2,340 research outputs found

### Transverse Evolution Operator for the Gross-Pitaevskii Equation in Semiclassical Approximation

The Gross-Pitaevskii equation with a local cubic nonlinearity that describes
a many-dimensional system in an external field is considered in the framework
of the complex WKB-Maslov method. Analytic asymptotic solutions are constructed
in semiclassical approximation in a small parameter $\hbar$, $\hbar\to 0$, in
the class of functions concentrated in the neighborhood of an unclosed surface
associated with the phase curve that describes the evolution of surface vertex.
The functions of this class are of the one-soliton form along the direction of
the surface normal. The general constructions are illustrated by examples.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA

### Exact Solutions and Symmetry Operators for the Nonlocal Gross-Pitaevskii Equation with Quadratic Potential

The complex WKB-Maslov method is used to consider an approach to the
semiclassical integrability of the multidimensional Gross-Pitaevskii equation
with an external field and nonlocal nonlinearity previously developed by the
authors. Although the WKB-Maslov method is approximate in essence, it leads to
exact solution of the Gross-Pitaevskii equation with an external and a nonlocal
quadratic potential. For this equation, an exact solution of the Cauchy problem
is constructed in the class of trajectory concentrated functions. A nonlinear
evolution operator is found in explicit form and symmetry operators (mapping a
solution of the equation into another solution) are obtained for the equation
under consideration. General constructions are illustrated by examples.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA

### The Dirac equation in an external electromagnetic field: symmetry algebra and exact integration

Integration of the Dirac equation with an external electromagnetic field is
explored in the framework of the method of separation of variables and of the
method of noncommutative integration. We have found a new type of solutions
that are not obtained by separation of variables for several external
electromagnetic fields. We have considered an example of crossed electric and
magnetic fields of a special type for which the Dirac equation admits a
nonlocal symmetry operato

### An application of the Maslov complex germ method to the 1D nonlocal Fisher-KPP equation

A semiclassical approximation approach based on the Maslov complex germ
method is considered in detail for the 1D nonlocal
Fisher-Kolmogorov-Petrovskii-Piskunov equation under the supposition of weak
diffusion. In terms of the semiclassical formalism developed, the original
nonlinear equation is reduced to an associated linear partial differential
equation and some algebraic equations for the coefficients of the linear
equation with a given accuracy of the asymptotic parameter. The solutions of
the nonlinear equation are constructed from the solutions of both the linear
equation and the algebraic equations. The solutions of the linear problem are
found with the use of symmetry operators. A countable family of the leading
terms of the semiclassical asymptotics is constructed in explicit form.
The semiclassical asymptotics are valid by construction in a finite time
interval. We construct asymptotics which are different from the semiclassical
ones and can describe evolution of the solutions of the
Fisher-Kolmogorov-Petrovskii-Piskunov equation at large times. In the example
considered, an initial unimodal distribution becomes multimodal, which can be
treated as an example of a space structure.Comment: 28 pages, version accepted for publication in Int. J. Geom. Methods
Mod. Phy

### The Shapovalov determinant for the Poisson superalgebras

Among simple Z-graded Lie superalgebras of polynomial growth, there are
several which have no Cartan matrix but, nevertheless, have a quadratic even
Casimir element C_{2}: these are the Lie superalgebra k^L(1|6) of vector fields
on the (1|6)-dimensional supercircle preserving the contact form, and the
series: the finite dimensional Lie superalgebra sh(0|2k) of special Hamiltonian
fields in 2k odd indeterminates, and the Kac--Moody version of sh(0|2k). Using
C_{2} we compute N. Shapovalov determinant for k^L(1|6) and sh(0|2k), and for
the Poisson superalgebras po(0|2k) associated with sh(0|2k). A. Shapovalov
described irreducible finite dimensional representations of po(0|n) and
sh(0|n); we generalize his result for Verma modules: give criteria for
irreducibility of the Verma modules over po(0|2k) and sh(0|2k)

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