ELFT HÍRADÓ

["Steinczinger Zsuzsanna" <steinczinger.zsuzsa AT wigner.hun-ren.hu>]

Kedves  Kollégák!

 

 

 

Továbbítok egy meghívót a Rényi Alfréd Kutatóintézetben szeptember 21-én 
hétfőn délután, Lax Péter 100. születésnapja tiszteletére rendezett 
eseményre, amelyen Forgács Péter a Részecskefizikai és Térelméleti 
Kutatócsoport emeritus kutatója is előadó lesz. 

 

 

Az esemény honlapja itt érhető el:

https://renyi.hu/en/events/egyeb/lax-100-colloquium-honour-peter-d-laxs-centennial-anniversary

 

A részvétel ingyenes, de regisztrációhoz kötött. 

 


LAX-100: A Colloquium in Honour of Peter D. Lax's Centennial Anniversary


To celebrate the 100th anniversary of  
<https://en.wikipedia.org/wiki/Peter_Lax> Péter Lax’s birth, we are 
organizing a special institute colloquium consisting of three presentations 
on September 21, 2026.



  _____  



Schedule:


14:15-14:20


*         <https://renyi.hu/en/staff/andras-stipsicz> András Stipsicz (Rényi 
Institute, Budapest)
Opening


14:20-15:10


*         
<https://www.mpg.de/18604789/mathematics-in-the-sciences-szekelyhidi> László 
Székelyhidi Jr. (Max Planck Institute, Leipzig)
The uniqueness problem for hyperbolic systems


15:10-15:40

*       coffee break

15:40-16:30


*         <https://mta.hu/koztestuleti_tagok?PersonId=12302> Péter Forgács 
(Wigner Institute, Budapest)
Solitons, integrability and Lax pairs


16:30-16:40

*       short break

16:40-17:30


*         <https://renyi.hu/en/staff/balazs-maga> Balázs Maga and  
<https://renyi.hu/en/staff/balint-toth> Bálint Tóth (Rényi Institute, 
Budapest)
Hyperbolic conservation laws and hydrodynamic limits


17:45-19:00

*       reception



  _____  



Abstracts of the talks:



 <https://www.mpg.de/18604789/mathematics-in-the-sciences-szekelyhidi> 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.

 <https://mta.hu/koztestuleti_tagok?PersonId=12302> 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.


 <https://renyi.hu/en/staff/balazs-maga> Balázs Maga and  
<https://renyi.hu/en/staff/balint-toth> Bálint Tóth (Rényi Institute, 
Budapest)


Title: Hyperbolic conservation laws and hydrodynamic limits


Abstract: Stochastic interacting particle systems model microscopic 
"physical" dynamics probabilistically, through random local transitions of 
the configurations, conserving some meaningful physical observables (like 
number of particles of various types, momentum, etc.). Under an appropriate 
("Eulerian") space-time scaling — the hydrodynamic limit — the large-scale 
collective behaviour of the system becomes deterministic and expectedly can 
be described by a system of partial differential equations involving the 
local densities of the conserved observables. This passage from stochastic 
microscopic dynamics to deterministic macroscopic evolution provides a 
striking mathematical explanation of how continuum laws emerge from 
interacting discrete particle systems -- in line with Hilbert's sixth problem.
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.