• ETRI Journal
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• 제23권1호
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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.

• 대한수학회보
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• 제54권4호
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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.

• Nonlinear Functional Analysis and Applications
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• 제26권1호
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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.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제17권4호
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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.

• 대한수학회지
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• 제47권3호
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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.

• Interaction and multiscale mechanics
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• 제5권2호
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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.

• 대한조선학회논문집
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• 제28권1호
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• pp.53-59
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• 1991

본 논문에서는 포텐셜유동이란 가정하에 정현변조하는 비선형 입사파가 쐐기에 의하여 산란하는 문제를 마하반사의 관점에서 다척도전개기법을 이용하여 해석하였다. 산란파의 진폭전개는 선형항을 포함한 3차 $Schr\ddot{o}dinger$ 방정식으로 기술할 수 있음을 밝혔다. 즉, 비선형성을 나타내는 3차항과 분산성을 표시하는 선형항이 진폭전개의 복원력으로 작용함을 규명하였다. 패기의 반각이 17.55$^{\circ}$와 24.09$^{\circ}$인 2가지 모형에 대하여 입사각의 기울기와 변조비를 바꾸어 가며 계산을 수행하였다. 수치계산에서 얻어진 stem파의 진폭비와 폭은 실험에서 관측된 현상을 잘 반영하고 있으나, stem파는 입사파의 기울기가 매우 큰 경우에만 나타났다. 또한 분산성의 영향은 매우 미약하여 정현변조의 경우에는 비선형성이 지배적이란 결론에 도달하였다.

• Kyungpook Mathematical Journal
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• 제48권1호
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• pp.101-108
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• 2008

In this paper we study the existence of global small solutions of the Cauchy problem for the non-isotropically perturbed nonlinear Schr$\"{o}$dinger equation: $iu_t\;+\;{\Delta}u\;+\;{\mid}u{\mid}^{\alpha}u\;+\;a{\Sigma}_i^d\;u_{x_ix_ix_ix_i}$ = 0, where a is real constant, 1 $\leq$ d < n is a integer is a positive constant, and x = $(x_1,x_2,\cdots,x_n)\;\in\;R^n$. For some admissible ${\alpha}$ we show the existence of global(almost global) solutions and we calculate the regularity of those solutions.

• 한국항해항만학회지
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• 제30권3호
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• pp.181-187
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• 2006

선형파 이론에 의한 파랑스펙트럼 분포에 의해서는 30m 크기의 파랑은 현실적으로 거의 발생 불가능하다고 인식되어 왔다. 그러나 최근의 위성 영상을 이용한 조사에 의해 3주간의 기간 동안 25m 이상의 거대파가 10개 이상 관측됨에 따라 실해역에서 빈번히 마주칠 수 있는 현상임이 입증되었으며 이에 따라 지금까지 원인 불명으로 치부되어 왔던 많은 해양 재난이 거대파에 의해 발생했던 것으로 추정되고 있다. 거대파의 발생 원인으로는 파군 형성과 관련한 파고분포 특성의 변화, 전파하는 파군의 비선형 공명간섭 등이 제기되고 있으나, 그 출현의 복잡성과 자료의 부족 등으로 아직 명확하게 해명되지 못하고 있다. 본 연구에서는 실해역에서 발생하는 거대파의 실태 및 선형 및 비선형 파랑집중 이론에 근거한 거대파 발생 기구를 고찰하였으며, 비선형 파랑전파를 모사할 수 있는 수치모형을 개발하여 비선형 파랑 집중에 의한 거대 파랑의 형성을 모사하였다.

• Journal of the Optical Society of Korea
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• 제19권2호
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• pp.130-135
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• 2015

Based on the extended nonlinear Schr$\ddot{o}$dinger equation, the influences of the filter effect on pulse splitting in a passively mode-locked erbium-doped fiber laser with positive dispersion cavity are investigated theoretically. Numerical results show that, as the bandwidth of the spectral filter decreases, the nonlinear chirp appended to the pulse increases under the combined action of the filter effect of the super-Gaussian spectral filter and the self-phase modulation effect. On further decreasing the bandwidth, the wave breaking of the pulse takes place. In addition, by varying the pump power of the laser or the profile of the spectral filter, the influences of the filter effect on pulse splitting also change accordingly.