Geometric Flows of Curves, Two-Component Camassa-Holm Equation and Generalized Heisenberg Ferromagnet Equation

In this paper, we study the generalized Heisenberg ferromagnet equation, namely, the M-CVI equation. This equation is integrable. The integrable motion of the space curves induced by the M-CVI equation is presented. Using this result, the Lakshmanan (geometrical) equivalence between the M-CVI equation and the two-component Camassa-Holm equation is established.

Generalized Ricci flow on nilpotent Lie groups

We define solitons for the generalized Ricci flow on an exact Courant algebroid. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the generalized Ricci flow on the Heisenberg group.

Symmetries and solutions to dissipative hyperbolic geometric flow

Based on the Lie-symmetric method, we study the solutions of dissipative hyperbolic geometric flows on Riemann surfaces; In the process of simplification, the mixed equations are produced. And the hyperbolic equations are obtained under limited conditions. Considering the Cauchy problem of the hyperbolic equation, the existence and uniqueness conditions of the global solutions are obtained. Finally, the phenomenon of blow up is discussed.