• Bulletin of the Korean Mathematical Society
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• v.56 no.6
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• pp.1447-1465
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• 2019

In this paper, the relativistic Vlasov-Klein-Gordon system in one dimension is investigated. This non-linear dynamics system consists of a transport equation for the distribution function combined with Klein-Gordon equation. Without any assumption of continuity or compact support of any initial particle density $f_0$, we prove the existence and uniqueness of the mild solution via the iteration method.

• Bulletin of the Korean Chemical Society
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• v.32 no.12
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• pp.4233-4238
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• 2011

Based on the exact quantization rule for the nonrelativistic Schrodinger equation, the exact quantization rule for the relativistic one-dimensional Klein-Gordon equation is suggested. Using the new quantization rule, the exact relativistic energies for exactly solvable potentials, e.g. harmonic oscillator, Morse, and Rosen-Morse II type potentials, are obtained. Consequently the new quantization rule is found to be exact for one-dimensional spinless relativistic quantum systems. Though the physical meanings of the new quantization rule have not been fully understood yet, the present formal derivation scheme may shed light on understanding relativistic quantum systems more deeply.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1227-1237
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• 2010

In this paper, a Klein-Gordon equation is solved by using the homotopy analysis method (HAM), homotopy perturbation method (HPM) and modified homotopy perturbation method (MHPM). The approximation solution of this equation is calculated in the form of series which its components are computed easily. The uniqueness of the solution and the convergence of the proposed methods are proved. The accuracy of these methods are compared by solving an example.

• Bulletin of the Korean Chemical Society
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• v.31 no.12
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• pp.3573-3578
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• 2010

Recently it is suggested that it may be possible to obtain the approximate or exact bound state solutions of nonrelativistic Schr$\ddot{o}$dinger equation from relativistic Klein-Gordon equation, which seems to be counter-intuitive. But the suggestion is further elaborated to propose a more detailed method for obtaining nonrelativistic solutions from relativistic solutions. We demonstrate the feasibility of the proposed method with the Morse potential as an example. This work shows that exact relativistic solutions can be a good starting point for obtaining nonrelativistic solutions even though a rigorous algebraic method is not found yet.

• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.281-294
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• 2020

The initial-boundary value problem for a class of semilinear Klein-Gordon equation with logarithmic nonlinearity in bounded domain is studied. The existence of global solution for this problem is proved by using potential well method, and obtain the exponential decay of global solution through introducing an appropriate Lyapunov function. Meanwhile, the blow-up of solution in the unstable set is also obtained.

• Honam Mathematical Journal
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• v.30 no.4
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• pp.631-635
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• 2008

In this paper we discuss on the formal linearization and exact solution of Klein-Gordon's equation (1) $u_{tt}-au_{xx}+bu-cu^3=0 a,b,c{\in}R^+$ So that we know an efficient method for constructing of particular solutions of some nonlinear partial differential equations is introduced.

• Bulletin of the Korean Chemical Society
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• v.35 no.12
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• pp.3443-3446
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• 2014

We present the solution of Klein Gordon equation with new generalized Morse-like potential using SUSYQM formalism. We obtained approximately the energy eigenvalues and the corresponding wave function in a closed form for any arbitrary l state. We computed the numerical results for some selected diatomic molecules.

• Honam Mathematical Journal
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• v.40 no.4
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• pp.771-783
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• 2018

This paper, we investigate a strongly damped nonlinear Klein-Gordon equation with nonlinearities of variable exponent type $$u_{tt}-{\Delta}u-{\Delta}u_t+m^2u+{\mid}u_t{\mid}^{p(x)-2}u_t={\mid}u{\mid}^{q(x)-2}u$$ associated with initial and Dirichlet boundary conditions in a bounded domain. We obtain a nonexistence of solutions if variable exponents p (.), q (.) and initial data satisfy some conditions.

• Journal of the Korean Mathematical Society
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• v.58 no.5
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• pp.1131-1145
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• 2021

In this article, we study the Klein-Gordon-Maxwell equations arising from a semilocal gauge field model. This model describes the interaction of two complex scalar fields and one gauge field, and generalizes the classical Klein-Gordon equation coupled with the Maxwell electrodynamics. We prove that there exist infinitely many standing wave solutions for p ∈ (2, 6) which are radially symmetric. Here, p comes from the exponent of the potential of scalar fields. We also prove the nonexistence of nontrivial solutions for the critical case p = 6.

• Bulletin of the Korean Chemical Society
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• v.34 no.11
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• pp.3425-3428
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• 2013

We solve the Klein-Gordon equation with the Morse empirical potential energy model. The bound state energy equation has been obtained in terms of the supersymmetric shape invariance approach. The relativistic vibrational transition frequencies for the $X^1{\sum}^+$ state of ScI molecule have been computed by using the Morse potential model. The calculated relativistic vibrational transition frequencies are in good agreement with the experimental RKR values.