arXiv subjects

Hiroyuki Hirayama

Publications and source records attributed to Hiroyuki Hirayama.

At least 19 recordsLinked to original sources

arXiv2026

On solitary wave solutions with two-frequency parameters to the three-component system of quadratic nonlinear Schrödinger equations

In the present paper, we consider the Cauchy problem of a system of three nonlinear Schrödinger equations with quadratic nonlinearity. We first prove the existence of ground states in the form of solitary wave solutions with two frequency parameters. We then show that the conditions on the frequency parameters for the existence of ground states depend on the resonance structure of the system. Next, we give the two results for global solutions. The first is an improvement of global well-posedness for initial data below the ground state threshold. The second is an improvement of global well-posedness for oscillating initial data. We also prove the orbital stability of the ground state sets with small speed parameter.

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arXiv2025

Strichartz estimates and its application to the well-posedness of the nonlinear Schrödinger equations on H-type groups

The aim of this article is to give the well-posedness results for the Cauchy problem of the nonlinear Schrödinger equation with power type nonlinearities on H-type groups. To do this, we prove the dispersive estimate and Strichartz estimate. Although these estimates are given by Hierro (2005), its complete proofs cannot be find. We correct the statement of these estimates, give the proofs, and apply to the nonlinear problem. Our well-posedness results are an improvement of the previous result by Bruno et al.

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arXiv2025

Well-posedness and ill-posedness for a system of periodic quadratic derivative nonlinear Schrödinger equations

We consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. For the nonperiodic setting, the authors proved some well-posedness results, which contain the scaling critical case for $d\geq 2$. In the present paper, we prove the well-posedness of this system for the periodic setting. In particular, well-posedness is proved at the scaling critical regularity for $d\geq 3$ under some conditions for the coefficients of the Laplacian. We also prove some ill-posedness results. As long as we use an iteration argument, our well-posedness results are optimal except for some critical cases.

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arXiv2024

Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations with low regularity periodic initial data

We consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. For the nonperiodic case, the author proved the small data global well-posedness and the scattering at the scaling critical regularity for $d\geq 2$ when the coefficients of Laplacian satisfy some condition. In the present paper, we prove the well-posedness of the system for the periodic case. In particular, well-posedness is proved at the scaling critical regularity for $d\geq 3$ under some condition for the coefficients of Laplacian.

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arXiv2024

Optimal Decay Estimate and Asymptotic Profile for Solutions to the Generalized Zakharov-Kuznetsov-Burgers Equation in 2D

We consider the Cauchy problem for the generalized Zakharov-Kuznetsov-Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative equations, which has a spatial anisotropic dissipative term $-μu_{xx}$. In this paper, we prove that the solution to this problem decays at the rate of $t^{-\frac{3}{4}}$ in the $L^{\infty}$-sense, provided that the initial data $u_{0}(x, y)$ satisfies $u_{0}\in L^{1}(\mathbb{R}^{2})$ and some appropriate regularity assumptions. Moreover, we investigate the more detailed large time behavior and obtain a lower bound of the $L^{\infty}$-norm of the solution. As a result, we prove that the given decay rate $t^{-\frac{3}{4}}$ of the solution to be optimal. Furthermore, combining the techniques used for the parabolic equations and for the Schr$\ddot{\mathrm{o}}$dinger equation, we derive the explicit asymptotic profile for the solution.

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arXiv2024

Results of existence and uniqueness for the Cauchy problem of semilinear heat equations on stratified Lie groups

The aim of this paper is to give existence and uniqueness results for solutions of the Cauchy problem for semilinear heat equations on stratified Lie groups $\mathbb{G}$ with the homogeneous dimension $N$. We consider the nonlinear function behaves like $|u|^α$ or $|u|^{α-1}u$ $(α>1)$ and the initial data $u_0$ belongs to the Sobolev spaces $L^p_s(\mathbb{G})$ for $1<p<\infty$ and $0<s<N/p$. Since stratified Lie groups $\mathbb{G}$ include the Euclidean space ${\mathbb R}^n$ as an example, our results are an extension of the existence and uniqueness results obtained by F. Ribaud on ${\mathbb R}^n$ to $\mathbb{G}$. It should be noted that our proof is very different from it given by Ribaud on ${\mathbb R}^n$. We adopt the generalized fractional chain rule on $\mathbb{G}$ to obtain the estimate for the nonlinear term, which is very different from the paracomposition technique adopted by Ribaud on ${\mathbb R}^n$. By using the generalized fractional chain rule on $\mathbb{G}$, we can avoid the discussion of Fourier analysis on $\mathbb{G}$ and make the proof more simple.

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arXiv2023

Variational problems for the system of nonlinear Schrödinger equations with derivative nonlinearities

We consider the Cauchy problem of the system of nonlinear Schrödinger equations with derivative nonlinearlity. This system was introduced by Colin-Colin (2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin-Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for $1$-dimension.

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arXiv2023

Large time behavior and optimal decay estimate for solutions to the generalized Kadomtsev--Petviashvili--Burgers equation in 2D

We consider the Cauchy problem for the generalized Kadomtsev--Petviashvili--Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropic dissipative term. Under some suitable regularity assumptions on the initial data $u_{0}$, especially the condition $\partial_{x}^{-1}u_{0} \in L^{1}(\mathbb{R}^{2})$, it is known that the solution to this problem decays at the rate of $t^{-\frac{7}{4}}$ in the $L^{\infty}$-sense. In this paper, we investigate the more detailed large time behavior of the solution and construct the approximate formula for the solution at $t\to \infty$. Moreover, we obtain a lower bound of the $L^{\infty}$-norm of the solution and prove that the decay rate $t^{-\frac{7}{4}}$ of the solution given in the previous work to be optimal.

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arXiv2022

Sharp well-posedness for the Cauchy problem of the two dimensional quadratic nonlinear Schrödinger equation with angular regularity

This paper is concerned with the Cauchy problem of the quadratic nonlinear Schrödinger equation in $\mathbb{R} \times \mathbb{R}^2$ with the nonlinearity $η|u|^2$ where $η\in \mathbb{C} \setminus \{0\}$ and low regularity initial data. If $s < -1/4$, the ill-posedness result in the Sobolev space $H^{s}(\mathbb{R}^2)$ is known. We will prove the well-posedness in $H^s(\mathbb{R}^2)$ for $-1/2 < s < -1/4$ by assuming some angular regularity on initial data. The key tools are the modified Fourier restriction norm and the convolution estimate on thickened hypersurfaces.

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arXiv2020

Well-posedness for the fourth-order Schrödinger equation with third order derivative nonlinearities

We study the Cauchy problem to the semilinear fourth-order Schrödinger equations: \begin{equation}\label{0-1}\tag{4NLS} \begin{cases} i\partial_t u+\partial_x^4u=G\left(\left\{\partial_x^{k}u\right\}_{k\le γ},\left\{\partial_x^{k}\bar{u}\right\}_{k\le γ}\right), & t>0,\ x\in \mathbb{R}, \\ \ \ \ u|_{t=0}=u_0\in H^s(\mathbb{R}), \end{cases} \end{equation} where $γ\in \{1,2,3\}$ and the unknown function $u=u(t,x)$ is complex valued. In this paper, we consider the nonlinearity $G$ of the polynomial \[ G(z)=G(z_1,\cdots,z_{2(γ+1)}) :=\sum_{m\le |α|\le l}C_αz^α, \] for $z\in \mathbb{C}^{2(γ+1)}$, where $m,l\in\mathbb{N}$ with $3\le m\le l$ and $C_α\in \mathbb{C}$ with $α\in (\mathbb{N}\cup \{0\})^{2(γ+1)}$ is a constant. The purpose of the present paper is to prove well-posedness of the problem (\ref{0-1}) in the lower order Sobolev space $H^s(\mathbb{R})$ or with more general nonlinearities than previous results. Our proof of the main results is based on the contraction mapping principle on a suitable function space employed by D. Pornnopparath (2018). To obtain the key linear and bilinear estimates, we construct a suitable decomposition of the Duhamel term introduced by I. Bejenaru, A. D. Ionescu, C. E. Kenig, and D. Tataru (2011). Moreover we discuss scattering of global solutions and the optimality for the regularity of our well-posedness results, namely we prove that the flow map is not smooth in several cases.

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arXiv2018

Well-posedness for KdV-type equations with quadratic nonlinearity

We consider the Cauchy problem of the KdV-type equation \[ \partial_t u + \frac{1}{3} \partial_x^3 u = c_1 u \partial_x^2u + c_2 (\partial_x u)^2, \quad u(0)=u_0. \] Pilod (2008) showed that the flow map of this Cauchy problem fails to be twice differentiable in the Sobolev space $H^s(\mathbb{R})$ for any $s \in \mathbb{R}$ if $c_1 \neq 0$. By using a gauge transformation, we point out that the contraction mapping theorem is applicable to the Cauchy problem if the initial data are in $H^2(\mathbb{R})$ with bounded primitives. Moreover, we prove that the Cauchy problem is locally well-posed in $H^1(\mathbb{R})$ with bounded primitives.

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arXiv2018

Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations with radial initial data

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations. This system was introduced by M. Colin and T. Colin (2004). The first and second authors obtained some well-posedness results in the Sobolev space $H^{s}(\mathbb{R}^d)$. We improve these results for conditional radial initial data by rewriting the system radial form.

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arXiv2018

Local and global well-posedness for the 2D Zakharov-Kuznetsov-Burgers equation in low regularity Sobolev space

In the present paper, we consider the Cauchy problem of the 2D Zakharov-Kuznetsov-Burgers (ZKB) equation, which has the dissipative term $-\partial_x^2u$. This is known that the 2D Zakharov-Kuznetsov equation is well-posed in $H^s(\mathbb{R}^2)$ for $s>1/2$, and the 2D nonlinear parabolic equation with quadratic derivative nonlinearity is well-posed in $H^s(\mathbb{R}^2)$ for $s\ge 0$. By using the Fourier restriction norm with dissipative effect, we prove the well-posedness for ZKB equation in $H^s(\mathbb{R}^2)$ for $s>-1/2$.

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arXiv2018

Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schrödinger equations

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations for the spatial dimension $d=2$ and $3$. This system was introduced by M. Colin and T. Colin (2004). The first author obtained some well-posedness results in the Sobolev space $H^{s}$. But under some condition for the coefficient of Laplacian, this result is not optimal. We improve the bilinear estimate by using the nonlinear version of the classical Loomis-Whitney inequality, and prove the well-posedness in $H^s$ for $s\ge 1/2$ if $d=2$, and $s>1/2$ if $d=3$.

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arXiv2015

Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity

We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity $(i\partial _t + Δ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|ξ| \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\R ^d)$ with $s> \max \left( \frac{d-1}{d} s_c , \frac{s_c}{2}, s_c - \frac{d}{2(d+1)} \right)$ for $d+m \ge 5$, where $s$ is below the scaling critical regularity $s_c := \frac{d}{2}-\frac{1}{m-1}$.

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arXiv2015

Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity

We consider the Cauchy problem for the fourth order nonlinear Schrödinger equation with derivative nonlinearity $(i\partial _t + Δ^2) u= \pm \partial (|u|^2u)$ on $\mathbb{R} ^d$, $d \ge 3$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|ξ| \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\mathbb{R} ^d)$ with $\max ( \frac{d-5}{2}, \frac{d-5}{6}) < s$, whose lower bound is below the scale critical regularity $s_c= \frac{d-3}{2}$.

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arXiv2015

Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity

In the present paper, we consider the Cauchy problem of fourth order nonlinear Schrödinger type equations with a derivative nonlinearity. In one dimensional case, we prove that the fourth order nonlinear Schrödinger equation with the derivative quartic nonlinearity $\partial _x (\overline{u}^4)$ is the small data global in time well-posed and scattering to a free solution. Furthermore, we show that the same result holds for the $d \ge 2$ and derivative polynomial type nonlinearity, for example $|\nabla | (u^m)$ with $(m-1)d \ge 4$.

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arXiv2013

Well-posedness and scattering for nonlinear Schrödinger equations with a derivative nonlinearity at the scaling critical regularity

In the present paper, we consider the Cauchy problem of nonlinear Schrödinger equations with a derivative nonlinearity which depends only on $\bar{u}$. The well-posedness of the equation at the scaling subcritical regularity was proved by A. Grünrock (2000). We prove the well-posedness of the equation and the scattering for the solution at the scaling critical regularity by using $U^{2}$ space and $V^{2}$ space which are applied to prove the well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (2009).

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