1. Asymptotic methods for the solution of linear and nonlinear equations of mathematical physics.
2. Semiclassical quantization of nonintegrable Hamiltonian systems.
3. Asymptotic methods for financial mathematics.
4. Reaction-diffusion equations.
Main publications:
Lisok A.L., Shapovalov A.V. and Trifonov A.Yu., “Symmetry and Intertwining Operators for the Nonlocal Gross–Pitaevskii Equation”, Sym., Integ. and Geom.: Meth. and Appl., 9 (2013), 066, 1–21
Belov V.V., Litvinets F.N. and Trifonov A Yu., “The semiclassical spectral series for a Hartree-type equation corresponding to a rest point of the Hamilton–Ehrenfest system”, Theor. Math. Phys., 150:1 (2007), 26–40
Belov V.V., Trifonov A.Yu. and Shapovalov A.V., “The trajectory-coherent approximation and the system of moments for the Hartree type equation”, Int. J. of Math. and Math. Scien., 32:6 (2002), 325–370
Bagrov V.G., Belov V.V., Trifonov A.Yu., “Semiclassical trajectory-coherent approximation in quantum mechanics: I. High order corrections to multidimensional time-dependent equations of Schrodinger type”, Ann. of Phys. (N.Y.), 246:2 (1996), 231–290
Bagrov V.G., Belov V.V., Yevseyevich A.A., Trifonov A.Yu., “Quasiclassical spectral series of the Dirac operators corresponding to quantized two-dimensional Lagrangian tori”, J. Phys. A: Math. Gen., 27:15 (1994), 5273–5306
E. A. Levchenko, A. Yu. Trifonov, A. V. Shapovalov, “Semiclassical approximation for the nonlocal multidimensionalfisher-kolmogorov-petrovskii-piskunov equation”, Computer Research and Modeling, 7:2 (2015), 205–219
2013
2.
E. A. Levchenko, A. Yu. Trifonov, A. V. Shapovalov, “Large-time asymptotic solutions of the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation”, Computer Research and Modeling, 5:4 (2013), 543–558
Aleksandr L. Lisok, Aleksandr V. Shapovalov, Andrey Yu. Trifonov, “Symmetry and Intertwining Operators for the Nonlocal Gross–Pitaevskii Equation”, SIGMA, 9 (2013), 066, 21 pp.
A. V. Borisov, A. Yu. Trifonov, A. V. Shapovalov, “Convection effect on two-dimensional dynamicsin the nonlocal reaction-diffusion model”, Computer Research and Modeling, 3:1 (2011), 55–61
2010
5.
R. O. Rezaev, A. Yu. Trifonov, A. V. Shapovalov, “The Einstein–Ehrenfest system of $(0,M)$-type and asymptotical solutions of the multidimensional nonlinear Fokker–Planck–Kolmogorov equation”, Computer Research and Modeling, 2:2 (2010), 151–160
6.
A. V. Borisov, A. Yu. Trifonov, A. V. Shapovalov, “Numerical modeling of population 2D-dynamics with nonlocal interaction”, Computer Research and Modeling, 2:1 (2010), 33–40
A. V. Borisov, A. Yu. Trifonov, A. V. Shapovalov, “Semiclassical solutions localized in a neighborhood of a circle for the Gross–Pitaevskii equation”, Computer Research and Modeling, 1:4 (2009), 359–365
8.
A. V. Shapovalov, A. Yu. Trifonov, E. A. Masalova, “Semiclassical asymptotics of nonlinear Fokker–Plank equation for distributions of asset returns”, Computer Research and Modeling, 1:1 (2009), 41–49
Alexander V. Shapovalov, Roman O. Rezaev, Andrey Yu. Trifonov, “Symmetry Operators for the Fokker–Plank–Kolmogorov Equation with Nonlocal Quadratic Nonlinearity”, SIGMA, 3 (2007), 005, 16 pp.
V. V. Belov, F. N. Litvinets, A. Yu. Trifonov, “Semiclassical spectral series of a Hartree-type operator corresponding
to a rest point of the classical Hamilton–Ehrenfest system”, TMF, 150:1 (2007), 26–40; Theoret. and Math. Phys., 150:1 (2007), 21–33
Alexey Borisov, Alexander Shapovalov, Andrey Trifonov, “Transverse Evolution Operator for the Gross–Pitaevskii Equation in Semiclassical Approximation”, SIGMA, 1 (2005), 019, 17 pp.
Alexander Shapovalov, Andrey Trifonov, Alexander Lisok, “Exact Solutions and Symmetry Operators for the Nonlocal Gross–Pitaevskii Equation with Quadratic Potential”, SIGMA, 1 (2005), 007, 14 pp.
A. L. Lisok, A. Yu. Trifonov, A. V. Shapovalov, “Symmetry operators of a Hartree-type equation with quadratic potential”, Sibirsk. Mat. Zh., 46:1 (2005), 149–165; Siberian Math. J., 46:1 (2005), 119–132
A. L. Lisok, A. Yu. Trifonov, A. V. Shapovalov, “Green's Function of a Hartree-Type Equation with a Quadratic Potential”, TMF, 141:2 (2004), 228–242; Theoret. and Math. Phys., 141:2 (2004), 1528–1541
V. V. Belov, A. Yu. Trifonov, A. V. Shapovalov, “Semiclassical Trajectory-Coherent Approximations of Hartree-Type Equations”, TMF, 130:3 (2002), 460–492; Theoret. and Math. Phys., 130:3 (2002), 391–418
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