On the critical nongauge invariant nonlinear Schrödinger equation
Pavel I. Naumkin Isahi Sánchez-Suárez
We consider the Cauchy problem for the critical nongauge invariant nonlinear Schrödinger equations

$iu_{t}+\frac{1}{2}$uxx$=i\mu\overline{u}^{\alpha}u^{\beta},\text{ } x\in\mathbf{R},\text{ }t>0,$
$\ \ \ \ \ \ \ \ u(0,x) =u_{0}(x) ,\text{ }x\in\mathbf{R,} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$

where $\beta>\alpha\geq0,$ $\alpha+\beta\geq2,$ $\mu=-i^{\frac{\omega}{2} }t^{\frac{\theta}{2}-1},$ $\omega=\beta-\alpha-1,$ $\theta=\alpha+\beta-1.$ We prove that there exists a unique solution $u\in\mathbf{C}( [ 0,\infty) ;\mathbf{H}^{1}\cap\mathbf{H}^{0,1}) $ of the Cauchy problem (1). Also we find the large time asymptotics of solutions.

keywords: Asymptotics of solutions Nonlinear Schrödinger equations.
Modified wave operator for Schrodinger type equations with subcritical dissipative nonlinearities
Nakao Hayashi Pavel I. Naumkin
We construct the modified wave operator for the nonlinear Schrödinger type equations

$u_{t}-\frac{i}{\beta }\| partial _{x} |^{\beta }u=i\lambda \ |u| ^{\rho -1}u,$

for $\( t,x ) \in \mathbf{R}\times \mathbf{R.}$ We find the solutions in the neighborhood of suitable approximate solutions provided that $\beta \geq 2$, $\Im \lambda >0$ and $\rho <3$ is sufficiently close to $3.$ Also we prove the time decay estimate of solutions

$\ ||u ( t )| |\ _{\mathbf{L}^{2}}\leq Ct^{\frac{1}{2} -\frac{1}{\rho -1}}.$

When we prove the existence of a modified scattering operator, then a natural inverse problem arises to reconstruct the parameters $\beta ,\lambda $ and $\rho $ from the modified scattering operator.

keywords: Wave operators. subcritical nonlinearities Nonlinear Schrödinger equations Asymptotics of solutions
Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited
Nakao Hayashi Pavel I. Naumkin
We continue to study the asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation

$ i u_t + u_{x x} + ia(|u|^2u)_x = 0, \quad (t,x) \in \mathbf{R}\times \mathbf{R},$

$ u(0,x) = u_0 (x), \quad x\in \mathbf{R},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$(DNLS)

where $a \in \mathbf{R}$. We prove that if $ ||u_0||_{ H^{1,\gamma}} + ||u_0||_{ H^{1+\gamma,0}}$ is sufficiently small with $\gamma > 1/2$, then the solution of (DNLS) satisfies the time decay estimate

$ ||u(t)||_{L^\infty} + ||u_x(t)||_{L^\infty}\le C(1+|t|)^{-1/2}, $

where $H^{m,s}= \{f\in \mathcal{S}'; ||f||_{m,s}= ||(1+|x|^2)^{s/2}(1-\partial_x^2)^{m/2}f||_{L^2} < \infty\}$, $m,s\in \mathbf{R}$. In the previous paper [4,Theorem 1.1] we showed the same result under the condition that $\gamma \ge 2$. Furthermore we show the asymptotic behavior in time of solutions involving the previous result [4,Theorem 1.2].

keywords: Asymptotic behavior nonlinear Schrödinger equation.
Smoothing effects for some derivative nonlinear Schrödinger equations
Nakao Hayashi Pavel I. Naumkin Patrick-Nicolas Pipolo
In this paper we study a smoothing property of solutions to the Cauchy problem for the nonlinear Schrödinger equations of derivative type:

$iu_t + u_{x x} = \mathcal N(u, \bar u, u_x, \bar u_x), \quad t \in \mathbf R,\ x\in \mathbf R;\quad u(0, x) = u_0(x),\ x\in \mathbf R,\qquad$ (A)

where $\mathcal N(u, \bar u, u_x, \bar u_x) = K_1|u|^2u+K_2|u|^2u_x +K_3u^2\bar u_x +K_4|u_x|^2u+K_5\bar u$ $u_x^2 +K_6|u_x|^2u_x$, the functions $K_j = K_j (|u|^2)$, $K_j(z)\in C^\infty ([0, \infty))$. If the nonlinear terms $\mathcal N =\frac{\bar{u} u_x^2}{1+|u|^2}$, then equation (A) appears in the classical pseudospin magnet model [16]. Our purpose in this paper is to consider the case when the nonlinearity $\mathcal N$ depends both on $u_x$ and $\bar u_x$. We prove that if the initial data $u_0\in H^{3, \infty}$ and the norms $||u_0||_{3, l}$ are sufficiently small for any $l\in N$, (when $\mathcal N$ depends on $\bar u_x$), then for some time $T > 0$ there exists a unique solution $u\in C^\infty ([-T, T]$\ $\{0\};\ C^\infty(\mathbb R))$ of the Cauchy problem (A). Here $H^{m, s} = \{\varphi \in \mathbf L^2;\ ||\varphi||_{m, s}<\infty \}$, $||\varphi||_{m, s}=||(1+x^2)^{s/2}(1-\partial_x^2)^{m/2}\varphi||_{\mathbf L^2}, \mathbf H^{m, \infty}=\cap_{s\geq 1} H^{m, s}.$

keywords: nonlinear Schrödinger derivative type. Smoothing effects
Nonlinear dispersive wave equations in two space dimensions
Nakao Hayashi Seishirou Kobayashi Pavel I. Naumkin
We study the global existence and time decay of solutions to nonlinear dispersive wave equations $ \partial_t^2 u+\frac{1}{\rho^2}( -\Delta) ^{\rho }u=F ( \partial _t u )$ in two space dimensions, where $F( \partial _t u) =\lambda \vert \partial _t u\vert ^{p-1}\partial _t u$ or $\lambda \vert \partial _t u \vert ^p$, $\lambda \in \mathbf{C,}$ with $ p > 2 $ for $0 < \rho <1,$ $p > 3$ for $\rho =1,$ and $p > 1+\rho $ for $1 < \rho <2.$ If $\rho =1,$ then the equation converts into the well-known nonlinear wave equation.
keywords: Dispersive nonlinear wave global in time of solutions asymptotic behavior system of equations vector field method.
Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation
Nakao Hayashi Elena I. Kaikina Pavel I. Naumkin
We study the Cauchy problem for a nonlinear Schrödinger equation which is the generalization of a one arising in plasma physics. We focus on the so called subcritical case and prove that when the initial datum is "small", the solution exists globally in time and decays in time just like in the linear case. For a certain range of the exponent in the nonlinear term, we prove that the solution is asymptotic to a "final state" and the nonexistence of asymptotically free solutions. The method used in this paper is based on some gauge transformation and on a certain phase function.
keywords: large time asymptotics. Derivative nonlinear Schrödinger equation

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