ABSTRACT An operator equation on a Banach space, which represents the operator analog of Burgers ... more ABSTRACT An operator equation on a Banach space, which represents the operator analog of Burgers equation, is here considered. The well known Cole-Hopf transformation, a particular case of the wider class of Bäcklund transformations, which connects the classical nonlinear Burgers equation to the linear heat equation, is extended to the case of operator valued equations. Then, since the operator Burgers equation admits a recursion operator, a whole hierarchy of Burgers operator equations is generated. Notably, each member of such a Burgers operator hierarchy is related, via Cole-Hopf transformation to the corresponding member of a heat operator hierarchy. Indeed, also the recursion operator admitted by the Burgers operator equation, is related, via Cole-Hopf transformation, to the (trivial) recursion operator admitted by the linear heat operator equation. Furthermore, the Burgers recursion operator is not Abelian, hence, the whole hierarchy does not enjoy commutativity properties.
Discrete and Continuous Dynamical Systems - Series S, 2012
ABSTRACT The dynamics of magneto-viscoelastic materials is described by a nonlinear system which ... more ABSTRACT The dynamics of magneto-viscoelastic materials is described by a nonlinear system which couples the equation of the magnetization, given in Gibert form, and the viscoelastic integro-differential equation for the displacements. We study the general three-dimensional case and establish a theorem for the existence of weak solutions. The existence is proved by compactness of the approximated penalty problem.
The present work continues work on KdV-type hierarchies presented by S. Carillo and C. Schiebold ... more The present work continues work on KdV-type hierarchies presented by S. Carillo and C. Schiebold [“Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods,” J. Math. Phys. 50, 073510 (2009)]10.1063/1.3155080. General solution formulas for the KdV and mKdV hierarchies are derived by means of Banach space techniques both in the scalar and matrix case. A detailed analysis is given of solitons, breathers, their countable superpositions as well as of multisoliton solutions for the matrix hierarchies.
Here, noncommutative hierarchies of nonlinear equations are studied. They represent a generalizat... more Here, noncommutative hierarchies of nonlinear equations are studied. They represent a generalization to the operator level of corresponding hierarchies of scalar equations, which can be obtained from the operator ones via a suitable projection. A key tool is the application of Bäcklund transformations to relate different operator-valued hierarchies. Indeed, in the case when hierarchies in 1+1-dimensions are considered, a “Bäcklund chart” depicts links relating, in particular, the Korteweg–de Vries (KdV) to the modified KdV (mKdV) hierarchy. Notably, analogous links connect the hierarchies of operator equations. The main result is the construction of an operator soliton solution depending on an infinite-dimensional parameter. To start with, the potential KdV hierarchy is considered. Then Bäcklund transformations are exploited to derive solution formulas in the case of KdV and mKdV hierarchies. It is remarked that hierarchies of matrix equations, of any dimension, are also incorporated in the present framework.
Waves and Stability in Continuous Media, Apr 1, 2008
... Korteweg-deVries (Kdv), the modified Korteweg-deVries (mKdv), the Harry Dym (Dym), on one sid... more ... Korteweg-deVries (Kdv), the modified Korteweg-deVries (mKdv), the Harry Dym (Dym), on one side, and the Caudrey-Dodd-Gibbon (cdg), the Kaup-Kupershmidt (kk) and the Kawamoto equations, on the other one, are all connected via two analogous Backlund Charts which ...
ABSTRACT An operator equation on a Banach space, which represents the operator analog of Burgers ... more ABSTRACT An operator equation on a Banach space, which represents the operator analog of Burgers equation, is here considered. The well known Cole-Hopf transformation, a particular case of the wider class of Bäcklund transformations, which connects the classical nonlinear Burgers equation to the linear heat equation, is extended to the case of operator valued equations. Then, since the operator Burgers equation admits a recursion operator, a whole hierarchy of Burgers operator equations is generated. Notably, each member of such a Burgers operator hierarchy is related, via Cole-Hopf transformation to the corresponding member of a heat operator hierarchy. Indeed, also the recursion operator admitted by the Burgers operator equation, is related, via Cole-Hopf transformation, to the (trivial) recursion operator admitted by the linear heat operator equation. Furthermore, the Burgers recursion operator is not Abelian, hence, the whole hierarchy does not enjoy commutativity properties.
Discrete and Continuous Dynamical Systems - Series S, 2012
ABSTRACT The dynamics of magneto-viscoelastic materials is described by a nonlinear system which ... more ABSTRACT The dynamics of magneto-viscoelastic materials is described by a nonlinear system which couples the equation of the magnetization, given in Gibert form, and the viscoelastic integro-differential equation for the displacements. We study the general three-dimensional case and establish a theorem for the existence of weak solutions. The existence is proved by compactness of the approximated penalty problem.
The present work continues work on KdV-type hierarchies presented by S. Carillo and C. Schiebold ... more The present work continues work on KdV-type hierarchies presented by S. Carillo and C. Schiebold [“Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods,” J. Math. Phys. 50, 073510 (2009)]10.1063/1.3155080. General solution formulas for the KdV and mKdV hierarchies are derived by means of Banach space techniques both in the scalar and matrix case. A detailed analysis is given of solitons, breathers, their countable superpositions as well as of multisoliton solutions for the matrix hierarchies.
Here, noncommutative hierarchies of nonlinear equations are studied. They represent a generalizat... more Here, noncommutative hierarchies of nonlinear equations are studied. They represent a generalization to the operator level of corresponding hierarchies of scalar equations, which can be obtained from the operator ones via a suitable projection. A key tool is the application of Bäcklund transformations to relate different operator-valued hierarchies. Indeed, in the case when hierarchies in 1+1-dimensions are considered, a “Bäcklund chart” depicts links relating, in particular, the Korteweg–de Vries (KdV) to the modified KdV (mKdV) hierarchy. Notably, analogous links connect the hierarchies of operator equations. The main result is the construction of an operator soliton solution depending on an infinite-dimensional parameter. To start with, the potential KdV hierarchy is considered. Then Bäcklund transformations are exploited to derive solution formulas in the case of KdV and mKdV hierarchies. It is remarked that hierarchies of matrix equations, of any dimension, are also incorporated in the present framework.
Waves and Stability in Continuous Media, Apr 1, 2008
... Korteweg-deVries (Kdv), the modified Korteweg-deVries (mKdv), the Harry Dym (Dym), on one sid... more ... Korteweg-deVries (Kdv), the modified Korteweg-deVries (mKdv), the Harry Dym (Dym), on one side, and the Caudrey-Dodd-Gibbon (cdg), the Kaup-Kupershmidt (kk) and the Kawamoto equations, on the other one, are all connected via two analogous Backlund Charts which ...
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