In this letter, we construct new meshy soliton structures by using two concrete (2+1)-dimensional integrable systems. The explicit expressions based on corresponding Cole-Hopf type transformations are obtained. Constraint equation ft+sum_{j=1}^{N} h_
j(y)f_{jx} = 0 shows that these meshy soliton structures can be linear or parabolic. Interaction between meshy soliton structure and Lump structure are also revealed.
We prove Liouville theorems for Dirac-harmonic maps from the Euclidean space $R^n$, the hyperbolic space $H^n$ and a Riemannian manifold $mathfrak{S^n}$ ($ngeq 3$) with the Schwarzschild metric to any Riemannian manifold $N$.
We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows that a di
sk with two corners admits a conformal metric with constant Gauss curvature and constant geodesic curvature on its boundary if and only if the two corners have the same angle. In fact, we can classify all the solutions in a more general situation, that of the 2-sphere cut by two planes.
We develop analytical methods for nonlinear Dirac equations. Examples of such equations include Dirac-harmonic maps with curvature term and the equations describing the generalized Weierstrass representation of surfaces in three-manifolds. We provide
the key analytical steps, i.e., small energy regularity and removable singularity theorems and energy identities for solutions.
Motivated by the supersymmetric extension of Liouville theory in the recent physics literature, we couple the standard Liouville functional with a spinor field term. The resulting functional is conformally invariant. We study geometric and analytic a
spects of the resulting Euler-Lagrange equations, culminating in a blow up analysis.
