In this paper, we investigate KP equation by the Hirota bilinear method and obtain its bilinear form successfully. On the basis of above bilinear form, a number of explicit solutions including one-solitary wave solution, two-solitary wave solution and their generalized form N-solitary wave solution are obtained successfully. Moreover, in view of the homoclinic breather limit method, we also express breather wave solutions and rogue wave solutions of KP-equation. Finally, with mathematical software Maple, we obtained overhead views, perspective views and wave propagation pattern of solutions in different parameter areas.

The motion of plane curves specified by hyperbolic inverse mean curvature with a constant force term is considered. We proved that this flow remains the convexity for any forced term. Furthermore, we give an example to understand how the constant forced term $c$ affects this hyperbolic inverse mean flow. Particularly, the asymptotic behavior of the flow under different initial conditions is discussed.

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.

This paper considers the initial value problem for a normal hyperbolic curvature flow derived by the cell-based mathematical models of tissue growth to account for the mechanistic influence of curvature on cell evolution. The equations satisfied by support functions under this flow is a hyperbolic Monge-Amp$\grave{e}$re equation. The equation for both perimeter and area of closed curves under the flow are also obtained. Based on this, we show that for a closed curve, if the initial velocity $v_{0}<0$, the solution of this flow converges to a point in finite time; if $v_{0}>0$, the solution of this flow exists for all $t\in[0,\infty)$.

This paper concerns the generalized hyperbolic mean curvature flow for spacelike curves in Minkowski $R^{1,1}$. Base on the derived quasilinear hyperbolic system, we investigate the formation of singularities in the motion of these curves. In particular, under the generalized hyperbolic mean curvature flow, we prove that the motion of periodic spacelike curves with small variation on one period and small initial velocity blows up in finite time. Some blowup results have been obtained and the estimates on the life-span of the solutions are given.