Related papers: On Transformations of the Rabelo Equations
We consider systems of partial differential equations of the form \begin{equation}\nonumber \left\{ \begin{array}{l} u_{xt}=F\left(u,u_x,v,v_x\right),\\ v_{xt}=G\left(u,u_x,v,v_x\right), \end{array} \right. \end{equation} describing…
We establish Liouville type theorems for degenerate conformally invariant equations.
We give the governing equations for multiple scalar fields in a flat Friedmann-Robertson-Walker (FRW) background spacetime on all scales, allowing for metric and field perturbations up to second order. We then derive the Klein-Gordon…
We consider on Riemannian manifolds solutions of the Leibenson equation \begin{equation*} \partial _{t}u=\Delta _{p}u^{q}. \end{equation*} This equation is also known as doubly nonlinear evolution equation. We prove gradient estimates for…
Integrable systems of the sine-Gordon/Liouville type, which arise from reducing the BPS equations for solutions invariant under 16 supersymmetries in Type IIB supergravity and M-theory, are shown to be special cases of an infinite family of…
Darboux transformation is constructed for superfields of the super sine-Gordon equation and the superfields of the associated linear problem. The Darboux transformation is shown to be related to the super B\"{a}cklund transformation and is…
We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients…
We examine the relation between two known classes of solutions of the sine--Gordon equation, which are expressed by theta functions on hyperelliptic Riemann surfaces. The first one is a consequence of the Fay's trisecant identity. The…
We consider the class of differential equations that describe pseudo-spherical surfaces of the form $u\_t=F(u,u\_x,u\_{xx})$ and $u\_{xt}=F(u, u\_x)$ given in Chern-Tenenblat \cite{ChernTenenblat} and Rabelo-Tenenblat…
We carry out the generalized symmetry classification of polylinear autonomous discrete equations defined on the square, which belong to a twelve-parametric class. The direct result of this classification is a list of equations containing no…
In this paper we classify the solutions to the geometric Neumann problem for the Liouville equation in the upper half-plane or an upper half-disk, with the energy condition given by finite area. As a result, we classify the conformal…
So far, there are described in the literature two ways to superize the Liouville equation: for a scalar field (for $N\leq 4$) and for a vector-valued field (analogs of the Leznov--Saveliev equations) for N=1. Both superizations are…
In this paper we study certain integrability properties of the supersymmetric sine-Gordon equation. We construct Lax pairs with their zero-curvature representations which are equivalent to the supersymmetric sine-Gordon equation. From the…
We give a survey of the following six closely related topics: (i) a general method for constructing a soliton hierarchy from a splitting of a loop algebra into positive and negative subalgebras, together with a sequence of commuting…
We study an integrable equation whose solutions define a triad of one-forms describing a surface with Gaussian curvature -1. We identify a local group of diffeomorphisms that preserve these solutions and establish conserved quantities. From…
A new form of a binary Darboux transformation is used to generate analytical solutions of a nonlinear Liouville-von Neumann equation. General theory is illustrated by explicit examples.
We establish three circles theorems for subharmonic functions on Riemannian manifolds with nonnegative Ricci curvature, as well as on gradient shrinking Ricci solitons with scalar curvature bounded from below by $\frac{n-2}{2}$. We also…
In this article, a numerical simulation of two dimensional nonlinear sine-Gordon equation with Neumann boundary condition is obtained by using a composite scheme referred to as a modified cubic B spline differential quadrature method. The…
In a recent paper, we established optimal Liouville-type theorems for conformally invariant second-order elliptic equations in the Euclidean space. In this work, we prove an optimal Liouville-type theorem for these equations in the…
We point out novel connections between complex PT-invariant solutions of several nonlinear equations such as $\phi^4$, $\phi^6$, sine-Gordon, hyperbolic sine-Gordon, double sine-Gordon, double hyperbolic sine-Gordon, mKdV, etc. We then use…