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Classification of system of nonlinear ordinary differential equations with superposition principles
Nonlinear
science
abstracts
413
solidification under the action of simultaneous heat conduction and solute The formulation, based directly on fundamental principles of modern diffusion. thermodynamics, is global, in the form of a pair of conservation laws valid over Thus it is the whole region occupied by the alloy in a distributional sense. especially convenient for numerical solution since it does not require tracking The wellof the interface, which, in fact, may develop into a mushy zone. posedness of the resulting coupled system of nonlinear parabolic equations will also be discussed in a particular case. DARBOUX AND BACKLUND TRANSFORMATIONS VS DRESSING METHOD, 0. Ragnisco, Dipartimento di Fisica, Universita di Roma "La Sapienza", Roma, ITALY, I.N.F.N. - Sezione di Roma. The original theory of Darboux transformations is revised, in order to elucidate the connections between Gauge-BZcklund Transformations and Dressing Method for some well-known soliton systems. THE FREDHOLM DETERMINAT METHOD FOR THE KdV FAMILY, Christoph Pappe, SFB 123, UniversitXt, Heidelberg, FRG. A method is presented to construct solutions of every nonlinear from evolution equation belonging to the KdV and modified KdV hierarchies This transformation is linearized equations. solutions of the corresponding defined by means of the Fredholm determinant of the Gel'fand-Levitan-Marcenko The inverse scattering operator. transform, Hirota's method, and integral Backlund transformations can be described in a quite simple and elegant way in this framework. A GENERALIZED CALOGERO-MOSER SYSTEM, J. Gibbons, Dipartimento di Fisica, Universita di Roma "La Sapienza", 00185, Roma, ITALY. A generalization of the many body problem of Calogero and Moser is presented, in which the particles have extra internal degrees of freedom; it is an integrable system, and the method of solving it (analogous to the method of Olshanetsky and Perelomov) will be given. CLASSIFICATION OF SYSTEMS OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH SUPERPOSITION PRINCIPLES, S. Schnider, Department of Mathematics, McGill University, Montreal, Quebec, CANADA; P. Winternitz, Centre de Recherche de Mathematiques Appliquges, Universitg de Mont&al, Montrgal, Quebec, CANADA. The theory of transitive primitive filtered Lie algebras is used to determine and the classify all systems of first order orginary nonlinear differential equations, the general solution of which can be expressed as a function of a finite number of particular solutions. Such a system can be associated with any Lie group - subgroup pair G Go. For example if we take G=SL(2n, R) and Go the group of block triangular matrices we obtain the matrix Riccati equation dW - = A + WB + CW + WDW dt
, W,A,B,C,DE
lRnxn
Its general solution can be expressed in terms of 5 particular solutions for aribtrary n>2. Systems of equations with superposition principles occur as Bfcklund transformations for partial differential equations corresponding to completely integrable systems.
science
abstracts
413
solidification under the action of simultaneous heat conduction and solute The formulation, based directly on fundamental principles of modern diffusion. thermodynamics, is global, in the form of a pair of conservation laws valid over Thus it is the whole region occupied by the alloy in a distributional sense. especially convenient for numerical solution since it does not require tracking The wellof the interface, which, in fact, may develop into a mushy zone. posedness of the resulting coupled system of nonlinear parabolic equations will also be discussed in a particular case. DARBOUX AND BACKLUND TRANSFORMATIONS VS DRESSING METHOD, 0. Ragnisco, Dipartimento di Fisica, Universita di Roma "La Sapienza", Roma, ITALY, I.N.F.N. - Sezione di Roma. The original theory of Darboux transformations is revised, in order to elucidate the connections between Gauge-BZcklund Transformations and Dressing Method for some well-known soliton systems. THE FREDHOLM DETERMINAT METHOD FOR THE KdV FAMILY, Christoph Pappe, SFB 123, UniversitXt, Heidelberg, FRG. A method is presented to construct solutions of every nonlinear from evolution equation belonging to the KdV and modified KdV hierarchies This transformation is linearized equations. solutions of the corresponding defined by means of the Fredholm determinant of the Gel'fand-Levitan-Marcenko The inverse scattering operator. transform, Hirota's method, and integral Backlund transformations can be described in a quite simple and elegant way in this framework. A GENERALIZED CALOGERO-MOSER SYSTEM, J. Gibbons, Dipartimento di Fisica, Universita di Roma "La Sapienza", 00185, Roma, ITALY. A generalization of the many body problem of Calogero and Moser is presented, in which the particles have extra internal degrees of freedom; it is an integrable system, and the method of solving it (analogous to the method of Olshanetsky and Perelomov) will be given. CLASSIFICATION OF SYSTEMS OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH SUPERPOSITION PRINCIPLES, S. Schnider, Department of Mathematics, McGill University, Montreal, Quebec, CANADA; P. Winternitz, Centre de Recherche de Mathematiques Appliquges, Universitg de Mont&al, Montrgal, Quebec, CANADA. The theory of transitive primitive filtered Lie algebras is used to determine and the classify all systems of first order orginary nonlinear differential equations, the general solution of which can be expressed as a function of a finite number of particular solutions. Such a system can be associated with any Lie group - subgroup pair G Go. For example if we take G=SL(2n, R) and Go the group of block triangular matrices we obtain the matrix Riccati equation dW - = A + WB + CW + WDW dt
, W,A,B,C,DE
lRnxn
Its general solution can be expressed in terms of 5 particular solutions for aribtrary n>2. Systems of equations with superposition principles occur as Bfcklund transformations for partial differential equations corresponding to completely integrable systems.