Nondegeneracy, Morse Index and Orbital Stability of the Lump Solution to the KP-I Equation

Let $Q(x,y)= 4 \frac{y^2-x^2+3}{ (x^2+y^2+3)^2}$ be the lump solution of the KP-I equation $$\partial_x^2 (\partial_x^2 u-u + 3 u^2)-\partial_y^2 u=0.$$ We show that this solution is rigid in the following senses: the only decaying solutions to the linearized operator $$\partial_x^2 (\partial_x^2 \phi -\phi + 6 Q \phi)-\partial_y^2 \phi=0$$ consist of the linear combinations of $\partial_x Q$ and $\partial_y Q$. Furthermore we show that the Morse index is exactly one and that it is orbital stable.Comment: 37 pages; Morse index and orbital stability added;comments welcom

Nondegeneracy of the traveling lump solution to the 2+12+1 Toda lattice

We consider the $2+1$ Toda system $$\frac{1}{4}\Delta q_{n}=e^{q_{n-1}-q_{n}}-e^{q_{n}-q_{n+1}}\text{ in }\mathbb{R}^{2},\ n\in\mathbb{Z}.$$ It has a traveling wave type solution $\left\{ Q_{n}\right\}$ satisfying $Q_{n+1}(x,y)=Q_{n}(x+\frac{1}{2\sqrt{2}},y)$, and is explicitly given by $$Q_{n}\left( x,y\right) =\ln\frac{\frac{1}{4}+\left( n-1+2\sqrt{2}x\right) ^{2}+4y^{2}}{\frac{1}{4}+\left( n+2\sqrt{2}x\right) ^{2}+4y^{2}}.$$ In this paper we prove that \{$Q_{n}$\} is nondegenerate.Comment: 25page

Multi-vortex traveling waves for the Gross-Pitaevskii equation and the Adler-Moser polynomials

For $N\leq34,$ we construct traveling waves with small speed for the Gross-Pitaevskii equation, by gluing $N(N+1)/2$ pairs of degree $\pm1$ vortices of the Ginzburg-Landau equation. The location of these vortices is symmetric in the plane and determined by the Adler-Moser polynomials, which has its origin in the study of Calogero-Moser system and rational solutions of the KdV equation. The construction still works for $N>34$, under the additional assumption that the corresponding Adler-Moser polynomial has no repeated root. It is expected that this assumption holds for any $N\in\mathbb{N}$.Comment: 23 pages; comments are welcom

On Helmholtz equation and Dancer's type entire solutions for nonlinear elliptic equations

Starting from a bound state (positive or sign-changing) solution to -\Delta \omega_m =|\omega_m|^{p-1} \omega_m -\omega_m \ \ \mbox{in}\ \R^n, \ \omega_m \in H^2 (\R^n) and solutions to the Helmholtz equation \Delta u_0 + \lambda u_0=0 \ \ \mbox{in} \ \R^n, \ \lambda>0, we build new Dancer's type entire solutions to the nonlinear scalar equation -\Delta u =|u|^{p-1} u-u \ \ \mbox{in} \ \R^{m+n}. Comment: 11 page

Global minimizers of the Allen-Cahn equation in dimension n≥8n\geq 8

We prove the existence of global minimizers of Allen-Cahn equation in dimensions $8$ and above. More precisely, given any strictly area-minimizing Lawson's cones, there are global minimizers whose nodal sets are asymptotic to the cones. As a consequence of Jerison-Monneau's program we establish the existence of many counter-examples to the De Giorgi's conjecture different from the Bombierie-De Giorgi-Giusti minimal graph.Comment: 21 page

On one phase free boundary problem in RN\mathbb{R}^{N}

We construct a smooth axially symmetric solution to the classical one phase free boundary problem in $\mathbb{R}^{N}$. Its free boundary is of \textquotedblleft catenoid\textquotedblright\ type. This is a higher dimensional analogy of the Hauswirth-Helein-Pacard solution in $\mathbb{R}^{2}$ (\cite{Pacard}). The existence of such solution is conjectured in \cite [Remark 2.4]{Pacard}.Comment: 42 page

Regularization of point vortices for the Euler equation in dimension two, part II

In this paper, we continue to construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure now is carried out by constructing solutions to the following elliptic problem {cases} -\ep^2 \Delta u=(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep})_+^p-(q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep}-u)_+^p, \quad & x\in\Omega, u=0, \quad & x\in\partial\Omega, {cases} where $p>1$, $\Omega\subset\mathbb{R}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $\Omega$ is a simply-connected smooth domain, then for any given non-degenerate critical point of Kirchhoff-Routh function $\mathcal{W}(x_1^+,...,x_m^+,x_1^-,...,x_n^-)$ with $\kappa^+_i=\kappa>0\,(i=1,...,m)$ and $\kappa^-_j=-\kappa\,(j=1,...,n)$, there is a stationary classical solution approximating stationary $m+n$ points vortex solution of incompressible Euler equations with total vorticity $(m-n)\kappa$.Comment: This paper is the continuation of the paper (arXiv:1208.3002v2), 35 page

Existence and instability of deformed catenoidal solutions for fractional Allen--Cahn equation

We develop a new infinite dimensional gluing method for fractional elliptic equations. As a model problem, we construct solutions of the fractional Allen--Cahn equation vanishing on a rotationally symmetric surface which resembles a catenoid and have sub-linear growth at infinity. Moreover, such solutions are unstable.Comment: This is a revised and expanded version of the previous article titled "A gluing construction for fractional elliptic equations. Part I: a model problem on the catenoid

On a free boundary problem and minimal surfaces

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension $8$, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that Savin's theorem is optimal.Comment: 34 page

Two-end solutions to the Allen-Cahn equation in R3\mathbb{R}^{3}

In this paper we consider the Allen-Cahn equation -\Delta u = u-u^3 \ \mbox{in} \ {\mathbb R}^3 We prove that for each $k\in\left( \sqrt{2},+\infty\right),$ there exists a solution to the equation which has growth rate $k$, i.e. $$\| u-H(\cdot -k \ln r + c_k) \|_{L^\infty} \to 0$$ The main ingredients of our proof consist: (1) compactness of solutions with growth $k$, (2) moduli space theory of analytical variety of formal dimension one.Comment: 55 pages; comments welcom