Symmetries and conservation laws of partial differential equations: Basic notions and results

The main notions and results which are necessary for finding higher symmetries and conservation laws for general systems of partial differential equations are given. These constitute the starting point for the subsequent papers of this volume. Some problems are also discussed.     

Scattering of solitons for the Schrödinger equation coupled to a particle

We establish soliton-like asymptotics for finite energy solutions to the Schrödinger equation coupled to a nonrelativistic classical particle. Any solution with initial state close to the solitary manifold converges to a sum of a travelling wave and an outgoing free wave. The convergence holds in global energy norm. The proof uses spectral theory and the symplectic projection onto the solitary manifold in the Hilbert phase space.     

On the variational representation of solutions to some quasilinear equations and systems of hyperbolic type on the basis of potential theory

We demonstrate a method that permits to obtain generalized solutions for some quasilinear equations and systems of hyperbolic type. The corresponding variational principle is constructed using the theory of equilibrium of a potential in an external field. Dedicated to the memory of B. M. Levitan Supported by RFBR grants Nos. 05-01-00522 and NSh-1551.2003.1, by Program No. 1 of the Branch of Mathematics, Russian Academy of Sciences, and by INTAS project No. 03-51-6637.     

Observation of the nonlinear effect in relativistic Thomson scattering of electron and laser beams

Thomson scattering of high-power laser and electron beams is a good test of electrodynamics in the high-field region. We demonstrated production of high-intensity X-rays in the head-on collision of a CO₂ laser and 60-MeV electron beams at Brookhaven National Laboratory, Accelerator Test Facility. The energy of an X-ray photon was limited at 6.5 keV in the linear (lowest order) Thomson scattering, but the nonlinear (higher order) process produces higher energy X-rays. We measured the angular distribution of the high-energy X-rays and confirmed that it agrees with theoretical predictions.     

Advantages of NIR radiation use for optical determination of skin horny layer thickness with embedded TiO2 nanoparticles during tape stripping procedure

The tape stripping technique is an experimental method frequently used for reconstruction of the in-depth distribution of various topically administered substances within the horny layer of human skin, e.g., compounds contained in sunscreens. Titanium dioxide (TiO₂) nanoparticles (25–200 nm in diameter) are one such compound. Optical techniques which apply blue light are found to be suitable for reconstruction. However, the presence of particles affects the light propagation within the skin and therefore causes incorrect determination of strip thickness, leading to an improper reconstructed distribution of nanoparticle concentration revealed from the experimental data. This study evaluates the errors emerging from the use of blue (400 nm) and NIR (800 nm) radiation and finds the use of longer wavelength light more advantageous. Particles of different diameters are considered, and it is revealed that the application of small particles (25–60 nm) results in the lowest rate of error.     

On construction of symmetries from integrals of hyperbolic partial differential systems

An algorithm is proposed which allows one to construct higher symmetries of arbitrary order for some special classes of hyperbolic systems possessing integrals. The Pohlmeyer-Lund-Regge system and the open two-dimensional Toda lattices are shown to belong to the class of systems where our algorithm is applicable. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 1, Geometry of Integrable Models, 2004.     

General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas

We show that the Lagrangian for interacting nonrelativistic particles can be coupled to an external gauge field and metric tensor in a way that exhibits a nonrelativistic version of general coordinate invariance. We explore the consequences of this invariance on the example of the degenerate Fermi gas at infinite scattering length, where conformal invariance also plays an important role. We find the most general effective Lagrangian consistent with both general coordinate and conformal invariance to leading and next-to-leading orders in the momentum expansion. At the leading order the Lagrangian contains one phenomenological constant and reproduces the results of the Thomas-Fermi theory and superfluid hydrodynamics. At the next-to-leading order there are two additional constants. We express various physical quantities through these constants.