Изучается асимптотическое поведение экспоненциального интеграла, в котором фазовая функция имеет вид специальной деформации ростка гиперболической унимодальной особенности $T_{4,4,4}$. Исследуемый интеграл удовлетворяет уравнению теплопроводности, его преобразование Коула-Хопфа дает решение векторного уравнения Бюргерса в четырехмерном пространстве-времени, а его главные асимптотические приближения выражаются через вещественные решения систем алгебраических уравнений третьей степени. Установленные аналитические результаты позволяют увидеть бифуркации асимптотической структуры, зависящей от величины параметра модуля особенности.

Рассматривается задача Коши для кубического нелинейного уравнения Шредингера с большим градиентом начальной функции и малым параметром дисперсии. Методом ренормализации строится асимптотическое решение в явном виде интегральной свертки. Устанавливается асимптотический аналог ренормгруппового свойства при масштабных преобразованиях, определяемых параметром дисперсии. В случае отрицательного коэффициента фокусировки получено уточняющее выражение для асимптотического решения через известные эллиптические специальные функции.

An asymptotic approximation, as time increases without limit, is constructed to the solution of the Cauchy problem for the heat equation in three-dimensional space. The locally integrable initial function, which does not necessarily tend to zero at infinity, is assumed to have powerlike asymptotics. The method of introduction of an auxiliary parameter, which also involves the regularization of singularities in integrals, plays the central role in the research. The asymptotic expression for the solution is shown to have the form of a series in negative half-integer powers of the time variable, with coefficients depending on self-similar variables and the logarithm of time; the leading term is found explicitly. Using the example of the Cauchy problem for the vector Burgers equation, it is shown that to perform an asymptotic analysis of the solution by the matching method one needs to construct an asymptotic approximation to a solution of the heat equation. Bibliography: 31 titles.

For a parabolic Hamilton–Jacobi type equation S_t+2^-1(S_x)^2+V(x,ε)=S_xx , a special asymptotic solution with a prescribed asymptotic expansion of the potential function is constructed. Since this asymptotic expansion is chosen for simplicity in the form of a series in natural powers of the small parameter ε , the asymptotic solution of the equation is presented in the form of a series of perturbation theory in integer powers of ε : S(x,t,ε)=∑_n=0^∞ε^nS_n(x,t) . The leading approximation of the solution is expressed in terms of an exponential integral as S_0(x,t)=-2ln_0^+∞exp(-σ^3 +tσ^2+xσ)dσ, where the versal deformation of the germ of the simple boundary singularity B_3 serves as the phase. The asymptotic behavior of this integral in the space variable at infinity is studied by the Laplace method. On the basis of an integral recurrence formula with the homogeneous initial condition for the remaining coefficients S_n(x,t) , an existence theorem is proved. Exponential estimates of these coefficients are also established; they provide the convergence of the corresponding integral convolutions. A successive growth is shown for the orders of smallness of the residuals remaining after the substitution of the partial sums of the asymptotic solution into the equation under consideration. In addition, it is proved that there exists a unique classical solution and the constructed asymptotic series is its asymptotic expansion. The statement of the problem under consideration is also discussed in the light of known approaches to studying the Hamilton–Jacobi equation. The connection of the obtained result with the general theory of singularities of differentiable maps is shown. .

A bisingular Cauchy problem for a quasilinear parabolic equation with a small parameter at the higher derivative is considered. The initial function depends on the space variable with another small parameter, and conditions are fulfilled under which the statement of the problem becomes a model of the evolution of nonlinear compression waves with a large initial gradient in physical systems in the presence of a small dissipation. In the limit case of the zero dissipation, when the equation under consideration becomes a first-order equation, there is a shock wave, whose origin is a singular point of the solution. Near the singular point, on the scales of the small value of dissipation, an asymptotic solution of the problem in the small parameters is constructed. With the help of the matching method on the basis of an earlier obtained asymptotic solution in a smaller region, it is established that the new asymptotic solution must have the form of a series in integer powers of the ratio of small parameters and its logarithm, and for the coefficients of this series a recurrence system of parabolic equations together with the corresponding asymptotic conditions of matching is obtained. After reducing this recurrence system of problems to integral relations, by applying the method of successive approximations and estimates of integral convolutions, the existence of necessary solutions is proved. In addition, it is shown that the constructed series is suitable in a transitional region of the multiscale evolution of the singularity between its initial stage and the boundary layer near the shock wave, and also in the particular case of the Burgers equation some explicit formulas are given.

The results obtained by Il’in and his school concerning the asymptotic behavior of solutions to the Cauchy problem for the quasi-linear parabolic equation with a small parameter multiplying the higher order derivative in the vicinity of singular points are presented. The equation under examination is of interest because it provides a model of the propagation of nonlinear waves in dissipative continuous media, and the importance of studying solutions in the vicinity of singular points is explained, in particular, by the fact that even though the singular events take a short time, they in many respects determine the subsequent evolution of the solutions. In this paper, we examine five types of singular points the emergence of which is caused by different initial data.

The long-time asymptotic behavior of the solution of the Cauchy problem for the evolutionary third-order Airy equation describing wave propagation in dispersive physical media is derived by using the auxiliary parameter method. For the solution in the form of the convolution of the initial data and the Airy function, an asymptotic Erdélyi expansion in inverse powers of the cube root of the time variable with the coefficients depending on a self-similar variable and the logarithm of time is obtained. To refine the asymptotics, a family of special function classes for its coefficients is introduced. It is pointed out how the used method is connected with the geometrical optics approach and how the obtained result can be applied to nonlinear third-order PDEs.

Установлено асимптотическое поведение на больших временах решения задачи Коши для уравнения Эйри - эволюционного уравнения третьего порядка. Предполагается, что начальная функция локально интегрируема по Лебегу и имеет степенную асимптотику на бесконечности. С использованием метода вспомогательного параметра и регуляризации особенностей для решения в виде интеграла свертки с функцией Эйри получен асимптотический ряд Эрдейи по обратным степеням кубического корня из переменной времени с коэффициентами, зависящими от автомодельной переменной и логарифма времени.

We consider the Cauchy problem for the multidimensional Burgers equation with a small dissipation parameter and use the matching method to construct an asymptotic solution near the singularity determined by the vector field structure at the initial instant. The method that we use allows tracing the evolution of the solution with a hierarchy of differently scaled structures and giving a rigorous mathematical definition of the asymptotic solution in the leading approximation. We discuss the relation of the considered problem to different models in fundamental and applied physics.

АСИМПТОТИЧЕСКОЕ РЕШЕНИЕ МНОГОМЕРНОГО УРАВНЕНИЯ БЮРГЕРСА ВБЛИЗИ СИНГУЛЯРНОСТИРассматривается задача Коши для многомерного уравнения Бюргерса с малым параметром диссипации.Методом согласования строится асимптотическое решение вблизи сингулярности, обусловленной структурой векторного поля в начальный момент времени.Использованный в

We consider the Cauchy problem for the Korteweg-de Vries equation with a small parameter at the higher derivative and a large gradient of the initial function.By means of the numerical and analytic methods we show that the formal asymptotics obtained by renormalization is an asymptotic solution to the KdV equation.We obtain the graphs of the asymptotic solutions including the case of non-monotone initial data.

For the heat equation in the plane, an asymptotic approximation of the solution of the Cauchy problem for large times is constructed in the case where the initial function has a power-like asymptotics at infinity. In addition to direct application to heat conduction and diffusion processes, the study of the asymptotic behavior of the solution of the problem under consideration is of independent interest for the asymptotic analysis.

The main problems formulated by A. M. Il'in and solved by his disciples working now in Yekaterinburg are considered. These problems are related to the method of matched asymptotic expansions used for finding asymptotic solutions of equations with a singular dependence on a small parameter. In addition to boundary value problems for equations of mathematical physics, we consider systems of nonlinear equations and systems of linear equations depending on two small parameters. We also consider problems of finding asymptotic expansions for fundamental solutions of parabolic equations and optimal control problems depending on a small parameter.

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.

The Cauchy problem for a quasi-linear parabolic equation with a small parameter multiplying a higher derivative is considered in two cases where the solution of the limit problem has a point of gradient catastrophe. The integrals determining the leading approximation correspond to the Lagrange singularity of type A 3 and the boundary singularity of type B 3. For another choice of the initial function, singular points corresponding to A 2n+1 and B 2n+1 with arbitrary n ≥ 1 are obtained.

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.

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.

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.

Asymptotics of a generalized solution of the steady-state Navier-Stokes system of equations in a bounded domain Omega of the three-dimensional space is studied under constraint on the generalized Reynolds number. By methods of functional analysis a theorem about approximation of the exact solution of the homogeneous boundary value problem by partial sums of the found series up to any degree of accuracy in the norm of space C((Omega) over bar) is proved. For the non-steady-state Navier-Stokes system of equations asymptotic approximation in the norm of space L-2(Omega) is proved.