Coupling Interactions and Trapped Effects for a Triple-Well Potential
ABSTRACT
Weak and strong coupling interactions and trapped effects have always played a significant role in understanding physical and chemical properties of materials. Triple-well anharmonic potential may be modeled for interpretation of energy spectra from the nuclear to macro molecular systems, and also crystalline systems. Exact periods of a trapped particle in each well of the potential are explicitly derived. For the extended Duffing system, it is predicted that infinite series of both frequency and spatial trajectory approach to exact results in the limit of weak-coupling cases (g→0).
Weak and strong coupling interactions and trapped effects have always played a significant role in understanding physical and chemical properties of materials. Triple-well anharmonic potential may be modeled for interpretation of energy spectra from the nuclear to macro molecular systems, and also crystalline systems. Exact periods of a trapped particle in each well of the potential are explicitly derived. For the extended Duffing system, it is predicted that infinite series of both frequency and spatial trajectory approach to exact results in the limit of weak-coupling cases (g→0).
Cite this paper
nullS. Aydın and M. Şimşek, "Coupling Interactions and Trapped Effects for a Triple-Well Potential," Journal of Modern Physics, Vol. 2 No. 8, 2011, pp. 898-907. doi: 10.4236/jmp.2011.28106.
nullS. Aydın and M. Şimşek, "Coupling Interactions and Trapped Effects for a Triple-Well Potential," Journal of Modern Physics, Vol. 2 No. 8, 2011, pp. 898-907. doi: 10.4236/jmp.2011.28106.
References
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[1] J. H. He, “An Elementary Introduction to Recently Developed Asymptotic Methods and Nanomechanics In Textile Engineering,” International Journal of Modern Physics B, Vol. 22, No. 21, 2008, pp. 3487-3578. doi:10.1142/S0217979208048668 [2] F. M. Fernandez and R. H. Tipping, “Accurate Calculation of Vibrational Resonances by Perturbation Theory,” Journal of Molecular Structure (Theochem), Vol. 488, No. 1-3, 1999, pp. 157-161. doi:10.1016/S0166-1280(98)00622-8 [3] F. J. Gomez and J. Sesma, “Bound States and “Resonances” in Quantum Anharmonic Oscillators,” Physics Letters A, Vol. 270, No. 1-2, 2000, pp. 20-26. doi:10.1016/S0375-9601(00)00290-5 [4] A. Pathak and S. Mandal, “Classical and Quantum Oscillators of Quartic Anharmonicities: Second-Order Solution,” Physics Letters A, Vol. 286, No. 4, 2001, pp. 261-276. doi:10.1016/S0375-9601(01)00401-7 [5] A. Pathak and S. Mandal, “Classical and Quantum Oscillators of Sextic and Octic Anharmonicities,” Physics Letters A, Vol. 298, No. 4, 2002, pp. 259-270. doi:10.1016/S0375-9601(02)00500-5 [6] Y. Meurice, “Arbitrarily Accurate Eigenvalues for One-Dimensional Polynomial Potentials,” Journal of Physics A: Mathematical and General, Vol. 35, No. 41, 2002, pp. 8831-8846. doi:10.1088/0305-4470/35/41/314 [7] I. A. Ivanov, “Transformation of the Asymptotic Perturbation Expansion for the Anharmonic Oscillator into a Convergent Expansion,” Physics Letters A, Vol. 322, No. 3-4, 2004, pp. 194-204. doi:10.1016/j.physleta.2004.01.014 [8] H. K. Khalil, “Nonlinear Systems,” Maxwell Macmillan, Toronto, 1992. [9] J. H. He, “Some new Approaches to Duffing Equation with Strongly and High Order Nonlinearity (II) Parametrized Perturbation Technique,” Communications in Nonlinear Science and Numerical Simulation, Vol. 4, No. 1, 1999, pp. 81-83. doi:10.1016/S1007-5704(99)90065-5 [10] J. Lin, “A New Approach to Duffing Equation with Strong and High Order Nonlinearity,” Communications in Nonlinear Science and Numerical Simulation, Vol. 4, No. 2, 1999, pp. 132-135. doi:10.1016/S1007-5704(99)90026-6 [11] J. H. He, “Variational iteration method – a Kind of Non-Linear Analytical Technique: Some Examples,” International Journal of Non-Linear Mechanics, Vol. 34, No. 4, 1999, pp. 699-708. doi:10.1016/S0020-7462(98)00048-1 [12] Y. Z. Chen, “Solution Of The Duffing Equation By Using Target Function Method,” Journal of Sound and Vibration, Vol. 256, No. 3, 2002, pp. 573-578. doi:10.1006/jsvi.2001.4221 [13] A. R. Chouikha, “Series Solutions of Some Anharmonic Motion Equations,” Journal of Mathematical Analysis and Applications, Vol. 272, No. 1, 2002, pp. 79-88. doi:10.1016/S0022-247X(02)00134-8 [14] M. El-Kady and E. M. E. Elbarbary, “A Chebyshev Expansion Method for Solving Nonlinear Optimal Control Problems,” Applied Mathematics and Computation, Vol. 129, No. 2-3, 2002, pp. 171-182. doi:10.1016/S0096-3003(01)00104-7 [15] C. W. Lim and B. S. Wu, “A New Analytical Approach to the Duffing-Harmonic Oscillator,” Physics Letters A, Vol. 311, No. 4-5, 2003, pp. 365-373. doi:10.1016/S0375-9601(03)00513-9 [16] Y. Z. Chen, “Evaluation of Motion of the Duffing Equation from Its General Properties,” Journal of Sound and Vibration, Vol. 264, No. 2, 2003, pp. 491-497. doi:10.1016/S0022-460X(02)01495-5 [17] H. R. Marzban and M. Razzaghi, “Numerical Solution of the Controlled Duffing Oscillator by Hybrid Functions,” Applied Mathematics and Computation, Vol. 140, No. 2-3, 2003, pp. 179-190. doi:10.1016/S0096-3003(02)00112-1 [18] T. Opatrny and K. K. Das, “Conditions for Vanishing Central-Well Population in Triple-Well Adiabatic Transport,” Physics Letters A, Vol. 79, No. 1, 2009, p. 02113. [19] V. B. Mandelzweig and F. Tabakin, “Quasilinearization Approach to Nonlinear Problems in Physics with Application to Nonlinear ODEs,” Computer Physics Communications, Vol. 141, No. 2, 2001, pp. 268-281. doi:10.1016/S0010-4655(01)00415-5 [20] J. I. Ramos, “Linearization methods in classical and quantum mechanics,” Computer Physics Communications, Vol. 153, No. 2, 2003, pp. 199-208. doi:10.1016/S0010-4655(03)00226-1 [21] J. L. Trueba, J. P. Baltanas and M. A. F. Sanjuan, “A generalized Perturbed Pendulum,” Chaos, Solitons & Fractals, Vol. 15, No. 5, 2003, pp. 911-924. doi:10.1016/S0960-0779(02)00210-2 [22] A. Post and W. Stuiver, “Modeling Non-Linear Oscillators: A New Approach,” International Journal of Non- Linear Mechanics, Vol. 39, No. 6, 2004, pp. 897-908. doi:10.1016/S0020-7462(03)00073-8 [23] A. Pelster, H. Kleinert and M. Schanz, “High-Order Variational Calculation for the Frequency of Time-Periodic Solutions,” Physical Review E, Vol. 67, No. 1, 2003, p. 016604. doi:10.1103/PhysRevE.67.016604 [24] C. M. Bender and T. T. Wu, “Analytic Structure of Energy Levels in a Field-Theory Model,” Physical Review Letters, Vol. 21, No. 6, 1968, pp. 406-409. doi:10.1103/PhysRevLett.21.406 [25] C. M. Bender and T. T. Wu, “Anharmonic Oscillator,” Physical Review, Vol. 184, No. 5, 1969, 1231-1260. doi:10.1103/PhysRev.184.1231 [26] J. J. Stoker, “Nonlinear Vibrations,” Interscience, New York, 1950. [27] N. Minorsky, “Nonlinear Oscillation,” Van Nostrand, Princeton, 1962. [28] A. H. Nayfeh, “Introduction to Perturbation Techniques,” John Wiley, New York, 1981. [29] J. H. He, “Modified Lindstedt–Poincare methods for some strongly non-linear oscillations: Part I: expansion of a constant,” International Journal of Non-Linear Mechanics, Vol. 37, No. 2, 2002, pp. 309-314. doi:10.1016/S0020-7462(00)00116-5 [30] J. H. He, “Modified Lindstedt–Poincare Methods for Some Strongly Non-Linear Oscillations: Part II: A New Transformation,” International Journal of Non-Linear Mechanics, Vol. 37, No. 2, 2002, pp. 315-320. doi:10.1016/S0020-7462(00)00117-7 [31] P. Amore and A. Aranda, “Presenting a New Method for the Solution of Nonlinear Problems,” Physics Letters A, Vol. 316, No. 3-4, 2003, pp. 218-225. doi:10.1016/j.physleta.2003.08.001 [32] L. S. Gradshteyn and L. M. Rhyzik, “Table of Integrals, Series and Products,” Academic Press, New York, 1980.