Martin David Kruskal, a world-renowned applied mathematician and mathematical physicist, died on December 26, 2006, after a stroke, the second in four months. He was a creative, innovative, and wonderful scientist who left his mark on many areas of mathematics, applied mathematics, and mathematical physics, and on many people, including the authors of this memoir. He was a loving husband and father, and is survived by Laura, his wife of 56 years, with whom he shared the joys of origami, travel, and food; their three children, Karen, Kerry, and Clyde; and five grandchildren. He had two siblings, both also well-known mathematicians: His older brother, William Kruskal (deceased), was a statistician at the University of Chicago, and his younger brother, Joseph Kruskal (retired), is a statistician and psychometrician who worked at Bell Laboratories. Kruskal received numerous awards and honors during his career, including election to the National Academy of Sciences (1980), the National Medal of Science (1993), and election to foreign membership in the Royal Society of London (1997).

Kruskal was born in New York City on September 28, 1925, and grew up in New Rochelle, New York. Although called David by most of his family, he was known to most research colleagues as Martin. His interest in mathematics started early, and he took every opportunity to teach math to his younger brother, Joe, and classmates in elementary school. After high school, Martin went off to the University of Chicago, where he received a BS in mathematics in 1945. One of the family's neighbors in New Rochelle was Richard Courant, who convinced him to go to New York University for his PhD. As a graduate student, Martin was known for showing a profound interest in many different topics, and some wondered whether he would ever settle down. Those many interests would serve him well throughout his productive and diverse research career. He obtained a PhD in 1952, under the guidance of Courant and Bernard Friedman. His thesis was titled "The Bridge Theorem for Minimal Surfaces."

In 1951, Kruskal's first job was to work with Lyman Spitzer on a classified project, called Project Matterhorn, which after de-classification became the Princeton Plasma Physics Laboratory. The objective of Project Matterhorn was to generate controlled nuclear fusion as a clean, safe source of energy. Spitzer, who had security badge No. 1, was astute enough to realize that such a complex project would require expertise in mathematical modeling and analysis, and he hired Kruskal, who then had badge No. 2.

Kruskal worked at the Plasma Physics Laboratory during the 1950s and much of the 1960s. Some of his most important research was carried out during this period, when he and his colleagues laid the theoretical foundations of controlled nuclear fusion and the undeveloped field of plasma physics. Kruskal applied his considerable, sophisticated knowledge of mathematics, together with his strong physical intuition, to arrive at many important results in both applied and abstract mathematics. His most noteworthy mathematical accomplishment during that time was the development of modern methods in asymptotics. He popularized these methods in his courses, not only by explaining them in a clear and luminous way, but also, and more importantly, by illustrating them with numerous interesting and intriguing problems and counterexamples. These drove the art of the application of these essential ideas into the minds of his students and colleagues, never to be forgotten.

His best known physical result during this period, aside from his work on fusion, was the development of the "Kruskal coordinates" in general relativity, sometimes referred to as the "Kruskal-Szekeres" coordinates [10], in which the nonphysical Schwarzschild singularity in black holes was removed. The most familiar fusion result that bears his name may be the "Kruskal-Shafranov" criterion for kink instabilities, which is still widely quoted by plasma physicists, many of whom no longer know of its authors' existence.

Kruskal did not receive full credit for much of this early work, perhaps because it was done in a classified environment. By the time of declassification, the work was already deeply embedded in the standard accepted physical and mathematical literature. Such knowledge, in general, may no longer be attributed to its authors, but is essential for our basic understanding of scientific work. This body of work serves as a lasting memorial to Kruskal's great mathematical and physical abilities.

To those of us who were mentored by Martin Kruskal, he seemed to be driven by two things: an insatiable curiosity about mysteries in the world around him that had not been explained scientifically, and a deep desire to get to the essence of problems from the very beginning and then let logic take over from there.

Martin Kruskal's unconventional and deep questioning of dominant paradigms was crucial in the discovery of solitons as solutions of the Korteweg-de Vries (KdV) equation. He and Zabusky coined the word "soliton" [17,18] to describe nonlinear solitary waves that have an additional special property: After two of these waves interact nonlinearly, they emerge from the interaction unchanged in shape and speed, in the asymptotic sense. These nonlinear waves act like linear waves in that they survive their interactions unscathed. They shift slightly in position, however, relative to the motions they would have executed had no interaction occurred.

Kruskal, with his colleagues Gardner, Greene, and Miura, showed the existence of an infinite number of conservation laws for the KdV equation, which led to the development of the inverse scattering method for exact solution of the KdV equation using direct and inverse quantum mechanical scattering theory [6,7]. For this work, they received the 2006 Leroy P. Steele Prize from the American Mathematical Society. This method and its generalizations are now called the inverse scattering transform (IST). These contributions have had an enormous impact on science, engineering, and mathematics and led to the new field of "nonlinear completely integrable systems." Such systems now have been recognized as widely applicable and useful models of science, occurring in fluid dynamics, particle physics, and optics, and the IST method is considered a significant and important method of mathematical physics.

It was Kruskal who recognized that it may be possible to solve other equations, such as the sine-Gordon equation (SGE), in a way similar to that used for the KdV equation. He showed that the SGE, like the KdV equation, has an infinite number of conservation laws [14]. This result also agreed with his numerical computations on the SGE [1], which suggested that it has soliton-type solutions with elastic interaction properties, whereas other equations of similar types do not.

A centrally important thread of Kruskal's work was his systematic development of asymptotic analysis and perturbation methods. Asymptotic analysis was essential in his work in plasma physics. Perturbation theory was critical in his work with Zabusky in transforming the Fermi-Pasta-Ulam problem into the KdV equation [18]. His extensive and deep knowledge of asymptotic methods led him to identify seven fun-damental principles of "asymptotology" [12,13].

One of the papers in asymptotic analysis, in particular, gave him great satisfaction: his asymptotic description [11] of Hamiltonian systems whose solutions are all nearly periodic. A major subject of this paper concerns adiabatic invariance to all orders, that is, the study of physical quantities that appear to be conserved at each order when expanded in an asymptotic limit. In many cases, however, the quantity is not actually conserved; rather, the delicate information about its dependence on time lies hidden in the asymptotic expansion "beyond all orders" [3].

Kruskal strove to develop transparent methods for capturing this hidden information. A crucial example on which he focused was a third-order model of the growth of needle-shaped crystals in the asymptotic limit of small surface tension. For a needle crystal solution to exist, the model had to have anti-symmetric solutions, which implied, in turn, that the curvature of the crystal had to vanish at the origin. The asymptotic expansion of this quantity appeared to be zero to all orders, until Kruskal and Segur developed [15] a method for capturing the information hidden beyond all orders. This showed that no such solution could exist. The method developed in that paper has been extended subsequently to many other settings and used by many researchers.

Asymptotics also motivated another abiding theme in Kruskal's work. Long-time asymptotics of solutions of completely integrable PDEs lead to ODEs in self-similar regions. Prominent among the reductions are the six Painlevé equations, known to be the only ones in a class of ODEs to have a characteristic property in the complex plane, now called the Painlevé property, that defines new transcendental functions as solutions. In addition to being reductions of soliton equations, these equations are now recognized as important in many fields, including random matrix theory and orthogonal polynomial theory.

The Painlevé property motivated Kruskal to become deeply interested in the Painlevé equations and their connection with integrability. Much of his work in the 1990s aimed to further our understanding of equations that have this property and of the true meaning of the loss of the property. He developed the so-called poly-Painlevé test [16] and showed how dense branching can result generically and how this implies nonintegrability [5]. His persistent questioning of classical methods led to a direct and simple method, developed with Joshi [9], to prove the Painlevé property. In Kruskal's opinion, if the Painlevé property is fundamental to integrability, then there should be direct analytic methods for describing the solutions of equations that have the property. This central belief led to new methods for describing the solutions of such equations in asymptotic limits [8], without the use of associated linear problems. Using methods from exponential asymptotics, Kruskal and O. Costin extended this approach to difference equations [4].

Kruskal questioned classical methods for finding reductions of PDEs. The reductions of PDEs are usually found and enumerated by Lie's classical theory of symmetry analysis. A commonly accepted belief in the field was that these results were complete, i.e., there were no reductions that could not be found by Lie's methods. However, Kruskal and Clarkson developed a direct method [2] for obtaining special solutions of the Boussinesq equation, some of which could not be obtained by classical symmetry analysis. This led to a renaissance in the field of non-classical symmetry methods.

Shortly after surreal numbers were discovered by Conway, Martin Kruskal started to work on this beautiful and deep topic, which became one of his favorite subjects. Surreal numbers vastly generalize usual numbers, preserving their properties while containing infinite and infinitesimal numbers. In substantial and brilliant contributions to an understanding of the foundation of surreal numbers, Kruskal elucidated their intimate structure and defined various operations and functions on them. In recent years, he worked passionately on the analog of calculus for surreal functions, a very difficult problem whose solution would affect many areas of mathematics. This passion was manifest in lectures he gave on surreal numbers, including the 1994 SIAM John von Neumann Lecture, in which he espoused "their still unrealized promise of highly relevant application to asymptotics."

Finally, it needs to be mentioned that with each of the scientific enterprises in which he was involved, Martin Kruskal showed a rare passion for understanding, which went well beyond proving or disproving a statement. He will be deeply missed by the people who were touched by him, and by the general applied mathematics community that he influenced.--Robert M. Miura, Department of Mathematical Sciences, New Jersey Institute of Technology; Mark J. Ablowitz, Department of Applied Mathematics, University of Colorado-Boulder; Ovidiu Costin, Department of Mathematics, Ohio State University; Nalini Joshi, School of Mathematics, University of Sydney; Russell Kulsrud, Plasma Physics Laboratory, Princeton University; and Norman J. Zabusky, Department of Physics of Complex Systems, Weizmann Institute of Science.

[1] M.J. Ablowitz, M.D. Kruskal, and J.F. Ladik, Solitary wave collisions,

[2] P.A. Clarkson and M.D. Kruskal, New similarity solutions of the Boussinesq equation,

[3] O. Costin, L. Dupaigne, and M.D. Kruskal, Borel summation of adiabatic invariants,

[4] O. Costin and M.D. Kruskal, Analytic methods for obstruction to integrability in discrete dynamical systems,

[5] R.D. Costin and M.D. Kruskal, Nonintegrability criteria for a class of differential equations with two regular singular points,

[6] C.S. Gardner, J.M. Greene, R.M. Miura, and M.D. Kruskal, Method for solving the Korteweg-de Vries equation,

[7] C.S. Gardner, J.M. Greene, M.D. Kruskal, and R.M. Miura, Korteweg-de Vries equation and generalizations. VI. Methods for exact solution,

[8] N. Joshi and M.D. Kruskal, The Painlevé connection problem: An asymptotic approach I,

[9] N. Joshi and M.D. Kruskal, A direct proof that the six Painlevé equations have no movable singularities except poles,

[10] M.D. Kruskal, Maximal extension of Schwarzschild metric,

[11] M.D. Kruskal, Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic,

[12] M.D. Kruskal,

[13] M.D. Kruskal,

[14] M.D. Kruskal,

[15] M.D. Kruskal and H. Segur, Asymptotics beyond all orders in a model of crystal growth,

[16] M.D. Kruskal and P.A. Clarkson, The Painlevé-Kowalevski and poly-Painlevé tests for integrability,

[17] N.J. Zabusky, Fermi-Pasta-Ulam, solitons and the fabric of nonlinear and computational science: History, synergetics, and visiometrics,

[18] N.J. Zabusky and M.D. Kruskal, Interaction of solitons in a collisionless plasma and recurrence of initial states,

April 11, 2007 © SIAM News

Kruskal was born in New York City on September 28, 1925, and grew up in New Rochelle, New York. Although called David by most of his family, he was known to most research colleagues as Martin. His interest in mathematics started early, and he took every opportunity to teach math to his younger brother, Joe, and classmates in elementary school. After high school, Martin went off to the University of Chicago, where he received a BS in mathematics in 1945. One of the family's neighbors in New Rochelle was Richard Courant, who convinced him to go to New York University for his PhD. As a graduate student, Martin was known for showing a profound interest in many different topics, and some wondered whether he would ever settle down. Those many interests would serve him well throughout his productive and diverse research career. He obtained a PhD in 1952, under the guidance of Courant and Bernard Friedman. His thesis was titled "The Bridge Theorem for Minimal Surfaces."

In 1951, Kruskal's first job was to work with Lyman Spitzer on a classified project, called Project Matterhorn, which after de-classification became the Princeton Plasma Physics Laboratory. The objective of Project Matterhorn was to generate controlled nuclear fusion as a clean, safe source of energy. Spitzer, who had security badge No. 1, was astute enough to realize that such a complex project would require expertise in mathematical modeling and analysis, and he hired Kruskal, who then had badge No. 2.

Kruskal worked at the Plasma Physics Laboratory during the 1950s and much of the 1960s. Some of his most important research was carried out during this period, when he and his colleagues laid the theoretical foundations of controlled nuclear fusion and the undeveloped field of plasma physics. Kruskal applied his considerable, sophisticated knowledge of mathematics, together with his strong physical intuition, to arrive at many important results in both applied and abstract mathematics. His most noteworthy mathematical accomplishment during that time was the development of modern methods in asymptotics. He popularized these methods in his courses, not only by explaining them in a clear and luminous way, but also, and more importantly, by illustrating them with numerous interesting and intriguing problems and counterexamples. These drove the art of the application of these essential ideas into the minds of his students and colleagues, never to be forgotten.

His best known physical result during this period, aside from his work on fusion, was the development of the "Kruskal coordinates" in general relativity, sometimes referred to as the "Kruskal-Szekeres" coordinates [10], in which the nonphysical Schwarzschild singularity in black holes was removed. The most familiar fusion result that bears his name may be the "Kruskal-Shafranov" criterion for kink instabilities, which is still widely quoted by plasma physicists, many of whom no longer know of its authors' existence.

Kruskal did not receive full credit for much of this early work, perhaps because it was done in a classified environment. By the time of declassification, the work was already deeply embedded in the standard accepted physical and mathematical literature. Such knowledge, in general, may no longer be attributed to its authors, but is essential for our basic understanding of scientific work. This body of work serves as a lasting memorial to Kruskal's great mathematical and physical abilities.

To those of us who were mentored by Martin Kruskal, he seemed to be driven by two things: an insatiable curiosity about mysteries in the world around him that had not been explained scientifically, and a deep desire to get to the essence of problems from the very beginning and then let logic take over from there.

Martin Kruskal's unconventional and deep questioning of dominant paradigms was crucial in the discovery of solitons as solutions of the Korteweg-de Vries (KdV) equation. He and Zabusky coined the word "soliton" [17,18] to describe nonlinear solitary waves that have an additional special property: After two of these waves interact nonlinearly, they emerge from the interaction unchanged in shape and speed, in the asymptotic sense. These nonlinear waves act like linear waves in that they survive their interactions unscathed. They shift slightly in position, however, relative to the motions they would have executed had no interaction occurred.

Kruskal, with his colleagues Gardner, Greene, and Miura, showed the existence of an infinite number of conservation laws for the KdV equation, which led to the development of the inverse scattering method for exact solution of the KdV equation using direct and inverse quantum mechanical scattering theory [6,7]. For this work, they received the 2006 Leroy P. Steele Prize from the American Mathematical Society. This method and its generalizations are now called the inverse scattering transform (IST). These contributions have had an enormous impact on science, engineering, and mathematics and led to the new field of "nonlinear completely integrable systems." Such systems now have been recognized as widely applicable and useful models of science, occurring in fluid dynamics, particle physics, and optics, and the IST method is considered a significant and important method of mathematical physics.

It was Kruskal who recognized that it may be possible to solve other equations, such as the sine-Gordon equation (SGE), in a way similar to that used for the KdV equation. He showed that the SGE, like the KdV equation, has an infinite number of conservation laws [14]. This result also agreed with his numerical computations on the SGE [1], which suggested that it has soliton-type solutions with elastic interaction properties, whereas other equations of similar types do not.

A centrally important thread of Kruskal's work was his systematic development of asymptotic analysis and perturbation methods. Asymptotic analysis was essential in his work in plasma physics. Perturbation theory was critical in his work with Zabusky in transforming the Fermi-Pasta-Ulam problem into the KdV equation [18]. His extensive and deep knowledge of asymptotic methods led him to identify seven fun-damental principles of "asymptotology" [12,13].

One of the papers in asymptotic analysis, in particular, gave him great satisfaction: his asymptotic description [11] of Hamiltonian systems whose solutions are all nearly periodic. A major subject of this paper concerns adiabatic invariance to all orders, that is, the study of physical quantities that appear to be conserved at each order when expanded in an asymptotic limit. In many cases, however, the quantity is not actually conserved; rather, the delicate information about its dependence on time lies hidden in the asymptotic expansion "beyond all orders" [3].

Kruskal strove to develop transparent methods for capturing this hidden information. A crucial example on which he focused was a third-order model of the growth of needle-shaped crystals in the asymptotic limit of small surface tension. For a needle crystal solution to exist, the model had to have anti-symmetric solutions, which implied, in turn, that the curvature of the crystal had to vanish at the origin. The asymptotic expansion of this quantity appeared to be zero to all orders, until Kruskal and Segur developed [15] a method for capturing the information hidden beyond all orders. This showed that no such solution could exist. The method developed in that paper has been extended subsequently to many other settings and used by many researchers.

Asymptotics also motivated another abiding theme in Kruskal's work. Long-time asymptotics of solutions of completely integrable PDEs lead to ODEs in self-similar regions. Prominent among the reductions are the six Painlevé equations, known to be the only ones in a class of ODEs to have a characteristic property in the complex plane, now called the Painlevé property, that defines new transcendental functions as solutions. In addition to being reductions of soliton equations, these equations are now recognized as important in many fields, including random matrix theory and orthogonal polynomial theory.

The Painlevé property motivated Kruskal to become deeply interested in the Painlevé equations and their connection with integrability. Much of his work in the 1990s aimed to further our understanding of equations that have this property and of the true meaning of the loss of the property. He developed the so-called poly-Painlevé test [16] and showed how dense branching can result generically and how this implies nonintegrability [5]. His persistent questioning of classical methods led to a direct and simple method, developed with Joshi [9], to prove the Painlevé property. In Kruskal's opinion, if the Painlevé property is fundamental to integrability, then there should be direct analytic methods for describing the solutions of equations that have the property. This central belief led to new methods for describing the solutions of such equations in asymptotic limits [8], without the use of associated linear problems. Using methods from exponential asymptotics, Kruskal and O. Costin extended this approach to difference equations [4].

Kruskal questioned classical methods for finding reductions of PDEs. The reductions of PDEs are usually found and enumerated by Lie's classical theory of symmetry analysis. A commonly accepted belief in the field was that these results were complete, i.e., there were no reductions that could not be found by Lie's methods. However, Kruskal and Clarkson developed a direct method [2] for obtaining special solutions of the Boussinesq equation, some of which could not be obtained by classical symmetry analysis. This led to a renaissance in the field of non-classical symmetry methods.

Shortly after surreal numbers were discovered by Conway, Martin Kruskal started to work on this beautiful and deep topic, which became one of his favorite subjects. Surreal numbers vastly generalize usual numbers, preserving their properties while containing infinite and infinitesimal numbers. In substantial and brilliant contributions to an understanding of the foundation of surreal numbers, Kruskal elucidated their intimate structure and defined various operations and functions on them. In recent years, he worked passionately on the analog of calculus for surreal functions, a very difficult problem whose solution would affect many areas of mathematics. This passion was manifest in lectures he gave on surreal numbers, including the 1994 SIAM John von Neumann Lecture, in which he espoused "their still unrealized promise of highly relevant application to asymptotics."

Finally, it needs to be mentioned that with each of the scientific enterprises in which he was involved, Martin Kruskal showed a rare passion for understanding, which went well beyond proving or disproving a statement. He will be deeply missed by the people who were touched by him, and by the general applied mathematics community that he influenced.--Robert M. Miura, Department of Mathematical Sciences, New Jersey Institute of Technology; Mark J. Ablowitz, Department of Applied Mathematics, University of Colorado-Boulder; Ovidiu Costin, Department of Mathematics, Ohio State University; Nalini Joshi, School of Mathematics, University of Sydney; Russell Kulsrud, Plasma Physics Laboratory, Princeton University; and Norman J. Zabusky, Department of Physics of Complex Systems, Weizmann Institute of Science.

**References**[1] M.J. Ablowitz, M.D. Kruskal, and J.F. Ladik, Solitary wave collisions,

*SIAM J. Appl. Math*.,**36**(1979), 428-437.[2] P.A. Clarkson and M.D. Kruskal, New similarity solutions of the Boussinesq equation,

*J. Math. Phys.*,**30**(1989), 2201-2213.[3] O. Costin, L. Dupaigne, and M.D. Kruskal, Borel summation of adiabatic invariants,

*Nonlinearity*,**17**(2004), 1509-1519.[4] O. Costin and M.D. Kruskal, Analytic methods for obstruction to integrability in discrete dynamical systems,

*Comm. Pure Appl. Math.*,**58**(2005), 723-749.[5] R.D. Costin and M.D. Kruskal, Nonintegrability criteria for a class of differential equations with two regular singular points,

*Nonlinearity*,**16**(2003), 1295-1317.[6] C.S. Gardner, J.M. Greene, R.M. Miura, and M.D. Kruskal, Method for solving the Korteweg-de Vries equation,

*Phys. Rev. Lett.*,**19**(1967), 1095-1097.[7] C.S. Gardner, J.M. Greene, M.D. Kruskal, and R.M. Miura, Korteweg-de Vries equation and generalizations. VI. Methods for exact solution,

*Comm. Pure Appl. Math.*,**XXVII**(1974), 97-133.[8] N. Joshi and M.D. Kruskal, The Painlevé connection problem: An asymptotic approach I,

*Stud. Appl. Math.*,**86**(1992), 315-376.[9] N. Joshi and M.D. Kruskal, A direct proof that the six Painlevé equations have no movable singularities except poles,

*Stud. Appl. Math.*,**93**(1994), 187-207.[10] M.D. Kruskal, Maximal extension of Schwarzschild metric,

*Phys. Rev.*,**119**(1960), 1743-1745.[11] M.D. Kruskal, Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic,

*J. Math. Phys.*,**3**(1962), 806-828.[12] M.D. Kruskal,

*Asymptotology, in Mathematical Models in Physical Sciences*, S. Drobot, ed., Prentice-Hall, Englewood Cliffs, New Jersey, 1963, 17-48.[13] M.D. Kruskal,

*Asymptotology in numerical computation: Progress and plans on the Fermi-Pasta-Ulam problem*, Proceedings of IBM Scientific Computing Symposium on Large-Scale Problems in Physics, IBM Data Processing Division, White Plains, New York, 1965, 43-62.[14] M.D. Kruskal,

*The Korteweg-de Vries equation and related evolution equations, in Nonlinear Wave Motion*, A.C. Newell, ed., AMS Lectures in Applied Mathematics, American Mathematical Society, Providence, Rhode Island, 15 (1974), 61-83.[15] M.D. Kruskal and H. Segur, Asymptotics beyond all orders in a model of crystal growth,

*Stud. Appl. Math.*,**85**(1991), 129-181.[16] M.D. Kruskal and P.A. Clarkson, The Painlevé-Kowalevski and poly-Painlevé tests for integrability,

*Stud. Appl. Math.*,**86**(1992), 87-165.[17] N.J. Zabusky, Fermi-Pasta-Ulam, solitons and the fabric of nonlinear and computational science: History, synergetics, and visiometrics,

*Chaos*,**15**(2005), 015102.1-015102.16.[18] N.J. Zabusky and M.D. Kruskal, Interaction of solitons in a collisionless plasma and recurrence of initial states,

*Phys. Rev. Lett.*,**15**(1965), 240-243.April 11, 2007 © SIAM News