The Gross–Pitaevskii Equation with a Nonlocal Interaction in a Semiclassical Approximation on a Curve

We propose an approach to constructing semiclassical solutions for the generalized multidimensional Gross–Pitaevskii equation with a nonlocal interaction term. The key property of the solutions is that they are concentrated on a one-dimensional manifold (curve) that evolves over time. The approach reduces the Cauchy problem for the nonlocal Gross–Pitaevskii equation to a similar problem for the associated linear equation. The geometric properties of the resulting solutions are related to Maslov’s complex germ, and the symmetry operators of the associated linear equation lead to the approximation of the symmetry operators for the nonlocal Gross–Pitaevskii equation.

Short-Wave Asymptotics for Gaussian Beams and Packets and Scalarization of Equations in Plasma Physics

We study Gaussian wave beam and wave packet types of solutions to the linearized cold plasma system in a toroidal domain (tokamak). Such solutions are constructed with help of Maslov’s complex germ theory (short-wave or semi-classical asymptotics with complex phases). The term “semi-classical” asymptotics is understood in a broad sense: asymptotic solutions of evolutionary and stationary partial differential equations from wave or quantum mechanics are expressed through solutions of the corresponding equations of classical mechanics. This, in particular, allows one to use useful geometric considerations. The small parameter of the expansion is h = λ / 2 π L where λ is the wavelength and L the dimension of the system. In order to apply the asymptotic algorithm, we need this parameter to be small, so we deal only with high-frequency waves, which are in the range of lower hybrid waves used to heat the plasma. The asymptotic solution appears to be a Gaussian wave packet divided by the square root of the determinant of an appropriate Jacobi matrix (“complex divergence”). When this determinant is zero, focal points appear. Our approach allows one to write out asymptotics near focal points. We also claim that this approach is very practical and leads to formulas that can be used for numerical simulations in software like Wolfram Mathematica, Maple, etc. For the particular case of high-frequency beams, we present a recipe for constructing beams and packets and show the results of their numerical implementation. We also propose ideas to treat the more difficult general case of arbitrary frequency. We also explain the main ideas of asymptotic theory used to obtain such formulas.

An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher–KPP equation

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) 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–KPP 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.

Uniparental Disomy May Be Associated with NPM Mutations in AML with a Normal Karyotype.

Acute myeloid leukemia (AML) is a heterogeneous group of neoplastic disorders characterized by an abnormal proliferation of the myeloid precursors and a maturation block. A large proportion of AML cases have either a normal karyotype or non-recurrent chromosomal alterations. Underlying genetic lesions of some of these cases have been characterized with the discovery of MLL-internal tandem duplications, activating FLT3 mutations and NPM mutations. Loss of heterozygosity (LOH) derives from the loss of one of the two alleles at a given locus and can be a sign of inactivation of tumor-suppressor genes. We performed a high-resolution genotype analysis on DNA obtained from 19 AML patients with a normal karyotype, both at diagnosis and in samples obtained in complete remission(assessed by multiparametric flow cytometry) using the 10K SNP Array (Affymetrix). Both LOH and copy number analysis, as well as visualization of these analysis were performed by means of the dChip software (M. Lin et al., Bioinformatics (2004), 20:1233–40). A mean call rate of 96.8%. SNP array-based LOH analysis revealed that 4 patients presented large regions of homozygosity at diagnosis which were absent from samples in complete remission. In all four patients copy number analysis indicated no gross chromosomal losses or gains, as was confirmed by conventional cytogenetic analysis. Therefore, it can concluded that the LOH observed in these four patients was due to the presence of uniparental disomy. Simultaneous analysis of FLT-3 internal tandem duplications (FLT-3/ITD), FLT3- D835 mutations, NPM mutations and MLL rearrangements was performed using conventional molecular methods. Two of these patients (UPN2 and UPN12) had FLT-3/ITD in association with NPM mutations. UPN4 had a mutated form of NPM whereas in patient UPN16 FLT-3 and NPM genes were in the germ line configuration. All four cases were negative for MLL rearrangements and FLT-3-D835 mutations. These results suggest that NPM and FLT3 mutations may be associated with acquired somatic recombinations. It remains to be investigated whether there are loci preferentially involved by these events. Uniparental disomy and genetic lesions in normal karyotype AML Patient LOH FLT3 NPM D835 MLL UPN2 13q Mutated Mutated Germ line Germ line UPN4 6pter-p12.212q13.12-qter Germ line Mutated Germ line Germ line UPN12 2p Mutated Mutated Germ line Germ line UPN16 complex Germ line Germ line Germ line Germ line