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

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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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APPROXIMATE SOLUTIONS OF SCHRÖDINGER EQUATION WITH A QUARTIC POTENTIAL

  • Jung, Soon-Mo;Kim, Byungbae
    • Nonlinear Functional Analysis and Applications
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    • v.26 no.1
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    • pp.157-164
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    • 2021
  • Recently we investigated a type of Hyers-Ulam stability of the Schrödinger equation with the symmetric parabolic wall potential that efficiently describes the quantum harmonic oscillations. In this paper we study a type of Hyers-Ulam stability of the Schrödinger equation when the potential barrier is a quartic wall in the solid crystal models.

A RANDOM DISPERSION SCHRÖDINGER EQUATION WITH NONLINEAR TIME-DEPENDENT LOSS/GAIN

  • Jian, Hui;Liu, Bin
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.4
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    • pp.1195-1219
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    • 2017
  • In this paper, the limit behavior of solution for the $Schr{\ddot{o}}dinger$ equation with random dispersion and time-dependent nonlinear loss/gain: $idu+{\frac{1}{{\varepsilon}}}m({\frac{t}{{\varepsilon}^2}}){\partial}_{xx}udt+{\mid}u{\mid}^{2{\sigma}}udt+i{\varepsilon}a(t){\mid}u{\mid}^{2{\sigma}_0}udt=0$ is studied. Combining stochastic Strichartz-type estimates with $L^2$ norm estimates, we first derive the global existence for $L^2$ and $H^1$ solution of the stochastic $Schr{\ddot{o}}dinger$ equation with white noise dispersion and time-dependent loss/gain: $idu+{\Delta}u{\circ}d{\beta}+{\mid}u{\mid}^{2{\sigma}}udt+ia(t){\mid}u{\mid}^{2{\sigma}_0}udt=0$. Secondly, we prove rigorously the global diffusion-approximation limit of the solution for the former as ${\varepsilon}{\rightarrow}0$ in one-dimensional $L^2$ subcritical and critical cases.

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.

Vortex Filament Equation and Non-linear Schrödinger Equation in S3

  • Zhang, Hongning;Wu, Faen
    • Kyungpook Mathematical Journal
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    • v.47 no.3
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    • pp.381-392
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    • 2007
  • In 1906, da Rios, a student of Leivi-Civita, wrote a master's thesis modeling the motion of a vortex in a viscous fluid by the motion of a curve propagating in $R^3$, in the direction of its binormal with a speed equal to its curvature. Much later, in 1971 Hasimoto showed the equivalence of this system with the non-linear Schr$\ddot{o}$dinger equation (NLS) $$q_t=i(q_{ss}+\frac{1}{2}{\mid}q{\mid}^2q$$. In this paper, we use the same idea as Terng used in her lecture notes but different technique to extend the above relation to the case of $R^3$, and obtained an analogous equation that $$q_t=i[q_{ss}+(\frac{1}{2}{\mid}q{\mid}^2+1)q]$$.

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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.

Radial basis function collocation method for a rotating Bose-Einstein condensation with vortex lattices

  • Shih, Y.T.;Tsai, C.C.;Chen, K.T.
    • Interaction and multiscale mechanics
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    • v.5 no.2
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    • pp.131-144
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    • 2012
  • We study a radial basis function collocation method (RBFCM) to discretize a coupled nonlinear Schr$\ddot{o}$dinger equation (CNLSE) that governs a two dimensional rotating Bose-Einstein condensate (BEC) with an angular momentum rotation term. We exploit a RBFCM-continuation method (RBFCM-CM) to trace the solution curve of the CNLSE. We compare the performance of the RBFCM-CM with the FEM-CM. We observe that the RBFCM-CM is very robust in a coarse grid for resolving the ground state solution with many vortices when the angular momentum rotation is close to the limit. Numerical results demonstrate the efficiency and accuracy of the RBFCM-CM for computing the superfluid density of the ground level of the BEC.

Mach Reflection of Sinusoidally- Modulated Nonlinear Stokes Waves by a Thin Wedge (쐐기에 의한 비선형파의 마하반사)

  • Hang-S. Choi;Won-S. Chee
    • Journal of the Society of Naval Architects of Korea
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    • v.28 no.1
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    • pp.53-59
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    • 1991
  • By employing multiple-scale expansion techniques, the diffraction of sinusoidally-modulated nonlinear Stokes waves by a stationary thin wedge has been studied within the framework of potential theory. It is found that the evolution of diffracted waves can be described by the Zakharov equation to the leading order and it can be replaced by the cubic $Schr\ddot{o}dinger$ equation with an additional linear term for stable modulations. Computations are made for the cubic $Schr\ddot{o}dinger$ equation with different values of nonlinear and dispersion parameters. Numerical results well reflect the experimental findings in the amplitude and width of generated stem waves. It is numerically confirmed that the nonlinearity dominates the wave field, while the dispersion hardly affects the wave evolution, and stem waves are likely to be formed for steep incident waves in the case of stable sinusoidal modulations.

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