Invariant Solutions Of Rank Research Articles

One class of solutions to the equations of ideal plasticity

Much attention is given to the study and solution of nonlinear differential equations in the modern mathematical literature. Despite this, there are not many methods for researching and solving such equations. These are point and contact transformations of equations, various methods of separating variables, the method of differential connections, the search for various symmetries and their use to construct solutions, as well as conservation laws. The paper considers a nonlinear differential equation describing the plastic flow of a prismatic rod. A group of point symmetries is found for this equation. The optimal system of one-dimensional subalgebras is calculated. Conservation laws corresponding to Noetherian symmetries are given, and it is also shown that there are infinitely many non-Noetherian conservation laws. Several new invariant solutions of rank one, i. e. depending on one independent variable, are constructed. It is shown how classes of new solutions can be constructed from two exact solutions, passing to a linear equation. Thus, in this short article, almost all methods of modern research of nonlinear differential equations are involved.

New submodels describing attenuation of the ultrasonic beams in 3-d cubic nonlinear medium in the absence of dissipation

For a three-dimensional generalized model of nonlinear hydroacoustics by Khokhlov–Zabolotskaya–Kuznetsov in a cubic nonlinear medium in the absence of dissipation, we studied the attenuation of ultrasonic beams after the formation of shock fronts. Earlier, in the work of one of the authors for this model, submodels were obtained and studied, described by non-stationary solutions, invariant with respect to some three-dimensional subgroups of the main ten-parameter group of the differential equation that defines this model. In our paper we obtained and studied four non-stationary submodels of this model that are invariant with respect to four-parameter subgroups of the main group of this differential equation. These submodels are new submodels and have not been previously noted in the literature. They are given by invariant solutions of rank 1. Among these 4 submodels, 2 submodels describe axisymmetric ultrasonic beams, the remaining 2 describe one-dimensional ultrasonic beams. The search for invariant solutions that define these 4 submodels is reduced to solving of nonlinear integral equations, the implicit solutions of which are obtained in the form of nonlinear algebraic equations containing transcendental functions. These submodels are used to study the propagation of ultrasonic beams, for which either the acoustic pressure and its rate of change or the acoustic pressure and its gradient are given at the initial time at a fixed point. Conditions are obtained that ensure the existence and uniqueness of solutions to boundary value problems describing these processes. This makes it possible to correctly carry out numerical calculations in the study of these processes. As a result of the numerical solution of these boundary value problems for some values of the parameters characterizing these processes, pressure distribution graphs were obtained. In all cases, ultrasonic beams are weaken monotonically with time and completely fade away in a finite time. At each point, the time of complete attenuation of ultrasonic beams is found.The obtained and studied new submodels are another step towards the creation of a database of physically significant submodels of the three-dimensional generalized model of nonlinear hydroacoustics by Khokhlov–Zabolotskaya–Kuznetsov in a cubic nonlinear medium in the absence of dissipation, which describes the attenuation of ultrasonic beams after the formation of shock fronts.

Invariant submodels describing a propagation of the ultrasonic beams in a cubically nonlinear medium without dissipation after self-focusing

Recently, for high-power ultrasonic beams, it was experimentally found that as a result of self-action, their self-focusing occurs. With self-focusing, a powerful ultrasonic beam is noticeably narrowed, has a nonlinear narrowing, and it is significantly amplified at the focus. A generalization of the three-dimensional Khokhlov–Zabolotskaya–Kuznetsov model in a cubic nonlinear medium in the presence of dissipation with a special nonlinearity coefficient describes the propagation of ultrasonic beams after self-focusing. This work is devoted to the study of submodels of this model without dissipation, which are invariant with respect to four-parameter subgroups of the main group of the equation of this model. These submodels are defined by invariant solutions of rank 0 or 1, which describe one-dimensional, plane and axisymmetric ultrasonic beams. Solutions of rank 0 and some solutions of rank 1 are found explicitly. Some of them, at each fixed moment of time, contain a destroying element in the form of an ultrasonic needle or an ultrasonic knife, which at each fixed moment of time are localized in a limited domain, on the surface of which the acoustic pressure is zero. These submodels describe one-dimensional, plane, or axisymmetric ultrasonic beams. Finding other invariant solutions of rank 1 is reduced to solving nonlinear integral equations. These invariant solutions are used to study the propagation of ultrasonic beams after self-focusing, for which either the acoustic pressure and its gradient, or the acoustic pressure and the rate of its change are given at the initial moment of time at a fixed point. Under certain additional conditions, the existence and uniqueness of the solutions to boundary value problems describing these processes are established. This makes it possible to correctly carry out numerical calculations when studying these processes. The graphs of the pressure distribution obtained as a result of the numerical solution of these boundary value problems for some values of the parameters characterizing the indicated processes are presented. These graphs show that for these submodels in the ultrasonic beams a monotonic increase in acoustic pressure occurs over time.

A study of the generalization of the three-dimensional model of nonlinear hydroacoustics of Khokhlov–Zabolotskaya–Kuznetsov in a cubic-nonlinear medium in the presence of dissipation

This article is devoted to the study of the generalization of the Khokhlov–Zabolotskaya–Kuznetsov model (3KZK) of nonlinear hydroacoustics in a cubic-nonlinear medium in the presence of dissipation. This model is described by third order quasilinear partial differential equation (3KZK). We obtained that the (3KZK) equation admits an eight parametric Lie group of the transformations. The submodels of the (3KZK) model are described by the invariant solutions of the (3KZK) equation. We have studied all essentially distinct, not linked by means of the point transformations, invariant solutions of rank 1 of this equation. The invariant solutions of rank 1 are found either explicitly, or their search is reduced to the solution of the nonlinear integro-differential equations. For example, we obtained the invariant solutions that we called by “First ultrasonic needle” and “Second ultrasonic needle”. The submodels described by this solutions have the following property: at each fixed moment of the time in the field of the existence of the solutions, near some point, the pressure increases and becomes infinite in this point. These submodels can be used, in particular, in medicine as a test in preparing for the operations with a help of an ultrasound. Also we obtained the invariant solutions that we called by “First ultrasonic knife” and “Second ultrasonic knife”. The submodels described by these solutions have the following property: at each fixed moment of the time in the field of the existence of the solution, near some plane the pressure increases and becomes infinite on this plane. The same as Ultrasonic needles, these submodels can be used, in particular, in medicine as a test in preparing for the operations with a help of an ultrasound. The presence of the arbitrary constants in the integro-differential equations, that determine invariant solutions of rank 1 provides a new opportunities for analytical and numerical study of the boundary value problems for the received submodels, and, thus, for the original (3KZK) model. With a help of these invariant solutions we researched a propagation of the intensive acoustic waves (one-dimensional, axisymmetric and planar) for which the acoustic pressure, speed and acceleration of its change are specified at the initial moment of the time at a fixed point. Under the certain additional conditions, we established the existence and the uniqueness of the solutions of boundary value problems, describing these wave processes. The graphs of the pressure distribution obtained as a result of numerical solution of these boundary problems are given. A mechanical relevance of the obtained solutions is as follows: 1) these solutions describe in a cubic-nonlinear medium with dissipation the nonlinear and diffraction effects in ultrasonic fields of a special kind, 2) these solutions can be used as a test solutions in the numerical calculations, performed in studies of ultrasonic fields generated by powerful emitters in a cubic-nonlinear medium with dissipation. Application of the obtained formulas generating the new solutions for the found solutions gives the families of the solutions containing the arbitrary constants.

Self-similar waves in a three-dimensional porous medium in the presence of non-stationary singular source or absorption

We study an important for applications a model of three-dimensional porous medium in the presence of non-stationary singular source or absorption. We obtained all invariant submodels of this model. These submodels are described by invariant solutions. We found that among the invariant solutions of rank 1, this model has only three types of essentially different (not connected by point transformations) self-similar waves. These waves include: a self-similar wave, propagating in a porous medium along one of the axes of coordinates, a plane self-similar circular wave and a self-similar spherically symmetric wave. We obtained containing arbitrary constant, integral equations describing these self-similar waves. We established the existence and uniqueness of self-similar waves for which at an initial instant of the time at a fixed point, or on a fixed circle, or on a fixed sphere, either a pressure and rate of its change, or a pressure and its gradient are given. The pressure distribution graphs, obtained as results of a numerical solution of these boundary value problem, are given. The obtained results can be used to study to describe the processes associated with a underground fluid or gas flow, with water filtration, with the engineering surveys in the construction of the buildings, and also with shale oil and gas production.

Invariant submodels and exact solutions of the generalization of the Leith model of the wave turbulence

A generalization of Leith’s model of the phenomenological theory of the wave turbulence is investigated. With the methods of group analysis, the basic models possessing non-trivial symmetries are obtained. For each model, all the invariant submodels are found. For nonlinear differential equations describing these models, formulas for the production of new solutions containing arbitrary constants are obtained. By virtue of these formulas, each solution generates a family of the new solutions. In an explicit form, some invariant solutions (not connected by point transformations) describing invariant submodels are found. The physical meaning of these solutions is obtained. In particular, with the help of these solutions the turbulent processes for which there are “destructive waves” both with fixed wave numbers and with varying wave numbers are described. On the example of an invariant solution of rank 1, it was shown that the search of the invariant solutions of rank 1, which cannot be found explicitly, can be reduced to solve the integral equations. For this solution, turbulent processes are investigated for which at the initial instant of a time and for a fixed value of the wave number either the turbulence energy and rate of its change or the turbulence energy and its gradient are given. Under certain conditions, the existence and uniqueness of the solutions of the boundary value problems describing these processes are established.

Invariant submodels and exact solutions of Khokhlov–Zabolotskaya–Kuznetsov model of nonlinear hydroacoustics with dissipation

We study three-dimensional Khokhlov–Zabolotskaya–Kuznetsov (KZK) model of the nonlinear hydroacoustics with dissipation. This model is described by third order quasilinear partial differential equation of the (KZK). We obtained that the (KZK) equation admits an infinite Lie group of the transformations, depending on the three arbitrary functions. This is due to the fact that in the (KZK) model the main direction of the wave’s propagation is singled out. The submodels of the (KZK) model.are described by the invariant solutions of the (KZK) equation. We studied essentially distinct, not linked by means of the point transformations, invariant solutions of rank 0 and 1 of this equation. Also we considered the invariant solutions of rank 2 and 3. The invariant solutions of rank 0 and 1 are found either explicitly, or their search is reduced to the solution of the nonlinear integro-differential equations. For example, we obtained the invariant solutions that we called by “Ultrasonic knife” and “Ultrasonic destroyer”. The submodel “Ultrasonic knife” have the following property: at each fixed moment of the time in the field of the existence of the solution near a some plane the pressure increases indefinitely and becomes infinite on this plane. The submodel “Ultrasonic destroyer” contains a countable number of “Ultrasonic knives”. The presence of the arbitrary constants in the integro-differential equations, that determine invariant solutions of rank 1 provides a new opportunities for analytical and numerical study of the boundary value problems for the received submodels, and, thus, for the original (KZK) model. With a help of these invariant solutions we researched a propagation of the intensive acoustic waves (one-dimensional, axisymmetric and planar) for which the acoustic pressure, speed and acceleration of its change, or the acoustic pressure , speed and acceleration of its change in the radial direction, or the acoustic pressure, speed and acceleration of its change in the direction of one of the axes are specified at the initial moment of the time at a fixed point. Under the certain additional conditions, we established the existence and the uniqueness of the solutions of boundary value problems, describing these wave processes. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters. Application of the obtained formula generating the new solutions for the found solutions gives families of the solutions containing three arbitrary functions.

Exact solutions of the equations of two-phase dynamics. Collapse of gas and particles in space

Under consideration is the system of partial differential equations describing the dynamics of a two-phase medium. Exact partially invariant solutions of rank 1 and defect 1 of this system are obtained with respect to some four-dimensional subalgebras. The phenomenon of collapse (an instantaneous source) in a two-phase medium is described.

Invariant solutions of the Westervelt model of nonlinear hydroacoustics without dissipation

We study three-dimensional Westervelt model of a nonlinear hydroacoustics without dissipation. We received all of its invariant submodels. We studied all invariant submodels described by the invariant solutions of rank 0 and 1. All invariant solutions of rank 0 and 1 are found either explicitly, or their search is reduced to the solution of the nonlinear integral equations. With a help of these invariant solutions we researched: (1) a propagation of the intensive acoustic waves (self-similar, axisymmetric, planar and one-dimensional) for which the acoustic pressure and a speed of its change, or the acoustic pressure and its derivative in the direction of one of the axes are specified at the initial moment of the time at a fixed point , (2) a spherically symmetric ultrasonic field for which the acoustic pressure and a speed of its change, or the acoustic pressure and its radial derivative are specified at the initial moment of the time at a fixed point. Solving of the boundary value problems describing these processes is reduced to the solving of nonlinear integral equations. We are established the existence and uniqueness of solutions of these boundary value problems under some additional conditions. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters. We found all the conservation laws of the first order for the Westerveld equation written in dimensionless variables.

Invariant submodels of the Westervelt model with dissipation

We study three-dimensional Westervelt model of nonlinear hydroacoustics with dissipation. We received all its invariant submodels. With the help of invariant solutions, we explored some wave processes, specifying their physical meaning. The boundary value problems describing these processes are reduced to the nonlinear integro-differential equations. We established the existence and uniqueness of the solutions of these boundary value problems under some additional conditions. Also we considered the invariant solutions of rank 2 and 3. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters.

The motion of the particles volume corresponding to invariant solution of rank 2 submodel of hydrodynamic type

Invariant submodel of rank 2 on the subalgebra consisting of the sum of transfers for hydrodynamic equations with the equation of state in the form of pressure as the sum of density and entropy functions, is presented. In terms of the Lagrangian coordinates from condition of nonhyperbolic submodel solutions depending on the four essential constants are obtained. For simplicity, we consider the solution depending on two constants. The trajectory of particles motion, the motion of parallelepiped of the same particles are studied using the Maple.

Simple waves of a seven-dimensional subalgebra of all translations in gas dynamics

A differentially invariant solution of rank 1 for gas dynamics equations is considered in a seven-dimensional subalgebra of all translations including Galilean translations. Exact solutions with a uniformly accelerated plane of the level of invariant functions are obtained. A nonisentropic wave depends on two arbitrary functions and constants. A general solution of an isentropic simple wave depends on the constants only.

Two-dimensional gas vortices and twisted gas jets

This paper studies an invariant solution of rank one of the equations of motion of a polytropic gas that describes two-dimensional gas vortices and twisted gas jets. Flow types are classified according to the governing parameter: vortices in the form of sources and sinks, unlimited expansion, and collapse.

Galilean-invariant axisymmetric self-similar submodel of gas dynamics without swirling

This paper deals with one insufficiently studied submodel of invariant solutions of rank 1 of the equations of gas dynamics. It is shown that, in cylindrical coordinates, the submodel without swirling reduces to a system of two ordinary differential equations. For the equation of state with additional invariance, a self-similar system is obtained. A pattern of phase trajectories is constructed, and particle motion is studied using asymptotic methods. The obtained solutions describe unsteady flows over axisymmetric bodies with possible strong discontinuities.

Regular, partially invariant solutions of rank 1 and defect 1 of equations of plane motion of a viscous heat-conducting gas

A system of the Navier-Stokes equations of two-dimensional motion of a viscous heat-conducting perfect gas with a polytropic equation of state is considered. Regular, partially invariant solutions of rank 1 and defect 1 are studied. A sufficient condition of their reducibility to invariant solutions of rank 1 is proved. All solutions of this class with a linear dependence of the velocity-vector components on spatial coordinates are examined. New examples of solutions that are not reducible to invariant solutions are obtained.

Irregular Partially Invariant Solutions of Rank 2 and Defect 1 to Equations of Gas Dynamics

We consider three-dimensional subalgebras admitted by the equations of gas dynamics having time as an invariant and containing no rotation operator. For such subalgebras we seek for irregular partially invariant solutions of rank 2 and defect 1. The representation for solutions has the form which generalizes motion of a gas with a linear velocity field. We show that partially invariant solutions exist for each subalgebra. We describe the set of these solutions. We find solutions with the indicated representation that are not partially invariant. The solutions reducible to invariant solutions are generalized to new submodels.

Regular Partially Invariant Solutions of Rank 0 and Defect 1 of Equations of Axisymmetric Motions of a Viscous Heat-Conducting Perfect Gas

All partially invariant solutions of rank 0 and defect 1 of the equations of axisymmetric motions of a viscous heat‐conducting perfect gas with a polytropic equation of state that are nonreduced to invariant solutions are described. The gas motions corresponding to these solutions in time and space are presented.

Exact solutions of the axisymmetric equations of motion of a viscous heat-conducting perfect gas described by systems of ordinary differential equations

All invariant solutions of rank 1 of the two-dimensional equations of motion of a heat-conducting perfect gas with a polytropic equation of state are described. A sufficient condition for reducibility of regular, partially invariant solutions of rank 1 and defect 1 to invariant solutions is given.

Invariant solutions of rank two of the equations of the rotationally symmetric motions of an inhomogeneous liquid

An optimal system of first-order algebras of the system of equations for the rotationally-symmetric unsteady motion of an inhomogeneous liquid is constructed. New exact solutions of certain factor systems are found which describe motions with free boundaries or internal non-linear waves.

Invariant solutions of rank 1 of the equations of plane motion of a viscous heat-conducting perfect gas

Invariant solutions of rank 1 of the equations of plane motion of a viscous heat-conducting perfect gas