相关论文: Long Range Scattering for the Modified Schr"odinge…
We study the theory of scattering for a class of Hartree type equations with long range interactions in space dimension n > 2, including Hartree equations with potential V(x) = lambda |x|^{- gamma}. For 0 < gamma < or =1 we prove the…
In this paper, we study the long time dynamics of small solutions to Schr\"odinger map flows from $\Bbb R$ to Riemannian surfaces. The results are threefold. (i) We prove that for general Riemannian surface targets the points with some…
We study the long-time behavior of the (critical) Kohn--Sham equation in two and three dimensions, i.e.,\[ \mathrm{i} \partial_t {\gamma} = \Big[-\frac{1}{2}\Delta + \lambda \, |\cdot|^{-1} \ast \rho_{{\gamma}} + \mu \,…
We consider the ultrahyperbolic equation in the Euclidean space. The behavior at the infinity of a certain class of solutions is studied. We examine the issue of existence of solutions to the scattering problem: for a given asymptotics at…
Using the recent analysis of the output of the low-energy resolvent of Schr\"odinger operators on asymptotically conic manifolds (including Euclidean space) when the potential is short-range, we produce detailed asymptotic expansions for…
In this paper, we study the long time asymptotic behavior for the Cauchy problem of the Novikov equation with $3\times 3$ matrix spectral problem \begin{align} &u_{t}-u_{txx}+4 u_{x}=3uu_xu_{xx}+u^2u_{xxx}, \nonumber &u(x,…
We consider the Cauchy problem for the focusing nonlinear Schr\"odinger equation with initial data approaching two different plane waves $A_j\mathrm{e}^{\mathrm{i}\phi_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ as $x\to\pm\infty$. Using…
In this paper, we consider the Cauchy problem {align*} \{{array}{ll}&i u_t+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u, \quad t\in\mathbb{R}, \quad x\in\mathbb{R}^N &u(0,x)=\phi(x)\in \Sigma, \quad x\in\mathbb{R}^N, {array}. {align*}…
We consider the Cauchy problem for a system of balance laws derived from a chemotaxis model with singular sensitivity in multiple space dimensions. Utilizing energy methods, we first prove the global well-posedness of classical solutions to…
In this paper, we consider the Cauchy problem for an integrable real nonlocal (also called reverse-space-time) mKdV equation with nonzero boundary conditions \begin{align*} &q_t(x,t)-6\sigma q(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0,…
We consider the nonlinear Schr\"odinger equation $iu_t + \Delta u= \lambda |u|^{\frac {2} {N}} u $ in all dimensions $N\ge 1$, where $\lambda \in {\mathbb C}$ and $\Im \lambda \le 0$. We construct a class of initial values for which the…
In the recent physics literature there have appeared contradictory statements concerning the behaviour of scattering solutions of the 3-dimensional Schroedinger equation at large times. We clarify the situation and point out that the issue…
We consider a two-component system of cubic nonlinear Schr\"odinger equations in one space dimension. We show that each component of the solutions to this system behaves like a free solution in the large time, but there is a strong…
We study the final state problem for the nonlinear Schr\"{o}dinger equation with a critical long-range nonlinearity and a long-range linear potential. Given a prescribed asymptotic profile which is different from the free evolution, we…
We consider the Cauchy problem for an evolution equation modeling bidirectional surface waves in a convecting fluid. Under small condition on the initial value, the existence and asymptotic behavior of global solutions in some time weighted…
We study the Cauchy problem for the defocusing nonlinear Schr\"odinger (NLS) equation under the assumption that the solution vanishes as $x \to + \infty$ and approaches an oscillatory plane wave as $x \to -\infty$. We first develop an…
The solutions of the Landau-Lifshitz equation with finite-gap behavior at infinity are considered. By means of the inverse scattering method the large-time asymptotics is obtained.
In this paper, as an improvement of the paper [K. Ishige, T. Kawakami and H. Michihisa, SIAM J. Math. Anal. 49 (2017) pp. 2167--2190], we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy…
In this study, we examine the asymptotic behavior of solutions to nonlinear Schr\"{o}dinger equations with time-dependent harmonic oscillators and prove the time-decay property of solutions in the case of a long range power type…
We study the theory of scattering for a class of Hartree type equations with long range interactions in space dimension n > 2, including Hartree equations with potential V(x) = lambda |x|^{- gamma} with gamma < 1. For 1/2 < gamma < 1 we…