Recent advances on Schrödinger equations with dissipative nonlinearities
We give a survey on recent developments on nonlinear Schrödinger equations with dissipative structure based on the authors' recent works.
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arXiv subjects
Yuji Sagawa
Publications and source records attributed to Yuji Sagawa.
Upper and lower $L^2$-decay bounds for a class of derivative nonlinear Schrödinger equations
We consider the initial value problem for cubic derivative nonlinear Schrödinger equations possessing weakly dissipative structure in one space dimension. We show that the small data solution decays like $O((\log t)^{-1/4})$ in $L^2$ as $t\to +\infty$. Furthermore, we find that this $L^2$-decay rate is optimal by giving a lower estimate of the same order.
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On the derivative nonlinear Schrödinger equation with weakly dissipative structure
We consider the initial value problem for cubic derivative nonlinear Schrödinger equation in one space dimension. Under a suitable weakly dissipative condition on the nonlinearity, we show that the small data solution has a logarithmic time decay in $L^2$.
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Large time asymptotics for a cubic nonlinear Schrödinger system in one space dimension, II
This is a sequel to the paper "Large time asymptotics for a cubic nonlinear Schrödinger system in one space dimension" by the same authors. We continue to study the Cauchy problem for the two-component system of cubic nonlinear Schrödinger equations in one space dimension. We provide criteria for large time decay or non-decay in $L^2$ of the small amplitude solutions in terms of the Fourier transforms of the initial data.
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Large time asymptotics for a cubic nonlinear Schrödinger system in one space dimension
We consider a two-component system of cubic nonlinear Schrödinger 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 restriction between the profiles of them. This turns out to be a consequence of non-trivial long-range nonlinear interactions.
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A sharp lower bound for the lifespan of small solutions to the Schrödinger equation with a subcritical power nonlinearity
Let $T_ε$ be the lifespan for the solution to the Schrödinger equation on $\mathbb{R}^d$ with a power nonlinearity $λ|u|^{2θ/d}u$ ($λ\in \mathbb{C}$, $0<θ<1$) and the initial data in the form $εφ(x)$. We provide a sharp lower bound estimate for $T_ε$ as $ε\to +0$ which can be written explicitly by $λ$, $d$, $θ$, $φ$ and $ε$. This is an improvement of the previous result by H.Sasaki [Adv. Diff. Eq. 14 (2009), 1021--1039].
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The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension
Consider the initial value problem for cubic derivative nonlinear Schrödinger equations in one space dimension. We provide a detailed lower bound estimate for the lifespan of the solution, which can be computed explicitly from the initial data and the nonlinear term. This is an extension and a refinement of the previous work by one of the authors where the gauge-invariant nonlinearity was treated.
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