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Papers by Philippe Yves Picard
Journal of Nonlinear Mathematical Physics, 2007
Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the m... more Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the method of the weak transversality method (WTM), which is based on Lie group theory. This analytical method makes use of the symmetry group of the MHD system in situations where the "classical" Lie approach of symmetry reductions is no longer applicable. Also, a brief physical interpretation of these solutions is given.
This thesis is devoted to the resolution of the equations of magnetohydrody namics in (3+ 1) dime... more This thesis is devoted to the resolution of the equations of magnetohydrody namics in (3+ 1) dimensions with two different approaches based on the methods of conditional and classical symrnetry reductions. In Chapter 1, we briefty introduce the equations that constitute magnetohy drodynamics. The fiow under consideration is assumed to be ideal, nonstationary and isentropic for a compressible conductive ftuid placed in a magnetic field J. The electrical conductivity of the fluid is assumed to be infinitely large. In Chapter 2, we present in Section 1 a version of the conditional symmetry method for resolving quasilinear hyperbolic systems of partial differential equations of the first order. The rank-one solutions (also called simple waves solutions) obtained by this method are expressed in terms of Riemann invariants. In Section 2, we use it to obtain simple waves of MHD system equations. Section 3 generalizes the above construction to the case of many simple waves described in terms of Riemann invariants. In Section 4 we present some double waves solutions admitted by the MHD equations. Finally, in Section 5 we impose some differential constraints in order to get some new classes of solutions. In Chapter 3, we have applied the symmetry reduction method to the MHD system of equations. The symmetry algebra and its classification by conju gacy classes of r-dimensional subalgebras (i r 4) was already known [17]. We restrict our study to the three dirnensional galilean similitude subalgebras that give systems of ordinary differential equation ftom which we obtain G-invariant solutions. Finally, Chapter 4 contains conclusions of the obtained resuits and possible futures developments.
The symmetry reduction method is used to obtain invariant and conditionally invariant solutions o... more The symmetry reduction method is used to obtain invariant and conditionally invariant solutions of magnetohydrodynamic equations in (3+1) dimensions. Detail description of the procedure for constructing such solutions is presented. These solutions represent simple and double Riemann waves.
Journal of Nonlinear Mathematical Physics, 2007
Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the m... more Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the method of the weak transversality method (WTM), which is based on Lie group theory. This analytical method makes use of the symmetry group of the MHD system in situations where the "classical" Lie approach of symmetry reductions is no longer applicable. Also, a brief physical interpretation of these solutions is given.
Journal of Nonlinear Mathematical Physics, 2004
We present a version of the conditional symmetry method in order to obtain multiple wave solution... more We present a version of the conditional symmetry method in order to obtain multiple wave solutions expressed in terms of Riemann invariants. We construct an abelian distribution of vector fields which are symmetries of the original system of PDEs subjected to certain first order differential constraints. The usefulness of our approach is demonstrated on simple and double wave solutions of MHD equations. The paper also contains a comparison of the conditional symmetry method with the generalized method of characteristics.
Journal of Mathematical Analysis and Applications, 2008
We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, th... more We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, the so-called invariant solutions, of the ideal magnetohydrodynamic equations in (3 + 1) dimensions. In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1 r 4) was already known. We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. Here, some examples of these solutions are presented with a brief physical interpretation.
arXiv: Mathematical Physics, 2005
In this paper we use the symmetry reduction method to obtain invariant solutions of the ideal mag... more In this paper we use the symmetry reduction method to obtain invariant solutions of the ideal magnetohydrodynamic equations in (3+1) dimensions. These equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras ($1\leq r\leq 4$) was already known. So we restrict our study to the three-dimensional Galilean-similitude subalgebras that give systems composed of ordinary differential equations. We present here several examples of these solutions. Some of these exact solutions show interesting physical interpretations.
This thesis is devoted to the resolution of the equations of magnetohydrodynamics in (3 + 1) dime... more This thesis is devoted to the resolution of the equations of magnetohydrodynamics in (3 + 1) dimensions with two different approaches based on the methods of conditional and classical symmetry reductions.
Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the m... more Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the method of the weak transversality method (WTM), which is based on Lie group theory. This analytical method makes use of the symmetry group of the MHD system in situations where the "classical" Lie approach of symmetry reductions is no longer applicable. Also, a brief physical interpretation of these solutions is given.
Some partially invariant solutions (PIS) with a defect structure δ = 1 are constructed for the id... more Some partially invariant solutions (PIS) with a defect structure δ = 1 are constructed for the ideal magnetohydrodynamic (MHD) equations. We use a general and systematic approach based on subgroup classification methods. In particular, we restrict our study to the three-dimensional subgroups, which have generic orbits of dimension 2 in the space of independent variables, that allow us to introduce the corresponding symmetry variables and then to reduce the initial equations to different nonequivalent classes of partial differential equations.This gives us some new partially invariant solutions, which are determined to be reducible with respect to the symmetry group. Some physical interpretations of the results are briefly discussed.
The symmetry reduction method is used to obtain invariant and conditionally invariant solutions o... more The symmetry reduction method is used to obtain invariant and conditionally invariant solutions of magnetohydrodynamic equations in (3+1) dimensions. Detail description of the procedure for constructing such solutions is presented. These solutions represent simple and double Riemann waves.
We present a version of the conditional symmetry method in order to obtain multiple wave solution... more We present a version of the conditional symmetry method in order to obtain multiple wave solutions expressed in terms of Riemann invariants. We construct an abelian distribution of vector fields which are symmetries of the original system of PDEs subjected to certain first order differential constraints. The usefulness of our approach is demonstrated on simple and double wave solutions of MHD equations. The paper also contains a comparison of the conditional symmetry method with the generalized method of characteristics.
We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, th... more We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, the so-called invariant solutions, of the ideal magnetohydrodynamic equations in (3 + 1) dimensions. In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1 r 4) was already known. We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. Here, some examples of these solutions are presented with a brief physical interpretation.
Journal of Nonlinear Mathematical Physics, 2007
Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the m... more Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the method of the weak transversality method (WTM), which is based on Lie group theory. This analytical method makes use of the symmetry group of the MHD system in situations where the "classical" Lie approach of symmetry reductions is no longer applicable. Also, a brief physical interpretation of these solutions is given.
This thesis is devoted to the resolution of the equations of magnetohydrody namics in (3+ 1) dime... more This thesis is devoted to the resolution of the equations of magnetohydrody namics in (3+ 1) dimensions with two different approaches based on the methods of conditional and classical symrnetry reductions. In Chapter 1, we briefty introduce the equations that constitute magnetohy drodynamics. The fiow under consideration is assumed to be ideal, nonstationary and isentropic for a compressible conductive ftuid placed in a magnetic field J. The electrical conductivity of the fluid is assumed to be infinitely large. In Chapter 2, we present in Section 1 a version of the conditional symmetry method for resolving quasilinear hyperbolic systems of partial differential equations of the first order. The rank-one solutions (also called simple waves solutions) obtained by this method are expressed in terms of Riemann invariants. In Section 2, we use it to obtain simple waves of MHD system equations. Section 3 generalizes the above construction to the case of many simple waves described in terms of Riemann invariants. In Section 4 we present some double waves solutions admitted by the MHD equations. Finally, in Section 5 we impose some differential constraints in order to get some new classes of solutions. In Chapter 3, we have applied the symmetry reduction method to the MHD system of equations. The symmetry algebra and its classification by conju gacy classes of r-dimensional subalgebras (i r 4) was already known [17]. We restrict our study to the three dirnensional galilean similitude subalgebras that give systems of ordinary differential equation ftom which we obtain G-invariant solutions. Finally, Chapter 4 contains conclusions of the obtained resuits and possible futures developments.
The symmetry reduction method is used to obtain invariant and conditionally invariant solutions o... more The symmetry reduction method is used to obtain invariant and conditionally invariant solutions of magnetohydrodynamic equations in (3+1) dimensions. Detail description of the procedure for constructing such solutions is presented. These solutions represent simple and double Riemann waves.
Journal of Nonlinear Mathematical Physics, 2007
Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the m... more Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the method of the weak transversality method (WTM), which is based on Lie group theory. This analytical method makes use of the symmetry group of the MHD system in situations where the "classical" Lie approach of symmetry reductions is no longer applicable. Also, a brief physical interpretation of these solutions is given.
Journal of Nonlinear Mathematical Physics, 2004
We present a version of the conditional symmetry method in order to obtain multiple wave solution... more We present a version of the conditional symmetry method in order to obtain multiple wave solutions expressed in terms of Riemann invariants. We construct an abelian distribution of vector fields which are symmetries of the original system of PDEs subjected to certain first order differential constraints. The usefulness of our approach is demonstrated on simple and double wave solutions of MHD equations. The paper also contains a comparison of the conditional symmetry method with the generalized method of characteristics.
Journal of Mathematical Analysis and Applications, 2008
We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, th... more We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, the so-called invariant solutions, of the ideal magnetohydrodynamic equations in (3 + 1) dimensions. In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1 r 4) was already known. We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. Here, some examples of these solutions are presented with a brief physical interpretation.
arXiv: Mathematical Physics, 2005
In this paper we use the symmetry reduction method to obtain invariant solutions of the ideal mag... more In this paper we use the symmetry reduction method to obtain invariant solutions of the ideal magnetohydrodynamic equations in (3+1) dimensions. These equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras ($1\leq r\leq 4$) was already known. So we restrict our study to the three-dimensional Galilean-similitude subalgebras that give systems composed of ordinary differential equations. We present here several examples of these solutions. Some of these exact solutions show interesting physical interpretations.
This thesis is devoted to the resolution of the equations of magnetohydrodynamics in (3 + 1) dime... more This thesis is devoted to the resolution of the equations of magnetohydrodynamics in (3 + 1) dimensions with two different approaches based on the methods of conditional and classical symmetry reductions.
Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the m... more Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the method of the weak transversality method (WTM), which is based on Lie group theory. This analytical method makes use of the symmetry group of the MHD system in situations where the "classical" Lie approach of symmetry reductions is no longer applicable. Also, a brief physical interpretation of these solutions is given.
Some partially invariant solutions (PIS) with a defect structure δ = 1 are constructed for the id... more Some partially invariant solutions (PIS) with a defect structure δ = 1 are constructed for the ideal magnetohydrodynamic (MHD) equations. We use a general and systematic approach based on subgroup classification methods. In particular, we restrict our study to the three-dimensional subgroups, which have generic orbits of dimension 2 in the space of independent variables, that allow us to introduce the corresponding symmetry variables and then to reduce the initial equations to different nonequivalent classes of partial differential equations.This gives us some new partially invariant solutions, which are determined to be reducible with respect to the symmetry group. Some physical interpretations of the results are briefly discussed.
The symmetry reduction method is used to obtain invariant and conditionally invariant solutions o... more The symmetry reduction method is used to obtain invariant and conditionally invariant solutions of magnetohydrodynamic equations in (3+1) dimensions. Detail description of the procedure for constructing such solutions is presented. These solutions represent simple and double Riemann waves.
We present a version of the conditional symmetry method in order to obtain multiple wave solution... more We present a version of the conditional symmetry method in order to obtain multiple wave solutions expressed in terms of Riemann invariants. We construct an abelian distribution of vector fields which are symmetries of the original system of PDEs subjected to certain first order differential constraints. The usefulness of our approach is demonstrated on simple and double wave solutions of MHD equations. The paper also contains a comparison of the conditional symmetry method with the generalized method of characteristics.
We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, th... more We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, the so-called invariant solutions, of the ideal magnetohydrodynamic equations in (3 + 1) dimensions. In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1 r 4) was already known. We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. Here, some examples of these solutions are presented with a brief physical interpretation.