arXiv subjects
Publications and source records attributed to Masaru Hamano.
In this paper, we consider nonlinear Schr\"odinger equation. Duyckaerts--Roudenko [Comm. Math. Phys. 334 (2015), no. 3, 1573--1615] investigated time behavior of solutions to the equation, whose mass-energy is greater than that of the ground state. We revisit the results and show by using the action. Moreover, we apply it to other equations without scale invariance.
We discuss the global existence of solutions to a system of stochastic Schrödinger equations with multiplicative noise. Our setting of the quadratic nonlinear terms in dimension 4 is $L^2$-critical. We treat the solutions under the ground state. We estimate the time derivative of the quantity of energy by using the cancellation of the cubic terms in the spatial derivative of the solution.
The time-global existence of solutions to a system of stochastic Schrödinger equations with multiplicative noise and the quadratic nonlinear terms are discussed in this paper. The same system in the deterministic treatment was studied in [18] where the mass and energy are conserved. In our stochastic situation, those are not conserved and which causes several difficulties in the arguments of composing a-priori estimate.
In this paper, we consider the energy critical nonlinear Schrödinger equation with a repulsive inverse square potential. In particular, we deal with radial initial data, whose energy is equal to the energy of static solution to the corresponding nonlinear Schrödinger equation without a potential. We investigate time behavior of the radial solutions with such initial data.
We consider the nonlinear Schrödinger equations on the star graph with the Kirchhoff boundary and the repulsive Dirac delta boundary at the origin. In the present paper, we show the scattering-blowup dichotomy result below the mass-energy of the ground state on the real line. The proof of the scattering part is based on a concentration compactness and rigidity argument. Our main contribution is to give a linear profile decomposition on the star graph by using a symmetrical decomposition.
This paper addresses the focusing cubic-quintic nonlinear Schrodinger equation in three space dimensions. Especially, we study the global dynamics of solutions whose energy and mass equal to those of the ground state in the sprits of Duyckaerts and Merle (2009). When we try to obtain the corresponding result, we meet several difficulties due to the cubic-quintic nonlinearity. We overcome them by using the one-pass theorem (no return theorem) developed by Nakanishi and Schlag (2012).
In this paper, we consider the mass-critical nonlinear Schrödinger equation in one dimension. Ogawa--Tsutsumi [19] proved a blow-up result for negative energy solution by using a scaling argument for initial data. By the reason, the method cannot be used to an equation with a linear potential. So, we modify the proof and get that for the equation with the linear potential.
In this paper we study long time dynamics (i.e., scattering and blow-up) of solutions for the focusing NLS with a repulsive inverse-power potential and with initial data lying exactly at the mass-energy threshold, namely, when $E_{V}(u_{0})M(u_{0})=E_{0}(Q)M(Q)$. Moreover, we prove failure of the uniform space-time bounds at the mass-energy threshold.
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.
In this paper, we consider a nonlinear Schrödinger equation with a repulsive inverse-power potential. It is known that the corresponding stationary problem has a "radial" ground state. Here, the "radial" ground state is a least energy solution among radial solutions to the stationary problem. We prove that if radial initial data below the "radial" ground state has positive virial functional, then the corresponding solution to the nonlinear Schrödinger equation scatters. In particular, we can treat not only short range potentials but also long range potentials.
In this paper, we consider the nonlinear Schrödinger equation with a repulsive inverse power potential. First, we show that some global well-posedness results and "blow-up or grow-up" results below the ground state without the potential. Then, we prove equivalence of the conditions on the initial data below the ground state without potential. We note that recently, we established existence of a radial ground state and characterized it by the virial functional for NLS with a general potential in two or higher space dimensions in [8]. Then, we also prove a global well-posedness result and a "blow-up or grow-up" result below the radial ground state with a repulsive inverse power potential obtained in [8].
In this paper, we consider the quadratic nonlinear Schrödinger system in three space dimensions. Our aim is to obtain sharp scattering criteria. Because of the mass-subcritical nature, it is difficult to do so in terms of conserved quantities. The corresponding single equation is studied by the second author and a sharp scattering criteria is established by introducing a distance from a trivial scattering solution, the zero solution. By the structure of the nonlinearity we are dealing with, the system admits a scattering solution which is a pair of zero solution and a linear solution. Taking this fact into account, we introduce a new optimizing quantity and give a sharp scattering criterion in terms of it.
We consider the quadratic nonlinear Schrödinger system (NLS system) \begin{align*}\begin{cases} i\partial_t u + Δu = v \overline{u}, \\ i\partial_t v+κΔv = u^2, \end{cases} \text{ on } I \times \mathbb{R}^5, \end{align*} where $κ>0$. The scattering below the standing wave solutions for NLS system was obtained by the first author when $κ= 1/2$. The condition of $κ=1/2$ is called mass-resonance. In this paper, we prove scattering below the standing wave solutions when $κ\neq 1/2$ under the radially symmetric assumption. Our proof is based on the concentration compactness and the rigidity by Kenig--Merle. Moreover, we discuss the concentration compactness and the rigidity for non-radial solutions.
In this paper, we consider the nonlinear Schrödinger equation with a real valued potential V=V(x). We study global behavior of solutions to the equation with a data below the ground state under some conditions for the potential V and prove a scattering result and a blowing-up result in mass-supercritical and energy-subcritical. Our proof of the blowing-up or growing-up result without radially symmetric assumption is based on the argument by Du-Wu-Zhang in [6]. We can exclude the possibility of the growing-up result by the argument in [23], [15], and [10] if "the data and the potential are radially symmetric" or "the data has finite variance".
In this paper we consider the nonlinear Schrödinger system (NLS) with quadratic interaction in five dimensions. We determine the global behavior of the solutions to the system with data below the ground state. Our proof of the scattering result is based on an argument by Kenig Merle [16]. In particular, the new part of this paper is to deal with asymmetric interaction. A blowing up or growing up result is proved by combining the argument by Du Wu Zhang in [6] and a variational characterization of minimizers. Moreover, we show a blowing-up result if the data has finite variance or is radial.