In this paper, we establish the soliton resolution for the energy critical wave equation with inverse square potential in the radial case and in all dimensions $N\geq3$. The structure of the radial linear operator $\mathcal{L}_a :=-Δ+\frac{a}{|x|^2}=A^*A$, is essential for the channel of energy, where $A$ is a first order differential operator and $A^*$ is its adjoint operator. Modulation and analysis of the multi-solitons are performed in the function spaces $\dot{H}^1_a(\Bbb R^N)\times L^2(\Bbb R^N)$ associated with $\mathcal{L}_a$.
In this paper we prove the soliton resolution conjecture for all times, for all solutions in the energy space, of the co-rotational wave map equation. To our knowledge this is the first such result for all initial data in the energy space for a wave-type equation. We also prove the corresponding results for radial solutions, which remain bounded in the energy norm, of the cubic (energy-critical) nonlinear wave equation in space dimension 4.
We consider the focusing energy-critical nonlinear wave equation for radially symmetric initial data in space dimensions $D \ge 4$. This equation has a unique (up to sign and scale) nontrivial, finite energy stationary solution $W$, called the ground state. We prove that every finite energy solution with bounded energy norm resolves, continuously in time, into a finite superposition of asymptotically decoupled copies of the ground state and free radiation.
Consider the energy-critical focusing wave equation in odd space dimension $N\geq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. In this paper we prove that any radial, bounded in the energy norm solution of the equation behaves asymptotically as a sum of modulated $W$s, decoupled by the scaling, and a radiation term. The proof essentially boils down to the fact that the equation does not have purely nonradiative multisoliton solutions. The proof overcomes the fundamental obstruction for the extension of the 3D case (treated in our…
In a recent paper, we developed an inverse scattering approach to the Boussinesq equation in the case when no solitons are present. In this paper, we extend this approach to include solutions with solitons.
We show how the inverse scattering transform can be used as a convenient tool to derive the long-time asymptotics of shock waves for the Korteweg-de Vries (KdV) equation in the soliton region. In particular, we improve the results previously obtained via the nonlinear steepest decent approach both with respect to the decay of the initial conditions as well as the region where they are valid.
We study solutions to the inverse mean curvature flow which evolve by homotheties of a given submanifold with arbitrary dimension and codimension. We first show that the closed ones are necessarily spherical minimal immersions and so we reveal the strong rigidity of the Clifford torus in this setting. Mainly we focus on the Lagrangian case, obtaining numerous examples and uniqueness results for some products of circles and spheres that generalize the Clifford torus to arbitrary dimension. We also characterize the pseudoumbilical ones in terms of soliton curves for the inverse curve shortening …
In this paper, we consider radial standing waves to a nonlinear Klein-Gordon equation with a repulsive inverse-square potential. It is known that existence of a "radial" ground state to the stationary problem of the nonlinear Klein-Gordon equation. Here, the "radial" ground state is a solution with the least energy among radial solutions to the stationary problem. We deal with stability and instability of the standing wave for the "radial" ground state.
Existence and uniqueness in Minkowski space of entire downward translating solitons with prescribed values at infinity for a scalar curvature flow equation. The radial case translates into an ordinary differential equation and the general case into a fully non-linear elliptic PDE on R^n.
We address stability of multi-solitons in the cubic NLS (nonlinear Schrödinger) equation on the line. By using the dressing transformation and the inverse scattering transform methods, we obtain the orbital stability of multi-solitons in the $L^2(\mathbb{R})$ space when the initial data is in a weighted $L^2(\mathbb{R})$ space.
In this article we consider the Cauchy problem for the cubic focusing nonlinear Schrö\-dinger (NLS) equation on the line with initial datum close to a particular $N$-soliton. Using inverse scattering and the $\bar{\partial}$ method we establish the decay of the $L^{\infty}$ norm of the residual term in time.
We prove the existence of global solutions to the DNLS equation with initial data in a large subset of $H^2(\mathbb R)\cap H^{1,1}(\mathbb R)$ containing a neighborhood of all solitons. We use the inverse scattering transform method, which was recently developed by D. Pelinovsky and Y. Shimabukuro, and an auto-Bäcklund transform in order to include solitons.
Consider the nonlinear Schrödinger equation (NLS) with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and, if the nonlinearity is focusing, then also solitons with positive large energy, which are unstable. In this paper we classify the global dynamics below the second lowest energy of solitons under small mass and radial symmetry constraints.
We consider the focusing energy-critical quintic nonlinear wave equation in three dimensional Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve in $ \dot H^s_x(\mathbb{R}^3) \times H^{s-1}_x(\mathbb{R}^3)$, for any $s > 1/2$. By randomizing radial initial data in $ \dot H^s_x(\mathbb{R}^3) \times H^{s-1}_x(\mathbb{R}^3)$ for $s > 5/6$, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton which give rise …
In this paper we consider global and non-global radial solutions of the focusing energy--critical wave equation on $\mathbb{R} \times \mathbb{R}^N$ where $N \geq 5$ is odd. We prove that if the solution remains bounded in the energy space as you approach the maximal forward time of existence, then along a sequence of times converging to the maximal forward time of existence, the solution decouples into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space. If, in addition, we assume a bound on the evolution that rules out formation of m…
We show that, for the 1d cubic NLS equation, widely separated equal amplitude in-phase solitons attract and opposite-phase solitons repel. Our result gives an exact description of the evolution of the two solitons valid until the solitons have moved a distance comparable to the logarithm of the initial separation. Our method does not use the inverse scattering theory and should be applicable to nonintegrable equations with local nonlinearities that support solitons with exponentially decaying tails. The result is presented as a special case of a general framework which also addresses, for exam…
The large-time behavior of solutions to the derivative nonlinear Schrödinger equation is established for initial conditions in some weighted Sobolev spaces under the assumption that the initial conditions do not support solitons. Our approach uses the inverse scattering setting and the nonlinear steepest descent method of Deift and Zhou as recast by Dieng and McLaughlin.
In this note we prove scattering for perturbations of solitons in the scaling space appropriate for the quartic nonlinearity, namely $\dot{H}^{-\frac16}$. The article relies strongly on refined estimates for a KdV equation linearized at the soliton. In contrast to the work of Tao (2006), we are able to work purely in the scaling space without additional regularity assumptions, allowing us to prove some results on the existence of inverse wave operators.
Inverse scattering for the focusing nonlinear Schrödinger equation with elliptic background and full soliton gas
Scuola Internazionale Superiore di Studi Avanzati (SISSA)（高等国际研究学院）；University of Central Florida（中佛罗里达大学）；China University of Mining and Technology（中国矿业大学）
We consider the quadratic semilinear wave equation in six dimensions. This energy critical problem admits a ground state solution, which is the unique (up to scaling) positive stationary solution. We prove that any spherically symmetric solution, that remains bounded in the energy norm, evolves asymptotically to a sum of decoupled modulated ground states, plus a radiation term. As a by-product of the approach we prove the non-existence of multisoliton solutions that do not emit any radiation. The proof follows the method initiated for large odd dimensions by the last three authors, reducing th…
In this work, a generalized nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation is introduced, and its integrability as an infinite dimensional Hamilton dynamic system is established. Motivated by the ideas of Ablowitz and Musslimani (2016 Nonlinearity 29 915), we successfully derive the inverse scattering transform (IST) of the nonlocal LPD equation. The direct scattering problem of the equation is first constructed, and some important symmetries of the eigenfunctions and the scattering data are discussed. By using a novel Left-Right Riemann-Hilbert (RH) problem, the inverse scattering proble…
We present a rigorous theory of a unified and simple inverse scattering transform (IST) for both focusing and defocusing real nonlocal (reverse-space-time) modified Korteweg-de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity. The IST problems for the nonlocal equations with NZBCs are more complicated then ones for the local equations with NZBCs. The suitable uniformization variable is introduced in order to make the direct and inverse problems be established on a complex plane instead of a two-sheeted Riemann surface. The direct scattering problem establishes the a…
We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow (cf. \cite{DLW}) in $\mathbb{R}^n\times \mathbb{R}$, $n\ge 2$, of the form $(r,y(r))$ or $(r(y),y)$ where $r=|x|$, $x\in\mathbb{R}^n$, is the radially symmetric coordinate and $y\in \mathbb{R}$. More precisely for any $\frac{1}{n}<λ<\frac{1}{n-1}$ and $μ<0$, we will give a new proof of the existence of a unique solution $r(y)\in C^2(μ,\infty)\cap C([μ,\infty))$ of the equation $\frac{r_{yy}(y)}{1+r_y(y)^2}=\frac{n-1}{r(y)}-\frac{1+r_y(y)^2}{λ(r(y)-yr_y(y))}$, $r(y)>0$, in $(μ,\infty)…
The theory of inverse scattering is developed to study the initial-value problem for the modified matrix Korteweg-de Vries (mmKdV) equation with the $2m\times2m$ $(m\geq 1)$ Lax pairs under the nonzero boundary conditions at infinity. In the direct problem, by introducing a suitable uniform transformation we establish the proper complex $z$-plane in order to discuss the Jost eigenfunctions, scattering matrix and their analyticity and symmetry of the equation. Moreover the asymptotic behavior of the Jost functions and scattering matrix needed in the inverse problem are analyzed via Wentzel-Kram…
We prove a removable singularities theorem for stationary Einstein equations, with useful implications for constructions of stationary solutions using soliton methods.
We use the inverse scattering transform, the auto-Backlund transformation and the steepest descent method of Deift and Zhou to obtain the asymptotic stability of the solitons in the cubic NLS (nonlinear Schrodinger) equation.
