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.     

Integration of the equations of gas dynamics for 2.5-dimensional solutions

We integrate the equations of gas dynamics in finite form for the solutions in which the thermodynamic parameters depend only on one spatial variable. The corresponding motion of gas represents the nonlinear superposition of the one-dimensional gas motion corresponding to the invariant system and the two-dimensional motion determined by noninvariant functions. These motions are called 2.5-dimensional. We reduce the invariant system to a first-order implicit ordinary differential equation. We study various solutions of the latter. We construct some continuous and discontinuous solutions to the equations of gas dynamics and give their physical interpretation.     

A hierarchy of submodels of differential equations

We consider a system of differential equations admitting a group of transformations. The Lie algebra of the group generates a hierarchy of submodels. This hierarchy can be chosen so that the solutions to each of submodels are solutions to some other submodel in the same hierarchy. For this we must calculate an optimal system of subalgebras and construct a graph of embedded subalgebras and then calculate the differential invariants and invariant differentiation operators for each subalgebra. The invariants of a superalgebra are functions of the invariants of the algebra. The invariant differentiation operators of a superalgebra are linear combinations of invariant differentiation operators of a subalgebra over the field of invariants of the subalgebra. The comparison of the representations of group solutions gives a relation between the solutions to the models of the superalgebra and the subalgebra. Some examples are given of embedded submodels for the equations of gas dynamics.     

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

It is shown that Lie group analysis of differential equations provides the exact solutions of two-dimensional stratified rotating Boussinesq equations which are a basic model in geophysical fluid dynamics. The exact solutions are obtained as group invariant solutions corresponding to the translation and dilation generators of the group of transformations admitted by the equations. The comparison with the previous analytic studies and experimental observations confirms that the anisotropic nature of the wave motion allows to associate these invariant solutions with uni-directional internal wave beams propagating through the medium. It is also shown that the direction of internal wave beam propagation is in the transverse direction to one of the invariants which corresponds to a linear combination of the translation symmetries. Furthermore, the amplitudes of a linear superposition of wave-like invariant solutions forming the internal gravity wave beams are arbitrary functions of that invariant. Analytic examples of the latitude-dependent invariant solutions associated with internal gravity wave beams that have different general profiles along the obtained invariant and propagating in the transverse direction are considered. The behavior of the invariant solutions near the critical latitude is illustrated.     

Integrable hydrodynamic submodels with a linear velocity field

Invariant and partially invariant solutions to the equations of gas dynamics with a linear velocity field are defined by a matrix satisfying a homogeneous integrable Riccati equation. The classification is carried out of solutions by the acceleration vector in the Lagrangian coordinates. Some example is given of an invariant solution for which the selected volume “collapses” to an interval.     

Symmetry analysis and optimal systems of generalized Chaplygin gas equations with a source term

The system of generalized Chaplygin gas equations with a coulomblike friction term has been investigated by using the famous Lie symmetry method. A direct and systematic algorithm based on the adjoint transformation and invariants of the admitted Lie algebras is then used to construct one- and two-dimensional optimal system of the Chaplygin gas equations. Inequivalent classes of group invariant solutions are then obtained using the one-dimensional optimal system. Further, the evolutionary behaviour of the weak discontinuity wave within the state characterized by one of the group invariant solutions is investigated in detail, and certain observations are noted in respect to their contrasting behaviour.     

Unsteady Solutions of Euler Equations Generated by Steady Solutions

Invariant solutions of partial differential equations are found by solving a reduced system involving one independent variable less. When the solutions are invariant with respect to the so-called projective group, the reduced system is simply the steady version of the original system. This feature enables us to generate unsteady solutions when steady solutions are known. The knowledge of an optimal system of subalgebras of the principal Lie algebra admitted by a system of differential equations provides a method of classifying H-invariant solutions as well as constructing systematically some transformations (essentially different transformations) mapping the given system to a suitable form. Here the transformations allowing to reduce the steady two-dimensional Euler equations of gas dynamics to an equivalent autonomous form are classified by means of the program SymboLie, after that an optimal system of two-dimensional subalgebras of the principal Lie algebra has been calculated. Some steady solutions of two-dimensional Euler equations are determined, and used to build unsteady solutions.     

The non-stationary invariant solution of the equations of gas dynamics describing the spreading of a gas into a vacuum

An invariant solution of the equations of gas dynamics, constructed on a one-dimensional subgroup (according to the classification in /1/) which is only allowed in the case of a polytropic gas with a special adiabatic index, is considered. The gas spreads out into a vacuum after a finite time. New solutions are constructed which describe one-dimensional flows from a source into a vacuum and the focussing of the gas within a sphere or a cylinder with shock waves. The spreading of a concentration of the gas with an arbitrary boundary when there is a contact discontinuity is also considered.

One-dimensional flows have been treated in detail in /2, 3/, mainly in the case of extended subgroups.     

DYNAMICS OF RELATIVISTIC FLUID FOR COMPRESSIBLE GAS

In this paper the relativistic fluid dynamics for compressible gas is studied.We show that the strict convexity of the negative thermodynamical entropy preserves invariant under the Lorentz transformation if and only if the local speed of sound in this gas is strictly less than that of light in the vacuum.A symmetric form for the equations of relativistic hydrodynamics is presented,and thus the local classical solutions to these equations can be deduced.At last,the non-relativistic limits of these local cla...     

Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow for large data: Application of the generalized invariant regions and the modified Godunov scheme

We study the motion of isentropic gas in nozzles. This is a major subject in fluid dynamics. In fact, the nozzle is utilized to increase the thrust of rocket engines. Moreover, the nozzle flow is closely related to astrophysics. These phenomena are governed by the compressible Euler equations, which are one of crucial equations in inhomogeneous conservation laws.In this paper, we consider its unsteady flow and devote to proving the global existence and stability of solutions to the Cauchy problem for the general nozzle. The theorem has been proved in Tsuge (2013). However, this result is limited to small data. Our aim in the present paper is to remove this restriction, that is, we consider large data. Although the subject is important in Mathematics, Physics and engineering, it remained open for a long time. The problem seems to rely on a bounded estimate of approximate solutions, because we have only method to investigate the behavior with respect to the time variable. To solve this, we first introduce a generalized invariant region. Compared with the existing ones, its upper and lower bounds are extended constants to functions of the space variable. However, we cannot apply the new invariant region to the traditional difference method. Therefore, we invent the modified Godunov scheme. The approximate solutions consist of some functions corresponding to the upper and lower bounds of the invariant regions. These methods enable us to investigate the behavior of approximate solutions with respect to the space variable. The ideas are also applicable to other nonlinear problems involving similar difficulties.     

On a field equation generating a new class of particular solutions to the Yang-Mills equations

In a pseudo-Euclidean space, a field equation (system of equations) is considered that is invariant under orthogonal (from the group O(p, q)) coordinate transformations and invariant under gauge transformations from the spinor group Pin(p, q). The solutions to the field equation are connected with a class of new particular solutions to the Yang-Mills equations.     

Symmetries of compatibility conditions for systems of differential equations

When symmetries of differential equations are applied, various types of associated systems of equations appear. Compatibility conditions of the associated systems expressed in the form of differential equations inherit Lie symmetries of the initial equations. Invariant solutions to compatibility systems are known as orbits of partially invariant and generic solutions involved in the Lie group foliation of differential equations and so on. In some cases Bäcklund transformations and differential substitutions connecting quotient equations for compatibility conditions and initial systems naturally arise. Besides, Ovsiannikov's orbit method for finding partially invariant solutions is essentially based on such symmetries.     

Classification of Differential Invariant Submodels

Calculation of differential invariants and invariant differentiation operators of a subalgebra of the Lie algebra admitted by a system of differential equations enables us to construct differential invariant submodels. We classify submodels for every subalgebra of an optimal system of subalgebras. Classification includes the invariant submodels and partially invariant submodels considered earlier. We give examples of classification for three-dimensional subalgebras admitted by the equations of gas dynamics.     

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.     

Patterns in parabolic problems with nonlinear boundary conditions

We obtain existence of asymptotically stable nonconstant equilibrium solutions for semilinear parabolic equations with nonlinear boundary conditions on small domains connected by thin channels. We prove the convergence of eigenvalues and eigenfunctions of the Laplace operator in such domains. This information is used to show that the asymptotic dynamics of the heat equation in this domain is equivalent to the asymptotic dynamics of a system of two ordinary differential equations diffusively (weakly) coupled. The main tools employed are the invariant manifold theory and a uniform trace theorem.     

Exact solutions of the time fractional nonlinear Schrödinger equation with two different methods

In the present paper, exact solutions of fractional nonlinear Schrödinger equations have been derived by using two methods: Lie group analysis and invariant subspace method via Riemann‐Liouvill derivative. In the sense of Lie point symmetry analysis method, all of the symmetries of the Schrödinger equations are obtained, and these operators are applied to find corresponding solutions. In one case, we show that Schrödinger equation can be reduced to an equation that is related to the Erdelyi‐Kober functional derivative. The invariant subspace method for constructing exact solutions is presented for considered equations.     

Solving a nonlinear pde that prices real options using utility based pricing methods

We provide group invariant solutions to two nonlinear differential equations associated with the valuing of real options with utility pricing theory. We achieve these through the use of the Lie theory of continuous groups, namely, the classical Lie point symmetries. These group invariant solutions, constructed through the use of the symmetries that also leave the boundary conditions invariant, are consistent with the results in the literature. Thus it may be shown that Lie symmetry algorithms underlie many ad hoc methods that are utilised to solve differential equations in finance.     

Well-posedness of modified Camassa-Holm equations

The Camassa-Holm equation can be viewed as the geodesic equation on some diffeomorphism group with respect to the invariant H¹ metric. We derive the geodesic equations on that group with respect to the invariant Hk metric, which we call the modified Camassa-Holm equation, and then study the well-posedness and dynamics of a modified Camassa-Holm equation on the unit circle S, which has some significant difference from that of Camassa-Holm equation, e.g., it does not admit finite time blowup solutions.     

Analysis of reaction-diffusion systems by the method of linear determining equations

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.     

A refined invariant subspace method and applications to evolution equations

The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations.The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit.A two-component nonlinear system of dissipative equations is analyzed to shed light on the resulting theory,and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differential equations and their corresponding exact solutions with generalized separated variables.