Wilhelm Fushchych was born on December 18, 1936 in the village Siltze of the Zakarpattya (Transcarpathian) Region of Ukraine. He has graduated from the Uzhgorod University (1958) and finished the post-graduate course at the Institute of Mathematics (1963). He defended his Ph.D. thesis in 1964, and the doctor (Dr. Sc.) thesis in 1971. He worked at the Institute of Mathematics of the National Academy of Sciences of Ukraine since 1963, and was the head of the Department of Applied Research since 1978. He was elected a Corresponding Member of the National Academy of Sciences of Ukraine in 1987.
The field of research interests of Wilhelm Fushchych included quantum field theory, representations of Lie groups and algebras, subgroup structure of Lie groups, group-theoretical analysis of differential equations, etc. However, the principal topic of his papers is symmetry in mathematical physics. Today this particular branch of mathematical physics is developing extremely quickly, and the papers and monographs by W. Fushchych have contributed considerably to this process.
In the series of papers published in the 70-ies, W. Fushchych solved a fundamental problem of mathematical physics which attracted much attention of such famous scientists as Wigner, Bargmann, Schwinger, Harish-Chandra and others. The essence of this problem is a description of multidimensional systems of differential and integro-differential equations, which are invariant with respect to the Galilei and Poincaré groups and satisfy some important physical requirements. To solve this problem, W. Fushchych suggested a new (non-Lagrangian) approach to construction and investigation of equations of motion in quantum mechanics. These equations do not lead to difficulties with causality violation, which are typical for other equations for higher spin particles. On the other hand, the equations found by W. Fushchych present convenient models for particles with higher spins, and these models can be exactly solved in many physically interesting cases. On the basis of this method, he derived new equations of motion for particles of arbitrary spin. In particular, three types of equations of motion for massless particles with spin 1/2 were found. One of them coincides with the equation which was discovered in 1929 by H. Weyl, and two others were not known before. The new equation of motion, found by W. Fushchych, is invariant with respect to the Galilei group and correctly describes spin-orbit and Darwin interactions. It presents an essentially new view on the nature of these interactions, earlier interpreted as relativistic effects which can be consistently described only in theories invariant with respect to the Poincaré group.
W. Fushchych has suggested equations of motion which have the symmetry intermediate between the Galilei and Poincaré groups. Mechanics based on these equations of motion provides the dependence of mass on velocity, but the relevant maximal velocity can exceed the velocity of light in vacuum. A new equation of motion for electromagnetic waves has also been suggested. For this equation the velocity of light in vacuum is not a constant but a nonlinear function of the strength of electric field.
In the papers by W. Fushchych and his students and coworkers, there were investigated and classified in detail irreducible representations of generalized Poincaré groups in multidimensional spaces, carried out the subgroup analysis of these groups and reduction by the subgroups found. He has discovered the equation which is simultaneously invariant under the Galilei and Lorentz transformations. On the basis of these results, the mathematical foundations of quantum mechanics for particles with variable mass and spin were constructed. Poincaré-invariant equations of motion in multidimensional spaces were found, and the reduction to equations in four-dimensional spaces was performed.
W. Fushchych was the first to discover hidden symmetries and new integrals of motion for a collection of equations of mathematical physics, which includes the Maxwell, Dirac, and Lamé equations. In particular, for the Maxwell equation he found an eight-parametrical group of integral symmetry transformations, which extends the well-known Heavyside-Larmor-Rainich symmetry. The related conservation laws are connected with the Stokes parameters of the polarization density matrix.
He established the dual symmetry of relativistic wave equations. It was proved that besides the Lorentz transformations these equations admit such a (nonlocal) transformation at transition to a new inertial reference frame, for which time is not changed. These symmetries in principle cannot be found by means of the classical Sophus Lie method. Thus, W. Fushchych has proposed a new view on the notion of time in the relativistic physics.
The non-Lie approach to investigation of symmetries of differential equations, suggested by W. Fushchych, was further developed in the papers of mathematicians from different countries.
W. Fushchych has formulated a new efficient approach (method of ansatzes) to integration of nonlinear multidimensional partial differential equations. In the framework of this method, he and his coworkers were the first to find wide classes of exact solutions for the nonlinear d'Alembert, Liouville, Schrödinger, Dirac, Maxwell-Dirac and many other equations.
Nobody succeeded in construction of solutions of such complex equations of mathematical physics before works of W. Fushchych.
W. Fushchych suggested a new concept: conditional symmetry. This approach opened new exciting facilities for integration of multidimensional nonlinear equations and helped to attain a deep understanding of mysterious relations among them. At present, conditional symmetries are actively studied by scientists of many countries.
There would be no exaggeration to say that W. Fushchych has opened a new branch of mathematical physics which he called symmetry analysis. As a matter of fact, this very term (and also such widely used terms as non-Lie symmetry, nonlocal symmetry, conditional symmetry, non-Lie method, antireduction, nonlinear mathematical physics) were introduced into mathematical usage by W. Fushchych.
W. Fushchych has published more than 330 papers in leading mathematical and physical journals. He together with coworkers have written 10 books, see the list. These books are widely cited by many researchers.
W. Fushchych created a large scientific school of symmetry analysis in Ukraine. There are 50 Candidates (Ph.D.) and 8 Doctors of Science (Dr.Sc.) among his former students. They now work in different cities of Ukraine and abroad.
W. Fushchych has founded a new international journal -
of Nonlinear Mathematical Physics". Working as Editor-in-Chief and
publisher of this journal, he made an essential contribution to the study
of symmetries in mathematical physics. He actively cooperated with researchers
from different countries of the world.