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Backlund transformation for equations which exhibit exotic soliton solutions
410
Nonlinear
science
abstracts
respectively, solutions and transformations which map solutions into new solutions of the original system of equations. It is natural, therefore, to consider as integrable (completely integrable) these equations when the metric gAB possesses n (correspondingly, 2n) linear independent Killing vector fields, which immediately provide integral of motion of the geodesic equations of (N, gAB)' The Einstein equations with two commuting Killing fields, the EinsteinMaxwell equations with two commuting Killing fields, and the SU(m> self-dual Yang-Mills equations are the systems of equations which I am aware of which can be described both as harmonic mappings and as Lax pairs. It is remarkable that all these systems of equations are completely integrable in the proposed sense (as harmonic mappings) as well in the sense of the inverse scattering method. Ref: B.C.X., J. Phys. A: Math. Gen. 15, L61-L66(1982) ON THE INVERSION OF THE COMMUTATION OPERATOR, M. Bruschi, 0. Ragnisco, Dipartimento di Fisica, Universita di Roma "La Sapienza", Roma, ITALY, Istituto di Fisica Nucleare - Sezione de Roma, ITALY. The explicit form of the inverse of the commutation operator [A,.] is exhibited, A being a matrix of arbitrary dimension with simple characteristic roots. BACKLUND TRANSFORMATIONS FOR EQUATIONS WHICH EXHIBIT EXOTIC SOLITON SOLUTIONS, D. Levi, 0. Ragnisco and A. Sym, Department of Physics-Rome University I, 00185, Roma, I.N.F.N. Sezione di Roma, ITALY; Instiute of Theoretical Physics-Warsaw University, Warsaw, POLAND. We present a set of results on the B?icklund transformation for Wadati Konno Ichikawa equation in which we perform a deep analysis of this system in an attempt to get a comprehensive understanding of its structure. We show that we are not able to write a Bgcklund transofrmation but we can construct, in an easy way, the N-soliton solutions via the dressing method. The technique used to construct the W.K.I. equation can be extended to obtain new equations with the same properties, some of which are explicitely derived. SOME COMMENTS ON THE HARRY DYM EQUATION, Luigi Martina, Dipartimento di Fisica dell;Universita della Calabria, Arcavacata di Rende (CS), ITALY. The isospectral eigenvalue problem for the Harry Dym equation Ut
+
u3uxxx= 0
is derived using a prolongation technique. The eigenvalue equation is then exploited
to find a recursion operator.
SPECIAL FAMILIES OF HAMILTONIAN STRUCTURES AND INTEGRABLE NONLINEAR Dipartimento di Fisica della IA EVOLUTION EQUATIONS, 0. Ragnisco, Universita di Roma, P.le A. Moro 2 00185 Roma, ITALY. The algebraic properties of well-coupled Hamiltonian structures are exploited to recover, through proper reductions, some well-known hierarchies of integrable Nonlinear Evolution Equations (AKNS-ZS hierarchy, DNLS hierarchy, WKI hierarchy, etc.). This work has been done in collaboration with F. Magri and C. Morosi of the University of Milano, Italy.
Nonlinear
science
abstracts
respectively, solutions and transformations which map solutions into new solutions of the original system of equations. It is natural, therefore, to consider as integrable (completely integrable) these equations when the metric gAB possesses n (correspondingly, 2n) linear independent Killing vector fields, which immediately provide integral of motion of the geodesic equations of (N, gAB)' The Einstein equations with two commuting Killing fields, the EinsteinMaxwell equations with two commuting Killing fields, and the SU(m> self-dual Yang-Mills equations are the systems of equations which I am aware of which can be described both as harmonic mappings and as Lax pairs. It is remarkable that all these systems of equations are completely integrable in the proposed sense (as harmonic mappings) as well in the sense of the inverse scattering method. Ref: B.C.X., J. Phys. A: Math. Gen. 15, L61-L66(1982) ON THE INVERSION OF THE COMMUTATION OPERATOR, M. Bruschi, 0. Ragnisco, Dipartimento di Fisica, Universita di Roma "La Sapienza", Roma, ITALY, Istituto di Fisica Nucleare - Sezione de Roma, ITALY. The explicit form of the inverse of the commutation operator [A,.] is exhibited, A being a matrix of arbitrary dimension with simple characteristic roots. BACKLUND TRANSFORMATIONS FOR EQUATIONS WHICH EXHIBIT EXOTIC SOLITON SOLUTIONS, D. Levi, 0. Ragnisco and A. Sym, Department of Physics-Rome University I, 00185, Roma, I.N.F.N. Sezione di Roma, ITALY; Instiute of Theoretical Physics-Warsaw University, Warsaw, POLAND. We present a set of results on the B?icklund transformation for Wadati Konno Ichikawa equation in which we perform a deep analysis of this system in an attempt to get a comprehensive understanding of its structure. We show that we are not able to write a Bgcklund transofrmation but we can construct, in an easy way, the N-soliton solutions via the dressing method. The technique used to construct the W.K.I. equation can be extended to obtain new equations with the same properties, some of which are explicitely derived. SOME COMMENTS ON THE HARRY DYM EQUATION, Luigi Martina, Dipartimento di Fisica dell;Universita della Calabria, Arcavacata di Rende (CS), ITALY. The isospectral eigenvalue problem for the Harry Dym equation Ut
+
u3uxxx= 0
is derived using a prolongation technique. The eigenvalue equation is then exploited
to find a recursion operator.
SPECIAL FAMILIES OF HAMILTONIAN STRUCTURES AND INTEGRABLE NONLINEAR Dipartimento di Fisica della IA EVOLUTION EQUATIONS, 0. Ragnisco, Universita di Roma, P.le A. Moro 2 00185 Roma, ITALY. The algebraic properties of well-coupled Hamiltonian structures are exploited to recover, through proper reductions, some well-known hierarchies of integrable Nonlinear Evolution Equations (AKNS-ZS hierarchy, DNLS hierarchy, WKI hierarchy, etc.). This work has been done in collaboration with F. Magri and C. Morosi of the University of Milano, Italy.