• Title/Summary/Keyword: Schr$\ddot{o}$dinger equations

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AN APPROACH TO SOLUTION OF THE SCHRÖDINGER EQUATION USING FOURIER-TYPE FUNCTIONALS

  • Chang, Seung Jun;Choi, Jae Gil;Chung, Hyun Soo
    • Journal of the Korean Mathematical Society
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    • v.50 no.2
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    • pp.259-274
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    • 2013
  • In this paper, we consider the Fourier-type functionals on Wiener space. We then establish the analytic Feynman integrals involving the ${\diamond}$-convolutions. Further, we give an approach to solution of the Schr$\ddot{o}$dinger equation via Fourier-type functionals. Finally, we use this approach to obtain solutions of the Schr$\ddot{o}$dinger equations for harmonic oscillator and double-well potential. The Schr$\ddot{o}$dinger equations for harmonic oscillator and double-well potential are meaningful subjects in quantum mechanics.

A Coupled Higher-Order Nonlinear $Schr{\ddot{o}}dinger$ Equation Including Higher-Order Bright and Dark Solitons

  • Kim, Jong-Bae
    • ETRI Journal
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    • v.23 no.1
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    • pp.9-15
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    • 2001
  • We suggest a generalized Lax pair on a Hermitian symmetric space to generate a new coupled higher-order nonlinear $Schr{\ddot{o}}dinger$ equation of a dual type which contains both bright and dark soliton equations depending on parameters in the Lax pair. Through the generalized ways of reduction and the scaling transformation for the coupled higher-order nonlinear $Schr{\ddot{o}}dinger$ equation, two integrable types of higher-order dark soliton equations and their extensions to vector equations are newly derived in addition to the corresponding equations of the known higher-order bright solitons. Analytical discussion on a general scalar solution of the higher-order dark soliton equation is then made in detail.

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UNIQUE CONTINUATION FOR SCHRӦDINGER EQUATIONS

  • Shin, Se Chul;Lee, Kyung Bok
    • Journal of the Chungcheong Mathematical Society
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    • v.15 no.2
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    • pp.25-34
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    • 2003
  • We prove a local unique continuation for Schr$\ddot{o}$dinger equations with time independent coefficients. The method of proof combines a technique of Fourier-Gauss transformation and a Carleman inequality for parabolic operator.

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EXISTENCE AND NON-EXISTENCE FOR SCHRÖDINGER EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS

  • Zou, Henghui
    • Journal of the Korean Mathematical Society
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    • v.47 no.3
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    • pp.547-572
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    • 2010
  • We study existence of positive solutions of the classical nonlinear Schr$\ddot{o}$dinger equation $-{\Delta}u\;+\;V(x)u\;-\;f(x,\;u)\;-\;H(x)u^{2*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$. In fact, we consider the following more general quasi-linear Schr$\ddot{o}$odinger equation $-div(|{\nabla}u|^{m-2}{\nabla}u)\;+\;V(x)u^{m-1}$ $-f(x,\;u)\;-\;H(x)u^{m^*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$, where m $\in$ (1, n) is a positive number and $m^*\;:=\;\frac{mn}{n-m}\;>\;0$, is the corresponding critical Sobolev embedding number in $\mathbb{R}^n$. Under appropriate conditions on the functions V(x), f(x, u) and H(x), existence and non-existence results of positive solutions have been established.

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR NONLINEAR SCHRÖDINGER-KIRCHHOFF-TYPE EQUATIONS

  • CHEN, HAIBO;LIU, HONGLIANG;XU, LIPING
    • Journal of the Korean Mathematical Society
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    • v.53 no.1
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    • pp.201-215
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    • 2016
  • In this paper, we consider the following $Schr{\ddot{o}}dinger$-Kirchhoff-type equations $\[a+b\({\int}_{{\mathbb{R}}^N}({\mid}{\nabla}u{\mid}^2+V(x){\mid}u{\mid}^2)dx\)\][-{\Delta}u+V(x)u]=f(x,u)$, in ${\mathbb{R}}^N$. Under certain assumptions on V and f, some new criteria on the existence and multiplicity of nontrivial solutions are established by the Morse theory with local linking and the genus properties in critical point theory. Some results from the previously literature are significantly extended and complemented.

HIGH ORDER EMBEDDED RUNGE-KUTTA SCHEME FOR ADAPTIVE STEP-SIZE CONTROL IN THE INTERACTION PICTURE METHOD

  • Balac, Stephane
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.17 no.4
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    • pp.238-266
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    • 2013
  • The Interaction Picture (IP) method is a valuable alternative to Split-step methods for solving certain types of partial differential equations such as the nonlinear Schr$\ddot{o}$dinger equation or the Gross-Pitaevskii equation. Although very similar to the Symmetric Split-step (SS) method in its inner computational structure, the IP method results from a change of unknown and therefore do not involve approximation such as the one resulting from the use of a splitting formula. In its standard form the IP method such as the SS method is used in conjunction with the classical 4th order Runge-Kutta (RK) scheme. However it appears to be relevant to look for RK scheme of higher order so as to improve the accuracy of the IP method. In this paper we investigate 5th order Embedded Runge-Kutta schemes suited to be used in conjunction with the IP method and designed to deliver a local error estimation for adaptive step size control.

SMALL DATA SCATTERING OF HARTREE TYPE FRACTIONAL SCHRÖDINGER EQUATIONS IN DIMENSION 2 AND 3

  • Cho, Yonggeun;Ozawa, Tohru
    • Journal of the Korean Mathematical Society
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    • v.55 no.2
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    • pp.373-390
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    • 2018
  • In this paper we study the small-data scattering of the d dimensional fractional $Schr{\ddot{o}}dinger$ equations with d = 2, 3, $L{\acute{e}}vy$ index 1 < ${\alpha}$ < 2 and Hartree type nonlinearity $F(u)={\mu}({\mid}x{\mid}^{-{\gamma}}{\ast}{\mid}u{\mid}^2)u$ with max(${\alpha}$, ${\frac{2d}{2d-1}}$) < ${\gamma}{\leq}2$, ${\gamma}$ < d. This equation is scaling-critical in ${\dot{H}}^{s_c}$, $s_c={\frac{{\gamma}-{\alpha}}{2}}$. We show that the solution scatters in $H^{s,1}$ for any s > $s_c$, where $H^{s,1}$ is a space of Sobolev type taking in angular regularity with norm defined by ${\parallel}{\varphi}{\parallel}_{H^{s,1}}={\parallel}{\varphi}{\parallel}_{H^s}+{\parallel}{\nabla}_{{\mathbb{S}}{\varphi}}{\parallel}_{H^s}$. For this purpose we use the recently developed Strichartz estimate which is $L^2$-averaged on the unit sphere ${\mathbb{S}}^{d-1}$ and utilize $U^p-V^p$ space argument.

ON THE ORBITAL STABILITY OF INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATIONS WITH SINGULAR POTENTIAL

  • Cho, Yonggeun;Lee, Misung
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.6
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    • pp.1601-1615
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    • 2019
  • We show the existence of ground state and orbital stability of standing waves of nonlinear $Schr{\ddot{o}}dinger$ equations with singular linear potential and essentially mass-subcritical power type nonlinearity. For this purpose we establish the existence of ground state in $H^1$. We do not assume symmetry or monotonicity. We also consider local and global well-posedness of Strichartz solutions of energy-subcritical equations. We improve the range of inhomogeneous coefficient in [5, 12] slightly in 3 dimensions.