Leírás
To celebrate the 100th anniversary of Péter Lax’s birth, we are organizing a special institute colloquium consisting of three presentations on September 21, 2026.
Schedule:
14:15-14:20
- András Stipsicz (Rényi Institute, Budapest)
Opening
14:20-15:10
- László Székelyhidi Jr. (Max Planck Institute, Leipzig)
The uniqueness problem for hyperbolic systems
15:10-15:40
- coffee break
15:40-16:30
- Péter Forgács (Wigner Institute, Budapest)
Solitons, integrability and Lax pairs
16:30-16:40
- short break
16:40-17:30
- Balázs Maga and Bálint Tóth (Rényi Institute, Budapest)
Hyperbolic conservation laws and hydrodynamic limits
17:45-19:00
- reception
Abstracts of the talks:
László Székelyhidi Jr. (Max Planck Institute, Leipzig)
Title: The uniqueness problem for hyperbolic systems
Abstract: There is no theory for the initial value problem for compressible flows in two space dimensions once shocks show up, much less in three space dimensions. This is a scientific scandal and a challenge.” In this statement, made in 2007, Peter Lax referred to one of the most vexing outstanding open problems in the theory of partial differential equations: to define a notion of solution for hyperbolic systems of conservation laws, which is both guaranteed to exist and is unique. Whilst in the case of one space dimension considerable progress has been achieved with entropy methods, in large part owing to the work of Lax in the 20. century, the higher dimensional case continues to resist all attempts. In this talk I will survey the work of Lax as well as some more recent attempts on this topic, and will try to explain why the problem is so difficult.
Péter Forgács (Wigner Research Centre for Physics, Budapest)
Title: Solitons, integrability and Lax pairs
Abstract: In "Integrals of nonlinear equations of evolution and solitary waves" [Comm. Pure Appl. Math. 21, 467–490 (1968)] Lax has demonstrated that the “miraculous” solution of Gardner, Greene, Kruskal and Miura (GGKM) of the Cauchy problem of the Korteweg–de Vries (KdV) equation, reposes on finding differential operators L and B associated to the function u(t,x) such that dL/dt = [B,L] if u(t,x) satisfies the KdV equation. A Lax pair is an isospectrality condition for the (self-adjoint) operator L when u(t,x) evolves in time according to the KdV equation. This insight of Lax has turned the pioneering breakthrough of GGKM, which at first appeared to be an isolated lucky case, into a profound principle. It opened up a whole new field on “solving” a number of important nonlinear partial differential equations (PDE) by reducing their solution to that of a series of linear ones. The talk introduces the KdV equation — as the universal model of weak dispersion balanced against weak nonlinearity for unidirectional waves propagating along a line, illustrates some elementary properties of its celebrated solitons. The method of inverse scattering transformation (IST) discovered by GGKM as well as Lax’s own reformulation leading to the generalization of the IST is sketched. The Lax equation has been reformulated as a zero curvature condition by Zakharov and Shabat, which lead to a Riemann-Hilbert holomorphic factorization problem. An important generalization of the KdV equation to two spatial dimensions, the Kadomtsev-Petviashvili equation, has been found to admit a Lax-pair, further enhancing the significance of such “solvable” or “integrable” systems in the spirit of Lax. Finally the generalized Lax-pairs for the self-dual Yang-Mills equations in four and three dimensional Euclidean space and some of their soliton-type solutions – mostly magnetic monopoles – shall be mentioned. The main contributions of the “Budapest group” (Zalán Horváth, László Palla and myself) in the 1980s to the solution of the “multi-monopole” problem, describing n magnetic monopoles in 3 dimensional space in static equilibrium shall be pointed out.
While microscopic evolution is well-defined for all times (at least for locally finite systems), smooth solutions of the interesting hyperbolic conservation laws may develop singularities, such as shocks, in finite time. This apparent discrepancy is resolved by weakening the notion of a solution of the PDE. This shift is not without danger though: such weak solutions are typically not unique, so an additional admissibility principle is needed to identify the solution corresponding to the underlying physical system. Péter Lax's theory of entropy solutions of systems of hyperbolic conservation laws plays a paramount role in properly stating the challenge of (Eulerian) hydrodynamic limits and also in the technical sense of realising them.
We illustrate the passage from interacting particle systems to hyperbolic systems of conservation laws through concrete examples. In homage to Péter Lax, we discuss entropy conditions, how they are used to select the physically meaningful weak solutions, and how some aspects of the PDE theory reflect the physical features of the underlying microscopic particle systems.