Remarks on nonlinear Schrdinger equations in one space dimension

We consider the initial value problem for nonlinear Schodinger equations :｛i8tu+ ½82u = F(u,8u,'ii,8'ii), (t,:z:) E ]R+ X IR, u(O,:z:)=uo(:z:), :z:EIR,｝where 8 = 82 = 8/8:z:, F : C4 -+ C is a polynomial having no constant nor linear terms. Without smallness condition on the data uo, it is shown that (t) have a unique local solution in time if uo is in H3•0 n H2•1 , where Hm,• = {/ E S'; 11/llm,a = 11(1 + :z:2 )½(1- )Zf' /ll2 < oo}, m,s E !R

Neumann problem for the Korteweg–de Vries equation

AbstractWe consider Neumann initial-boundary value problem for the Korteweg–de Vries equation on a half-line(0.1){ut+λuux+uxxx=0,t>0,x>0,u(x,0)=u0(x),x>0,ux(0,t)=0,t>0. We prove that if the initial data u0∈H10,214∩H21,72 and the norm ‖u0‖H10,214+‖u0‖H21,72⩽ε, where ε>0 is small enough Hps,k={f∈L2;‖f‖Hps,k=‖〈x〉k〈i∂x〉sf‖Lp<∞}, 〈x〉=1+x2 and λ∫0∞xu0(x)dx=λθ<0. Then there exists a unique solution u∈C([0,∞),H21,72)∩L2(0,∞;H22,3) of the initial-boundary value problem (0.1). Moreover there exists a constant C such that the solution has the following asymptoticsu(x,t)=Cθ(1+ηlogt)−1t−23Ai′(xt3)+O(ε2t−23(1+ηlogt)−65) for t→∞ uniformly with respect to x>0, where η=−9θλ∫0∞Ai′2(z)dz and Ai(q) is the Airy functionAi(q)=12πi∫−i∞i∞e−z3+zqdz=1πRe∫0∞e−iξ3+iξqdξ

Defect imaging for plate-like structures using diffuse acoustic wave generated by modulated laser

Authors have studied defect imaging technique for plate-like structures, which creates an image beyond diffraction limit and can be applied to a plate-like structure with a complex shape because this technique just uses variations of flexural vibration energy due to the differences in nominal bending stiffness at laser spots. However, this technique generates spurious images caused by the resonance as well as defect images. The current study described how the spurious images can be reduced using diffuse acoustic wave, and then showed that images of defects and adhesive regions can be obtained appropriately even in such complex structures as a flat plate with a complex shape, a curved plate, and a branch pipe.45TH ANNUAL REVIEW OF PROGRESS IN QUANTITATIVE NONDESTRUCTIVE EVALUATION, VOLUME 38, 15–19 July 2018, Vermont, USATakahiro Hayashi and Shogo Nakao, "Defect imaging for plate-like structures using diffuse acoustic wave generated by modulated laser", AIP Conference Proceedings 2102, 050003 (2019) https://doi.org/10.1063/1.509976

Remote defect imaging for plate-like structures based on the scanning laser source technique

In defect imaging with a scanning laser source technique, the use of a fixed receiver realizes stable measurements of flexural waves generated by laser at multiple rastering points. This study discussed the defect imaging by remote measurements using a laser Doppler vibrometer as a receiver. Narrow-band burst waves were generated by modulating laser pulse trains of a fiber laser to enhance signal to noise ratio in frequency domain. Averaging three images obtained at three different frequencies suppressed spurious distributions due to resonance. The experimental system equipped with these newly-devised means enabled us to visualize defects and adhesive objects in plate-like structures such as a plate with complex geometries and a branch pipe.44TH ANNUAL REVIEW OF PROGRESS IN QUANTITATIVE NONDESTRUCTIVE EVALUATION, VOLUME 37, 16–21 July 2017, Provo, Utah, USATakahiro Hayashi, Atsuya Maeda, and Shogo Nakao, "Remote defect imaging for plate-like structures based on the scanning laser source technique", AIP Conference Proceedings 1949, 090006 (2018) https://doi.org/10.1063/1.503156

(Semi)classical limit of the Hartree equation with harmonic potential

Nonlinear Schrodinger Equations (NLS) of the Hartree type occur in the modeling of quantum semiconductor devices. Their "semiclassical" limit of vanishing (scaled) Planck constant is both a mathematical challenge and practically relevant when coupling quantum models to classical models. With the aim of describing the semi-classical limit of the 3D Schrodinger--Poisson system with an additional harmonic potential, we study some semi-classical limits of the Hartree equation with harmonic potential in space dimension n>1. The harmonic potential is confining, and causes focusing periodically in time. We prove asymptotics in several cases, showing different possible nonlinear phenomena according to the interplay of the size of the initial data and the power of the Hartree potential. In the case of the 3D Schrodinger-Poisson system with harmonic potential, we can only give a formal computation since the need of modified scattering operators for this long range scattering case goes beyond current theory. We also deal with the case of an additional "local" nonlinearity given by a power of the local density - a model that is relevant when incorporating the Pauli principle in the simplest model given by the "Schrodinger-Poisson-X$\alpha$ equation". Further we discuss the connection of our WKB based analysis to the Wigner function approach to semiclassical limits.Comment: 26 page