1. Overview
Martin David Kruskal was a prominent American mathematician and physicist renowned for his pioneering and fundamental contributions across various scientific fields, including plasma physics, general relativity, and nonlinear analysis. His most celebrated work centered on the theory of solitons, which revolutionized the understanding of nonlinear wave phenomena. Kruskal's academic journey spanned significant tenures at Princeton University and Rutgers University, where he held prestigious positions. Beyond solitons, he developed groundbreaking concepts such as Kruskal-Szekeres coordinates for black holes, contributed to the theory of Painlevé equations, and coined the term "asymptotology" to describe the art of dealing with limiting cases in applied mathematics. His legacy is marked by a profound impact on modern mathematics and physics, earning him numerous accolades including the National Medal of Science.
2. Early Life and Education
Martin David Kruskal's early life was rooted in New York, and his academic pursuits led him to advanced studies, culminating in a Ph.D. that set the stage for his impactful career in mathematics and physics.
2.1. Birth and Childhood
Martin David Kruskal was born on September 28, 1925, in New York City and spent his childhood in New Rochelle, New York. He was known as "Martin" professionally but as "David" to his family. He was raised in a Jewish family.
2.2. Family Background
His father, Joseph B. Kruskal, Sr., was a successful fur wholesaler. His mother, Lillian Rose Vorhaus Kruskal Oppenheimer, became a notable figure in promoting the art of origami during the early days of television. She founded the Origami Center of America in New York City, which later became OrigamiUSA. Martin was one of five children. His two brothers, Joseph Kruskal (1928-2010) and William Kruskal (1919-2005), also became eminent mathematicians. Joseph Kruskal is known for discovering multidimensional scaling, the Kruskal tree theorem, and Kruskal's algorithm, while William Kruskal discovered the Kruskal-Wallis test.
2.3. Education
Kruskal began his higher education at the University of Chicago. He then pursued his doctoral studies at New York University, where he earned his Ph.D. in 1952. His dissertation, titled "The Bridge Theorem For Minimal Surfaces", was supervised by Richard Courant and Bernard Friedman.
3. Academic Career
Kruskal's professional career as a mathematician and physicist was predominantly marked by his long and influential tenures at leading academic and research institutions.
3.1. Princeton University
Kruskal spent a significant portion of his career at Princeton University. He joined the Plasma Physics Laboratory as a research scientist in 1951, contributing significantly to fusion power research. Over the years, he expanded his academic roles, becoming a professor of astronomy in 1961. In 1968, he founded and chaired the Program in Applied and Computational Mathematics. He was appointed a professor of mathematics in 1979. He retired from Princeton University in 1989.
3.2. Rutgers University
Following his retirement from Princeton, Kruskal joined the mathematics department at Rutgers University. There, he held the prestigious David Hilbert Chair of Mathematics, continuing his academic and research pursuits.
4. Research Contributions
Martin Kruskal's scientific work spans a remarkable breadth and depth across mathematics and physics, with his discoveries shaping several fundamental areas and creating entirely new fields of study.
4.1. Nonlinear Analysis and Partial Differential Equations
Kruskal maintained a lifelong interest in partial differential equations and nonlinear analysis. He developed foundational concepts concerning asymptotic expansions and adiabatic invariants, which proved crucial in various applied mathematical contexts. His Ph.D. dissertation focused on "The Bridge Theorem For Minimal Surfaces", indicating his early engagement with fundamental mathematical problems.
4.2. Plasma Physics
During the 1950s and early 1960s, Kruskal made substantial contributions to plasma physics, laying much of the groundwork for modern understanding in the field. His theory of adiabatic invariants was particularly vital for fusion research. Key concepts in plasma physics bearing his name include the Kruskal-Shafranov instability and the Bernstein-Greene-Kruskal (BGK) modes. Collaborating with I. B. Bernstein, E. A. Frieman, and R. M. Kulsrud, he developed the MHD (or magnetohydrodynamic) Energy Principle, a cornerstone of plasma stability analysis. His research extended to both laboratory plasmas and plasma astrophysics.
4.3. General Relativity
In 1960, Kruskal made a significant discovery in general relativity by fully mapping the classical spacetime structure of the simplest type of black hole. While the Schwarzschild solution previously described the region exterior to a black hole's event horizon, Kruskal (independently and in parallel with George Szekeres) found its maximal analytic continuation. He elegantly represented this using what are now known as Kruskal-Szekeres coordinates. This led to the astonishing revelation that the interior of a black hole functions as a "wormhole", connecting two identical, asymptotically flat universes. This marked the first concrete example of a wormhole solution within general relativity. Although this wormhole collapses into a singularity before any observer or signal can traverse from one universe to the other, its discovery later proved to be an important ingredient in understanding the thermal nature of black hole physics in the 1970s. Today, it remains a fundamental clue in efforts to comprehend quantum gravity.
4.4. Solitons and Integrable Systems
Kruskal's most widely recognized achievement was his seminal work in the 1960s on the integrability of certain nonlinear partial differential equations involving functions of one spatial variable and time. This groundbreaking research began with a pioneering computer simulation conducted by Kruskal and Norman Zabusky (with assistance from Harry Dym) of the Korteweg-de Vries equation (KdV). The KdV equation serves as an asymptotic model for the propagation of nonlinear dispersive waves. Kruskal and Zabusky made the remarkable discovery of a "solitary wave" solution to the KdV equation. This wave propagated without dispersion and astonishingly regained its original shape even after colliding with other such waves. Due to its particle-like behavior, they coined the term "soliton", which rapidly gained widespread acceptance in the scientific community.
This work was partly spurred by the near-recurrence paradox observed in an earlier computer simulation of a nonlinear lattice by Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou at Los Alamos National Laboratory in 1955. Their observations of long-term, nearly recurrent behavior in a one-dimensional chain of anharmonic oscillators contradicted the expected rapid thermalization. Kruskal and Zabusky's simulation of the KdV equation, which Kruskal derived as a continuum limit of that chain, revealed solitonic behavior, directly opposite to thermalization, thus explaining the core of the phenomenon.
The concept of solitary waves had been a 19th-century mystery, dating back to John Scott Russell's 1834 observation of what is now called a soliton propagating in a canal, which he famously chased on horseback. Despite his experimental observations, Scott Russell did not fully recognize them as solitons, focusing primarily on the "great wave of translation." His findings, presented in his 1844 Report on Waves to the British Association for the Advancement of Science, were met with skepticism by contemporary scientists like George Airy and George Stokes, whose linear water wave theories could not account for such phenomena. Later, Joseph Boussinesq (1871) and Lord Rayleigh (1876) published mathematical theories that justified Scott Russell's observations. In 1895, Diederik Korteweg and Gustav de Vries formulated the KdV equation to describe shallow water waves, but its essential properties remained largely unknown until the work of Kruskal and his collaborators in the 1960s.
The discovery of solitonic behavior suggested that the KdV equation possessed conservation laws beyond the obvious ones of mass, energy, and momentum. A fourth conservation law was found by Gerald Whitham, followed by a fifth discovered by Kruskal and Zabusky. Further new conservation laws were identified by Robert M. Miura, who also demonstrated the existence of numerous conservation laws for the related Modified Korteweg-de Vries (MKdV) equation. Miura's work, including the "Miura transformation" linking solutions of KdV and MKdV equations, provided a critical clue. This enabled Kruskal, along with Clifford S. Gardner, John M. Greene, and Miura (collectively known as GGKM), to discover a general technique for the exact solution of the KdV equation and a deeper understanding of its conservation laws. This technique, known as the inverse scattering method, was a surprising and elegant development that showed the KdV equation admitted an infinite number of Poisson-commuting conserved quantities, indicating its complete integrability. This discovery provided the modern foundation for understanding the soliton phenomenon: the solitary wave is recreated after interaction because this is the only way to satisfy all the underlying conservation laws. Shortly after the GGKM paper, Peter Lax famously interpreted the inverse scattering method in terms of isospectral deformations and Lax pairs.
The inverse scattering method has since seen an astonishing variety of generalizations and applications across diverse areas of mathematics and physics. Kruskal himself pioneered some of these generalizations, such as demonstrating the existence of infinitely many conserved quantities for the sine-Gordon equation. This work led to the discovery of an inverse scattering method for the sine-Gordon equation by M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur (AKNS). The sine-Gordon equation, a relativistic wave equation in 1+1 dimensions that also exhibits the soliton phenomenon, became an important model in solvable relativistic field theory. Preceding AKNS, Zakharov and Shabat had already discovered an inverse scattering method for the nonlinear Schrödinger equation.
Solitons are now recognized as ubiquitous in nature, appearing in fields ranging from physics to biology. In 1986, Kruskal and Zabusky were jointly awarded the Howard N. Potts Gold Medal by the Franklin Institute for their contributions to mathematical physics, early creative combinations of analysis and computation, and, most notably, their seminal work on the properties of solitons. When awarding the 2006 Steele Prize to Gardner, Greene, Kruskal, and Miura, the American Mathematical Society stated that prior to their work, "there was no general theory for the exact solution of any important class of nonlinear differential equations." The AMS further noted that in applied mathematics, "solitons and their descendants (kinks, anti-kinks, instantons, and breathers) have entered and changed such diverse fields as nonlinear optics, plasma physics, and ocean, atmospheric, and planetary sciences. Nonlinearity has undergone a revolution: from a nuisance to be eliminated, to a new tool to be exploited."
4.5. Painlevé Equations
In the 1980s, Kruskal developed a keen interest in the Painlevé equations. These equations frequently emerge as symmetry reductions of soliton equations, and Kruskal was fascinated by the deep connection he perceived between the characteristic properties of these equations and completely integrable systems. Much of his subsequent research was motivated by a desire to understand this relationship more profoundly and to develop new, direct, and simpler methods for their analysis. Kruskal was known for his innovative approach, often seeking alternatives to standard methods in differential equations.
The six Painlevé equations are defined by a specific property known as the Painlevé property: their solutions are single-valued around all singularities whose locations depend on the initial conditions. Kruskal believed that this defining property should be the starting point for deriving all necessary information about their solutions, without the need for additional, unnecessary structures. His early work in this area included an asymptotic study of the Painlevé equations with Nalini Joshi, which was notable for not requiring the use of associated linear problems, an unusual approach at the time. His persistent critical examination of classical results led to a direct and simplified method, also developed with Joshi, to prove the Painlevé property of the Painlevé equations.
4.6. Surreal Numbers and Asymptotology
In the latter part of his career, Kruskal dedicated significant attention to the theory of surreal numbers. Surreal numbers, defined constructively, encompass all the basic properties and operations of the real numbers, extending them to include various types of infinities and infinitesimals. Kruskal made valuable contributions to the foundational aspects of this theory, to the definition of surreal functions, and to the analysis of their structural properties. He identified a remarkable connection between surreal numbers, asymptotics, and exponential asymptotics. A major open question, posed by Conway, Kruskal, and Norton in the late 1970s, concerned whether sufficiently well-behaved surreal functions possessed definite integrals. While the question was answered negatively in its full generality (as hoped by Conway et al.) by Costin, Friedman, and Ehrlich in 2015, their analysis did show that definite integrals exist for a sufficiently broad class of surreal functions consistent with Kruskal's broader vision of asymptotic analysis. At the time of his death, Kruskal was collaborating with O. Costin on a book about surreal analysis.
Kruskal famously coined the term "asymptotology" to describe what he called the "art of dealing with applied mathematical systems in limiting cases." He formulated seven fundamental Principles of Asymptotology: The Principle of Simplification, The Principle of Recursion, The Principle of Interpretation, The Principle of Wild Behaviour, The Principle of Annihilation, The Principle of Maximal Balance, and The Principle of Mathematical Nonsense. While the term "asymptotology" is not as universally adopted as "soliton," Kruskal argued for its recognition as a distinct branch of knowledge, situated uniquely between science and art. His proposed framework has proven highly fruitful in the development and application of asymptotic methods.
4.7. Recreational Mathematics
Beyond his rigorous scientific pursuits, Kruskal was also known for his contributions to recreational mathematics. He invented the "Kruskal count", a magical effect that puzzled professional magicians due to its basis in a mathematical phenomenon rather than sleight of hand. Additionally, he designed several origami models, including a special envelope for sending secret messages that, while easy to unfold, was difficult to refold in a way that concealed the prior act of opening.
5. Personal Life
Martin Kruskal's personal life was marked by a long marriage and a family with notable intellectual pursuits, extending his own passion for mathematics and science into other domains.
5.1. Marriage and Family
Martin Kruskal was married to Laura Kruskal for 56 years. Laura was a lecturer and writer on origami, and the originator of many new origami models. His mother, Lillian, also had a strong influence on the art of origami, promoting it and founding the Origami Center of America. Martin and Laura had three children: Karen, an attorney; Kerry, an author of children's books; and Clyde Kruskal, a computer scientist at the University of Maryland. His brothers, Joseph and William, were also distinguished mathematicians, contributing significantly to various mathematical fields.
6. Awards and Honors
Martin David Kruskal received numerous significant awards and honors throughout his distinguished career, recognizing his profound impact on mathematics and physics.
His accolades include:
- Gibbs Lecturer, American Mathematical Society (1979)
- Dannie Heineman Prize, American Physical Society (1983)
- Howard N. Potts Gold Medal, Franklin Institute (1986)
- Award in Applied Mathematics and Numerical Analysis, National Academy of Sciences (1989)
- National Medal of Science (1993)
- John von Neumann Lectureship, SIAM (1994)
- Honorary DSc, Heriot-Watt University (2000)
- Maxwell Prize, Council For Industrial And Applied Mathematics (2003)
- Steele Prize, American Mathematical Society (2006)
Kruskal was also elected to several prestigious academic bodies:
- Member of the National Academy of Sciences (1980)
- Member of the American Academy of Arts and Sciences (1983)
- Foreign Member of the Royal Society (1997)
- Foreign Member of the Russian Academy of Sciences (2000)
- Fellow of the Royal Society of Edinburgh (2001)
7. Death
Martin David Kruskal passed away on December 26, 2006.
8. Legacy and Impact
Martin Kruskal's scientific legacy is profound and far-reaching, fundamentally altering numerous fields of mathematics and physics. His most revolutionary contribution was the discovery of solitons and the development of the inverse scattering method, which provided the first general theory for the exact solution of important classes of nonlinear differential equations. This work transformed the understanding of nonlinearity from a mere nuisance to an exploitable tool, influencing diverse areas such as nonlinear optics, plasma physics, and ocean, atmospheric, and planetary sciences.
His work on general relativity, particularly the Kruskal-Szekeres coordinates and the revelation of the "wormhole" nature of black holes, provided crucial insights that continue to inform research in quantum gravity and black hole thermodynamics. His contributions to plasma physics laid essential groundwork for fusion power research and the understanding of plasma behavior.
Kruskal's intellectual curiosity extended to developing new concepts like "asymptotology", offering a structured approach to solving problems in limiting cases. His later research on Painlevé equations and surreal numbers demonstrated his continuous pursuit of fundamental mathematical truths and innovative methodologies. The American Mathematical Society recognized the transformative impact of his work, stating that solitons and their descendants have "entered and changed" various scientific disciplines. As noted by eminent mathematician Philip A. Griffiths, Kruskal's discovery of KdV equation integrability beautifully showcased the unity of mathematics, connecting computation, analysis, algebraic geometry, representation theory, and elementary geometry. Kruskal's influence as a leader in nonlinear science and the principal architect of soliton theory was formally acknowledged with the National Medal of Science in 1993, solidifying his status as a visionary in modern science.