The article introduces contact germs that transform solutions of some partial differential equations into solutions of other equations. Parametric symmetries of differential equations generalizing point and contact symmetries are defined. New transformations and symmetries may depend on derivatives of arbitrary but finite order. The stationary Schrödinger equations, acoustics and gas dynamics equations are considered as examples.

We consider one-dimensional second-order partial differential equations describing waves in inhomogeneous and nonlinear media. Contact transformations and Euler differential substitution are used to construct general solutions. General and partial solutions of some nonstationary continuum mechanics models are found.

In this paper we consider mappings of jet spaces that preserve the module of canonical Pfaffian forms, but are not generally invertible. These mappings are called contact. A lemma on the prolongation of contact mappings is proved. Conditions are found for which mappings transform solutions of some partial differential equations into ones of other equations. Examples of contact mappings of differential equations are given. We consider contact mappings depending on a parameter and give example of differential equation invariant under the maps.

A generalized method of separation of variables is used to obtain new particular solutions for the stream function describing two-dimensional stationary motions of an ideal fluid. Patterns of streamlines are given. The proof of the stability of some of the solutions is based on a theorem due to Arnol'd /1/.

Methods to obtain solutions describing traveling waves in strongly inhomogeneous media are discussed within a linear wave equation with a variable speed (speed of sound). It is shown that there is a wide range of propagation speed variations that allow for the existence of waves that do not reflect despite the strong inhomogeneity of the medium. In this case, the shape of the wave and its characteristics change with distance. These waves can transfer energy over long distances without loss.

In this paper we consider mappings of jet spaces that preserve the module of canonical Pfaffian forms, but are not generally invertible. These mappings are called contact. A lemma on the prolongation of contact mappings is proved. Conditions are found under which these mappings transform solutions of some partial differential equations into solutions of other equations. Examples of contact mappings of differential equations are given. We consider contact mappings depending on a parameter and give example of differential equation invariant under the maps.

We consider a system of two-dimensional Euler equations describing the motions of an inviscid incompressible fluid. It reduces to one non-linear equation with partial derivatives of the third order. A group of point transformations allowed by this equation is found. Some invariant solutions and solutions not related to invariance are constructed. The solutions found describe vortices, jet streams, and vortex-like formations.

The problem of the existence of traveling waves in inhomogeneous fluid is very important for enabling an explanation of long-distance wave propagations such as tsunamis and storm waves. The present paper discusses new solutions to the variable-coefficient wave equations describing traveling waves in fluid layers of variable depths (1D shallow-water theory). Such solutions are obtained by using the transformation methods when variable-coefficient equations can be reduced to the constant coefficient equation when the existence of traveling waves is evident. It is shown that there is a wide class of monotonic bottom profiles (discrete set) that allow the existence of traveling waves that are not reflected in a strongly inhomogeneous water medium. Their temporal shape changes with distance, mainly near the water–land boundary (shoreline). Traveling waves can transfer the wave energy over a long distance that is often observed at the transoceanic propagation of tsunami waves.

The Euler equations describing two-dimensional steady flows of an inviscid fluid are studied. These equations are reduced to one equation for the stream function and then, using the Hirota function, solutions of three nonlinear elliptic equations are found. The solutions found are interpreted as sources in a rotating fluid, jets, chains of sources and sinks, vortex structures. We propose a new simple method for constructing solutions in the form of rational expressions of elliptic functions. It is shown that the flux of fluid across a closed curve is quantized in the case of the elliptic Sin-Gordon equation.

The one-dimensional system of equations of isentropic gas dynamics is considered.First-order invariants of characteristics of this system are classified.Second-order invariants of characteristics are classified for polytropic processes.The infinite sequence of Darboux integrable systems is described.The approach to construction of smooth solutions without gradient catastrophe is proposed.Examples of solutions without gradient catastrophe are presented.

As it is known, the problem of finding traveling waves in 1D nonlinear and dispersive media may be reduced to solving a system of ordinary differential equations. If the order of the system is large, then the internal wave structure can be very complicated and even random. If the medium is inhomogeneous, it is natural to expect the absence of solutions in the form of traveling waves due to the reflection and multiple reflection effects. If, however, the medium parameters change slowly (in comparison to the wavelength), and the reflection is weak, it becomes possible to construct an approximate solution in the form of a traveling wave with a variable amplitude and phase by using asymptotic methods (WKB, geometric optics or acoustics). For the media with a monotonic change in parameters, such solutions demonstrate the highest gain and the ability to transmit a signal over long distances without distortion. It turns out to be possible to find exact solutions in the form of traveling waves with variable amplitude and phase in highly inhomogeneous media under certain assumptions on the medium parameters. Our paper reviews possible approaches to finding travelling reflectionless waves in the shallow water channels with variable cross-sections and currents. The basic equations are the classical 1D nonlinear shallow-water equations for water displacement and velocity averaged through the cross-channel. Mathematical procedure to get the solutions in the form of travelling reflectionless waves is based on the transformation of the original equations with variable coefficients to the constant-coefficient PDE. We first demonstrate this procedure using the example of the classical linear wave equation with variable coefficient when it can be reduced to the Klein-Gordon equation with constant coefficients. This gives rise to an ordinary second-order differential equation for finding a variable coefficient (the wave speed), so that traveling waves exist in a wide class of inhomogeneous water channels. The second procedure is the reducing of variable-coefficient 1D wave equation to the spherical symmetric wave equation in the odd-dimensional space, where waves traveling to and from the center are separated. More complicated procedure is developed for the channels with non-uniform current. In conclusion, we discuss the effectiveness of this procedure in the framework of Boussinesq systems. The study is supported by grants RFBR (20-05-00162, 21-55-15008, 19-35-60022), President of the RF for the state support of Leading Scientific Schools of the RF (Grant No. NSH-70.2022.1.5). Recent publications: * Didenkulova I. and Pelinovsky E. On shallow water rogue wave formation in strongly inhomogeneous channels. Journal of Physics A: Mathematical and Theoretical, 2016, vol. 49, 194001. * Pelinovsky E., Didenkulova I., Shurgalina E., and Aseeva N. Nonlinear wave dynamics in self-consistent water channels. J Phys. A, 2017, vol. 50, 505501. * Pelinovsky E., Talipova T., Didenkulova I., Didenkulova E. Interfacial long traveling waves in a two-layer fluid with variable depth. Studies in Applied Mathematics, 2019, vol. 142, No. 4, 513–527. * Churilov S.M., Stepanyants Yu.A. Reflectionless wave propagation on shallow water with variable bathymetry and current. J. Fluid Mech., 2022, vol. 931, A15.

The dynamics of passive scalar in swirling turbulent far wakes with varied values of the total excess momentum and angular momentum are described within a second-order mathematical model. The model includes averaged equations of momenta, turbulence energy balance, dissipation rate transfer, averaged concentration, and dispersion of its turbulent fluctuations in the far wake approximation. The closure of the mathematical model is based on Rodi’s algebraic model of the Reynolds stresses. At large distances, a self-similar solution based on numerical experiments is obtained for problems of the dynamics of passive scalar in turbulent wakes behind a self-propelled body and in a momentumless swirling turbulent wake. The group-theoretical analysis of the mathematical model under study is carried out. The model is reduced to a system of ordinary differential equations, which is solved numerically by the shooting method. The resulting solution is compared with the self-similar solution found by direct numerical integration of the differential equations of the model at large distances from the body. Good agreement is obtained. The problem of asymptotic behavior of characteristics of passive admixture in a swirling turbulent wake behind a sphere with nonzero values of the total excess momentum and angular momentum is also considered. With application of the group-theoretical analysis, it is shown that there is no physically meaningful self-similar solution to the equations of passive admixture dynamics.

In this work we find general solutions to some classes of linear wave equations with variable coefficients. Such equations describe the oscillations of rods, acoustic waves, and also some models of gas dynamics are reduced to these equations. To construct general solutions, we employ special types of Euler-Darboux transformations, namely, Levi type transformations. These transformations are first order differential substitutions. For constructing each transformation, we need to solve two linear second order ordinary differential equations. The solutions of one of these equations are determined by the solutions of the other equations by means of a differential substitution and Liouville formula. In the general case, it is not easy to solve these ordinary differential equations. However, it is possible to provide some formula for the superposition of the transformation of Levy type. Starting with a classical wave equation with constant coefficients and employing the found transformations, we can construct infinite series of equations possessing explicit general solutions. By means of Matveev method we obtain limiting forms of iterated transformations. We provide a series of particular examples of the equations possessing general solutions.

A group of point transformations admitted by the three-dimensional Kadomtsev — Petviashvili equation is calculated. An example of an invariant solution is given. Exact solutions for the equation under study in the form of double waves are revealed. The resulting solutions are expressed in terms of elementary functions and describe an interaction between a pair of solitons. Smooth bounded rational solutions are also constructed.

In this paper, we consider the problem of formal iteration. We construct an area preserving mapping which does not have any square root. This leads to a counterexample to Moser's existence theorem for an interpolation problem. We give examples of formal transformation groups such that the iteration problem has a solution for every element of the groups.

The flow in a plane momentumless turbulent wake is described with the use of a second-order mathematical model based on the Rodi’s algebraic model of Reynolds stresses. In view of the properties of the plane momentumless turbulent wake, the mathematical model is an analog of the two-parameter  $$e \sim \varepsilon$$  turbulence model in the far wake approximation with a modified empirical constant in the diffusion term of the equations. For moderate distances from the body, the results predicted by this model agree well with the known experimental data of Cimbala and Park (1990). At large distances from the body, a self-similar solution based on numerical experiments is obtained. A group-theoretical analysis of the mathematical model of the wake is performed. The model is reduced to a system of ordinary differential equations, which is solved numerically by the shooting method. The self-similar solution derived from the group-theoretical analysis is found to be in good agreement with that obtained by means of direct numerical integration of the differential equations of the model at large distances from the body.

We propose a new algebraic approach to study compatibility of partial differential equations. The approach uses concepts from commutative algebra, algebraic geometry and Gr¨obner bases to clarify crucial notions concerning compatibility such as passivity and reducibility. One obtains sufficient condi- tions for a differential system to be passive and proves that such systems generate manifolds in the jet space. Some examples of constructions of passive systems associated with the sinh-Cordon equation are given

The flow in swirling turbulent wakes with varying total excess momentum and angular momentum is described using two second-order mathematical models. The first one includes averaged equations of momenta, turbulence energy balance, and dissipation rate in the far-wake approximation. The closure of the mathematical model relies on Rodi's algebraic model for Reynolds stresses. The second model is based on simplified representations of the turbulent viscosity coefficients. For small distances, the calculated profiles of averaged motion velocities and turbulence energy are in good agreement with the experimental data of Lavrent'ev Institute of Hydrodynamics of SB RAS. At large distances, numerical experiments have yielded a self-similar solution of problems of dynamics of turbulent wake behind a self-propelled body and momentumless swirling turbulent wake. Group-theoretical analysis of the simplified mathematical model has been done. The model had been reduced to a system of ordinary differential equations, which was solved numerically using asymptotic expansions. The solution obtained was compared with the self-similar solution found by direct numerical integration of the differential equations of the model at large distances from the body, and good agreement was observed. In addition, the problem of asymptotic behavior of swirling turbulent wake behind a sphere with non-zero values of total excess momentum and angular momentum was considered. The group-theoretical analysis has shown the absence of physically meaningful self-similar solutions to the equations of the turbulence model under consideration.

In this paper, we consider two Boussinesq models that describe propagation of small-amplitude long water waves. Exact solutions of the classical Boussinesq equation that represent the interaction of wave packets and waves on solitons are found. We use the Hirota representation and computer algebra methods. Moreover, we find various solutions for one of the variants of the Boussinesq system. In particular, these solutions can be interpreted as the fusion and decay of solitary waves, as well as the interaction of more complex structures.

The classical Boussinesq equation describing gravity waves in shallow waters is under consideration. Hirota’s bilinear representation is used to construct exact solutions describing wave packets, waves on solitons, and “dancing” waves. The principle of multiplying the solutions of the Hirota equation is formulated, which helps constructing more complex structures made of solitons, wave packets, and other types of waves.