Arlind, J., Bordemann, M., Hoppe, J., and Lee, C.. “Goldfish geodesics and Hamiltonian reduction of matrix dynamics.” Lett. Math. Phys. 84: 89–98. (2008).CrossRefGoogle Scholar

Bihun, O. and Calogero, F.. “Solvable many-body models of goldfish type with one-, two- and three-body forces.” SIGMA 9(059) (18 pages). (2013). http://arxiv.org/abs/1310.2335.Google Scholar

Bihun, O. and Calogero, F.. “Novel solvable many-body problems.” J. Nonlinear Math. Phys. 23: 190–212. (2016). DOI: 10.1080/14029251.2016.1161260.CrossRefGoogle Scholar

Bihun, O. and Calogero, F.. “A new solvable many-body problem of goldfish type.” J. Nonlinear Math. Phys. 23: 28–46. (2016); arXiv:13749 [math-ph].Google Scholar

Bihun, O. and Calogero, F.. “Generations of monic polynomials such that the coefficients of each polynomial of the next generation coincide with the zeros of a polynomial of the current generation, and new solvable many-body problems.” Lett. Math. Phys. 106(7), 1011–1031. (2016). DOI: 10.1007/s11005–016–0836–8.CrossRefGoogle Scholar

Bihun, O. and Calogero, F.. “Generations of solvable discrete-time dynamical systems,” J. Math. Phys. 58: 052701 (2017). DOI: 10.1063/1.4928959.CrossRefGoogle Scholar

Bihun, O. and Calogero, F.. “Time-dependent polynomials with one double root, and related new solvable systems of nonlinear evolution equations,” Qual. Theory Dyn. Syst. (submitted November 2017).CrossRefGoogle Scholar

Bihun, O. and Calogero, F.. “Zeros of time-dependent superpositions of orthogonal polynomials, and related new solvable systems of nonlinear evolution equations.” (in preparation).Google Scholar

Bihun, O. and Fulghesu, D.. “Polynomials whose coefficients coincide with their zeros.” Aequationes mathematicae 92(3), 453–470 (2018). DOI: 10.1007/s000100–018–0546–7. arXiv:1705.02057v1[Math.CA]5May2017.CrossRefGoogle Scholar

Bruschi, M. and Calogero, F.. “Novel solvable variants of the goldfish many-body model.” J. Math. Phys. 47: 022703 (2006) (25 pages).Google Scholar

Bruschi, M. and Calogero, F.. “Goldfishing: a new solvable many-body problem.” J. Math. Phys. 47: 042901:1–35. (2006).Google Scholar

Bruschi, M. and Calogero, F.. “A convenient expression of the time-derivative of arbitrary order k, of the zero zn(t) of a time-dependent polynomial pN(z; t) of arbitrary degree N in z, and solvable dynamical systems.” J. Nonlinear Math. Phys. 23: 474–485. (2016).CrossRefGoogle Scholar

Bruschi, M., Calogero, F., and Leyvraz, F.. “A large class of solvable discrete-time many-body problems.” J. Math. Phys. 55: 082703 (9 pages) (2014). DOI: http://dx.doi.org/10.1063/1.4891760.CrossRefGoogle Scholar

Bruschi, M. and Calogero, F.. “A convenient expression of the time-derivative of arbitrary order k, of the zero zn(t) of a time-dependent polynomial pN(z; t) of arbitrary degree N in z, and solvable dynamical systems.” J. Nonlinear Math. Phys. 23: 474–485. (2016).CrossRefGoogle Scholar

Calogero, F.. “Solution of the one-dimensional N-body problem with quadratic and inversely-quadratic pair potentials.” J. Math. Phys. 12: 419–436. Erratum: 1996. Journal of Mathematical Physics 37: J. Math. Phys. 37, 3646. (1996).CrossRefGoogle Scholar

Calogero, F.. “Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations, and related ‘solvable’ many body problems.” Nuovo Cimento 43B: 177–241. (1978).CrossRefGoogle Scholar

Calogero, F.. “Why are certain nonlinear PDEs both widely applicable and integrable?” In What is integrability?, edited by Zakharov, V. E., 1–62. Heidelberg: Springer. (1991).Google Scholar

Calogero, F.. “Integrable nonlinear evolution equations and dynamical systems in multidimensions.” in: KdV’95, Proceedings of the International Symposium held in Amsterdam, The Netherlands, April 23–26,1995 (edited by M. Hazewinkel, H.W. Capel and E.M. de Jager).CrossRefGoogle Scholar

Calogero, F.. “A class of integrable Hamiltonian systems whose solutions are (perhaps) all completely periodic.” J. Math. Phys. 38: 5711–5719. (1997).CrossRefGoogle Scholar

Calogero, F.. Classical many-body problems amenable to exact treatments, Lecture Notes in Physics Monograph m66. Heidelberg: Springer. (2001). (749 pages).CrossRefGoogle Scholar

Calogero, F.. “The ‘neatest’ many-body problem amenable to exact treatments (a ‘goldfish’?).” Physica D: 152–153, 78–84. (2001).CrossRefGoogle Scholar

Calogero, F.. “On the quantization of two other nonlinear harmonic oscillators.” 2003. Phys. Lett. A 319: 240–245. (2003).CrossRefGoogle Scholar

Calogero, F.. “Solution of the goldfish N-body problem in the plane with (only) nearest-neighbor coupling constants all equal to minus one half.” J. Nonlinear Math. Phys. 11: 102–112. (2004).CrossRefGoogle Scholar

Calogero, F.. “On the quantization of yet another two nonlinear harmonic oscillators.” J. Nonlinear Math. Phys. 11: 1–6. (2004).CrossRefGoogle Scholar

Calogero, F.. “A technique to identify solvable dynamical systems, and a solvable generalization of the goldfish many-body problem.” J. Math. Phys. 45: 2266–2279. (2004).CrossRefGoogle Scholar

Calogero, F.. “A technique to identify solvable dynamical systems, and another solvable extension of the goldfish many-body problem.” J. Math. Phys. 45: 4661– 4678. (2004).Google Scholar

Calogero, F.. Isochronous systems. Oxford University Press: Oxford. (2008) (264 pages) (marginally updated paperback version, 2012).CrossRefGoogle Scholar

Calogero, F.. “Two new solvable dynamical systems of goldfish type.” J. Nonlinear Math. Phys. 17: 397–414. (2010). DOI: 10.1142/S1402925110000970.CrossRefGoogle Scholar

Calogero, F.. “Discrete-time goldfishing.” SIGMA 7: 082, 35 pages (2011).CrossRefGoogle Scholar

Calogero, F.. “A new goldfish model.” Theor. Math. Phys. 167: 714–724. (2011).CrossRefGoogle Scholar

Calogero, F.. “On a technique to identify solvable discrete-time many-body problems.” Theor. Math. Phys. 172: 1052–1072. (2012).CrossRefGoogle Scholar

Calogero, F.. “New solvable many-body model of goldfish type.” J. Nonlinear Math. Phys. 19: 1250006. (19 pages) (2012). DOI: 10.1142/S1402925112500064.Google Scholar

Calogero, F.. “Another new solvable model of goldfish type.” SIGMA 8:046 (17 pages) (2012).Google Scholar

Calogero, F.. “New solvable variants of the goldfish many-body problem”, Studies Appl. Math. 137(1), 123–139 (2016); DOI: 10.1111/sapm.12096.CrossRefGoogle Scholar

Calogero, F.. “Nonlinear differential algorithm to compute all the zeros of a generic polynomial.” J. Math. Phys. 57: 083508 (4 pages) (2016). DOI: 10.1063/1.4960821.Google Scholar

Calogero, F.. “Comment on ‘Nonlinear differential algorithm to compute all the zeros of a generic polynomial’ [J. Math. Phys. 57: 083508 (2016)].” J. Math. Phys. 57: 104101 (4 pages) (2016).Google Scholar

Calogero, F.. “A solvable N-body problem of goldfish type featuring N2 arbitrary coupling constants.” J. Nonlinear Math. Phys. 23: 300–305 (2016). DOI: 10.1080/14029251.2016.1175823.CrossRefGoogle Scholar

Calogero, F.. “Novel isochronous N-body problems featuring N arbitrary rational coupling constants.” J. Math. Phys. 57: 072901 (2016). DOI:10.1063/1.4954851.CrossRefGoogle Scholar

Calogero, F.. “Three new classes of solvable N-body problems of goldfish type with many arbitrary coupling constants.” Symmetry 8: 53. (2016). DOI:10.3390/sym8070053.CrossRefGoogle Scholar

Calogero, F.. “Yet another class of new solvable N-body problems of goldfish type.” Qualit. Theory Dyn. Syst. 16: (3), 561–577 (2017). DOI: 10.1007/s12346–016–0215-y.CrossRefGoogle Scholar

Calogero, F.. “New solvable dynamical systems.” J. Nonlinear Math. Phys. 23: 486– 493. (2016).Google Scholar

Calogero, F.. “Integrable Hamiltonian N-body problems of goldfish type featuring N arbitrary functions.” J. Nonlinear Math. Phys. 24(1): 1–6. (2017).CrossRefGoogle Scholar

Calogero, F.. “New C-integrable and S-integrable systems of nonlinear partial differential equation.” J. Nonlinear Math. Phys. 24(1): 142–148. (2017).CrossRefGoogle Scholar

Calogero, F.. “Novel differential algorithm to evaluate all the zeros of any generic polynomial.” J. Nonlinear Math. Phys. 24: 469–472. (2017).CrossRefGoogle Scholar

Calogero, F.. “Zeros of entire functions and related systems of infinitely many nonlinearly coupled evolution equations.” Theor. Math. Phys. (in press).Google Scholar

Calogero, F. and Degasperis, A.. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations. North Holland: Amsterdam. (1982). pp.514.Google Scholar

Calogero, F. and Degasperis, A.. “On the quantization of Newton-equivalent Hamiltonians.” Amer. J. Phys. 72: 1202–1203. (2004).CrossRefGoogle Scholar

Calogero, F. and De Lillo, S., “The Eckhaus PDE iyt + yxx + 2 (|y|2)x y +|y|4 y = 0.” Inverse Problems 3: 633–681. (1987); Erratum 4: 571. (1988).CrossRefGoogle Scholar

Calogero, F. and Françoise, J.P.. “Hamiltonian character of the motion of the zeros of a polynomial whose coefficients oscillate over time.” J. Phys. A: Math. Gen. 30: 211–218. (1997).CrossRefGoogle Scholar

Calogero, F. and Françoise, J.P.. “A novel solvable many-body problem with elliptic interactions.” Int. Math. Res. Notices 15: 775–786. (2000).CrossRefGoogle Scholar

Calogero, F., Françoise, J.P., and Sommacal, M.. “Solvable nonlinear evolution PDEs in multidimensional space involving trigonometric functions.” J. Phys. A: Math. Theor. 40: F363–F368. (2007).CrossRefGoogle Scholar

Calogero, F., Françoise, J.P., and Sommacal, M.. “Solvable nonlinear evolution PDEs in multidimensional space involving elliptic functions.” J. Phys. A: Math. Theor. 40: F705–F711. (2007).CrossRefGoogle Scholar

Calogero, F. and Gómez-Ullate, D.. “Two novel classes of solvable many-body problems of goldfish type with constraints.” J. Phys. A: Math. Theor. 40: 5335–5353. (2007).CrossRefGoogle Scholar

Calogero, F. and Gómez-Ullate, D.. “Asymptotically isochronous systems.” J. Nonlinear Math. Phys. 15: 410–426. (2008).CrossRefGoogle Scholar

Calogero, F., Gómez-Ullate, D., Santini, P. M., and Sommacal, M.. “The transition from regular to irregular motions, explained as travel on Riemann surfaces.” J. Phys. A: Math. Gen. 38: 8873–8896. (2005). arXiv:nlin.SI/0507024 v1 13 Jul 2005.CrossRefGoogle Scholar

Calogero, F., Gómez-Ullate, D., Santini, P. M., and Sommacal, M.. “Towards a theory of chaos explained as travel on Riemann surfaces.” J. Phys. A.: Math. Theor. 42: 015205 (26 pages) (2009). DOI: 10.1088/1751–8113/42/1/015205.CrossRefGoogle Scholar

Calogero, F. and Graffi, S.. “On the quantization of a nonlinear Hamiltonian oscillator.” Phys. Lett. A 313: 356–362. (2003).CrossRefGoogle Scholar

Calogero, F. and Iona, S.. “Novel solvable extensions of the goldfish many-body model.” J. Math. Phys. 46: 103515 (2005).CrossRefGoogle Scholar

Calogero, F. and Langmann, E.. “Goldfishing by gauge theory.” J. Math. Phys. 47, 082702: 1–23 (2006).CrossRefGoogle Scholar

Calogero, F. and Leyvraz, F.. “On a class of Hamiltonians with (classical) isochronous motions and (quantal) equispaced spectra.” J. Phys. A: Math. Gen. 39: 11803–11824. (2006).CrossRefGoogle Scholar

Calogero, F. and Leyvraz, F.. “Isochronous extension of the Hamiltonian describing free motion in the Poincaré half-plane: classical and quantal treatments.” J. Math. Phys. 48, 092903: 1–15. (2007).CrossRefGoogle Scholar

Calogero, F. and Leyvraz, F.. “On a new technique to manufacture isochronous Hamiltonian systems: classical and quantal treatments.” J. Nonlinear Math. Phys. 14: 505–529. (2007).Google Scholar

Calogero, F. and Leyvraz, F.. “General technique to produce isochronous Hamiltonians.” J. Phys. A: Math. Theor. 40: 12931–12944. (2007).CrossRefGoogle Scholar

Calogero, F. and Leyvraz, F.. “New solvable discrete-time many-body problem featuring several arbitrary parameters.” J. Math. Phys. 53: 082702 (19 pages) (2012).CrossRefGoogle Scholar

Calogero, F. and Leyvraz, F.. “New solvable discrete-time many-body problem featuring several arbitrary parameters. II.” J. Math. Phys. 54: 102702 (15 pages) (2013). DOI: http://dx.doi.org/10.1063/1.4822419.CrossRefGoogle Scholar

Calogero, F. and Leyvraz, F.. “A nonautonomous yet solvable discrete-time N-body problem.” J. Phys. A: Math. Theor. 47, 105203 (10 pages) (2014).CrossRefGoogle Scholar

Calogero, F. and Leyvraz, F.. “The peculiar (monic) polynomials the zeros of which equal their coefficients.” J. Nonlinear Math. Phys. 24: 541–551 (2017). DOI:10.1080/14029251.2017.1375690.Google Scholar

Calogero, F. and Leyvraz, F.. “Examples of Hamiltonians isochronous in configuration space only, and their quantization.” J. Math. Phys. (in press).Google Scholar

Calogero, F. and Sommacal, M.. “Solvable nonlinear evolution PDEs in multidimensional space.” SIGMA 2: 088 (2006) (17 pages), nlin.SI/0612019.Google Scholar

Calogero, F. and Yi, Ge. “A new class of solvable many-body problems.” SIGMA 8: 066 (29 pages) (2012).Google Scholar

Di Scala, A. J. and Maciá, O.. “Finiteness of Ulam polynomials.” La matematica e le sue applicazioni, Quaderni del Dipartimento di Matematica, Politecnico Torino, 11: July 2008. Turin, Italy; arXiv:00904.0133v1[math.AG]1April2009.Google Scholar

Erdélyi, A. (editor). Higher transcendental functions, vol. II. McGraw-Hill: New York. (1953).Google Scholar

Gallavotti, G. and Marchioro, C.. “On the computation of an integral.” J. Math. Anal. Appl. 44: 661–675. (1973).CrossRefGoogle Scholar

Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M.. “Method for Solving the Korteweg-deVries Equation.” Phys. Rev. Lett. 19: 1095–1097 (1967). DOI: 10.1103/PhysRevLett.19.1095.CrossRefGoogle Scholar

Gómez-Ullate, D., Santini, P. M., Sommacal, M., and Calogero, F.. “Understanding complex dynamics by means of an associated Riemann surface.” Physica D. 241: 1291–1305: (2012).CrossRefGoogle Scholar

Gómez-Ullate, D. and Sommacal, M.. “Periods of the goldfish many-body problem.” J. Nonlinear Math. Phys. 12, Suppl. 1: 351–362. (2005).CrossRefGoogle Scholar

Grinevich, P. G. and Santini, P. M.. “Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve μ2 = νn −1, n ∈ Z: Ergodicity, isochrony and fractals.” Physica D 232: 22–32. (2007).CrossRefGoogle Scholar

Guillot, A.. “The Painleve’ property for quasi homogeneous systems and a many-body problem in the plane.” Comm. Math. Phys. 256: 181–194. (2005).CrossRefGoogle Scholar

Jairuk, U., Yoo-Kong, S., and Tanasittikosol, M.. “On the Lagrangian structure of Calogero’s goldfish model.” arXiv:1409.7168 [nlin.SI] (2014).CrossRefGoogle Scholar

Kundu, A.. “Landau–Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations.” J. Math. Phys. 25: 3433–3438. (1984).CrossRefGoogle Scholar

Leyvraz, F.. “An approach for obtaining integrable Hamiltonians from Poisson-commuting polynomial families.” J. Math. Phys. 58: 072902 (2017).CrossRefGoogle Scholar

Mathematica, Wolfram Research, Inc., Version 11.0. Champaign, IL, USA. (2016).Google Scholar

Marikhin, V. G.. “Electron-levels dynamics in the presence of impurities and the Ruijsenaars-Schneider model.” JETP Lett. 77:44 (2003). DOI:10.1134/1.1561980.126CrossRefGoogle Scholar

Moser, J.. “Three integrable Hamiltonian systems connected with isospectral deformations.” Adv. Math. 16: 197–220. (1975).CrossRefGoogle Scholar

Nucci, M. C.. “Calogero’s ‘goldfish’ is indeed a school of free particles.” J. Phys. A: Math. Gen. 37: 11391–11400. (2004).CrossRefGoogle Scholar

Olshanetsky, M. A. and Perelomov, A. M.. “Explicit solution of the Calogero model in the classical case and geodetic flows on symmetric spaces of zero curvature.” Lett. Nuovo Cimento 16: 333–339. (1976).CrossRefGoogle Scholar

Salmon, G.. Lessons Introductory to the Higher Modern Algebra. Cambridge University Press: Cambridge, U. (1885).Google Scholar

Sabatier, P. C. (ed.). Inverse Methods in Action. Springer: Berlin. (1990).CrossRefGoogle Scholar

Sommacal, M.. “The transition from regular to irregular motion, explained as travel on Riemann surfaces.” PhD thesis, SISSA, Trieste. (2005).Google Scholar

Stein, P. R.. “On polynomial equations with coefficients equal to their roots.” A. Math. Monthly 73(3): 272–274. (1966).CrossRefGoogle Scholar

Suris, Y. B.. “Time discretization of F. Calogero’s ‘Goldfish’. J. Nonlinear Math. Phys. 12, Suppl. 1: 633–647. (2005).CrossRefGoogle Scholar

Ulam, S. M.. A collection of mathematical problems. Interscience: New York. 1960 (see pages 30–31).Google Scholar

Zakharov, V. E.. “On the dressing method.” in [88], pp. 602–623 (see p. 622).CrossRefGoogle Scholar