Martin Kruskal
Encyclopedia
Martin David Kruskal was an American mathematician
and physicist
. He made fundamental contributions in many areas of mathematics and science, ranging from plasma physics to general relativity and from nonlinear analysis to asymptotic analysis. His single most celebrated contribution was the discovery and theory of solitons.
He was a student at the University of Chicago
and at New York University
, where he completed his Ph.D. under Richard Courant
in 1952. He spent much of his career at Princeton University
, as a research scientist at the Plasma Physics Laboratory starting in 1951, and then as a professor of astronomy (1961), founder and chair of the Program in Applied and Computational Mathematics (1968), and professor of mathematics (1979). He retired from Princeton University
in 1989 and joined the mathematics department of Rutgers University
, holding the David Hilbert Chair of Mathematics.
Apart from his research, Kruskal was known as a mentor of younger scientists. He worked tirelessly and always aimed not just to prove a result but to understand it thoroughly. And he was notable for his playfulness. He invented the Kruskal Count, a magical effect that has been known to perplex professional magicians because – as he liked to say – it was based not on sleight of hand but on a mathematical phenomenon.
and grew up in New Rochelle
. He was generally known as Martin to the world and David to his family. His father, Joseph B. Kruskal, Sr., was a successful fur wholesaler. His mother, Lillian Rose Vorhaus Kruskal Oppenheimer, became a noted promoter of the art of origami
during the early era of television and founded the Origami Center of America in New York City, which later became OrigamiUSA. He was one of five children. His two brothers, both eminent mathematicians, were Joseph Kruskal
(born 1928; discoverer of multidimensional scaling
, the Kruskal tree theorem, and Kruskal's algorithm
) and William Kruskal
(1919–2005; discoverer of the Kruskal–Wallis test).
Martin Kruskal was married to Laura Kruskal, his wife of 56 years. Laura is well-known as a lecturer and writer about origami and originator of many new models. Martin, who had a great love of games, puzzles, and word play of all kinds, also invented several quite unusual origami models including an envelope for sending secret messages (anyone who unfolded the envelope to read the message would have great difficulty refolding it to conceal the deed).
Martin and Laura traveled extensively to scientific meetings and to visit Martin’s many scientific collaborators. Laura used to call Martin “my ticket to the world.” Wherever they went, Martin would be hard at work and Laura would often keep busy teaching origami workshops in schools and institutions for elderly people and people with disabilities. Martin and Laura had a great love of traveling and hiking.
Their three children are Karen, Kerry, and Clyde, who are known respectively as an attorney, an author of children’s books, and a mathematician.
His Ph.D. dissertation, written under the direction of Richard Courant
and Bernard Friedman at New York University
, was on the topic “The Bridge Theorem For Minimal Surfaces.” He received his Ph.D. in 1952.
In the 1950s and early 1960s, he worked largely on plasma physics, developing many ideas that are now fundamental in the field. His theory of adiabatic invariants was important in fusion research. Important concepts of plasma physics that bear his name include the Kruskal–Shafranov instability and the Bernstein–Greene–Kruskal (BGK) modes. With I. B. Bernstein, E. A. Frieman, and R. M. Kulsrud, he developed the MHD (or magnetohydrodynamic) Energy Principle. His interests extended to plasma astrophysics as well as laboratory plasmas. Martin Kruskal's work in plasma physics is considered by some to be his most outstanding.
In 1960, Kruskal discovered the full classical spacetime structure of the simplest type of black hole in General Relativity. A spherically symmetric black hole can be described by the Schwarzschild solution, which was discovered in the early days of General Relativity. However, in its original form, this solution only describes the region exterior to the horizon of the black hole. Kruskal (in parallel with George Szekeres
) discovered the maximal analytic continuation of the Schwarzschild solution
, which he exhibited elegantly using what are now called Kruskal–Szekeres coordinates.
This led Kruskal to the astonishing discovery that the interior of the black hole looks like a “wormhole
” connecting two identical, asymptotically flat universes. This was the first real example of a wormhole solution in General Relativity. The wormhole collapses to a singularity before any observer or signal can travel from one universe to the other. This is now believed to be the general fate of wormholes in General Relativity. In the 1970s, when the thermal nature of black hole physics was discovered, the wormhole property of the Schwarzschild solution turned out to be an important ingredient. Nowadays, it is considered a fundamental clue in attempts to understand quantum gravity.
Kruskal's most widely known work was the discovery in the 1960s of the integrability of certain nonlinear partial differential equations involving functions of one spatial variable as well as time. These developments began with a pioneering computer simulation by Kruskal and Norman Zabusky
(with some assistance from Gary Deem) of a nonlinear equation known as the Korteweg–de Vries equation
(KdV). The KdV equation is an asymptotic model of the propagation of nonlinear dispersive waves. But Kruskal and Zabusky made the startling discovery of a “solitary wave” solution of the KdV equation that propagates nondispersively and even regains its shape after a collision with other such waves. Because of the particle-like properties of such a wave, they named it a "soliton
,” a term that caught on almost immediately.
This work was partly motivated by the near-recurrence paradox that had been observed in a very early computer simulation of a nonlinear lattice by Enrico Fermi, John Pasta, and Stanislaw Ulam, at Los Alamos in 1955. Those authors had observed long-time nearly recurrent behavior of a one-dimensional chain of anharmonic oscillators, in contrast to the rapid thermalization that had been expected. Kruskal and Zabusky simulated the KdV equation, which Kruskal had obtained as a continuum limit of that one-dimensional chain, and found solitonic behavior, which is the opposite of thermalization. That turned out to be the heart of the phenomenon.
Solitary wave phenomena had been a 19th century mystery dating back to work by John Scott Russell
who, in 1834, observed what we now call a soliton, propagating in a canal, and chased it on horseback. In spite of his observations of solitons in wave tank experiments, Scott Russell never recognized them as such, because of his focus on the “great wave of translation,” the largest amplitude solitary wave. His experimental observations, presented in his Report on Waves to the British Association for the Advancement of Science in 1844, were viewed with skepticism by George Airy and George Stokes
because their linear water wave theories were unable to explain them. Joseph Boussinesq (1871) and Lord Rayleigh (1876) published mathematical theories justifying Scott Russell’s observations. In 1895, Diederik Korteweg
and Gustav de Vries
formulated the KdV equation to describe shallow water waves (such as the waves in the canal observed by Russell), but the essential properties of this equation were not understood until the work of Kruskal and his collaborators in the 1960s.
Solitonic behavior suggested that the KdV equation must have conservation laws beyond the obvious conservation laws of mass, energy, and momentum. A fourth conservation law was discovered by Gerald Whitham and a fifth one by Kruskal and Zabusky. Several new conservation laws were discovered by hand by Robert Miura, who also showed that many conservation laws existed for a related equation known as the Modified Korteweg–de Vries (MKdV) equation. With these conservation laws, Miura showed a connection (called the Miura transformation) between solutions of the KdV and MKdV equations. This was a clue that enabled Kruskal, with Clifford S. Gardner, John M. Greene, and Miura, to discover a general technique for exact solution of the KdV equation and understanding of its conservation laws. This was the inverse scattering method, a surprising and elegant method that demonstrates that the KdV equation admits an infinite number of Poisson-commuting conserved quantities and is completely integrable. This discovery gave the modern basis for understanding of the soliton phenomenon: the solitary wave is recreated in the outgoing state because this is the only way to satisfy all of the conservation laws.
The inverse scattering method has had an astonishing variety of generalizations and applications in different areas of mathematics and physics. Kruskal himself pioneered some of the generalizations, such as the existence of infinitely many conserved quantities for the sine-Gordon equation
. This led to the discovery of an inverse scattering method for that equation by M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur. The sine-Gordon equation is a relativistic wave equation in 1+1 dimensions that also exhibits the soliton phenomenon and which became an important model of solvable relativistic field theory.
Solitons are now known to be ubiquitous in nature, from physics to biology. In 1986, Kruskal and Zabusky shared the Howard N. Potts Gold Medal
from the Franklin Institute “for contributions to mathematical physics and early creative combinations of analysis and computation, but most especially for seminal work in the properties of solitons.” In awarding the 2006 Steele Prize to Gardner, Greene, Kruskal, and Miura, the American Mathematical Society stated that before their work “there was no general theory for the exact solution of any important class of nonlinear differential equations.” The AMS added, “In applications of 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.”
Kruskal received the National Medal of Science
in 1993 “for his influence as a leader in nonlinear science for more than two decades as the principal architect of the theory of soliton solutions of nonlinear equations of evolution.”
In an article surveying the state of mathematics at the turn of the millennium, the eminent mathematician Philip A. Griffiths wrote that the discovery of integrability of the KdV equation “exhibited in the most beautiful way the unity of mathematics. It involved developments in computation, and in mathematical analysis, which is the traditional way to study differential equations. It turns out that one can understand the solutions to these differential equations through certain very elegant constructions in algebraic geometry. The solutions are also intimately related to representation theory, in that these equations turn out to have an infinite number of hidden symmetries. Finally, they relate back to problems in elementary geometry.”
In the 1980s, Kruskal developed an acute interest in the Painlevé
equations. They frequently arise as symmetry reductions of soliton equations, and Kruskal was intrigued by the intimate relationship that appeared to exist between the properties characterizing these equations and completely integrable systems. Much of his subsequent research was driven by a desire to understand this relationship and to develop new direct and simple methods for studying the Painlevé equations. Kruskal was rarely satisfied with the standard approaches to differential equations.
The six Painlevé equations
have a characteristic property called the Painlevé property: their solutions are single-valued around all singularities whose locations depend on the initial conditions. In Kruskal’s opinion, since this property defines the Painlevé equations, one should be able to start with this, without any additional unnecessary structures, to work out all the required information about their solutions. The first result was an asymptotic study of the Painlevé equations with Nalini Joshi
, unusual at the time in that it did not require the use of associated linear problems. His persistent questioning of classical results led to a direct and simple method, also developed with Joshi, to prove the Painlevé property of the Painlevé equations.
In the later part of his career, one of Kruskal's chief interests was the theory of surreal numbers. Surreal numbers, which are defined constructively, have all the basic properties and operations of the real numbers. They include the real numbers alongside many types of infinities and infinitesimals. Kruskal contributed to the foundation of the theory, to defining surreal functions, and to analyzing their structure. He discovered a remarkable link between surreal numbers, asymptotics, and exponential asymptotics. A major open question is whether sufficiently well behaved surreal functions possess definite integrals. A positive answer would have deep implications in analysis. Over the last years of his life, Kruskal worked with much dedication on defining surreal integration. Much was learned but important parts of this project remain unfinished. At the time of his death, he was in the process of writing a book on surreal analysis with O. Costin.
Kruskal coined the term Asymptotology
to describe the “art of dealing with applied mathematical systems in limiting cases”. He formulated seven Principles of Asymptotology: 1. The Principle of Simplification; 2. The Principle of Recursion; 3. The Principle of Interpretation; 4. The Principle of Wild Behaviour; 5. The Principle of Annihilation; 6. The Principle of Maximal Balance; 7. The Principle of Mathematical Nonsense.
The term asymptotology
is not so widely used as the term soliton
. Asymptotic methods of various types have been successfully used since almost the birth of science itself. Nevertheless, Kruskal tried to show that asymptotology is a special branch of knowledge, intermediate, in some sense, between science and art. His proposal has been found to be very fruitful.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
. He made fundamental contributions in many areas of mathematics and science, ranging from plasma physics to general relativity and from nonlinear analysis to asymptotic analysis. His single most celebrated contribution was the discovery and theory of solitons.
He was a student at the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
and at New York University
New York University
New York University is a private, nonsectarian research university based in New York City. NYU's main campus is situated in the Greenwich Village section of Manhattan...
, where he completed his Ph.D. under Richard Courant
Richard Courant
Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...
in 1952. He spent much of his career at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, as a research scientist at the Plasma Physics Laboratory starting in 1951, and then as a professor of astronomy (1961), founder and chair of the Program in Applied and Computational Mathematics (1968), and professor of mathematics (1979). He retired from Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
in 1989 and joined the mathematics department of Rutgers University
Rutgers University
Rutgers, The State University of New Jersey , is the largest institution for higher education in New Jersey, United States. It was originally chartered as Queen's College in 1766. It is the eighth-oldest college in the United States and one of the nine Colonial colleges founded before the American...
, holding the David Hilbert Chair of Mathematics.
Apart from his research, Kruskal was known as a mentor of younger scientists. He worked tirelessly and always aimed not just to prove a result but to understand it thoroughly. And he was notable for his playfulness. He invented the Kruskal Count, a magical effect that has been known to perplex professional magicians because – as he liked to say – it was based not on sleight of hand but on a mathematical phenomenon.
Personal
Martin David Kruskal was born in New York CityNew York City
New York is the most populous city in the United States and the center of the New York Metropolitan Area, one of the most populous metropolitan areas in the world. New York exerts a significant impact upon global commerce, finance, media, art, fashion, research, technology, education, and...
and grew up in New Rochelle
New Rochelle, New York
New Rochelle is a city in Westchester County, New York, United States, in the southeastern portion of the state.The town was settled by refugee Huguenots in 1688 who were fleeing persecution in France...
. He was generally known as Martin to the world and David to his family. His father, Joseph B. Kruskal, Sr., was a successful fur wholesaler. His mother, Lillian Rose Vorhaus Kruskal Oppenheimer, became a noted promoter of the art of origami
Origami
is the traditional Japanese art of paper folding, which started in the 17th century AD at the latest and was popularized outside Japan in the mid-1900s. It has since then evolved into a modern art form...
during the early era of television and founded the Origami Center of America in New York City, which later became OrigamiUSA. He was one of five children. His two brothers, both eminent mathematicians, were Joseph Kruskal
Joseph Kruskal
Joseph Bernard Kruskal, Jr. was an American mathematician, statistician, computer scientist and psychometrician. He was a student at the University of Chicago and at Princeton University, where he completed his Ph.D. in 1954, nominally under Albert W...
(born 1928; discoverer of multidimensional scaling
Multidimensional scaling
Multidimensional scaling is a set of related statistical techniques often used in information visualization for exploring similarities or dissimilarities in data. MDS is a special case of ordination. An MDS algorithm starts with a matrix of item–item similarities, then assigns a location to each...
, the Kruskal tree theorem, and Kruskal's algorithm
Kruskal's algorithm
Kruskal's algorithm is an algorithm in graph theory that finds a minimum spanning tree for a connected weighted graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized...
) and William Kruskal
William Kruskal
William Henry Kruskal was an American mathematician and statistician. He is best known for having formulated the Kruskal–Wallis one-way analysis of variance , a widely-used nonparametric statistical method.Kruskal was born in New York City to a successful fur wholesaler...
(1919–2005; discoverer of the Kruskal–Wallis test).
Martin Kruskal was married to Laura Kruskal, his wife of 56 years. Laura is well-known as a lecturer and writer about origami and originator of many new models. Martin, who had a great love of games, puzzles, and word play of all kinds, also invented several quite unusual origami models including an envelope for sending secret messages (anyone who unfolded the envelope to read the message would have great difficulty refolding it to conceal the deed).
Martin and Laura traveled extensively to scientific meetings and to visit Martin’s many scientific collaborators. Laura used to call Martin “my ticket to the world.” Wherever they went, Martin would be hard at work and Laura would often keep busy teaching origami workshops in schools and institutions for elderly people and people with disabilities. Martin and Laura had a great love of traveling and hiking.
Their three children are Karen, Kerry, and Clyde, who are known respectively as an attorney, an author of children’s books, and a mathematician.
Scientific work
Martin Kruskal's scientific interests covered a wide range of topics in pure mathematics and applications of mathematics to the sciences. He had lifelong interests in many topics in partial differential equations and nonlinear analysis and developed fundamental ideas about asymptotic expansions, adiabatic invariants, and numerous related topics.His Ph.D. dissertation, written under the direction of Richard Courant
Richard Courant
Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...
and Bernard Friedman at New York University
New York University
New York University is a private, nonsectarian research university based in New York City. NYU's main campus is situated in the Greenwich Village section of Manhattan...
, was on the topic “The Bridge Theorem For Minimal Surfaces.” He received his Ph.D. in 1952.
In the 1950s and early 1960s, he worked largely on plasma physics, developing many ideas that are now fundamental in the field. His theory of adiabatic invariants was important in fusion research. Important concepts of plasma physics that bear his name include the Kruskal–Shafranov instability and the Bernstein–Greene–Kruskal (BGK) modes. With I. B. Bernstein, E. A. Frieman, and R. M. Kulsrud, he developed the MHD (or magnetohydrodynamic) Energy Principle. His interests extended to plasma astrophysics as well as laboratory plasmas. Martin Kruskal's work in plasma physics is considered by some to be his most outstanding.
In 1960, Kruskal discovered the full classical spacetime structure of the simplest type of black hole in General Relativity. A spherically symmetric black hole can be described by the Schwarzschild solution, which was discovered in the early days of General Relativity. However, in its original form, this solution only describes the region exterior to the horizon of the black hole. Kruskal (in parallel with George Szekeres
George Szekeres
George Szekeres AM was a Hungarian-Australian mathematician.-Early years:Szekeres was born in Budapest, Hungary as Szekeres György and received his degree in chemistry at the Technical University of Budapest. He worked six years in Budapest as an analytical chemist. He married Esther Klein in 1936...
) discovered the maximal analytic continuation of the Schwarzschild solution
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...
, which he exhibited elegantly using what are now called Kruskal–Szekeres coordinates.
This led Kruskal to the astonishing discovery that the interior of the black hole looks like a “wormhole
Wormhole
In physics, a wormhole is a hypothetical topological feature of spacetime that would be, fundamentally, a "shortcut" through spacetime. For a simple visual explanation of a wormhole, consider spacetime visualized as a two-dimensional surface. If this surface is folded along a third dimension, it...
” connecting two identical, asymptotically flat universes. This was the first real example of a wormhole solution in General Relativity. The wormhole collapses to a singularity before any observer or signal can travel from one universe to the other. This is now believed to be the general fate of wormholes in General Relativity. In the 1970s, when the thermal nature of black hole physics was discovered, the wormhole property of the Schwarzschild solution turned out to be an important ingredient. Nowadays, it is considered a fundamental clue in attempts to understand quantum gravity.
Kruskal's most widely known work was the discovery in the 1960s of the integrability of certain nonlinear partial differential equations involving functions of one spatial variable as well as time. These developments began with a pioneering computer simulation by Kruskal and Norman Zabusky
Norman Zabusky
Norman J. Zabusky is an American physicist, who is noted for the discovery of the soliton in the Korteweg–de Vries equation, in work completed with Martin Kruskal. This result early in his career was followed by an extensive body of work in computational fluid dynamics, which led him more...
(with some assistance from Gary Deem) of a nonlinear equation known as the Korteweg–de Vries equation
Korteweg–de Vries equation
In mathematics, the Korteweg–de Vries equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified...
(KdV). The KdV equation is an asymptotic model of the propagation of nonlinear dispersive waves. But Kruskal and Zabusky made the startling discovery of a “solitary wave” solution of the KdV equation that propagates nondispersively and even regains its shape after a collision with other such waves. Because of the particle-like properties of such a wave, they named it a "soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
,” a term that caught on almost immediately.
This work was partly motivated by the near-recurrence paradox that had been observed in a very early computer simulation of a nonlinear lattice by Enrico Fermi, John Pasta, and Stanislaw Ulam, at Los Alamos in 1955. Those authors had observed long-time nearly recurrent behavior of a one-dimensional chain of anharmonic oscillators, in contrast to the rapid thermalization that had been expected. Kruskal and Zabusky simulated the KdV equation, which Kruskal had obtained as a continuum limit of that one-dimensional chain, and found solitonic behavior, which is the opposite of thermalization. That turned out to be the heart of the phenomenon.
Solitary wave phenomena had been a 19th century mystery dating back to work by John Scott Russell
John Scott Russell
John Scott Russell was a Scottish naval engineer who built the Great Eastern in collaboration with Isambard Kingdom Brunel, and made the discovery that gave birth to the modern study of solitons.-Personal life:John Scott Russell was born John Russell on 9 May 1808 in Parkhead, Glasgow, the son of...
who, in 1834, observed what we now call a soliton, propagating in a canal, and chased it on horseback. In spite of his observations of solitons in wave tank experiments, Scott Russell never recognized them as such, because of his focus on the “great wave of translation,” the largest amplitude solitary wave. His experimental observations, presented in his Report on Waves to the British Association for the Advancement of Science in 1844, were viewed with skepticism by George Airy and George Stokes
George Stokes
George Stokes is the name of:* Sir George Stokes, 1st Baronet , Irish mathematician and physicist* George Henry Stokes , Member of Canadian Parliament* George Thomas Stokes , Irish historian...
because their linear water wave theories were unable to explain them. Joseph Boussinesq (1871) and Lord Rayleigh (1876) published mathematical theories justifying Scott Russell’s observations. In 1895, Diederik Korteweg
Diederik Korteweg
Diederik Johannes Korteweg was a Dutch mathematician. He is now remembered as the joint discoverer of the Korteweg–de Vries equation.-Early life and education:...
and Gustav de Vries
Gustav de Vries
Gustav de Vries was a Dutch mathematician, who is best remembered for his work on the Korteweg–de Vries equation with Diederik Korteweg. He was born on 22 January 1866 in Amsterdam, and studied at the University of Amsterdam with the distinguished physical chemist Johannes van der Waals and with...
formulated the KdV equation to describe shallow water waves (such as the waves in the canal observed by Russell), but the essential properties of this equation were not understood until the work of Kruskal and his collaborators in the 1960s.
Solitonic behavior suggested that the KdV equation must have conservation laws beyond the obvious conservation laws of mass, energy, and momentum. A fourth conservation law was discovered by Gerald Whitham and a fifth one by Kruskal and Zabusky. Several new conservation laws were discovered by hand by Robert Miura, who also showed that many conservation laws existed for a related equation known as the Modified Korteweg–de Vries (MKdV) equation. With these conservation laws, Miura showed a connection (called the Miura transformation) between solutions of the KdV and MKdV equations. This was a clue that enabled Kruskal, with Clifford S. Gardner, John M. Greene, and Miura, to discover a general technique for exact solution of the KdV equation and understanding of its conservation laws. This was the inverse scattering method, a surprising and elegant method that demonstrates that the KdV equation admits an infinite number of Poisson-commuting conserved quantities and is completely integrable. This discovery gave the modern basis for understanding of the soliton phenomenon: the solitary wave is recreated in the outgoing state because this is the only way to satisfy all of the conservation laws.
The inverse scattering method has had an astonishing variety of generalizations and applications in different areas of mathematics and physics. Kruskal himself pioneered some of the generalizations, such as the existence of infinitely many conserved quantities for the sine-Gordon equation
Sine-Gordon equation
The sine–Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally considered in the nineteenth century in the course of study of surfaces of constant negative...
. This led to the discovery of an inverse scattering method for that equation by M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur. The sine-Gordon equation is a relativistic wave equation in 1+1 dimensions that also exhibits the soliton phenomenon and which became an important model of solvable relativistic field theory.
Solitons are now known to be ubiquitous in nature, from physics to biology. In 1986, Kruskal and Zabusky shared the Howard N. Potts Gold Medal
Howard N. Potts Medal
The Howard N. Potts Medal was a science and engineering award presented by the Franklin Institute, of Philadelphia, Pennsylvania, USA.Also see The Franklin Institute Awards.-Laureates:Following people received the Howard N. Potts Medal:...
from the Franklin Institute “for contributions to mathematical physics and early creative combinations of analysis and computation, but most especially for seminal work in the properties of solitons.” In awarding the 2006 Steele Prize to Gardner, Greene, Kruskal, and Miura, the American Mathematical Society stated that before their work “there was no general theory for the exact solution of any important class of nonlinear differential equations.” The AMS added, “In applications of 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.”
Kruskal received the National Medal of Science
National Medal of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social sciences, biology, chemistry, engineering, mathematics and...
in 1993 “for his influence as a leader in nonlinear science for more than two decades as the principal architect of the theory of soliton solutions of nonlinear equations of evolution.”
In an article surveying the state of mathematics at the turn of the millennium, the eminent mathematician Philip A. Griffiths wrote that the discovery of integrability of the KdV equation “exhibited in the most beautiful way the unity of mathematics. It involved developments in computation, and in mathematical analysis, which is the traditional way to study differential equations. It turns out that one can understand the solutions to these differential equations through certain very elegant constructions in algebraic geometry. The solutions are also intimately related to representation theory, in that these equations turn out to have an infinite number of hidden symmetries. Finally, they relate back to problems in elementary geometry.”
In the 1980s, Kruskal developed an acute interest in the Painlevé
Paul Painlevé
Paul Painlevé was a French mathematician and politician. He served twice as Prime Minister of the Third Republic: 12 September – 13 November 1917 and 17 April – 22 November 1925.-Early life:Painlevé was born in Paris....
equations. They frequently arise as symmetry reductions of soliton equations, and Kruskal was intrigued by the intimate relationship that appeared to exist between the properties characterizing these equations and completely integrable systems. Much of his subsequent research was driven by a desire to understand this relationship and to develop new direct and simple methods for studying the Painlevé equations. Kruskal was rarely satisfied with the standard approaches to differential equations.
The six Painlevé equations
Painlevé transcendents
In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property , but which are not generally solvable in terms of elementary functions...
have a characteristic property called the Painlevé property: their solutions are single-valued around all singularities whose locations depend on the initial conditions. In Kruskal’s opinion, since this property defines the Painlevé equations, one should be able to start with this, without any additional unnecessary structures, to work out all the required information about their solutions. The first result was an asymptotic study of the Painlevé equations with Nalini Joshi
Nalini Joshi
Nalini Joshi is an Australian mathematician.Joshi is currently a Professor in the School of Mathematics and Statistics at the University of Sydney , and President of the Australian Mathematical Society...
, unusual at the time in that it did not require the use of associated linear problems. His persistent questioning of classical results led to a direct and simple method, also developed with Joshi, to prove the Painlevé property of the Painlevé equations.
In the later part of his career, one of Kruskal's chief interests was the theory of surreal numbers. Surreal numbers, which are defined constructively, have all the basic properties and operations of the real numbers. They include the real numbers alongside many types of infinities and infinitesimals. Kruskal contributed to the foundation of the theory, to defining surreal functions, and to analyzing their structure. He discovered a remarkable link between surreal numbers, asymptotics, and exponential asymptotics. A major open question is whether sufficiently well behaved surreal functions possess definite integrals. A positive answer would have deep implications in analysis. Over the last years of his life, Kruskal worked with much dedication on defining surreal integration. Much was learned but important parts of this project remain unfinished. At the time of his death, he was in the process of writing a book on surreal analysis with O. Costin.
Kruskal coined the term Asymptotology
Asymptotology
Asymptotology has been defined as “the art of dealing with applied mathematical systems in limiting cases” as well as “the science about the synthesis of simplicity and exactness by means of localization.-Principles:...
to describe the “art of dealing with applied mathematical systems in limiting cases”. He formulated seven Principles of Asymptotology: 1. The Principle of Simplification; 2. The Principle of Recursion; 3. The Principle of Interpretation; 4. The Principle of Wild Behaviour; 5. The Principle of Annihilation; 6. The Principle of Maximal Balance; 7. The Principle of Mathematical Nonsense.
The term asymptotology
Asymptotology
Asymptotology has been defined as “the art of dealing with applied mathematical systems in limiting cases” as well as “the science about the synthesis of simplicity and exactness by means of localization.-Principles:...
is not so widely used as the term soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
. Asymptotic methods of various types have been successfully used since almost the birth of science itself. Nevertheless, Kruskal tried to show that asymptotology is a special branch of knowledge, intermediate, in some sense, between science and art. His proposal has been found to be very fruitful.
Selected awards and honors
- Gibbs Lecturer, American Mathematical Society (1979);
- Dannie Heinemann Prize, American Physical Society (1983);
- Howard N. Potts Gold MedalHoward N. Potts MedalThe Howard N. Potts Medal was a science and engineering award presented by the Franklin Institute, of Philadelphia, Pennsylvania, USA.Also see The Franklin Institute Awards.-Laureates:Following people received the Howard N. Potts 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)
Memberships
- Member of the National Academy of Sciences (1980) and the American Academy of Arts and Sciences (1983)
- Foreign member of the Royal Society of London (1997), the Royal Society of Edinburgh (2001), and the Russian Academy of Arts and Sciences.
External links
- In Memoriam: Martin David Kruskal
- “Fermi–Pasta–Ulam, Solitons and the Fabric of Nonlinear and Computational Science: History, Synergetics, and Visiometrics”, CHAOS 15, 1 2005,. DOI: 10.1063/1.1861554 American Inst. of Physics
- Solitons, Singularities, Surreals and Such: A Conference in Honor of Martin Kruskal's Eightieth Birthday
- NY Times Obituary, 01-13-07
- Princeton University Weekly Bulletin Obituary, 02-05-07
- Chris Eilbeck/Heriot–Watt University, Edinburgh, UK
- Los Angeles Times Obituary, 01-06-07
- Obituaries – Martin David Kruskal Society for Industrial and Applied Mathematics, April 11, 2007