相关论文: Vanishing estimates for fully bubbling solutions o…

It is well known that the study of $SU(n+1)$ Toda systems is important not only to Chern-Simons models in Physics, but also to the understanding of holomorphic curves, harmonic sequences or harmonic maps from Riemann surfaces to $\mathbb…

In this article we study bubbling solutions of regular $SU(3)$ Toda systems defined on a Riemann surface. There are two major difficulties corresponding to the profile of bubbling solutions: partial blowup phenomenon and bubble…

In this paper, we study general Toda systems with homogeneous Neumann boundary conditions on Riemann surfaces. Assuming the surface satisfies the ``$k$-symmetric'' condition, we construct a family of bubbling solutions using singular…

In this note, we consider blow-up for solutions of the SU(3) Toda system on a compact surface \Sigma. In particular, we give a complete proof of the compactness result stated by Jost, Lin and Wang and we extend it to the case of…

In this article we prove that for locally defined singular SU(n+1) Toda systems in R^2, the profile of fully bubbling solutions near the singular source can be accurately approximated by global solutions. The main ingredients of our new…

For Toda systems with Cartan matrix either $B_2$ or $G_2$, we prove that the local mass of blowup solutions at its blowup points converges to a finite set. Further more this finite set can be completely determined for $B_2$ Toda systems,…

For regular $SU(3)$ Toda systems defined on Riemann surface, we initiate the study of bubbling solutions if parameters $(\rho_1^k,\rho_2^k)$ are both tending to critical positions: $(\rho_1^k,\rho_2^k)\to (4\pi, 4\pi N)$ or $(4\pi N, 4\pi)$…

We consider the $SU(n+1)$ Toda system $$(S_\lambda) \quad \left\{ \begin{aligned} & \Delta u_1 + 2\lambda e^{u_1} - \lambda e^{u_2}- \dots - \lambda e^{u_k} = 0\quad \hbox{in}\ \Omega,\\ & \Delta u_2 - \lambda e^{u_1} + 2\lambda e^{u_2} -…

We consider the so-called Toda system in a smooth planar domain under homogeneous Dirichlet boundary conditions. We prove the existence of a continuum of solutions for which both components blow-up at the same point. This blow-up behavior…

For all rank two Toda systems with an arbitrary singular source, we use a unified approach to prove: (i) The pair of local masses $(\sigma_1,\sigma_2)$ at each blowup point has the expression $$\sigma_i=2(N_{i1}\mu_1+N_{i2}\mu_2+N_{i3}),$$…

In this paper we consider the so-called Toda System in planar domains under Dirichlet boundary condition. We show the existence of continua of solutions for which one component is blowing up at a certain number of points. The proofs use…

In this paper, we study the degree counting formula of the rank two Toda system with simple singular source when $\rho_1\in(0,4\pi)\cup(4\pi,8\pi)$ and $\rho_2\notin 4\pi\mathbb{N}.$ The key step is to derive the degree formula of the…

For a singular Liouville equation, it is plausible that a non-simple blowup phenomenon occurs around a quantized singular pole. The presence of complex blowup profiles of bubbling solutions presents substantial challenges in applications.…

In this paper, we continue to consider the 2-dimensional (open) Toda system (Toda lattice) for $SU(N+1)$. We give a much more precise bubbling behavior of solutions and study its existence in some critical cases

In this paper, we study the blow-up analysis of an affine Toda system corresponding to minimal surfaces into ${\mathbb S}^4$ [19]. This system is an integrable system which is a natural generalization of sinh-Gordon equation [18]. By…

We introduce the super-Toda system on Riemann surfaces and study the blow-up analysis for a sequence of solutions to the super-Toda system on a closed Riemann surface with uniformly bounded energy. In particular, we show the energy…

Under spherical symmetry, with double-null coordinates $(u,v)$, we study the gravitational collapse of the Einstein--scalar field system with a positive cosmological constant. The spacetime singularities arise when area radius $r$ vanishes…

For Liouville equations with singular sources, the interpretation of the equation and its impact are most significant if the singular sources are quantized: the strength of each Dirac mass is a mutliple of $4\pi$. However the study of…

This work studies the partial blow-up phenomena for the $SU(3)$ Toda system on compact Riemann surfaces with smooth boundary. We consider the following coupled Liouville system with Neumann boundary conditions: $$ -\Delta_g u_1 =…

In several fields of Physics, Chemistry and Ecology, some models are described by Liouville systems. In this article we first prove a uniqueness result for a Liouville system in $\mathbb R^2$. Then we establish an uniform estimate for…