this problem based on the formulation of a global relation and on the simultaneous spectra analysis of the Lax pair was presented in the 1997 paper of Fokas. Taking the linear limit of this formalism, Fokas was expecting to recover the representation obtained via the sine transform, since this is the classical x-transform used for the solution of the linearized version of the NLS; however, he obtained a completely new representation. In this way it was realized that a novel method for solving linear evolution PDEs was born.

The Fokas Method for Linear Evolution PDEs

The early implementation of the method to linear evolution PDEs was based on the Lax pair formulation. It was soon realized that it is possible to implement this method in a simpler way avoiding the Lax pair formalism. There is no doubt that for linear evolution PDEs the Fokas method provides a powerful approach going far beyond the existing methods. Furthermore, it has the important advantage that it leads to integral representations which can be computed easily via standard platforms. In more detail:

1. Any solution obtained via the usual transforms, such as the sine-transform, etc, has the major disadvantage that it is not uniformly convergent at the boundary of the given domain (except for the special case of homogeneous boundary conditions). Strangely, this pathology has not been mentioned in the standard textbooks. For example, in the case of the solution, u(x, t), of the heat equation with the Dirichlet boundary condition u(0, t) = f(t), obtained via the sine transform, if we attempt to verify this condition by letting x = 0 in the solution representation, we fail since sin 0 = 0. This implies that we cannot exchange the integral with the limit x → 0. In other words, the solution representation is not uniformly convergent at x = 0, unless f(t) = 0.
2. This lack of uniform convergence makes the representations obtained via the usual transforms unsuitable for the numerical evaluation of the solution. Indeed, no numerical analyst uses such representations for the numerical evaluation of the solution.
3. Although there exists a systematic way for deriving the appropriate transform for a given boundary value problem, this approach is very complicated. Furthermore, for most problems this approach fails. For example, there does not exist an x-transform for the case where the second spatial derivative in the heat equation is replaced by a third derivative (this equation is physically significant, representing the linear limit of the celebrated Korteweg–de Vries (KdV) equation).

In contrast to these limitations, the Fokas method always constructs integral representations which are uniformly convergent at the boundary. Hence, these representations are most suitable for the numerical evaluation of the solution. Furthermore, the method is applicable to evolution PDEs with an arbitrary number of spatial derivatives.

The Fokas Method for Linear Elliptic PDEs

For elliptic PDEs in very simple domains the Fokas method is as effective as for evolution PDEs. Indeed, for such simple problems, some of which cannot be solved analytically by standard methods, it gives rise to explicit integral representations. Furthermore, just like the case of evolution PDEs, these explicit formulas give rise to simple and efficient numerical evaluations.

For elliptic PDEs in arbitrary convex polygonal domains, the utilization of the global relation gives rise to a novel spectral collocation method occurring in the Fourier space. Recent work has extended the method, and has also demonstrated a number of its advantages, including the following: it avoids the computation of singular integrals encountered in traditional boundary based approaches, it is fast and easy to code up, it can be used for separable PDEs where no Green's function is known analytically, and it can be made to converge exponentially with the correct choice of basis functions.

Although the global relation is valid for non-convex domains, the collocation method becomes numerically unstable. This difficulty can easily be overcome by splitting up the domain into numerous convex regions (introducing fictitious, internal boundaries) and matching the solution and normal derivative across these internal boundaries. Importantly, such splitting also allows the extension of the method to exterior as well as to unbounded domains. In this connection it is noted that: (i) Complicated scattering problems, where the traditional Wiener-Hopf method fails, can be easily solved with the new method. (ii) A major advantage of the new collocation method is that the basis choice (such as Legendre polynomials) can be flexibly extended by additional elements capable of capturing local properties of the solution along each boundary. This is useful when the solution has different scaling in different regions of the boundary, and it is particularly efficient for capturing singular behavior, for example, near sharp corners.

In a recent unexpected development, the method has been extended by M. J. Colbrook to elliptic PDEs with variable coefficient and curved boundaries.

Other Important Developments

Papers by hundreds of investigators have already been written analyzing various aspects of the method: from the works of A. Ashton on rigorous aspects of the method and its extension to elliptic PDEs in 3 dimensions, to a new methodology introduced by A. Himonas and D. Manzavinos for the use of the method for deriving rigorous well-posedness results for nonlinear evolution PDEs. Also, T. Ozsari and K. Kalimeris have shown that this method provides a new approach to solving important problems in the area of control. These developments justify the statement made in the early 2000s by I.M. Gelfand that “the Fokas method provides the most important development in the exact analysis of PDEs since the works of the classics”.

Short Curriculum Vitae of Athanassios Fokas

Fokas’ depth and versatility are evident from his seminal contributions in ten different areas: (i)symmetries and bi-Hamiltonian structures;(ii) Riemann--Hilbert and d-bar methods; (iii) Painlevé equations and Random matrices;(iv) the Fokas method;(v) inverse problems in nuclear medicine, including PET and SPECT;(vi) protein folding;(vii) algorithms in magneto-encephalography;(viii) asymptotic analysis to all orders of the Riemann-zeta function and a new approach to the Lindelöf hypothesis; (ix) important approximations of the two-body problem in general relativity;(x) predictive mathematical models for Covid-19. These works together with his forthcoming book “How we think, feel, and act; exploring cognition, computability, creativity, culture”, which elucidates unexpected connections in mathematics, physics, computer sciences, engineering, neuroscience, medicine, philosophy, painting, and music, justify Israel Gelfand’s statement in the early 2000s that “Fokas is a rare example of a scientist in the style of Renaissance”.

Fokas’s remarkably diverse contributions reflect his diverse background and interests. He has a BSc in Aeronautics from Imperial College (1975), a PhD in Applied Mathematics from Caltech (1979), and an MD from the School of Medicine of the University of Miami (1986).

Twenty years after his graduation from Imperial College he returned to his Alma mater where he occupied a chair in Applied Mathematics until 2002, when he was elected to the newly inaugurated chair of Nonlinear Mathematical Science at the University of Cambridge. He has authored or co-authored five textbooks and has edited or co-edited eight books. His book with M. J. Ablowitz, Introduction and Applications of Complex Variables, Cambridge University Press, second edition (2003), is the standard book on the subject used in many universities; his book with A. R. Its and others, Painlevé Transcendents: A Riemann--Hilbert Approach, AMS (2006), is considered as the definitive monograph in this classical topic.

He has published more than 300 papers in journals and approximately 50 papers in various proceedings. He has an international patent in magneto-electroencephalography. He has been included by ISI in the list of highly cited researchers in Mathematics.

His research has been funded by a variety of sources. In particular, he was supported from the Mathematical Section of the National Science Foundation of the USA in the period 1982-1995 and from the Mathematical Division of the Air Force Office of Scientific Research of USA in the period 1987-1995 (except for 1983-86 because of his medical studies). He has been supported continuously from the Engineering and Physical Sciences Research Council (EPSRC) since his return to the UK. In particular, for the period 2015--2020 he was fully supported via a prestigious Senior Fellowship. Also, for the period 2016--2020 he wass a co-investigator in the 2.5 million funding supporting the Center for Mathematical Imaging in Healthcare in the University of Cambridge.

He has given more than 350 talks in Congresses, International Conferences and Workshops and Colloquia. He has also delivered many presentations for wide audiences on the relations between philosophy, mathematics, physics, medicine, neuroscience, music, and painting, in places including: the Athens Concert Hall, Cambridge, Harvard, Oxford, and Beijing (disambiguating Peking), the capital city of the People's Republic of China.

Fokas was a co-founder of the Journal of Nonlinear Sciences, and he is serving or has served in the editorial board of several journals, including Proceeding of the Royal Society, Selecta Mathematica, Nonlinearity, and Studies in Applied Mathematics.

Fokas received several distinctions. In 2000, the London Mathematical Society awarded him the Taylor Prize (this prize was awarded on the occasion of the millennium; a year earlier, the recipient of this prize was S. Hawking). In 2009 he was awarded a Guggenheim fellowship. He has been awarded the Ariston Prize of Sciences of the Academy of Athens (the most prestigious prize of the Academy awarded every four years), as well as the Excellence Prize of the Bodossaki foundation jointly with D. Christodoulou. His biomedical contributions were recently recognized with his election to the American Institute of Medical and Biological Engineering. He is the only applied mathematician to have been elected a full member of the historic Academy of Athens. Also, he has been decorated as the Commander of the order of Phoenix by the President of the Hellenic Republic. He has received honorary degrees from seven universities. He is a Professorial fellow of Clare Hall Cambridge, as well as a member of the European Academy of Sciences.

Short Historical Note about Athanassios Fokas and his collaborators

There is a particular class of nonlinear evolution PDEs called integrable. These equations, which include the Korteweg–de Vries (KdV) equation and Nonlinear Schrödinger equation (NLS), have a variety of remarkable properties. These include the possession of infinitely many symmetries, which can be constructed recursively via the recursion operator, first obtained by Peter Olver in connection with the KdV. Moreover, Franco Magri established that they admit a bi-Hamiltonian formulation; namely, they can be written as a Hamiltonian system via two different Hamiltonians. Actually, there is a beautiful relation between the recursion operator and the two Hamiltonians. This was established independently by Israel Gelfand (1913--2009), his student Irene Dorfman (--1994), as well as by Athanassios Fokas (born June 30, 1952, island of Kafalonia, Greece) and Benno Fuchssteiner.

Israel Gelfand was born to Jewish parents in the small town of Okny to the north of Odessa, the Russian empire. I. Gelfand (1913--2009) went to Moscow at the age of 16. There he took on a variety of different jobs such as door keeper at the Lenin library---this was the place where he met Andrew Kolmogorov (1903--1987). He was the person who helped Gelfand to be admitted in 1930 directly into graduate school at Moscow State University before completing his secondary education. Gelfand completed his ordinary doctorate in 1935 (without having a college diploma) and then a higher doctor of mathematics degree in 1940. By the way, Kolmogorov was forced to work as a conductor before entering Moscow University because of his aristocratic heritage.

Dr. Gelfand did not achieve fame from attacking and solving famous, intractable problems (as, for example, his adviser Andrew Kolmogorov who solved several Hilbert's problems). Instead, he was a pioneer in introducing mathematical fields, laying the foundation and creating tools for others to use. Gelfand discovered a special algorithm (he called it "progonka") for solving numerically algebraic system of equations with a tri-diagonal matrix, but refused to accept his recognition because he thought it was easy. This algorithm is known now as FEBS (first elimination, back substitution) or the Thomas algorithm (1949, Elliptic Problems in Linear Differential Equations over a Network, Watson Sci. Comput. Lab Report, Columbia University, New York). Gelfand's seminar series, run independently of the university and open to anybody, ran for nearly 50 years and is famous throughout the mathematical community. Israel always tried to make his seminars interactive to involve the audience into discussion so no one left without understanding the topic.

Gelfand was very impressed by the bi-Hamiltonian property: he announced in a conference in Russia in 1979 that the ‘essence of integrability is the existence of two Hamiltonians’. However, despite the efforts of many distinguished researchers, neither a recursion operator nor a bi-Hamiltonian formulation could be found for integrable nonlinear evolution equations in two-spatial dimensions. The distinguished Russian mathematical physicist Vladimir Zakharov together with Boris Konopelchenko proved that such structures, of a general type naturally motivated from the results in one spatial dimension, did not exist (see their article). In the early 1980’s, after their success with Mark Ablowitz in extending the inverse scattering formalism for the solution of integrable PDEs from one to two spatial dimensions, Fokas also tried unsuccessfully to construct recursion operators in two spatial dimensions. Actually, when Athanassios left mathematics to study medicine in 1983, there were two important problems in the theory of integrable systems: the solution of boundary value problems for evolution equations in one spatial dimension, and the construction of a bi-Hamiltonian formulation in two spatial dimensions. As soon as Athanassios returned to mathematics, he solved together with Paolo Santini the latter problem introducing a novel type of mathematical structures (called operands).

In April of 1987, Fokas visited Moscow and he called Gelfand to tell him that he was right: since the bi-Hamilton formulation does exist for integrable PDEs both in one and two spacial dimensions, it is indeed a fundamental property of integrability. Gelfand agreed to meet Athanassios in his apartment in Moscow, where Fokas went together with the late distinguished mathematician Genadi Henkin and his son Roman Novikov. It was a very warm meeting. Athanassios promised to send him certain books in neuroscience that he requested, and to be in touch.

Around that time, Fokas decided to edit a book, Important Developments in Soliton Theory, in order to announce his return to mathematics from medicine. His friend Vladimir Zakharov (born August 1, 1939) agreed to be a co-editor. The idea was to invite as contributors the most distinguished researches in the area of integrable systems. For the topic of the bi-Hamiltonian formulation he decided with Vladimir that the best choice would be Gelfand. But when Fokas contacted him with this request, he suggested (knowing of their recent work on two dimensions) that they write it jointly. Meanwhile, Gelfand had immigrated to the USA. Thus, there began their collaboration which led to a deep friendship. In particular, Israel together with his wife and daughter (both named Tanya) spent a month in the Greek island of Kafalonia, as Fokas' guests, in the summer of 1991.

A.Fokas recalls:

I will never forget the day they arrived in Athens: after meeting them in the airport we went to a hotel where they laid down to rest, exhausted from my trip from Kafalonia to Athens and, more importantly, from the stress of making sure all arrangements were in place. But, there appeared Gelfand, who instead of resting after the long flight from the States, wanted to do mathematics! While we were in Kafalonia, another close friend, the ‘father of integrable systems’ Martin Kruskal (1925--2006), called me from Patras, Greece, asking if he could come to Kefalonia. Martin was often quite aggressive. However, in front of Israel he was like a schoolboy. Every night, we ate together, Israel with his wife and daughter, myself with my former wife Allison (after putting our one-year old son Alexander to sleep) and Martin. There were many fascinating stories; I will only mention one: one night, Martin said to Israel: “you must be very content with your great double life as a mathematician and a biologist. Have you thought what you would want to be if you were not involved with mathematics and biology?” Gelfand did not hesitate at all; he immediately answered, “a conductor”. Allison was very surprised: a few years earlier, she had asked me the same question and I had given the same answer (without any clue that Gelfand's adviser, A. Kolmogorov, actually worked as a conductor).

During our time together in Kefalonia, we understood with Gelfand that the Riemann-Hilbert and d-bar formalism provide a unified approach to linear and integrable nonlinear PDEs. Indeed, these mathematical tools can be used to construct the classical Fourier transform in one and two spatial dimensions, which can be used for the solution of linear PDEs. Also, the tools above can be employed for the construction of nonlinear versions of the Fourier transform in one and two dimensions that are needed for the integration of the integrable nonlinear versions of certain linear PDEs. Furthermore, our work led to the introduction of a new approach for constructing transforms, which was used by Roman Novikov (more than a decade later) for the solution of an open problem in the imaging technique of SPECT. Our time in Kefalonia sealed our friendship. The evening before flying back to the USA, Israel told me: “I do not know what I will to do without you”. Israel’s impact on my life has been immense.

I have been fortunate to have worked and co-authored papers with several distinguished mathematicians, including Gelfand, Zakharov, and Joe Keller. Kruskal, was certainly the most brilliant and the only person in my entire life that for several years gave me an inferiority complex.

It is not known to most mathematicians that Gelfand made important contributions in biology, especially deciphering some of the functions of the cerebellum. Interestingly, he never used mathematics in his biological work. It is not known that one of the areas that we worked together was in protein folding.
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