Long-time asymptotics of solutions of the heat equation
The long-time asymptotics of solutions of the Cauchy problem for the heat equation are constructed in the case when the initial function at infinity has power asymptotics.
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arXiv subjects
Sergei V. Zakharov
Publications and source records attributed to Sergei V. Zakharov.
Large-gradient and Lagrange singularities of solutions of a quasilinear parabolic equation
Asymptotic solutions of a quasilinear parabolic equation with a small parameter at the higher derivative are constructed near large-gradient and Lagrange singularities of A-type, which represent interest for studying processes of shock waves formation in physical media with a small nonzero viscosity.
math.AP↗
Asymptotics of the convolution of the Airy function and a function of the power-like behavior
The asymptotic behavior of the convolution-integral of a special form of the Airy function and a function of the power-like behavior at infinity is obtained. The integral under consideration is the solution of the Cauchy problem for an evolutionary third-order partial differential equation used in the theory of wave propagation in physical media with dispersion. The obtained result can be applied to studying asymptotics of solutions of the KdV equation by the matching method.
math.AP↗
Formation of compression waves with multiscale asymptotics in the Burgers and KdV models
The Cauchy problem for the Burgers equation with a small dissipation and an initial weak discontinuity and the Cauchy problem with a large initial gradient for a quasilinear parabolic equation and for the Korteweg-de Vries (KdV) equation are considered. Multiscale asymptotics of solutions corresponding to shock waves are constructed. Some results can also be applied to rarefaction waves.
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Two-parameter asymptotics in the Cauchy problem for a parabolic equation
The Cauchy problem for a quasi-linear parabolic equation with a small parameter at a higher derivative is considered. The initial step-like function contains another small parameter. Formal asymptotic solutions of the problem in small parameters are constructed.
math.AP↗
Renormalized asymptotic solutions of the Burgers equation and the Korteweg-de Vries equation
The Cauchy problem for the Burgers equation and the Korteweg-de Vries equation is considered. Uniform renormalized asymptotic solutions are constructed in cases of a large initial gradient and a perturbed initial weak discontinuity.
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Asymptotics of solutions of a parabolic equation near singular points
Results of investigation of the asymptotic behavior of solutions to the Cauchy problems for a quasi-linear parabolic equation with a small parameter at a higher derivative near singular points of limit solutions are presented. Interest to the problem under consideration is explained by its applications to a wide class of physical systems and probabilistic processes such as acoustic waves in fluid and gas, hydrodynamical turbulence and nonlinear diffusion. The following cases are considered: a singularity generated by a jump discontinuity of the initial function, collision of two shock waves, gradient catastrophe, transition of a weak discontinuity into a shock wave, a singularity generated by a large initial gradient.
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