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We consider nonlinear systems of first order partial differential equations admitting at least two one-parameter Lie groups of transformations with commuting infinitesimal operators. Under suitable conditions it is possible to introduce a variable transformation based on canonical variables which reduces the model in point to autonomous form. Remarkably, the transformed system may admit costant solutions to which three correspond non-costant solutions of the original model. The results are specialized to the case of first order quasilinear systems admitting either dilatation or spiral groups of transformations and a systematic procedure to characterize special exact solutions is given. At the end of the paper the equations of axi-symmetric gas dynamics are considered.

In this paper it is shown an algorithm leading to linearization of nonlinear systems of partial differential equations admitting infinite-parameter lie groups of point trasformations. The linearization is carried out through the introduction of canonical variables. The canonical variables are privileged variables allowing to write a general lie group of point trasformations in its simplets form (a translation in only one variable). Various relevant examples of application of the procedure are considered

We consider quasilinear hyperbolic systems are left invariant by two Lie groups of transformations having commuting infinitesimal operators, it is shown how to reduce them to autonomous form. A physical example admitting two groups of transformations is considered; the application of the procedure allows us to characterize an exact particular solution. Finally, some considerations about the linearization of the model are pointed out

Motivated by many physical applications, we consider a general first order system of nonlinear partial differential equations involving two independent variables x, t and a vector field u (x,t). We suppose that the system admits two one parameter Lie groups of transformations with commuting infinitesimal operators. Then, by introducing canonical variables, it is possible to show, under suitable conditions, that the original governing system may be written in autonomous form. Of course, constant solutions of the transformed system are nonconstant solutions of the original system. By using the above mentioned result, we are able to point out a systematic procedure that allows us to build up special nonconstant solutions provided that the governing system admits two commuting infinitesimal operators. The results obtained in [1], in the case when the system under consideration is invariant with respect to the stretching group of transformation can be recovered as a special case of our procedure. In dealing with waves propagating in media having cylindrical or spherical symmetry, as well as in regions with inhomogeneities, we are led, in the one-dimensional case, to consider hyperbolic systems where the independent variables appear explicity in the coefficients of the system. Consequently, it becomes a suitable procedure to transform the system in autonomous form characterizing, in the meantime, exact solutions. Moreover, it is possible to study discontinuity and shock waves propagating in such special nonconstant states as propagation problems in constant states of the transformed system

The paper claims to give a systematic approach allowing one to obtain invertible variable transformations mapping nonlinear partial differential equations either into autonomous or linear form provided that some suitable conditions are satisfed. The procedure makes use of Lie group analysis so that some symmetries are required in order to obtain the required transformations, which are related to canonical variables. A linearization procedure is given in the last part of the paper valid for systems of partial differential equations that can be reduced to a single nonlinear equation linearly degenerate. The procedures are explained with some examples of physical interest.

In this paper a procedure is given allowing to obtain instability condition for quasi linear hyperbolic system in conservative form which are compatible with a supplementary conservative law. An application of the method to the Euler equation deriving form a Lagrangian is also carried out

an algorithm is given to linearize nonlinear first order systems of partial differential equations admitting an infinite-parameter lie group of point trasformations. The associated infinitesimal operator must be a linear combination of commuting operators which individually are not necessarily admitted by the basic system, whereas the cofficents of the combination are given by arbitrary solutions of suitable linear system. The procedure is based on the introduction of the canonical variables corresponding to the commuting operators. Within such a framework we reformulate a theorem already proved by Kumei and Bluman [SIAM J. appl. math. 42 (1982)]. The paper concludes with some illustrative examples of the proposed algorithm

In this paper it is shown to transform to autonomous form a general nonautonomous system of partial differential equations in n independent variables provides it is left invariant by n Lie groups of point transformations generating a n-dimensional Abelian Lie algebra. Two examples of physical interest are given to illustrate how the algorithm works

The formation of nonlinear shocks does not occur for nonlinear hyperbolic partial differential equations which are 'completely exceptional'. In some sense the solutions of these equations exhibit a behaviour similar to that of linear equations. We present a procedure which allows to transform, under suitable conditions, such type of equations to linear canonical form. The procedure is valid for more than one space dimension and allows also to characterize classes of linearizable systems

In this paper we consider a system of partial differential equations describing the resonant interaction between a long wave and a short wave. We show that the system at hand possesses the Painleve' property: as a consequence, an auto-Backlund transform is constructed. Moreover, we determine the classical as well as the non-classical symmetry groups allowing us to obtain various similarity reductions of the governing system.

Motivated by many physical applications, we consider a general first order system of nonlinear partial differential equations involving two independent variables x, t and a vector field u (x,t). We suppose that the system admits two one parameter Lie groups of transformations with commuting infinitesimal operators. Then, by introducing canonical variables, it is possible to show, under suitable conditions, that the original governing system may be written in autonomous form. Of course, constant solutions of the transformed system are nonconstant solutions of the original system. By using the above mentioned result, we are able to point out a systematic procedure that allows us to build up special nonconstant solutions provided that the governing system admits two commuting infinitesimal operators. The results obtained in [1], in the case when the system under consideration is invariant with respect to the stretching group of transformation can be recovered as a special case of our procedure. In dealing with waves propagating in media having cylindrical or spherical symmetry, as well as in regions with inhomogeneities, we are led, in the one-dimensional case, to consider hyperbolic systems where the independent variables appear explicity in the coefficients of the system. Consequently, it becomes a suitable procedure to transform the system in autonomous form characterizing, in the meantime, exact solutions. Moreover, it is possible to study discontinuity and shock waves propagating in such special nonconstant states as propagation problems in constant states of the transformed system

In this paper we consider a system of partial differential equations describing the resonant interaction between a long wave and a short wave. We show that the system at hand possesses the Painleve' property: as a consequence, an auto-Backlund transform is constructed. Moreover, we determine the classical as well as the non-classical symmetry groups allowing us to obtain various similarity reductions of the governing system.

In this paper it is shown to transform to autonomous form a general nonautonomous system of partial differential equations in n independent variables provides it is left invariant by n Lie groups of point transformations generating a n-dimensional Abelian Lie algebra. Two examples of physical interest are given to illustrate how the algorithm works

We consider nonlinear systems of first order partial differential equations admitting at least two one-parameter Lie groups of transformations with commuting infinitesimal operators. Under suitable conditions it is possible to introduce a variable transformation based on canonical variables which reduces the model in point to autonomous form. Remarkably, the transformed system may admit costant solutions to which three correspond non-costant solutions of the original model. The results are specialized to the case of first order quasilinear systems admitting either dilatation or spiral groups of transformations and a systematic procedure to characterize special exact solutions is given. At the end of the paper the equations of axi-symmetric gas dynamics are considered.

We consider quasilinear hyperbolic systems are left invariant by two Lie groups of transformations having commuting infinitesimal operators, it is shown how to reduce them to autonomous form. A physical example admitting two groups of transformations is considered; the application of the procedure allows us to characterize an exact particular solution. Finally, some considerations about the linearization of the model are pointed out

In this paper a procedure is given allowing to obtain instability condition for quasi linear hyperbolic system in conservative form which are compatible with a supplementary conservative law. An application of the method to the Euler equation deriving form a Lagrangian is also carried out

an algorithm is given to linearize nonlinear first order systems of partial differential equations admitting an infinite-parameter lie group of point trasformations. The associated infinitesimal operator must be a linear combination of commuting operators which individually are not necessarily admitted by the basic system, whereas the cofficents of the combination are given by arbitrary solutions of suitable linear system. The procedure is based on the introduction of the canonical variables corresponding to the commuting operators. Within such a framework we reformulate a theorem already proved by Kumei and Bluman [SIAM J. appl. math. 42 (1982)]. The paper concludes with some illustrative examples of the proposed algorithm

The formation of nonlinear shocks does not occur for nonlinear hyperbolic partial differential equations which are 'completely exceptional'. In some sense the solutions of these equations exhibit a behaviour similar to that of linear equations. We present a procedure which allows to transform, under suitable conditions, such type of equations to linear canonical form. The procedure is valid for more than one space dimension and allows also to characterize classes of linearizable systems

In this paper it is shown an algorithm leading to linearization of nonlinear systems of partial differential equations admitting infinite-parameter lie groups of point trasformations. The linearization is carried out through the introduction of canonical variables. The canonical variables are privileged variables allowing to write a general lie group of point trasformations in its simplets form (a translation in only one variable). Various relevant examples of application of the procedure are considered

The paper claims to give a systematic approach allowing one to obtain invertible variable transformations mapping nonlinear partial differential equations either into autonomous or linear form provided that some suitable conditions are satisfed. The procedure makes use of Lie group analysis so that some symmetries are required in order to obtain the required transformations, which are related to canonical variables. A linearization procedure is given in the last part of the paper valid for systems of partial differential equations that can be reduced to a single nonlinear equation linearly degenerate. The procedures are explained with some examples of physical interest.