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Journal articles on the topic 'Coupled evolution equations'

Author: Grafiati

Published: 14 September 2024

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1

Maruszewski, Bogdan. "Coupled evolution equations of deformable semiconductors." International Journal of Engineering Science 25, no. 2 (January 1987): 145–53. http://dx.doi.org/10.1016/0020-7225(87)90002-4.

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2

Yusufoğlu, Elcin, and Ahmet Bekir. "Exact solutions of coupled nonlinear evolution equations." Chaos, Solitons & Fractals 37, no. 3 (August 2008): 842–48. http://dx.doi.org/10.1016/j.chaos.2006.09.074.

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3

Nakagiri, Shin-ichi, and Jun-hong Ha. "COUPLED SINE-GORDON EQUATIONS AS NONLINEAR SECOND ORDER EVOLUTION EQUATIONS." Taiwanese Journal of Mathematics 5, no. 2 (June 2001): 297–315. http://dx.doi.org/10.11650/twjm/1500407338.

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4

Khan, K., and M. A. Akbar. "Solitary Wave Solutions of Some Coupled Nonlinear Evolution Equations." Journal of Scientific Research 6, no. 2 (April 23, 2014): 273–84. http://dx.doi.org/10.3329/jsr.v6i2.16671.

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In this article, the modified simple equation (MSE) method has been executed to find the traveling wave solutions of the coupled (1+1)-dimensional Broer-Kaup (BK) equations and the dispersive long wave (DLW) equations. The efficiency of the method for finding exact solutions has been demonstrated. It has been shown that the method is direct, effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this procedure reduces the large volume of calculations. Keywords: MSE method; NLEE; BK equations; DLW equations; Solitary wave solutions. © 2014 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v6i2.16671 J. Sci. Res. 6 (2), 273-284 (2014)

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5

Malfliet, W. "Travelling-wave solutions of coupled nonlinear evolution equations." Mathematics and Computers in Simulation 62, no. 1-2 (February 2003): 101–8. http://dx.doi.org/10.1016/s0378-4754(02)00182-9.

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6

Alabau, F., P. Cannarsa, and V. Komornik. "Indirect internal stabilization of weakly coupled evolution equations." Journal of Evolution Equations 2, no. 2 (May 1, 2002): 127–50. http://dx.doi.org/10.1007/s00028-002-8083-0.

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7

RYDER, E., and D. F. PARKER. "Coupled evolution equations for axially inhomogeneous optical fibres." IMA Journal of Applied Mathematics 49, no. 3 (1992): 293–309. http://dx.doi.org/10.1093/imamat/49.3.293.

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8

Zhao, Dan, and Zhaqilao. "Darboux transformation approach for two new coupled nonlinear evolution equations." Modern Physics Letters B 34, no. 01 (December 6, 2019): 2050004. http://dx.doi.org/10.1142/s0217984920500049.

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A new coupled Burgers equation and a new coupled KdV equation which are associated with [Formula: see text] matrix spectial problem are investigated for complete integrability and covariant property. For integrability, Lax pair and conservation laws of the two new coupled equations with four potentials are established. For covariant property, Darboux transformation (DT) is used to construct explicit solutions of the two new coupled equations.

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9

Khan, Kamruzzaman, and M. Ali Akbar. "Traveling Wave Solutions of Some Coupled Nonlinear Evolution Equations." ISRN Mathematical Physics 2013 (May 20, 2013): 1–8. http://dx.doi.org/10.1155/2013/685736.

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The modified simple equation (MSE) method is executed to find the traveling wave solutions for the coupled Konno-Oono equations and the variant Boussinesq equations. The efficiency of this method for finding exact solutions and traveling wave solutions has been demonstrated. It has been shown that the proposed method is direct, effective, and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this procedure reduces the large volume of calculations.

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10

Wan, Qian, and Ti-Jun Xiao. "Exponential Stability of Two Coupled Second-Order Evolution Equations." Advances in Difference Equations 2011 (2011): 1–14. http://dx.doi.org/10.1155/2011/879649.

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11

Arafa, A. A. M., and S. Z. Rida. "Numerical solutions for some generalized coupled nonlinear evolution equations." Mathematical and Computer Modelling 56, no. 11-12 (December 2012): 268–77. http://dx.doi.org/10.1016/j.mcm.2011.12.046.

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12

Seadawy, A. R., and K. El-Rashidy. "Traveling wave solutions for some coupled nonlinear evolution equations." Mathematical and Computer Modelling 57, no. 5-6 (March 2013): 1371–79. http://dx.doi.org/10.1016/j.mcm.2012.11.026.

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13

Liu, Wenyuan, Wei Xia, and Shengping Shen. "Fully Coupling Chemomechanical Yield Theory Based on Evolution Equations." International Journal of Applied Mechanics 08, no. 04 (June 2016): 1650058. http://dx.doi.org/10.1142/s1758825116500587.

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Coupling chemomechanical yield is one of the key issues in the oxidation of metal and polymer matrix materials. In this paper, the evolving equations for fully coupled thermal–chemical–mechanical processes were derived using the theory of thermodynamics. Then, the coupled chemomechanical yield condition and flow rule were directly obtained from the evolution equations by extending the von Mises theory of plasticity. The coupled yield condition reveals that only the chemical reactions or diffusions may lead to the yield of material even without the mechanical stress, which significantly differs from the previous works. In addition, the currently proposed yield condition combined with the flow rule as a new criterion, may be applicable in predicting the durability of materials within the allowed plastic deformation resulted from time dependent diffusion and chemical reactions. Particular attention is paid to the isothermal systems and isotropic materials for simplicity. Finally, three examples on how the deformation and reaction evolves simultaneously were given to show the applications of the present theory.

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14

Hassaballa, Abaker A., Fathea M. O. Birkea, Ahmed M. A. Adam, Ali Satty, Elzain A. E. Gumma, Emad A.-B. Abdel-Salam, Eltayeb A. Yousif, and Mohamed I. Nouh. "Multiple and Singular Soliton Solutions for Space–Time Fractional Coupled Modified Korteweg–De Vries Equations." International Journal of Analysis and Applications 22 (April 22, 2024): 68. http://dx.doi.org/10.28924/2291-8639-22-2024-68.

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The focus of this paper is on the nonlinear coupled evolution equations, specifically within the context of the fractional coupled modified Korteweg–de Vries (mKdV) equation, employing the conformable fractional derivative (CFD) approach. The primary objective of this paper is to thoroughly investigate the applicability of the Hirota bilinear method for deriving analytical solutions to the fractional mKdV equations. A range of exact analytical solutions for the fractional coupled mKdV equations is obtained. The findings in general indicate that the Hirota bilinear method is an effective approach for resolving the complexities associated with the fractional coupled mKdV equations.

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15

Han, Ding, Bing Gen Zhan, and Xiao Ming Huang. "Fatigue Analysis of the Asphalt Mixture Beam Using Damage Evolution Equations." Advanced Materials Research 163-167 (December 2010): 3332–35. http://dx.doi.org/10.4028/www.scientific.net/amr.163-167.3332.

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Damage evolution equations of asphalt mixture specimen beams were analyzed using the fracture toughness index and the ultimate tension strain index, respectively. The fracture toughness of the asphalt mixture was calculated by FEM. Damage evolution equations controlled apart by the stress and the strain were given. Their coefficients were back-calculated using partial fatigue tests data. The fully coupled stress-damage method and the fully coupled strain-damage method were used. The life prediction precision of each equation was verified by residual fatigue tests data. The results show that FEM is effective to calculate the fracture toughness of the asphalt mixture. Two damage evolution equations have better life prediction abilities comparing with the S-N equation used usually.

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16

Elwakil, Elsayed Abd Elaty, and Mohamed Aly Abdou. "New Applications of the Homotopy Analysis Method." Zeitschrift für Naturforschung A 63, no. 7-8 (August 1, 2008): 385–92. http://dx.doi.org/10.1515/zna-2008-7-801.

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An analytical technique, namely the homotopy analysis method (HAM), is applied using a computerized symbolic computation to find the approximate and exact solutions of nonlinear evolution equations arising in mathematical physics. The HAM is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The validity and reliability of the method is tested by application on three nonlinear problems, namely theWhitham-Broer-Kaup equations, coupled Korteweg-de Vries equation and coupled Burger’s equations. Comparisons are made between the results of the HAM with the exact solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics.

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17

Roy, P. K. "An integrable system governed by coupled non-linear evolution equations." Il Nuovo Cimento A 109, no. 11 (November 1996): 1613–15. http://dx.doi.org/10.1007/bf02778246.

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18

Abdelkawy, M. A., A. H. Bhrawy, E. Zerrad, and A. Biswas. "Application of Tanh Method to Complex Coupled Nonlinear Evolution Equations." Acta Physica Polonica A 129, no. 3 (March 2016): 278–83. http://dx.doi.org/10.12693/aphyspola.129.278.

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19

Schneider, Guido. "Justification of mean-field coupled modulation equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 127, no. 3 (1997): 639–50. http://dx.doi.org/10.1017/s0308210500029942.

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We are interested in reflection symmetric (x↦–x) evolution problems on the infinite line. In the systems which we have in mind, a trivial ground state loses stability and bifurcates into a temporally oscillating, spatial periodic pattern. A famous example of such a system is the Taylor-Couette problem in the case of strongly counter-rotating cylinders. In this paper, we consider a system of coupled Kuramoto–Shivashinsky equations as a model problem for such a system. We are interested in solutions which are slow modulations in time and in space of the bifurcating pattern. Multiple scaling analysis is used in the existing literature to derive mean-field coupled Ginzburg–Landau equations as approximation equations for the problem. The aim of this paper is to give exact estimates between the solutions of the coupled Kuramoto–Shivashinsky equations and the associated approximations.

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20

Junker, Philipp, and Daniel Balzani. "An extended Hamilton principle as unifying theory for coupled problems and dissipative microstructure evolution." Continuum Mechanics and Thermodynamics 33, no. 4 (June 7, 2021): 1931–56. http://dx.doi.org/10.1007/s00161-021-01017-z.

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AbstractAn established strategy for material modeling is provided by energy-based principles such that evolution equations in terms of ordinary differential equations can be derived. However, there exist a variety of material models that also need to take into account non-local effects to capture microstructure evolution. In this case, the evolution of microstructure is described by a partial differential equation. In this contribution, we present how Hamilton’s principle provides a physically sound strategy for the derivation of transient field equations for all state variables. Therefore, we begin with a demonstration how Hamilton’s principle generalizes the principle of stationary action for rigid bodies. Furthermore, we show that the basic idea behind Hamilton’s principle is not restricted to isothermal mechanical processes. In contrast, we propose an extended Hamilton principle which is applicable to coupled problems and dissipative microstructure evolution. As example, we demonstrate how the field equations for all state variables for thermo-mechanically coupled problems, i.e., displacements, temperature, and internal variables, result from the stationarity of the extended Hamilton functional. The relation to other principles, as the principle of virtual work and Onsager’s principle, is given. Finally, exemplary material models demonstrate how to use the extended Hamilton principle for thermo-mechanically coupled elastic, gradient-enhanced, rate-dependent, and rate-independent materials.

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21

Debsarma, S., S. Senapati, and K. P. Das. "Nonlinear Evolution Equations for Broader Bandwidth Wave Packets in Crossing Sea States." International Journal of Oceanography 2014 (June 9, 2014): 1–9. http://dx.doi.org/10.1155/2014/597895.

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Two coupled nonlinear equations are derived describing the evolution of two broader bandwidth surface gravity wave packets propagating in two different directions in deep water. The equations, being derived for broader bandwidth wave packets, are applicable to more realistic ocean wave spectra in crossing sea states. The two coupled evolution equations derived here have been used to investigate the instability of two uniform wave trains propagating in two different directions. We have shown in figures the behaviour of the growth rate of instability of these uniform wave trains for unidirectional as well as for bidirectional perturbations. The figures drawn here confirm the fact that modulational instability in crossing sea states with broader bandwidth wave packets can lead to the formation of freak waves.

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22

Alzaidy, J. F. "Extended Mapping Method and Its Applications to Nonlinear Evolution Equations." Journal of Applied Mathematics 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/597983.

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We use extended mapping method and auxiliary equation method for finding new periodic wave solutions of nonlinear evolution equations in mathematical physics, and we obtain some new periodic wave solution for the Boussinesq system and the coupled KdV equations. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

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23

MA, WEN-XIU. "AKNS Type Reduced Integrable Hierarchies with Hamiltonian Formulations." Romanian Journal of Physics 68, no. 9-10 (December 15, 2023): 116. http://dx.doi.org/10.59277/romjphys.2023.68.116.

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"The aim of this paper is to generate a kind of integrable hierarchies of four-component evolution equations with Hamiltonian structures, from a kind of reduced Ablowitz-Kaup-Newell-Segur (AKNS) matrix spectral problems. The zero curvature formulation is the basic tool and the trace identity is the key to establishing Hamiltonian structures. Two examples of Hamiltonian equations in the resulting inte- grable hierarchies are added to the category of coupled integrable nonlinear Schr¨odinger equations and coupled integable modified Korteweg-de Vries equations."

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24

Hua, Yuan, Bao Hua Lv, and Tai Quan Zhou. "Parametric Variational Principle for Solving Coupled Damage Problem." Key Engineering Materials 348-349 (September 2007): 813–16. http://dx.doi.org/10.4028/www.scientific.net/kem.348-349.813.

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The parametric variational principle adopts the extreme variational idea in the modern control theory and uses state equations deduced from the constitutive law to control the functional variation, which is an effective solution to the nonlinear equations. Based on the fundamental equations of elasto-plasticity coupled damage problem, the potential functional of elasto-plasticity is constructed. Also the state equations with approximation of damage evolution equation and load functions are constructed in the paper. The solution of elasto-plasticity damage problem can be deduced to solve problem of the minimum potential energy function under the restriction of state equations. Thus the parametric variational principle for coupled damage is proposed. The variational principle has the virtue of definite physical meaning and the finite element equations are presented in the article to facilitate the application of parametric variatioal principle, which is easy to program on computer. Using the method mentioned in the article, a numerical calculation is carried out and the calculation result shows that the method is efficient for solving elasto-plasticity damage problem.

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25

Guesmia, Aissa. "Asymptotic behavior for coupled abstract evolution equations with one infinite memory." Applicable Analysis 94, no. 1 (March 12, 2014): 184–217. http://dx.doi.org/10.1080/00036811.2014.890708.

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26

Tsang, S. C., and K. W. Chow. "The evolution of periodic waves of the coupled nonlinear Schrödinger equations." Mathematics and Computers in Simulation 66, no. 6 (August 2004): 551–64. http://dx.doi.org/10.1016/j.matcom.2004.04.002.

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27

da Silva Alves, Margareth, and Octavio Paulo Vera Villagrán. "Smoothing properties for a coupled system of nonlinear evolution dispersive equations." Indagationes Mathematicae 20, no. 2 (2009): 285–327. http://dx.doi.org/10.1016/s0019-3577(09)80015-3.

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28

Xiao, Ti-Jun, and Jin Liang. "Coupled second order semilinear evolution equations indirectly damped via memory effects." Journal of Differential Equations 254, no. 5 (March 2013): 2128–57. http://dx.doi.org/10.1016/j.jde.2012.11.019.

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29

Ghose, Chandana, and A. Roy Chowdhury. "Periodic inverse problem for a new hierarchy of coupled evolution equations." International Journal of Theoretical Physics 30, no. 7 (July 1991): 1033–39. http://dx.doi.org/10.1007/bf00673994.

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30

Bekir, Ahmet. "Applications of the extended tanh method for coupled nonlinear evolution equations." Communications in Nonlinear Science and Numerical Simulation 13, no. 9 (November 2008): 1748–57. http://dx.doi.org/10.1016/j.cnsns.2007.05.001.

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31

Hereman, Willy. "Exact solitary wave solutions of coupled nonlinear evolution equations using MACSYMA." Computer Physics Communications 65, no. 1-3 (April 1991): 143–50. http://dx.doi.org/10.1016/0010-4655(91)90166-i.

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32

Xu, Siqi, Xianguo Geng, and Bo Xue. "An extension of the coupled derivative nonlinear Schrödinger hierarchy." Modern Physics Letters B 32, no. 02 (January 20, 2018): 1850016. http://dx.doi.org/10.1142/s0217984918500161.

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In this paper, a 3 × 3 matrix spectral problem with six potentials is considered. With the help of the compatibility condition, a hierarchy of new nonlinear evolution equations which can be reduced to the coupled derivative nonlinear Schrödinger (CDNLS) equations is obtained. By use of the trace identity, it is proved that all the members in this new hierarchy have generalized bi-Hamiltonian structures. Moreover, infinitely many conservation laws of this hierarchy are constructed.

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33

Carrington, M. E., R. Kobes, G. Kunstatter, D. Pickering, and E. Vaz. "Equilibration in an interacting field theory." Canadian Journal of Physics 80, no. 9 (September 1, 2002): 987–93. http://dx.doi.org/10.1139/p02-065.

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We use a combination of perturbation theory and numerical techniques to study the equilibration of two interacting fields that are initially at thermal equilibrium at different temperatures. Using standard rules of quantum field theory, we examine the master equations that describe the time evolution of the distribution functions for the two coupled systems. By making a few reasonable assumptions we reduce the resulting coupled integral/differential equations to a pair of differential equations that can be solved numerically relatively easily and which give physically sensible results. PACS No.: 11.10W

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34

Erjaee, G. H., and M. Alnasr. "Phase Synchronization in Coupled Sprott Chaotic Systems Presented by Fractional Differential Equations." Discrete Dynamics in Nature and Society 2009 (2009): 1–10. http://dx.doi.org/10.1155/2009/753746.

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Phase synchronization occurs whenever a linearized system describing the evolution of the difference between coupled chaotic systems has at least one eigenvalue with zero real part. We illustrate numerical phase synchronization results and stability analysis for some coupled Sprott chaotic systems presented by fractional differential equations.

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35

KHANI, F., M. T. DARVISHI, A. FARMANY, and L. KAVITHA. "NEW EXACT SOLUTIONS OF COUPLED (2+1)-DIMENSIONAL NONLINEAR SYSTEMS OF SCHRÖDINGER EQUATIONS." ANZIAM Journal 52, no. 1 (July 2010): 110–21. http://dx.doi.org/10.1017/s1446181111000563.

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AbstractThe Exp-function method is applied to construct a new type of solution of the coupled (2+1)-dimensional nonlinear system of Schrödinger equations. It is shown that the method provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

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36

Yaşar, Emrullah, and Sait San. "A Procedure to Construct Conservation Laws of Nonlinear Evolution Equations." Zeitschrift für Naturforschung A 71, no. 5 (May 1, 2016): 475–80. http://dx.doi.org/10.1515/zna-2016-0057.

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AbstractIn this article, we established abundant local conservation laws to some nonlinear evolution equations by a new combined approach, which is a union of multiplier and Ibragimov’s new conservation theorem method. One can conclude that the solutions of the adjoint equations corresponding to the new conservation theorem can be obtained via multiplier functions. Many new families of conservation laws of the Pochammer–Chree (PC) equation and the Kaup–Boussinesq type of coupled KdV system are successfully obtained. The combined method presents a wider applicability for handling the conservation laws of nonlinear wave equations. The conserved vectors obtained here can be important for the explanation of some practical physical problems, reductions, and solutions of the underlying equations.

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37

Shah, Ijaz, Ghazala Anwar, H. A. Shah, T. Abdullah, and M. Anis Alam. "Chaotic Evolution of a Parametric Instability in a Piezoelectric Semiconductor Plasma." International Journal of Bifurcation and Chaos 07, no. 05 (May 1997): 1103–13. http://dx.doi.org/10.1142/s021812749700090x.

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We consider the chaotic evolution of three parametrically interacting waves in a piezoelectric semiconductor plasma. The evolution equation has been derived using the coupled mode theory. The waves are an extraordinary wave, ordinary wave and upper hybrid acoustic wave. The extraordinary wave is a growing wave and the other two are damped modes. The equations are transformed and put in a standard form and made dimensionless by suitable substitution of new variables. Finally a set of three equations is obtained. The real and imaginary parts of the equations are then separated. The three imaginary equations can be reduced to one resulting in a system of four equations. Analytical investigation are made of the stationary points. The evolution of equations for various parametric values is done numerically. Time series analysis and phase space plots of the chaotic evolution of the amplitude are also investigated numerically. Lyapunov exponents are computed and the results are presented graphically.

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38

Gao, Yi-Tian, and Bo Tian. "Notiz: A Symbolic Computation-Based Method and Two Nonlinear Evolution Equations for Water Waves." Zeitschrift für Naturforschung A 52, no. 3 (March 1, 1997): 295–96. http://dx.doi.org/10.1515/zna-1997-0311.

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Abstract A symbolic-computation-based method, which has been newly proposed, is considered for a (2+1)-dimensional generalization of shallow water wave equations and a coupled set of the (2 +1)-dimensional integrable dispersive long wave equations. New sets of soliton-like solutions are constructed, along with solitary waves.

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39

Li, Bang Qing, and Yu Lan Ma. "Exact Solutions for Coupled mKdV Equations by a New Symbolic Computation Method." Applied Mechanics and Materials 20-23 (January 2010): 184–89. http://dx.doi.org/10.4028/www.scientific.net/amm.20-23.184.

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By introducing (G′/G)-expansion method and symbolic computation software MAPLE, two types of new exact solutions are constructed for coupled mKdV equations. The solutions included trigonometric function solutions and hyperbolic function solutions. The procedure is concise and straightforward, and the method is also helpful to find exact solutions for other nonlinear evolution equations.

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40

Mohammed Djaouti, Abdelhamid. "Weakly Coupled System of Semi-Linear Fractional θ-Evolution Equations with Special Cauchy Conditions." Symmetry 15, no. 7 (June 30, 2023): 1341. http://dx.doi.org/10.3390/sym15071341.

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In this paper, we consider a weakly system of fractional θ-evolution equations. Using the fixed-point theorem, a global-in-time existence of small data solutions to the Cauchy problem is proved for one single equation. Using these results, we prove the global existence for the system under some mixed symmetrical conditions that describe the interaction between the equations of the system.

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41

KUZEMSKY, A. L. "GENERALIZED KINETIC AND EVOLUTION EQUATIONS IN THE APPROACH OF THE NONEQUILIBRIUM STATISTICAL OPERATOR." International Journal of Modern Physics B 19, no. 06 (March 10, 2005): 1029–59. http://dx.doi.org/10.1142/s0217979205029419.

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The method of the nonequilibrium statistical operator developed by D. N. Zubarev is employed to analyze and derive generalized transport and kinetic equations. The degrees of freedom in solids can often be represented as a few interacting subsystems (electrons, spins, phonons, nuclear spins, etc.). Perturbation of one subsystem may produce a nonequilibrium state which is then relaxed to an equilibrium state due to the interaction between particles or with a thermal bath. The generalized kinetic equations were derived for a system weakly coupled to a thermal bath to elucidate the nature of transport and relaxation processes. It was shown that the "collision term" had the same functional form as for the generalized kinetic equations for the system with small interactions among particles. The applicability of the general formalism to physically relevant situations is investigated. It is shown that some known generalized kinetic equations (e.g. kinetic equation for magnons, Peierls equation for phonons) naturally emerges within the NSO formalism. The relaxation of a small dynamic subsystem in contact with a thermal bath is considered on the basis of the derived equations. The Schrödinger-type equation for the average amplitude describing the energy shift and damping of a particle in a thermal bath and the coupled kinetic equation describing the dynamic and statistical aspects of the motion are derived and analyzed. The equations derived can help in the understanding of the origin of irreversible behavior in quantum phenomena.

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42

Singh, I., and Sh Kumar. "Modified Wavelet Method for Solving Two-dimensional Coupled System of Evolution Equations." Iranian Journal of Mathematical Sciences and Informatics 17, no. 1 (April 1, 2022): 239–59. http://dx.doi.org/10.52547/ijmsi.17.1.239.

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43

Cai, Zihan, Yan Liu, and Baiping Ouyang. "Decay properties for evolution-parabolic coupled systems related to thermoelastic plate equations." AIMS Mathematics 7, no. 1 (2021): 260–75. http://dx.doi.org/10.3934/math.2022017.

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<abstract><p>In this paper, we consider the Cauchy problem for a family of evolution-parabolic coupled systems, which are related to the classical thermoelastic plate equations containing non-local operators. By using diagonalization procedure and WKB analysis, we derive representation of solutions in the phase space. Then, sharp decay properties in a framework of $ L^p-L^q $ are investigated via these representations. Particularly, some thresholds for the regularity-loss type decay properties are found.</p></abstract>

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44

Block, Martin M., Loyal Durand, Phuoc Ha, and Douglas W. McKay. "Decoupling the NLO coupled DGLAP evolution equations: an analytic solution to pQCD." European Physical Journal C 69, no. 3-4 (August 21, 2010): 425–31. http://dx.doi.org/10.1140/epjc/s10052-010-1413-4.

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45

Kuetche, V. K., T. B. Bouetou, and T. C. Kofane. "On exact N-loop soliton solution to nonlinear coupled dispersionless evolution equations." Physics Letters A 372, no. 5 (January 2008): 665–69. http://dx.doi.org/10.1016/j.physleta.2007.08.023.

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46

Zhaqilao. "On Nth-order rogue wave solution to nonlinear coupled dispersionless evolution equations." Physics Letters A 376, no. 45 (October 2012): 3121–28. http://dx.doi.org/10.1016/j.physleta.2012.09.050.

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47

Jin, Kun-Peng, Jin Liang, and Ti-Jun Xiao. "Coupled second order evolution equations with fading memory: Optimal energy decay rate." Journal of Differential Equations 257, no. 5 (September 2014): 1501–28. http://dx.doi.org/10.1016/j.jde.2014.05.018.

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48

Yezzi, Anthony, Andy Tsai, and Alan Willsky. "A Fully Global Approach to Image Segmentation via Coupled Curve Evolution Equations." Journal of Visual Communication and Image Representation 13, no. 1-2 (March 2002): 195–216. http://dx.doi.org/10.1006/jvci.2001.0500.

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49

Borisov, V. E., A. V. Ivanov, B. V. Kritsky, and E. B. Savenkov. "Numerical Algorithms for Simulation of a Fluid-Filed Fracture Evolution in a Poroelastic Medium." PNRPU Mechanics Bulletin, no. 2 (December 15, 2021): 24–35. http://dx.doi.org/10.15593/perm.mech/2021.2.03.

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Abstract:

The paper deals with the computational framework for the numerical simulation of the three dimensional fluid-filled fracture evolution in a poroelastic medium. The model consists of several groups of equations including the Biot poroelastic model to describe a bulk medium behavior, Reynold’s lubrication equations to describe a flow inside fracture and corresponding bulk/fracture interface conditions. The geometric model of the fracture assumes that it is described as an arbitrary sufficiently smooth surface with a boundary. Main attention is paid to describing numerical algorithms for particular problems (poroelasticity, fracture fluid flow, fracture evolution) as well as an algorithm for the coupled problem solution. An implicit fracture mid-surface representation approach based on the closest point projection operator is a particular feature of the proposed algorithms. Such a representation is used to describe the fracture mid-surface in the poroelastic solver, Reynold’s lubrication equation solver and for simulation of fracture evolutions. The poroelastic solver is based on a special variant of X-FEM algorithms, which uses the closest point representation of the fracture. To solve Reynold’s lubrication equations, which model the fluid flow in fracture, a finite element version of the closet point projection method for PDEs surface is used. As a result, the algorithm for the coupled problem is purely Eulerian and uses the same finite element mesh to solve equations defined in the bulk and on the fracture mid-surface. Finally, we present results of the numerical simulations which demonstrate possibilities of the proposed numerical techniques, in particular, a problem in a media with a heterogeneous distribution of transport, elastic and toughness properties.

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Landim, Ricardo C. G. "Coupled tachyonic dark energy: A dynamical analysis." International Journal of Modern Physics D 24, no. 11 (September 6, 2015): 1550085. http://dx.doi.org/10.1142/s0218271815500856.

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In this paper, we present a dynamical analysis for a coupled tachyonic dark energy with dark matter. The tachyonic field ϕ is considered in the presence of barotropic fluids (matter and radiation) and the autonomous system due to the evolution equations is studied. The three cosmological eras (radiation, matter and dark energy) are described through the critical points, for a generic potential V(ϕ).

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