A novel third order nonlinear evolution equation is introduced. It is connected, via Baecklund transformations, with the Korteweg-deVries (KdV), modified Korteweg-deVries (mKdV) equation and other third order nonlinear evolution equations. Hence, it is termed KdV-type equation. In particular, the Baecklund transformation which links the KdV singularity manifold equation with the introduced equation is used to show that the latter, further to enjoy a nontrivial invariance property, admits a...
Topics: Nonlinear Sciences, Mathematical Physics, Analysis of PDEs, Exactly Solvable and Integrable...
Source: http://arxiv.org/abs/1702.06874

Classes of third order non-Abelian evolution equations linked to that of Korteweg-de Vries-type are investigated and their connections represented in a non-commutative B\"acklund chart, generalizing results in [Fuchssteiner B., Carillo S., Phys. A 154 (1989), 467-510]. The recursion operators are shown to be hereditary, thereby allowing the results to be extended to hierarchies. The present study is devoted to operator nonlinear evolution equations: general results are presented. The...
Topics: Analysis of PDEs, Mathematical Physics, Mathematics
Source: http://arxiv.org/abs/1512.02386

The recursion operators admitted by different operator Burgers equations, in the framework of the study of nonlinear evolution equations, are here con- sidered. Specifically, evolution equations wherein the unknown is an operator acting on a Banach space are investigated. Here, the mirror non-Abelian Burgers equation is considered: it can be written as $r_t = r_{xx} + 2r_x r$. The structural properties of the admitted recursion operator are studied; thus, it is proved to be a strong symmetry...
Topics: Analysis of PDEs, Mathematical Physics, Mathematics
Source: http://arxiv.org/abs/1606.07270

The existence of solutions to a one-dimensional problem arising in magneto-viscoelasticity is here considered. Specifically, a non-linear system of integro-differential equations is analyzed, it is obtained coupling an integro-differential equation modeling the viscoelastic behaviour, in which the kernel represents the relaxation function, with the non-linear partial differential equations modeling the presence of a magnetic field. The case under investigation generalizes a previous study since...
Topics: Analysis of PDEs, Mathematical Physics, Mathematics
Source: http://arxiv.org/abs/1601.06276

A noncommutative KdV-type equation is introduced extending the Baecklund chart in [S. Carillo, M. Lo Schiavo, and C. Schiebold, SIGMA 12 (2016)]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in [P.J. Olver and V.V. Sokolov Commun. Math. Phys. 193 (1998), Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies and an explicit solution class are derived.
Topics: Mathematical Physics, Analysis of PDEs, Mathematics
Source: http://arxiv.org/abs/1704.03208