# Seminars at the Landau Institute scientific council

## Razmernost’ fraktal’nogo interfeisa v prostranstvennoi zadache dilemmy uznika.

25 May in 11:30

L.N. Shchur

V poslednie dvadtsat’ let polucheno mnogo rezul’tatov po klassifikatsii kriticheskogo povedeniya dvumernykh sistem na osnove kriticheskogo povedeniya interfeisov. Nezavisimo ot razvitiya etoi nauki (Evolyutsiya Shrama-Levnera), poyavilis’ utverzhdeniya, osnovannye na chislennom modelirovanii, o tom, chto protekanie v nekotorykh statisticheskikh sistemakh demonstriruet svoistva, tipichnye dlya fazovogo perekhoda pervogo roda. My issledovali geometricheskie svoistva interfeisov v zadache, kotoraya osnovana na klassicheskoi zadache iz teorii igr, zadache dilemmy uznika. V issleduemoi nami modeli igroki raspolagayutsya v uzlakh kvadratnoi reshetki. Kazhdyi igrok na kazhdom shage po vremeni mozhet menyat’ svoyu strategiyu v zavisimosti ot ego okruzheniya. V opredelennom diapazone parametra podscheta vybora strategii takaya deterministicheskaya dinamika privodit k kvazi-statsionarnomu rezhimu, s chetko vyrazhennoi strukturoi. Struktura sostoit iz dvukh tipov klasterov, ravnoi chislu vozmozhnykh strategii. Eti klastery imeyut konechnuyu massu, to est’ ikh razmernost’ ravna dvum. V sluchae perekhoda pervogo roda, dlina granitsy klastera rastet lineino s razmerom sistemy. V sluchae perekhoda vtorogo roda, dlina granitsy rastet bystree lineinoi, no medlennee, chem ploshchad’. V nashem sluchae, okazalos’, chto asimptoticheski razmernost’ interfeisa rastet kvadratichno. Eto dovol’no neozhidannyi rezul’tat. Izvestno, chto regulyarnye fraktaly mogut imet’ razmernost’ 2. Dlya sluchainykh fraktalov takoi sluchai ranee byl neizvesten.

## Spetsial’naya geometriya v strunnykh kompaktifikatsiyakh i kiral’nye algebry

1 June in 11:30

Konstantin Aleshkin

V doklade ya planiruyu izlozhit’ nedavnie uspekhi v vychislenii geometrii prostranstv moduleimnogoobrazii Kalabi-Yau, neobkhodimykh dlya opisaniya nizkoenergetichnogo predela teorii strun. Sovmestno s A. Belavinym nami byli polucheny obshchie formuly dlya normalizatsii konstant svyazi nizkoenergetichnoi teorii (spetsial’naya geometriya). A imenno, ispol’zuya svyaz’ s kiral’nymi algebrami, my napisali yavnye formuly v vide ryadov dlya spetsial’noi geometrii dlya klassov mnogoobrazii Kalabi-Yau tipa Ferma.

## Dynamics of Poles in 2D Hydrodynamics with Free Surface: New Constants of Motion

8 June in 11:30

A. I. Dyachenko, S. A. Dyachenko, __P. M. Lushnikov__ and V. E. Zakharov

We consider Euler equations for potential flow of ideal incompressible fluid with a free surface and infinite depth in two dimensional geometry. We admit a presence of gravity forces and surface tension. A time-dependent conformal mapping z(w,t) of the lower complex half-plane of the variable w into the area filled with fluid is performed with the real line of w mapped into the free fluid's surface. We study the dynamics of singularities of both z(w,t) and the complex fluid potential Pi(w,t) in the upper complex half-plane of w. We show the existence of solutions with an arbitrary finite number N of simple complex poles in z_w(w,t) and Pi_w(w,t) which are the derivatives of z(w,t) and Pi(w,t) over w. These poles are often coupled with branch points located at other points of the upper half-plane of w. We find that the residues of the simple poles of z_w(w,t) are new, previously unknown constants of motion, provided surface tension is zero. All these constants of motion commute with each other in the sense of underlying Hamiltonian dynamics. In absence of both gravity and surface tension, the residues of simple poles of Pi_w(w,t) are also the constants of motion. For nonzero gravity and zero surface tension, the residues of poles of any order of Pi_w(w,t) are the trivial linear functions of time. Nonzero surface tension allows residues of poles of even order to be compatible with the fluid dynamics. We also found solutions with N higher order poles. In all above cases the number of independent real integrals of motion is 4N for zero gravity and 4N-1 for nonzero gravity. We suggest that the existence of these nontrivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics in deep water.

## Quantum Many-Body Physics of Qubits

22 June in 11:30

L. Glazman (Yale University)

The ongoing development of superconducting qubits has brought some basic questions of many-body physics to the research forefront, and in some cases helped solving them. I will address two effects in quantum condensed matter highlighted by the development of a fluxonium qubit. The first one is the so-called cosine-phi problem stemming from the seminal paper of Brian Josephson: the predicted there phase dependence of the dissipative current across a Josephson junction was observed in a fluxonium, after nearly 50 years of unsuccessful attempts by other techniques. The second one is the dynamics of a weakly-pinned charge density wave: we predict that the dynamics may be revealed in measurements of microwave reflection off a superinductor, which is a key element of the fluxonium.

Seminars are held on Fridays in the conference hall of Landau Institute for Theoretical Physics in Chernogolovka, beginning at 11:30.

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