• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.851-863
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• 2018

We derive lower bounds of the scalar curvature on complete non-compact gradient Yamabe solitons under some integral curvature conditions. Based on this, we prove that potential functions of Yamabe solitons have at most quadratic growth for distance function. We also obtain a finite topological type property on complete shrinking gradient Yamabe solitons under suitable scalar curvature assumptions.

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1265-1280
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• 2023

The aim of this paper is to investigate Betchov-Da Rios equation by using null Cartan, pseudo null and partially null curve in Minkowski spacetime. Time derivative formulas of frame of s parameter null Cartan, pseudo null and partially null curve are examined, respectively. By using the obtained derivative formulas, new results are given about the solution of Betchov-Da Rios equation. The differential geometric properties of these solutions are obtained with respect to Lorentzian causal character of s parameter curve. For a solution of Betchov-Da Rios equation, it is seen that null Cartan s parameter curves are space curves in three-dimensional Minkowski space. Then all points of the soliton surface are flat points of the surface for null Cartan and partially null curve. Thus, it is seen from the results obtained that there is no surface corresponding to the solution of Betchov-Da Rios equation by using the pseudo null s parameter curve.

• The Transactions of the Korean Institute of Electrical Engineers C
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• v.49 no.1
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• pp.52-58
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• 2000

Signal propagation in nonlinear optical fiber is analyzed numerically by using SS-FEM (Split-Step Finite Element Method). By adopting cubic element function in FEM, soliton equation of which exact solution was well known, has been solved. Also, accuracy of numerical results and computing times are compared with those of Fourier method, and we have found that solution obtained from using FEM was very relatively accurate. Especially, to reduce CPU time in matrix computation in each step, the matrix imposed by the boundary condition is approximated as a sparse matrix. As a result, computation time was shortened even with the same or better accuracy when compared to those of the conventional FEM and Fourier method.

• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1673-1685
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• 2023

Inspired by Hongjie Dong and Qi S. Zhang's article [3], we find that the analyticity in time for a smooth solution of the heat equation with exponential quadratic growth in the space variable can be extended to any complete noncompact Riemannian manifolds with Bakry-Émery Ricci curvature bounded below and the potential function being of at most quadratic growth. Therefore, our result holds on all gradient Ricci solitons. As a corollary, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with the similar growth condition. In addition, we also consider the solution in certain L^p spaces with p ∈ [2, +∞) and prove its analyticity with respect to time.

• Journal of Astronomy and Space Sciences
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• v.21 no.3
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• pp.201-208
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• 2004

In this paper we have examined the effect of dust charge density on nonlinear ion acoustic solitary wave which propagates obliquely with respect to the external magnetic field in a dusty plasma. For the dusty charge density below a critical value, the Sagdeev potential $\Psi1(n)$ has a singular point in the region n < 1, where n is the ion number density divided by its equilibrium number density. If there exists a dust charge density over the critical value, the Sagdeev potential becomes a finite function in the region n < 1, which means that there may exist the rarefactive ion acoustic solitary wave. By expanding the Sagdeev potential in the small amplitude limit up to on4 near n=1, we find the solution of ion acoustic solitary wave. Therefore we suggest that the dust charge density plays an important role in generating the rarefactive solitary wave.