Computational schemes based on the continuous analogueof Newton’s method in the numerical study of complexphysical systems at JINR
Вычислительные схемы на основе непрерывного аналога метода Ньютона для численного исследования сложных физических систем в ОИЯИ

Additional data

Submitted: 12.01.2026; Accepted: 27.02.2026; Published 12.03.2026;
Views: 0; Downloaded: 0

How to Cite

Elena Zemlyanaya, Ochbadrakh Chuluunbaatar. "Computational schemes based on the continuous analogueof Newton’s method in the numerical study of complexphysical systems at JINR" Natural Sci. Rev. 3 200605 (2026)
https://doi.org/10.54546/NaturalSciRev.200605
BibTex
Elena Zemlyanaya1,2,3, Ochbadrakh Chuluunbaatar1,4,5,a
  • 1Joint Institute for Nuclear Research, 141980 Dubna, Russia
  • 2Dubna State University, 141980 Dubna, Russia
  • 3Branch of Lomonosov Moscow State University in Dubna, 141980 Dubna, Russia
  • 4Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences, 13330 Ulaanbaatar, Mongolia
  • 5School of Applied Sciences, Mongolian University of Science and Technology, 14191 Ulaanbaatar, Mongolia
  • achuka@jinr.ru
DOI: 10.54546/NaturalSciRev.200605
Keywords: Newtonian iteration, computer modeling, nonlinear problem, complex physical system
Topics: Physics , Mathematical physics , Mathematical and Computer Sciences , Mathematical Modelling , Numerical Methods
PDF

Abstract

The Continuous analogue of Newton's method (CANM), developed at JINR since the 1970s, is one of most important areas of research at Laboratory of Computing Techniques (LCTA) -- Meshcheryakov Laboratory of Information Technologies (MLIT). CANM and its generalization are powerful tools for the effective numerical solution of nonlinear problems within a wide range of complex physical systems studied at JINR. This review article provides a general framework for the CANM-based approach, the main stages in the development and applications of CANM for solving various types of nonlinear problems that have been on the agenda in different years. The results of the development and application of CANM-based iterative methods, obtained over the past 20 years, are presented in more detail. 

Acknowledgements

The authors are grateful to the Directorate of JINR and MLIT for the honourable proposal to write this review article on the occasion of the 70th Anniversary of the Joint Institute for Nuclear Research and the 60th Anniversary of the Meshcheryakov Laboratory of Information Technologies of JINR. We thank S. I. Vinitsky, A. A. Gusev, Yu. V. Popov, B. B. Joulakian, I. V. Barashenkov, N. A. Alexeeva, A. A. Bogolubskaya, M. V. Bashashin, V. L. Derbov, and A. V. Mitin in collaboration with whom a number of results presented in Sections 3 and 4 were obtained. We are grateful to T. P. Puzynina, T. Zhanlav, and V. T. Thach for their contribution to the development of new algorithms for calculating the CANM iteration parameter described in Sections 2.2.1, 2.2.2, 2.4.1, and 2.4.2., We express our gratitude to T. A. Strizh, V. S. Melezhik, P. G. Akishin, Yu. L. Kalinovsky, A. A. Gusev, Yu. V. Popov, and B. B. Joulakian for their useful comments and recommendations during the writing of this review article. Most of the numerical calculations presented in Sections 3 and 4 were performed on the computational resources of the JINR Multifunctional Information and Computing Complex (MICC), including the HybriLIT heterogeneous computing platform and the “Govorun” supercomputer. We are thankful to the MICC and HybriLIT teams for their support., We would like to highlight the significant contribution of Professor Igor Viktorovich Puzynin to the field of theoretical research and practical developments of CANM and express our gratitude to him.

References

[1] E. P. Zhidkov, G. I. Makarenko, I. V. Puzynin, Continuous analog of the Newton method in non-linear physical problems, Soviet Journal of Particles and Nuclei 4 (1) (1973) 53.

[2] I. V. Puzynin, I. V. Amirkhanov, E. Zemlyanaya, V. N. Pervushin, T. P. Puzynina, T. A. Strizh, V. D. Lakhno, The generalized continuous analog of Newton’s method for the numerical study of some nonlinear quantum-field models, Physics of Particles and Nuclei 30 (1) (1999) 87–110. doi:10.1134/1.953099.

[3] E. P. Zhidkov, I. V. Puzynin, A method for introducing a parameter when solving boundary value problems for second-order nonlinear ordinary differential equations, USSR Computational Mathematics and Mathematical Physics 7 (5) (1967) 157–170.

[4] M. K. Gavurin, Nonlinear functional equations and continuous analogues of iterative methods, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika 5 (6) (1958) 18–31, (in Russian).

[5] I. V. Puzynin, T. L. Boyadzhiev, S. I. Vinitskii, E. V. Zemlyanaya, T. P. Puzynina, O. Chuluunbaatar, Methods of computational physics for investigation of models of complex physical systems, Physics of Particles and Nuclei 38 (1) (2007) 70–116. doi:10.1134/s1063779607010030.

[6] F. Gareev, S. A. Goncharov, E. P. Zhidkov, I. V. Puzynin, B. N. Khoromskii, R. M. Yamaleev, The scattering problem in quantum mechanics as an eigenvalue problem, USSR Computational Mathematics and Mathematical Physics 17 (2) (1977) 116–128.

[7] J. Bang, F. A. Gareev, I. V. Puzynin, R. M. Jamalejev, Single-particle quasistationary states in spherical and deformed nuclei, Nuclear Physics A 261 (1) (1976) 59–76.

[8] L. I. Ponomarev, I. V. Puzynin, T. P. Puzynina, L. N. Somov, The scattering problem in quantum mechanics as an eigenvalue problem, Annals of Physics 110 (2) (1978) 274–286.

[9] A. T. Filippov, I. V. Puzynin, D. P. Mavlo, Numerical methods for solving some singular boundary value problems of quantum mechanics and quantum field theory. I, Journal of Computational Physics 22 (2) (1976) 150–170. doi:10.1016/0021-9991(76)90073-5.

[10] V. S. Melezhik, Continuous analog of Newton method in the multichannel scattering problem, Journal of Computational Physics 65 (1) (1986) 1–17. doi:10.1016/0021-9991(86)90001-x.

[11] V. S. Melezhik, New method for solving multidimensional scattering problem, Journal of Com- putational Physics 92 (1) (1991) 67–81. doi:10.1016/0021-9991(91)90292-s.

[12] S. I. Vinitskii, I. V. Puzynin, Yu. S. Smirnov, High-precision calculations of the multichannel scattering problem for processes involving mesic atoms, Physics of Atomic Nuclei 55 (12) (1992) 1830–1838.

[13] A. T. Filippov, Yu. S. Gal’pern, T. L. Boyadjiev, I. V. Puzynin, Critical currents in Josephson junctions with microinhomogeneities attracting solitons, Physics Letters A 120 (1) (1987) 47–50. doi:10.1016/0375-9601(87)90262-3.

[14] I. V. Barashenkov, T. L. Boyadjiev, I. V. Puzynin, T. Zhanlav, Stability of moving bubbles in a system of interacting bosons, Physics Letters A 135 (2) (1989) 125–128. doi:10.1016/0375-9601(89)90658-0.

[15] L. I. Ponomarev, Muon catalysed fusion, Contemporary Physics 31 (4) (1990) 219–245. doi: 10.1080/00107519008222019.

[16] S. I. Vinitskii, V. S. Melezhik, L. I. Ponomarev, I. V. Puzynin, T. P. Puzynina, L. N. Somov, N. F. Truskova, Calculation of the energy levels of μ-mesic molecules of hydrogen isotopes in the adiabatic representation of the three-body problem, Soviet Journal of Experimental and Theoretical Physics 52 (1980) 353–360.

[17] V. S. Melezhik, I. V. Puzynin, T. P. Puzynina, L. N. Somov, Numerical solution of a system of integrodifferential equations arising from the quantum mechanical three-body problem with Coulomb interaction, Journal of Computational Physics 54 (2) (1984) 221–236. doi:10.1016/0021-9991(84)90115-3.

[18] A. D. Gocheva, V. V. Gusev, V. S. Melezhik, L. I. Ponomarev, I. V. Puzynin, T. P. Puzynina, L. N. Somov, S. I. Vinitsky, High accuracy energy-level calculations of the rotational-vibrational weakly bound states of ddμ and dtμ mesic molecules, Physics Letters B 153 (6) (1985) 349–352. doi:10.1016/0370-2693(85)90470-8.

[19] T. Zhanlav, I. V. Puzynin, An evolutionary Newton procedure for solving nonlinear equations, Computational Mathematics and Mathematical Physics 32 (1) (1992) 1–9.

[20] I. V. Puzynin, T. P. Puzynina, V. T. Thach, SLIPM – Maple programm for numerical solution of Sturm-Liouville partial problem with the help of the continuous analogue of Newton’s method, Bulletin of Peoples’ Friendship University of Russia. Mathematics, Information sciences. Physics 2 (2) (2010) 90–98, (in Russian).

[21] V. T. Thach, T. P. Puzynina, SLIPH4M — Program for numerical solution of the Sturm–Liouville partial problem, Journal of Software and Systems 95 (3) (2011) 75–80, (in Russian).

[22] T. Zhanlav, O. Chuluunbaatar, New Developments of Newton-Type Iterations for Solving Non- linear Problems, Springer Nature Switzerland, 2024. doi:10.1007/978-3-031-63361-4.

[23] I. V. Barashenkov, E. V. Zemlyanaya, Travelling solitons in the externally driven nonlinear Schr¨odinger equation, Journal of Physics A: Mathematical and Theoretical 44 (46) (2011) 465211. doi:10.1088/1751-8113/44/46/465211.

[24] I. V. Barashenkov, E. V. Zemlyanaya, T. C. van Heerden, Time-periodic solitons in a damped- driven nonlinear Schr¨odinger equation, Physical Review E 83 (5) (2011) 056609. doi:10.1103/ physreve.83.056609.

[25] I. V. Barashenkov, E. V. Zemlyanaya, Soliton complexity in the damped-driven nonlinear Schr¨odinger equation: Stationary to periodic to quasiperiodic complexes, Physical Review E 83 (5) (2011) 056610. doi:10.1103/physreve.83.056610.

[26] N. V. Alexeeva, I. V. Barashenkov, A. A. Bogolubskaya, E. V. Zemlyanaya, Understanding oscillons: Standing waves in a ball, Physical Review D 107 (7) (2023) 076023. doi:10.1103/physrevd.107.076023.

[27] P. Kh. Atanasova, T. L. Boyadjiev, Yu. M. Shukrinov, E. V. Zemlyanaya, Numerical modeling of long Josephson junctions in the frame of double sine-Gordon equation, Mathematical Models and Computer Simulations 3 (3) (2011) 389–398. doi:10.1134/s2070048211030033.

[28] P. Kh. Atanasova, T. L. Boyadjiev, E. V. Zemlyanaya, Yu. M. Shukrinov, Numerical study of magnetic flux in the LJJ model with double sine-Gordon equation, Lecture Notes in Computer Sciences 6046 (2011) 347–352. doi:10.1007/978-3-642-18466-6_41.

[29] P. Kh. Atanasova, T. L. Boyadjiev, Yu. M. Shukrinov, E. V. Zemlyanaya, P. Seidel, Influence of Josephson current second harmonic on stability of magnetic flux in long junctions, Journal of Physics: Conference Series 248 (2010) 012044. doi:10.1088/1742-6596/248/1/012044.

[30] P. Kh. Atanasova, E. V. Zemlyanaya, Yu. M. Shukrinov, Interconnection between static regimes in the LJJs described by the double sine-Gordon equation, Journal of Physics: Conference Series 393 (2012) 012023. doi:10.1088/1742-6596/393/1/012023.

[31] P. Kh. Atanasova, E. V. Zemlyanaya, Numerical investigation of bifurcations in long Josephson junctions with second harmonic in the current-phase relation, Comptes Rendus de l’Acad´emie Bulgare des Sciences 68 (12) (2015) 1483–1490.

[32] A. A. Gusev, L. L. Hai, O. Chuluunbaatar, V. Ulziibayar, S. I. Vinitsky, V. L. Derbov, A. G´o´zd´z, V. A. Rostovtsev, Symbolic-numeric solution of boundary-value problems for the Schr¨odinger equation using the finite element method: Scattering problem and resonance states, Lecture Notes in Computer Science 9301 (2015) 182–197. doi:10.1007/978-3-319-24021-3_14.

[33] A. A. Gusev, V. P. Gerdt, L. L. Hai, V. L. Derbov, S. I. Vinitsky, O. Chuluunbaatar, Symbolic- numeric algorithms for solving bvps for a system of odes of the second order: Multichannel scattering and eigenvalue problems, Lecture Notes in Computer Science 9890 (2016) 212–227. doi:10.1007/978-3-319-45641-6_14.

[34] A. A. Gusev, S. I. Vinitsky, O. Chuluunbaatar, A. G´o´zd´z, V. L. Derbov, P. M. Krassovitskiy, Adiabatic representation for atomic dimers and trimers in collinear configuration, Physics of Atomic Nuclei 81 (6) (2018) 945–970. doi:10.1134/s1063778818060169.

[35] A. A. Gusev, O. Chuluunbaatar, V. Derbov, R. Nazmitdinov, S. Vinitsky, P. Wen, C. Lin, H. M. Jia, L. L. Hai, Symbolic-numerical algorithm for solving the problem of heavy ion collisions in an optical model with a complex potential, Lecture Notes in Computer Science 14139 (2023) 128–140. doi:10.1007/978-3-031-41724-5_7.

[36] V. L. Derbov, G. Chuluunbaatar, A. A. Gusev, O. Chuluunbaatar, S. I. Vinitsky, A. G´o´zd´z, P. M. Krassovitskiy, I. Filikhin, A. V. Mitin, Spectrum of beryllium dimer in ground X1Σ+g state, Journal of Quantitative Spectroscopy and Radiative Transfer 262 (2021) 107529. doi: 10.1016/j.jqsrt.2021.107529.

[37] I. Hristov, R. Hristova, I. Puzynin, T. Puzynina, Z. Sharipov, Z. Tukhliev, New families of periodic orbits for the planar three-body problem computed with high precision, CEUR Workshop Proceedings 3191 (2022) 45–50.

[38] I. Hristov, R. Hristova, I. Puzynin, T. Puzynina, Z. Sharipov, Z. Tukhliev, Searching for new nontrivial choreographies for the planar three-body problem, Physics of Particles and Nuclei 55 (3) (2024) 495–497. doi:10.1134/s1063779624030444.

[39] R. V. Polyakova, I. P. Yudin, Continuous analogue of Newton method in beam dynamics problems, Bulletin of Peoples’ Friendship University of Russia. Mathematics, Information sciences. Physics (2) (2011) 83–91.

[40] T. T´oth, M. Dovica, J. Buˇsa, J. Buˇsa Jr., Verification of coordinate measuring machine using a gauge block, Measurment Science Review 24 (6) (2024) 225–233. doi:10.2478/msr-2024-0030.

[41] B. Batgerel, E. V. Zemlyanaya, I. V. Puzynin, NINE: computer code for numerical solution of the boundary problems for nonlinear differential equations on the basis of CANM, Computer Research and Modeling 2 (2) (2012) 315–324, (in Russian). doi:10.20537/

2076-7633-2012-4-2-315-324.

[42] M. Bordag, D. N. Voskresensky, Generation of a scalar vortex in a rotational frame, Physical Review D 112 (8) (2025) 085026. doi:10.1103/y2lc-27pn.

[43] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag Berlin, 1995.

[44] J. M. Ortega, W. C. Reinboldt, Iterative solution of nonlinear equations in several variables, New-York-London, Academic Press, 1970.

[45] D. F. Davidenko, On application of method of variation of parameter to the theory of nonlinear functional equations, Ukrainian Mathematical Journal 7 (1) (1955) 18–28, (in Russian).

[46] D. A. Kirzhnits, N. Z. Takibaev, New approach to 3 and more body problem, Soviet Journal of Nuclear Physics 25 (1) (1977) 370–376.

[47] G. I. Marchuk, Splitting Methods, Moscow: Nauka, 1988, (in Russian).

[48] G. I. Marchuk, V. V. Shaidurov, Difference Methods and Their Extrapolations, Springer New York, 1983. doi:10.1007/978-1-4613-8224-9.

[49] L. Alexandrov, Regularized trajectories of Newton-type approximation for solving of nonlinear equations, Differencial’nye Uravnenija 13 (7) (1977) 1281–1292, (in Russian).

[50] E. K. Blum, A. F. Chang, A numerical method for the solution of the double eigenvalue problem, IMA Journal of Applied Mathematics 22 (1) (1978) 29–42. doi:10.1093/imamat/22.1.29.

[51] L. Bracci, G. Fiorentini, Mesic molecules and muon catalysed fusion, Physics Reports 86 (4) (1982) 169–216. doi:10.1016/0370-1573(82)90095-3.

[52] I. V. Puzynin, S. I. Vinitsky, Energy levels of mesic molecule, Journal of Muon Catalyzed Fusion 3 (1988) 307–320.

[53] T. Zhanlav, I. V. Puzynin, The convergence of iterations based on a continuous analogue of Newton’s method, Computational Mathematics and Mathematical Physics 32 (6) (1992) 729–737.

[54] O. Yu. Kul’chitskii, L. I. Shimelevich, Determination of the initial approximation for Newton’s method, USSR Computational Mathematics and Mathematical Physics 14 (4) (1974) 188–190. doi:10.1016/0041-5553(74)90083-4.

[55] I. V. Puzynin, T. P. Puzynina, T. A. Strizh, SLIPH4 — Program for numerical solution of

Sturm–Liouville problem, JINR Communication P11-87-332, Dubna, (in Russian) (1987).

[56] E. V. Zemlyanaya, I. V. Puzynin, T. P. Puzynina, PROGS2H4 — Program for solution of boundary problem for system of differential equations, JINR Communication P11-97-414, Dubna, (in Russian) (1997).

[57] I. V. Puzynin, T. P. Puzynina, Program of approximate solution of Sturm–Liouville problem using the continuous analog of Newton’s method, Collection of Scientific Papers in Collaboration of JINR (Dubna, USSR) and Central Research Institute for Physics (Hungary, Budapest) (1974) 93–112.

[58] K. A. Lebedev, A method of finding the initial approximation for Newton’s method, Computational Mathematics and Mathematical Physics 36 (3) (1996) 283–289.

[59] V. V. Ermakov, N. N. Kalitkin, The optimal step and regularization for Newton’s method, USSR Computational Mathematics and Mathematical Physics 21 (2) (1981) 235–242. doi:10.1016/0041-5553(81)90022-7.

[60] S. I. Vinitskii, L. I. Ponomarev, I. V. Puzynin, T. P. Puzynina, Newton’s process in perturbation theory with continuous inclusion of interaction, JINR Communication P4-10942, Dubna, (in Russian) (1977).

[61] I. V. Amirkhanov, E. V. Zemlyanaya, T. P. Puzynina, SNIDE – Program package for solution of characteristic constant problems for integral-differential equation on the base of CANM, JINR Communication P11-91-87, Dubna, (in Russian) (1991).

[62] S. I. Vinitskii, I. V. Puzynin, Yu. S. Smirnov, Solution of the scattering problem on the basis of multiparameter Newton schemes – single-channel scattering, Soviet Journal of Nuclear Physics 52 (10) (1990) 746–754.

[63] I. V. Barashenkov, Yu. S. Smirnov, N. V. Alexeeva, Bifurcation to multisoliton complexes in the ac-driven, damped nonlinear Schr¨odinger equation, Physical Review E 57 (2) (1998) 2350–2364. doi:10.1103/physreve.57.2350.

[64] G. N. Chuev, V. D. Lakhno, Perspectives of Polarons, World Scientific, 1996. doi:10.1142/3208.

[65] P. G. Akishin, I. V. Puzynin, Yu. S. Smirnov, The Newtonian continuation method for numerical study of 3D polaron problem, Lecture Notes in Computer Science 1196 (1997) 1–8. doi:10.1007/3-540-62598-4_72.

[66] I. V. Amirkhanov, I. V. Puzynin, T. P. Puzynina, T. A. Strizh, E. V. Zemlyanaya, CANM in numerical investgation of QCD problems, Lecture Notes in Computer Science 1196 (1997) 9–16. doi:10.1007/3-540-62598-4_73.

[67] L. V. Kantorovich, Approximate solution of functional equations, Uspehi Mathematicheskih Nauk 11 (6) (1956) 99–116, (in Russian).

[68] T. P. Puzynina, SLIPS2 — Program for numerical solution of Sturm-Liouville problem for system of differential equations, JINR Communication Р11-89-728, Dubna, (in Russian) (1989).

[69] E. V. Zemlyanaya, SYSINT(SYSINTM) — Program complex for numerical solution of characteristic constant problems for system of integral equations, JINR Communications Р11-94-120, Dubna, (in Russian) (1994).

[70] E. V. Zemlyanaya, I. V. Barashenkov, Numerical study of travelling solitons in parametrically driven, damped nonlinear Shcr¨odinger equation, Matematicheskoe Modelirovanie 17 (1) (2005) 65–78, (in Russian).

[71] G. Chuluunbaatar, A. A. Gusev, O. Chuluunbaatar, S. I. Vinitsky, L. L. Hai, KANTBP 4M — Program for solving the scattering problem for a system of ordinary second-order differential equations, EPJ Web of Conferences 226 (2020) 02008. doi:10.1051/epjconf/202022602008.

[72] G. Chuluunbaatar, A. A. Gusev, V. L. Derbov, S. I. Vinitsky, L. L. Hai, O. Chuluunbaatar, V. V. Gerdt, A Maple implementation of the finite element method for solving boundary-value problems for systems of second-order ordinary differential equations, Maple Conference 2020 1414 (2021) 152–166. doi:10.1007/978-3-030-81698-8_11.

[73] E. V. Zemlyanaya, I. V. Barashenkov, Numerical study of the multisoliton complexes in the damped-driven NLS, Matematicheskoe Modelirovanie 16 (3) (2004) 3–14, (in Russian).

[74] I. V. Barashenkov, E. V. Zemlyanaya, M. B´ar, Traveling solitons in the parametrically driven nonlinear Schr¨odinger equation, Physical Review E 64 (1) (2001) 016603. doi:10.1103/physreve. 64.016603.

[75] E. V. Zemlyanaya, I. V. Barashenkov, Traveling solitons in the damped-driven nonlinear Schr¨odinger equation, SIAM Journal on Applied Mathematics 64 (3) (2004) 800–818. doi: 10.1137/s0036139903424837.

[76] I. V. Barashenkov, M. M. Bogdan, V. I. Korobov, Stability diagram of the phase-locked solitons in the parametrically driven, damped nonlinear Schr¨odinger equation, Europhysics Letters (EPL) 15 (2) (1991) 113–118. doi:10.1209/0295-5075/15/2/001.

[77] I. V. Barashenkov, E. V. Zemlyanaya, Stable complexes of parametrically driven, damped nonlinear Schr¨odinger solitons, Physical Review Letters 83 (13) (1999) 2568–2571. doi:10.1103/physrevlett.83.2568.

[78] N. V. Alexeeva, I. V. Barashenkov, D. E. Pelinovsky, Dynamics of the parametrically driven NLS solitons beyond the onset of the oscillatory instability, Nonlinearity 12 (1) (1999) 103–140. doi:10.1088/0951-7715/12/1/007.

[79] M. Bondila, I. V. Barashenkov, M. M. Bogdan, Topography of attractors of the parametrically driven nonlinear Schr¨odinger equation, Physica D: Nonlinear Phenomena 87 (1–4) (1995) 314–320. doi:10.1016/0167-2789(95)00126-o.

[80] E. V. Zemlyanaya, I. V. Barashenkov, N. V. Alexeeva, Temporally-periodic solitons of the para- metrically driven damped nonlinear Schr¨odinger equation, Lecture Notes in Computer Science 5434 (2009) 139–150. doi:10.1007/978-3-642-00464-3_13.

[81] E. V. Zemlyanaya, N. V. Alexeeva, Oscillating solitons of the driven, damped nonlinear Schr¨odinger equation, Theoretical and Mathematical Physics 159 (3) (2009) 870–876. doi: 10.1007/s11232-009-0075-6.

[82] E. Zemlyanaya, N. Alexeeva, Numerical study of stationary, time-periodic, and quasiperiodic two-soliton complexes in the damped-driven nonlinear Schr¨odinger equation, Lecture Notes in Computer Science 7125 (2012) 240–245. doi:10.1007/978-3-642-28212-6_27.

[83] N. V. Alexeeva, E. V. Zemlyanaya, Breathers in a damped-driven nonlinear Schr¨odinger equation, Theoretical and Mathematical Physics 168 (1) (2011) 858–864. doi:10.1007/s11232-011-0069-z.

[84] E. V. Zemlyanaya, N. V. Alexeeva, Numerical study of time-periodic solitons in the damped- driven NLS, International Journal of Numerical Analysis and Modeling, Series B. 2 (2–3) (2011) 248–261.

[85] R. Seydel, From equilibrium to chaos. Practical bifurcation and stability analysis, Elsevier Science Publishing Co., 1988.

[86] A. A. Samarskii, P. N. Vabishchevich, Computational Heat Transfer, Volume 1: Mathematical Modelling, Viley, 1996.

[87] E. L. Allgower, K. Georg, Numerical Continuation Methods, Springer Berlin Heidelberg, 1990. doi:10.1007/978-3-642-61257-2.

[88] I. V. Barashenkov, S. R. Woodford, E. V. Zemlyanaya, Parametrically driven dark solitons, Physical Review Letters 90 (5) (2003) 054103. doi:10.1103/physrevlett.90.054103.

[89] I. V. Barashenkov, S. R. Woodford, E. V. Zemlyanaya, Interactions of parametrically driven dark solitons. I. N´eel-N´eel and Bloch-Bloch interactions, Physical Review E 75 (2) (2007) 026604. doi:10.1103/physreve.75.026604.

[90] I. V. Barashenkov, N. V. Alexeeva, E. V. Zemlyanaya, Two- and three-dimensional oscillons in nonlinear Faraday resonance, Physical Review Letters 89 (10) (2002) 104101. doi:10.1103/physrevlett.89.104101.

[91] N. V. Alexeeva, E. V. Zemlyanaya, Nodal two-dimensional solitons in nonlinear parametric resonance, Lecture Notes in Computer Science 3401 (2005) 91–99. doi:10.1007/978-3-540-31852-1_9.

[92] N. A. Voronov, I. Y. Kobzarev, N. B. Konyukhova, Possibility of the existence of X mesons of a new type, Journal of Experimental and Theoretical Physics Letters 22 (1975) 290–291.

[93] I. L. Bogolyubskii, V. G. Makhankov, Lifetime of pulsating solitons in certain classical models, Journal of Experimental and Theoretical Physics Letters 24 (1976) 12–14.

[94] I. L. Bogolyubskii, V. G. Makhankov, Dynamics of spherically symmetrical pulsons of large amplitude, Journal of Experimental and Theoretical Physics Letters 25 (1977) 107–110.

[95] M. Gleiser, Pseudostable bubbles, Physical Review D 49 (6) (1994) 2978–2981. doi:10.1103/physrevd.49.2978.

[96] E. J. Copeland, M. Gleiser, H.-R. M¨uller, Oscillons: Resonant configurations during bubble collapse, Physical Review D 52 (4) (1995) 1920–1933. doi:10.1103/physrevd.52.1920.

[97] E. P. Honda, M. W. Choptuik, Fine structure of oscillons in the spherically symmetric φ4 Klein–Gordon model, Physical Review D 65 (8) (2002) 084037. doi:10.1103/physrevd.65.084037.

[98] G. Fodor, P. Forg´acs, P. Grandcl´ement, I. R´acz, Oscillons and quasi-breathers in the ϕ4 Klein–Gordon model, Physical Review D 74 (12) (2006) 124003. doi:10.1103/physrevd.74.124003.

[99] I. V. Barashenkov, N. V. Alexeeva, Variational formalism for the Klein-Gordon oscillon, Physical Review D 108 (9) (2023) 096022. doi:10.1103/physrevd.108.096022.

[100] N. V. Alexeeva, I. V. Barashenkov, A. Dika, R. De Sousa, The energy-frequency diagram of the (1+1)-dimensional ϕ4 oscillon, Journal of High Energy Physics 2024 (10) (2024) 136. doi:10.1007/jhep10(2024)136.

[101] E. V. Zemlyanaya, A. A. Bogolubskaya, M. V. Bashashin, N. V. Alexeeva, The ϕ4 oscillons in a ball: Numerical approach and parallel implementation, Physics of Particles and Nuclei 55 (3)(2024) 505–508. doi:10.1134/s106377962403095x.

[102] E. V. Zemlyanaya, A. A. Bogolubskaya, M. V. Bashashin, N. V. Alexeeva, Numerical study of the ϕ4 standing waves in a ball of finite radius, Discrete and Continuous Models and Applied Computational Science 32 (1) (2024) 106–111. doi:10.22363/2658-4670-2024-32-1-106-111.

[103] E. V. Zemlyanaya, A. A. Bogolubskaya, M. V. Bashashin, N. V. Alexeeva, ϕ4 oscillons as standing waves in a ball: A numerical study, Physics of Particles and Nuclei 56 (6) (2025) 1655–1659. doi:10.1134/S1063779625701047.

[104] C. Chicone, Ordinary Differential Equations with Applications. Texts in Applied Mathematics, Vol. 34, Springer, New York, 2006. doi:10.1007/0-387-35794-7.

[105] L. U. Ancarani, T. Montagnese, C. Dal Cappello, Role of the helium ground state in (e, 3e) processes, Physical Review A 70 (1) (2004) 012711. doi:10.1103/physreva.70.012711.

[106] S. Jones, J. H. Macek, D. H. Madison, Test of the Pluvinage wave function for the helium ground state, Physical Review A 70 (1) (2004) 012712. doi:10.1103/physreva.70.012712.

[107] S. Jones, J. H. Macek, D. H. Madison, Three-Coulomb-wave Pluvinage model for Compton double ionization of helium in the region of the cross-section maximum, Physical Review A 72 (1) (2005) 012718. doi:10.1103/physreva.72.012718.

[108] S. Jones, D. H. Madison, Role of the ground state in electron-atom double ionization, Physical Review Letters 91 (7) (2003) 073201. doi:10.1103/physrevlett.91.073201.

[109] M. Brauner, J. S. Briggs, H. Klar, Triply-differential cross sections for ionisation of hydrogen atoms by electrons and positrons, Journal of Physics B: Atomic, Molecular and Optical Physics 22 (14) (1989) 2265–2287. doi:10.1088/0953-4075/22/14/010.

[110] P. Pluvinage, Fonction d’onde approch´ee `a un param`etre pour l’´etat fondamental des atomes `a deux ´electrons, Annales de Physique 12 (5) (1950) 145–152. doi:10.1051/anphys/195012050145.

[111] P. Pluvinage, Nouvelle famille de solutions approch´ees pour certaines ´equations de Schr¨odinger non s´eparables. application `a l’´etat fondamental de l’h´elium, Journal de Physique et le Radium 12 (8) (1951) 789–792. doi:10.1051/jphysrad:01951001208078900.

[112] T. Kato, On the eigenfunctions of many-particle systems in quantum mechanics, Communications on Pure and Applied Mathematics 10 (2) (1957) 151–177. doi:10.1002/cpa.3160100201.

[113] A. Lahmam-Bennani, I. Taouil, A. Duguet, M. Lecas, L. Avaldi, J. Berakdar, Origin of dips and peaks in the absolute fully resolved cross sections for the electron-impact double ionization of He, Physical Review A 59 (5) (1999) 3548–3555. doi:10.1103/physreva.59.3548.

[114] A. M. Frolov, Two-stage strategy for high-precision variational calculations, Physical Review A 57 (4) (1998) 2436–2439. doi:10.1103/physreva.57.2436.

[115] V. I. Korobov, Nonrelativistic ionization energy for the helium ground state, Physical Review A 66 (2) (2002) 024501. doi:10.1103/physreva.66.024501.

[116] Yu. V. Popov, L. U. Ancarani, Rigorous mathematical study of the He bound states, Physical Review A 62 (4) (2000) 042702. doi:10.1103/physreva.62.042702.

[117] C. L. Sech, Accurate analytic wavefunctions for two-electron atoms, Journal of Physics B: Atomic, Molecular and Optical Physics 30 (2) (1997) L47–L50. doi:10.1088/0953-4075/30/2/003.

[118] P. E. Gill, W. Murray, Newton-type methods for unconstrained and linearly constrained optimization, Mathematical Programming 7 (1) (1974) 311–350. doi:10.1007/bf01585529.

[119] O. Chuluunbaatar, I. V. Puzynin, S. I. Vinitsky, Uncoupled correlated calculations of helium isoelectronic bound states, Journal of Physics B: Atomic, Molecular and Optical Physics 34 (14) (2001) L425–L432. doi:10.1088/0953-4075/34/14/101.

[120] E. A. Hylleraas, B. Undheim, Numerische berechnung der 2S-terme von ortho- und par-helium, Zeitschrift f¨ur Physik 65 (11–12) (1930) 759–772. doi:10.1007/bf01397263.

[121] E. Clementi, C. Roetti, Roothaan-Hartree-Fock atomic wavefunctions, Atomic Data and Nuclear Data Tables 14 (3–4) (1974) 177–478. doi:10.1016/s0092-640x(74)80016-1.

[122] E. A. Hylleraas, Neue berechnung der energie des heliums im grundzustande, sowie des tiefsten terms von ortho-helium, Zeitschrift f¨ur Physik 54 (5–6) (1929) 347–366. doi:10.1007/bf01375457.

[123] C. Eckart, The theory and calculation of screening constants, Physical Review 36 (5) (1930) 878–892. doi:10.1103/physrev.36.878.

[124] S. Chandrasekhar, Some remarks on the negative hydrogen ion and its absorption coefficient, The Astrophysical Journal 100 (1944) 176–180. doi:10.1086/144654.

[125] J. N. Silverman, O. Platas, F. A. Matsen, Simple configuration-interaction wave functions. i. two-electron ions: A numerical study, The Journal of Chemical Physics 32 (5) (1960) 1402–1406.doi:10.1063/1.1730930.

[126] J. Mitroy, I. E. McCarthy, E. Weigold, A natural orbital analysis of the helium (e, 2e) spectrum, Journal of Physics B: Atomic and Molecular Physics 18 (20) (1985) 4149–4157. doi:10.1088/0022-3700/18/20/017.

[127] R. A. Bonham, D. A. Kohl, Simple correlated wavefunctions for the ground state of heliumlike atoms, The Journal of Chemical Physics 45 (7) (1966) 2471–2473. doi:10.1063/1.1727963.

[128] M. S. Sch¨offler, O. Chuluunbaatar, Yu. V. Popov, S. Houamer, J. Titze, T. Jahnke, L. Ph. H. Schmidt, O. Jagutzki, A. G. Galstyan, A. A. Gusev, Transfer ionization and its sensitivity to the ground-state wave function, Physical Review A 87 (3) (2013) 032715. doi:10.1103/physreva.87.032715.

[129] M. S. Sch¨offler, O. Chuluunbaatar, S. Houamer, A. Galstyan, J. N. Titze, L. Ph. H. Schmidt, T. Jahnke, H. Schmidt-B¨ocking, R. D¨orner, Yu. V. Popov, A. A. Gusev, C. Dal Cappello, Two-dimensional electron-momentum distributions for transfer ionization in fast proton-helium collisions, Physical Review A 88 (4) (2013) 042710. doi:10.1103/physreva.88.042710.

[130] M. S. Sch¨offler, H.-K. Kim, O. Chuluunbaatar, S. Houamer, A. G. Galstyan, J. N. Titze, T. Jahnke, L. Ph. H. Schmidt, H. Schmidt-B¨ocking, R. D¨orner, Yu. V. Popov, A. A. Bulychev, Transfer excitation reactions in fast proton-helium collisions, Physical Review A 89 (3) (2014) 032707. doi:10.1103/physreva.89.032707.

[131] H. Gassert, O. Chuluunbaatar, M. Waitz, F. Trinter, H.-K. Kim, T. Bauer, A. Laucke, Ch. M¨uller, J. Voigtsberger, M. Weller, J. Rist, M. Pitzer, S. Zeller, T. Jahnke, L. Ph. H. Schmidt, J. B. Williams, S. A. Zaytsev, A. A. Bulychev, K. A. Kouzakov, H. Schmidt-B¨ocking, R. D¨orner, Yu. V. Popov, M. S. Sch¨offler, Agreement of experiment and theory on the single ionization of helium by fast proton impact, Physical Review Letters 116 (7) (2016) 073201. doi:10.1103/physrevlett.116.073201.

[132] O. Chuluunbaatar, S. A. Zaytsev, K. A. Kouzakov, A. Galstyan, V. L. Shablov, Yu. V. Popov, Fully differential cross sections for singly ionizing 1-Mev p +He collisions at small momentum transfer: Beyond the first born approximation, Physical Review A 96 (4) (2017) 042716. doi:10.1103/physreva.96.042716.

[133] O. Chuluunbaatar, K. A. Kouzakov, S. A. Zaytsev, A. S. Zaytsev, V. L. Shablov, Yu. V. Popov, H. Gassert, M. Waitz, H.-K. Kim, T. Bauer, A. Laucke, Ch. M¨uller, J. Voigtsberger, M. Weller, J. Rist, K. Pahl, M. Honig, M. Pitzer, S. Zeller, T. Jahnke, L. Ph. H. Schmidt, H. Schmidt-B¨ocking, R. D¨orner, M. S. Sch¨offler, Single ionization of helium by fast proton impact in different kinematical regimes, Physical Review A 99 (6) (2019) 062711. doi:10.1103/physreva.99.062711.

[134] A. S. Zaytsev, D. S. Zaytseva, S. A. Zaytsev, L. U. Ancarani, O. Chuluunbaatar, K. A. Kouzakov, Yu. V. Popov, Single ionization of helium by protons of various energies in the parabolic quasi-

Sturmians approach, Atoms 11 (10) (2023) 124. doi:10.3390/atoms11100124.

[135] M. Kircher, F. Trinter, S. Grundmann, I. Vela-Perez, S. Brennecke, N. Eicke, J. Rist, S. Eckart, S. Houamer, O. Chuluunbaatar, Yu. V. Popov, I. P. Volobuev, K. Bagschik, M. N. Piancastelli, M. Lein, T. Jahnke, M. S. Sch¨offler, R. D¨orner, Kinematically complete experimental study of Compton scattering at helium atoms near the threshold, Nature Physics 16 (7) (2020) 756–760. doi:10.1038/s41567-020-0880-2.

[136] O. Chuluunbaatar, S. Houamer, Yu. V. Popov, I. Volobuev, M. Kircher, R. D¨orner, Compton ionization of atoms as a method of dynamical spectroscopy, Journal of Quantitative Spectroscopy and Radiative Transfer 272 (2021) 107820. doi:10.1016/j.jqsrt.2021.107820.

[137] M. Kircher, F. Trinter, S. Grundmann, G. Kastirke, M. Weller, I. Vela-Perez, A. Khan, C. Janke, M. Waitz, S. Zeller, T. Mletzko, D. Kirchner, V. Honkim¨aki, S. Houamer, O. Chuluunbaatar, Yu. V. Popov, I. P. Volobuev, M. S. Sch¨offler, L. Ph. H. Schmidt, T. Jahnke, R. D¨orner, Ion and electron momentum distributions from single and double ionization of helium induced by Compton scattering, Physical Review Letters 128 (5) (2022) 053001. doi: 10.1103/physrevlett.128.053001.

[138] S. A. Zaytsev, V. A. Knyr, Yu. V. Popov, A. Lahmam-Bennani, Application of the J-matrix method to Faddeev-Merkuriev equations for (e, 2e) reactions: Beyond pseudostates, Physical Review A 75 (2) (2007) 022718. doi:10.1103/physreva.75.022718.

[139] X. M. Tong, C. D. Lin, Empirical formula for static field ionization rates of atoms and molecules by lasers in the barrier-suppression regime, Journal of Physics B: Atomic, Molecular and Optical Physics 38 (15) (2005) 2593–2600. doi:10.1088/0953-4075/38/15/001.

[140] K. Patkowski, V. ˇSpirko, K. Szalewicz, On the elusive twelfth vibrational state of beryllium dimer, Science 326 (5958) (2009) 1382–1384. doi:10.1126/science.1181017.

[141] A. V. Mitin, Ab initio calculations of weakly bonded He2 and Be2 molecules by MRCI method with pseudo-natural molecular orbitals, International Journal of Quantum Chemistry 111 (11) (2010) 2560–2567. doi:10.1002/qua.22691.

[142] J. Koput, The ground-state potential energy function of a beryllium dimer determined using the single-reference coupled-cluster approach, Physical Chemistry Chemical Physics 13 (45) (2011) 20311. doi:10.1039/c1cp22417d.

[143] A. V. Mitin, Unusual chemical bonding in the beryllium dimer and its twelve vibrational levels, Chemical Physics Letters 682 (2017) 30–33. doi:10.1016/j.cplett.2017.05.071.

[144] V. V. Meshkov, A. V. Stolyarov, M. C. Heaven, C. Haugen, R. J. LeRoy, Direct-potential-fit analyses yield improved empirical potentials for the ground X1Σ+g state of Be2, The Journal of Chemical Physics 140 (6) (2014) 064315. doi:10.1063/1.4864355.

[145] M. Lesiuk, M. Przybytek, J. G. Balcerzak, M. Musial, R. Moszynski, Ab initio potential energy curve for the ground state of beryllium dimer, Journal of Chemical Theory and Computation 15 (4) (2019) 2470–2480. doi:10.1021/acs.jctc.8b00845.

[146] S. G. Porsev, A. Derevianko, High-accuracy calculations of dipole, quadrupole, and octupole electric dynamic polarizabilities and van der waals coefficients C6, C8, and C10 for alkaline-earth dimers, Journal of Experimental and Theoretical Physics 102 (2) (2006) 195205. doi:10.1134/s1063776106020014.

[147] X. W. Sheng, X. Y. Kuang, P. Li, K. T. Tang, Analyzing and modeling the interaction potential of the ground-state beryllium dimer, Physical Review A 88 (2) (2013) 022517. doi:10.1103/physreva.88.022517.

[148] A. V. Mitin, A. A. Gusev, G. Chuluunbaatar, O. Chuluunbaatar, S. I. Vinitsky, V. L. Derbov, H. L. Luong, Dual nature of chemical bond and vibration-rotation spectrum of the Be2 molecule in the ground X1Σ+g state, Chemical and Materials Sciences: Research Findings 2 (2025) 100–133. doi:10.9734/bpi/cmsrf/v2/4945.

[149] J. M. Merritt, V. E. Bondybey, M. C. Heaven, Beryllium dimer-caught in the act of bonding, Science 324 (5934) (2009) 1548–1551. doi:10.1126/science.1174326.

[150] A. A. Gusev, O. Chuluunbaatar, S. I. Vinitsky, A. G. Abrashkevich, KANTBP 3.0: New version of a program for computing energy levels, reflection and transmission matrices, and corresponding wave functions in the coupled-channel adiabatic approach, Computer Physics Communications 185 (12) (2014) 3341–3343. doi:10.1016/j.cpc.2014.08.002.

[151] V. I. Kukulin, V. M. Krasnopol’sky, J. Hor´aˇcek, Theory of Resonances, Springer Netherlands, 1989. doi:10.1007/978-94-015-7817-2.

[152] A. J. F. Siegert, On the derivation of the dispersion formula for nuclear reactions, Physical Review 56 (8) (1939) 750–752. doi:10.1103/physrev.56.750.

[153] M. L. Goldberger, K. M. Watson, Collision Theory, John Wiley, New York, 1964.

[154] K. A. Walsh, Beryllium Chemistry and Processing, ASM International, 2009. doi:10.31399/asm.tb.bcp.9781627082983.

[155] A. Allouche, C. Linsmeier, Quantum study of tungsten interaction with beryllium (0001), Journal

of Physics: Conference Series 117 (2008) 012002. doi:10.1088/1742-6596/117/1/012002.

[156] A. Lasa, D. Dasgupta, M. J. Baldwin, M. A. Cusentino, P. Hatton, D. Perez, B. P. Uberuaga, L. Yang, B. D. Wirth, Assessment of the literature about Be-W mixed material layer formation in the fusion reactor environment, Materials Research Express 11 (3) (2024) 032002. doi:10.1088/2053-1591/ad2c3c.

[157] V. M. Shabaev, I. I. Tupitsyn, V. A. Yerokhin, G. Plunien, G. Soff, Dual kinetic balance approach to basis-set expansions for the Dirac equation, Physical Review Letters 93 (13) (2004) 130405. doi:10.1103/physrevlett.93.130405.

[158] W. Schwarz, H. Wallmeier, Basis set expansions of relativistic molecular wave equations, Molecular Physics 46 (5) (1982) 1045–1061. doi:10.1080/00268978200101771.

[159] W. Kutzelnigg, Basis set expansion of the Dirac operator without variational collapse, International Journal of Quantum Chemistry 25 (1) (1984) 107–129. doi:10.1002/qua.560250112.

[160] S. P. Goldman, Variational representation of the Dirac-Coulomb Hamiltonian with no spurious roots, Physical Review A 31 (6) (1985) 3541–3549. doi:10.1103/physreva.31.3541.

[161] J. D. Talman, Minimax principle for the Dirac equation, Physical Review Letters 57 (9) (1986) 1091–1094. doi:10.1103/physrevlett.57.1091.

[162] A. Kolakowska, J. D. Talman, K. Aashamar, Minimax variational approach to the relativistic two-electron problem, Physical Review A 53 (1) (1996) 168–177. doi:10.1103/physreva.53.168.

[163] A. Kolakowska, Application of the minimax principle to the Dirac-Coulomb problem, Journalof Physics B: Atomic, Molecular and Optical Physics 29 (20) (1996) 4515–4527. doi:10.1088/0953-4075/29/20/010.

[164] L. LaJohn, J. D. Talman, Variational solution of the single-particle Dirac equation in the field of two nuclei using relativistically adapted slater basis functions, Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta) 99 (5) (1998) 351–356. doi:10.1007/s002140050346.

[165] G. Chuluunbaatar, Computational schemes for solving quantum-mechanical problems, Candidate thesis, Peoples’ Friendship University of Russia, Moscow, (in Russian) (2022).

[166] J. L. Zhou, A. L. Tits, User’s guide for FFSQP Version 3.5: A Fortran code for solving constrained nonlinear minimax optimization problems, generating iterates satisfying all inequality and linear constraints (1992).

[167] L. Adolphs, H. Daneshmand, A. Lucchi, T. Hofmann, Local saddle point optimization: A curvature exploitation approach, Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics 89 (2019) 486–495.

[168] W. Johnson, K. Cheng, M. Chen, Accurate relativistic calculations including qed contributions for few-electron systems, Relativistic Electronic Structure Theory - Part 2. Applications 14 (2004) 120–187. doi:10.1016/s1380-7323(04)80030-x.

[169] O. Chuluunbaatar, B. B. Joulakian, G. Chuluunbaatar, J. Buˇsa Jr., G. O. Koshcheev, Accurate calculations for the Dirac electron in the field of two-center Coulomb field: Application to heavy ions, Chemical Physics Letters 784 (2021) 139099. doi:10.1016/j.cplett.2021.139099.

[170] I. I. Tupitsyn, D. V. Mironova, Relativistic calculations of ground states of single-electron diatomic molecular ions, Optics and Spectroscopy 117 (3) (2014) 351–357. doi:10.1134/s0030400x14090252.

[171] V. D. Lakhno, I. V. Amirkhanov, A. V. Volokhova, E. V. Zemlyanaya, I. V. Puzynin, T. P. Puzynina, V. S. Rikhvitskii, M. V. Bashashin, Dynamic model of the polaron for studying electron hydration, Physics of Particles and Nuclei 54 (5) (2023) 869–883. doi:10.1134/

s1063779623050167.

[172] A. Friesen, Y. Kalinovsky, A. Khmelev, Mesons in nonlocal model with four-dimensional separable kernel, The European Physical Journal A 62 (1) (2026) article number 6. doi:10.1140/epja/s10050-025-01772-6.

[173] I. V. Barashenkov, D. E. Pelinovsky, E. V. Zemlyanaya, Vibrations and oscillatory instabilities of gap solitons, Physical Review Letters 80 (23) (1998) 5117–5120. doi:10.1103/physrevlett.80.5117.

[174] I. V. Barashenkov, E. V. Zemlyanaya, Oscillatory instabilities of gap solitons: a numerical study, Computer Physics Communications 126 (1–2) (2000) 22–27. doi:10.1016/s0010-4655(99)00241-6.

[175] A. N. Vystavkin, Yu. F. Drachevskii, V. P. Koshelets, I. L. Serpuchenko, First observation of static bound states of fluxons in long Josephson junctions with inhomogeneities, Soviet Journal of Low Temperature Physics 14 (6) (1988) 357–358. doi:10.1063/10.0031962.

[176] A. Abrashkevich, I. V. Puzynin, CANM, a program for numerical solution of a system of nonlinear equations using the continuous analog of Newton’s method, Computer Physics Communications 156 (2) (2004) 154–170. doi:10.1016/s0010-4655(03)00461-2.