Benno Fuchssteiner | Paderborn - Academia.edu (original) (raw)
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Papers by Benno Fuchssteiner
Deutsche Texte der Salierzeit - Neuanfänge und Kontinuitäten im 11. Jahrhundert, 2010
We exhibit a simple and straight forward method for generating completely integrable nonlinear ev... more We exhibit a simple and straight forward method for generating completely integrable nonlinear evolution equations with time-dependent coefficients. For the equations under consideration, we relate the solutions to those of equations given by vector fields which are independent of time, thus explicit links between equations are obtained. As application of the proposed method we show that the linear superposition, with arbitrary time dependent coefficients, of different members of an integrable hierarchy is again integrable. Furthermore, it turns out that for some integrable equations (like the KdV, the BO or the KP) the resolvent operator of lower order flows can be explicitly obtained from that of any higher order flow. We completely classify (demonstrated for the KdV) those flows which can be generated by Lie homomorphisms coming from first order problems. Many well known equations which can be found in the literature are of that type. As an application of such a first order link we give a direct link from KdV to the cylindrical KdV, and from there to the KP with nontrivial dependence on the second spatial variable.
Topics in Soliton Theory and Exactly Solvable Nonlinear Equations
Lettere Al Nuovo Cimento, 1980
Eprint Arxiv Solv Int 9605009, May 1, 1996
A kind of Bargmann symmetry constraints involved in Lax pairs and adjoint Lax pairs is proposed f... more A kind of Bargmann symmetry constraints involved in Lax pairs and adjoint Lax pairs is proposed for soliton hierarchy. The Lax pairs and adjoint Lax pairs are nonlinearized into a hierarchy of commutative finite dimensional integrable Hamiltonian systems and explicit integrals of motion may also be generated. The corresponding binary nonlinearization procedure leads to a sort of involutive solutions to every system in soliton hierarchy which are all of finite gap. An illustrative example is given in the case of AKNS soliton hierarchy.
Progress of Theoretical Physics, Nov 1, 1987
Mathematische Zeitschrift
Lettere al Nuovo Cimento, 1980
Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, 1993
Here ordinary differential equations of third and higher order are considered; in particular, a c... more Here ordinary differential equations of third and higher order are considered; in particular, a class of equations which can be solved by quadratures is exploited. Indeed, crucial to obtain our result is the property of the Riccati equation, according to which, given one particular solution, then its general solution can be determined explicitly. Thus, what we term the "Riccati" Property is introduced to point out that the members of such a class are differential equations which are of a generalized form of Riccati equation. Trivial examples of differential equations which enjoy the Riccati Property are all linear second order ordinary differential equations. Here some further examples of ordinary differential equations which enjoy the same Property are considered. In particular, on the basis of group invariance requirements, a method to construct ordinary differential equations which enjoy the Riccati Property is given. Remarkably, it follows that ordinary differential equations enjoying the Riccati Property are related to nonlinear evolution equations which admit a hereditary recursion operator. Finally, further connections with nonlinear evolution equations are mentioned.
Deutsche Texte der Salierzeit - Neuanfänge und Kontinuitäten im 11. Jahrhundert, 2010
We exhibit a simple and straight forward method for generating completely integrable nonlinear ev... more We exhibit a simple and straight forward method for generating completely integrable nonlinear evolution equations with time-dependent coefficients. For the equations under consideration, we relate the solutions to those of equations given by vector fields which are independent of time, thus explicit links between equations are obtained. As application of the proposed method we show that the linear superposition, with arbitrary time dependent coefficients, of different members of an integrable hierarchy is again integrable. Furthermore, it turns out that for some integrable equations (like the KdV, the BO or the KP) the resolvent operator of lower order flows can be explicitly obtained from that of any higher order flow. We completely classify (demonstrated for the KdV) those flows which can be generated by Lie homomorphisms coming from first order problems. Many well known equations which can be found in the literature are of that type. As an application of such a first order link we give a direct link from KdV to the cylindrical KdV, and from there to the KP with nontrivial dependence on the second spatial variable.
Topics in Soliton Theory and Exactly Solvable Nonlinear Equations
Lettere Al Nuovo Cimento, 1980
Eprint Arxiv Solv Int 9605009, May 1, 1996
A kind of Bargmann symmetry constraints involved in Lax pairs and adjoint Lax pairs is proposed f... more A kind of Bargmann symmetry constraints involved in Lax pairs and adjoint Lax pairs is proposed for soliton hierarchy. The Lax pairs and adjoint Lax pairs are nonlinearized into a hierarchy of commutative finite dimensional integrable Hamiltonian systems and explicit integrals of motion may also be generated. The corresponding binary nonlinearization procedure leads to a sort of involutive solutions to every system in soliton hierarchy which are all of finite gap. An illustrative example is given in the case of AKNS soliton hierarchy.
Progress of Theoretical Physics, Nov 1, 1987
Mathematische Zeitschrift
Lettere al Nuovo Cimento, 1980
Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, 1993
Here ordinary differential equations of third and higher order are considered; in particular, a c... more Here ordinary differential equations of third and higher order are considered; in particular, a class of equations which can be solved by quadratures is exploited. Indeed, crucial to obtain our result is the property of the Riccati equation, according to which, given one particular solution, then its general solution can be determined explicitly. Thus, what we term the "Riccati" Property is introduced to point out that the members of such a class are differential equations which are of a generalized form of Riccati equation. Trivial examples of differential equations which enjoy the Riccati Property are all linear second order ordinary differential equations. Here some further examples of ordinary differential equations which enjoy the same Property are considered. In particular, on the basis of group invariance requirements, a method to construct ordinary differential equations which enjoy the Riccati Property is given. Remarkably, it follows that ordinary differential equations enjoying the Riccati Property are related to nonlinear evolution equations which admit a hereditary recursion operator. Finally, further connections with nonlinear evolution equations are mentioned.