Professor Nalini Joshi

FAA

F07 - Carslaw Building
The University of Sydney

Telephone 9351 2172
Fax 9351 4534

Website Personal web page

Biographical details

Nalini Joshi is a Georgina Sweet Australian Laureate Fellow in mathematics at the University of Sydney. She develops mathematical methods to study solutions of integrable systems, which arise as universal models in physics, such as the Painleve equations.

She obtained her PhD from Princeton University and has held the Chair of Applied Mathematics at the University of Sydney since 2002.

Research interests

Nalini's research interests lie in non-linear differential and difference equations, with a particular focus on asymptotic methods. Currently, Nalini is creating a geometric framework to reveal properties of critical solutions of nonlinear models that reflect universal structures in physical models.

Nonlinear integrable systems arise in a wide variety of fields, including ion channel transport in biology, random matrix theory, string theory and orthogonal polynomial theory. Many such systems appear as universal mathematical models in certain contexts. In the case of one continuous dimension, the systems of interest are called the Painlevé equations, while in two or more dimensions, the equations are soliton equations.

Their discrete versions have been at the leading edge of developments in recent times. However, the emphasis in the field has been focused on the identification of integrable discrete versions of the Painlevé equations and soliton equations. These equations provide an insight into physical models, such as random matrix theory. However, our knowledge of their solutions remains extremely limited.

One major problem in this field lies in the fact that the mathematical tools used widely to analyse differential equations are not available for analysing discrete equations. One way (possibly the only way) to overcome this problem is to analyse solutions as functions of initial values. Duistermaat and Joshi (J. J. Duistermaat and N. Joshi, “Okamoto's Space for the First Painlevé Equation in Boutroux Coordinates”, Archive for Rational Mechanics and Analysis, 202 (2011), 707785) showed how to analyse solutions of the first Painlevé equation in the space of its initial values. This approach connected the analytic properties of solutions to properties of algebraic curves that they parametrize, for the first time in the field of integrable systems.

Specific research areas: Integrable systems, the Painlevé equations, geometric asymptotics, nonlinear dynamics, slow-fast systems, nonlinear waves, chaos, perturbation theory

Teaching and supervision

See Nalini's course on integrable systems. Nalini is currently supervising several PhD projects in this area, which propose to extend the mathematical toolbox to describe solutions of non-linear differential and discrete equations.

Awards and honours

Fellow of the Australian Academy of Science (elected 2008)

International links

United Kingdom

(The University of Leeds) I collaborate with Professor Frank W. Nijhoff, who is the Professor of Mathematical Physics. Currently, I am co-writing a book with him and Professor Jarmo Hietarinta from the University of Turku in Finland.

United Kingdom

(The University of Leeds) Visiting Professor

Selected grants

2013

  • Critical solutions of nonlinear systems; Joshi N; Australian Research Council (ARC)/Discovery Projects (DP).

2012

  • Geometric construction of critical solutions of nonlinear systems; Joshi N; Australian Research Council (ARC)/Australian Laureate Fellowships (FL).

2011

  • Geometry and Analysis of Discrete Integrable Systems; Dullin H, Joshi N; Australian Research Council (ARC)/Discovery Projects (DP).

2010

  • Random and integrable models in mathematical physics (RIMMP); Grava T, Klein C, Joshi N; European Commission (Belgium)/Marie Curie Action: International Research Staff Exchange (IRSES).

2009

  • Integrable Lattice Equations; Joshi N; Australian Research Council (ARC)/Discovery Projects (DP).

2008

  • Integrable Lattice Equations; Joshi N; University of Sydney/Bridging Support.

2006

  • New Directions in Non-linear Mathematical Asymptotics; Joshi N; Australian Research Council (ARC)/Discovery Projects (DP).

2005

  • Integrable functional and delay differential equations; Cosgrove C, Joshi N; Australian Research Council (ARC)/Discovery Projects (DP).

2003

  • Mathematical biosciences network; Joshi N; Australian Research Council (ARC)/Special Research Initiatives (SRI).
  • Global Behaviour of Integrable Complex Systems; Joshi N; Australian Research Council (ARC)/Discovery Projects (DP).

2002

  • Singularities And Classifications Of Integrable Systems; Joshi N, Cosgrove C; Australian Research Council (ARC)/Discovery Projects (DP).

1997

  • Asymptotics and integrability of nonlinear differential and difference equations; Joshi N; Australian Research Council (ARC)/Australian Senior Research Fellowship.

Selected publications

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Book Chapters

  • Kruskal, M., Joshi, N., Halburd, R. (2004). Analytical And Asymptotic Methods For Nonlinear Singularity Analysis: A Review And Extensions Of Tests For The Painlevé Property. In Basil Grammaticos, Yvette Kosmann-Schwarzbach, K. M. Tamizhmani (Eds.), Integrability of Nonlinear Systems, (pp. 175-208). Berlin: Springer.
  • Joshi, N. (2003). Hunting mathematical butterflies. In Ball, Akhmediev (Eds.), Nonlinear Dynamics: from Lasers to Butterflies, (pp. 77-114). USA: World Scientific Publishing.

Journals

  • Howes, P., Joshi, N. (2014). Global Asymptotics of the Second Painlevé Equation in Okamoto’s Space. Constructive Approximation, 39(1), 11-41. [More Information]
  • Atkinson, J., Joshi, N. (2013). Singular-Boundary Reductions of Type-Q ABS Equations. International Mathematics Research Notices, 7, 1451-1481. [More Information]
  • Joshi, N., Shi, Y. (2012). Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem. II. Hypergeometric solutions. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 468(2146), 3247-3264. [More Information]
  • Atkinson, J., Joshi, N. (2012). The Schwarzian variable associated with discrete KdV-type equations. Nonlinearity, 25(6), 1851-1866. [More Information]
  • Joshi, N., Shi, Y. (2011). Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions. Proceedings of the Royal Society of London. Mathematical, Physical and Engineering Sciences, 467, 3443-3468. [More Information]
  • Duistermaat, J., Joshi, N. (2011). Okamoto's Space for the First Painlevé Equation in Boutroux Coordinates. Archive for Rational Mechanics and Analysis, 202(3), 707-785. [More Information]
  • Wood, L., Vu, T., Bower, M., Brown, N., Skalicky, J., Donovan, D., Loch, B., Joshi, N., Bloom, W. (2011). Professional development for teaching in higher education. International Journal of Mathematical Education in Science and Technology, 42(7), 997-1009. [More Information]
  • Butler, S., Joshi, N. (2010). An inverse scattering transform for the lattice potential KdV equation. Inverse Problems, 26(115012), 1-28. [More Information]
  • Kassotakis, P., Joshi, N. (2010). Integrable Non-QRT Mappings of the Plane. Letters in Mathematical Physics, 91(1), 71-81. [More Information]
  • Ramani, A., Grammaticos, B., Joshi, N. (2010). Second-degree discrete Painlevé equations conceal first-degree ones. Journal of Physics A: Mathematical and Theoretical, 43(17), 1-9. [More Information]
  • Joshi, N., Spicer, P. (2009). Direct "Delay" Reductions of the Toda Hierarchy. Journal of the Physical Society of Japan, 78(9), 094006-1-094006-5. [More Information]
  • Joshi, N. (2009). Direct 'delay' reductions of the Toda equation. Journal of Physics A: Mathematical and Theoretical, 42(2), 1-8. [More Information]
  • Joshi, N., Morrison, T. (2009). Existence and uniqueness of Tronquee solutions of the fourth-order Jimbo-Miwa second Painleve equation. Proceedings of the American Mathematical Society, 137(6), 2005-2014.
  • Joshi, N., Lafortune, S., Ramani, A. (2009). Hirota bilinear formalism and ultra-discrete singularity analysis. Nonlinearity, 22(4), 871-887.
  • Joshi, N., Kitaev, A., Treharne, P. (2009). On the linearization of the first and second Painleve equations. Journal of Physics A: Mathematical and Theoretical, 42(5), 055208-1-055208-18.
  • Joshi, N., Morrison, T. (2008). New exact solutions of spatially and temporally varying reaction-diffusion equations. Analysis and Applications, 6(4), 371-381.
  • Field, C., Joshi, N., Nijhoff, F. (2008). q-difference equations of kd V type and Chazy-type second degree difference equations. Journal of Physics A: Mathematical and Theoretical, 41, 332005-332018.
  • Hay, M., Hietarinta, J., Joshi, N., Nijhoff, F. (2007). A Lax pair for a lattice modified KdV equation, reductions to q-Painlevé equations and associated Lax pairs. Journal of Physics A: Mathematical and Theoretical, 40(2), F61-F73. [More Information]
  • Joshi, N., Kitaev, A., Treharne, P. (2007). On the linearization of the Painleve' III-VI equations and reductions of the three-wave resonant system. Journal of Mathematical Physics, 48(10), 103512-1-103512-42.
  • Joshi, N., Ormerod, C. (2007). The general theory of linear difference equations over the max-plus semi-ring. Studies in Applied Mathematics, 118(1), 85-97.
  • Clarkson, P., Joshi, N., Mazzocco, M. (2007). The Lax Pair for the mKdV Hierarchy. Seminaires et Congres, 14(Theories Asymptotiques et Equations de Painleve), 53-64.
  • Ramani, A., Joshi, N., Grammaticos, B., Tamizhmani, T. (2006). Deconstructing an integrable lattice equation. Journal of Physics A: Mathematical and General, 39(8), L145-L149.
  • Joshi, N., Grammaticos, B., Tamizhmani, T., Ramani, A. (2006). From integrable lattices to non-QRT mappings. Letters in Mathematical Physics, 78(1), 27-37. [More Information]
  • Joshi, N., Pickering, A. (2006). Generalized Halphen systems. Proceedings of the Royal Society of Edinburgh Section A (Mathematics), 136A (6), 1287-1301.
  • Joshi, N., Kajiwara, K., Mazzocco, M. (2006). Generating function associated with the Hankel determinant formula for the solutions of the Painlevé IV equation. Funkcialaj Ekvacioj, Serio Internacia, 49(3), 451-468.
  • Joshi, N., Lafortune, S. (2006). Integrable ultra-discrete equations and singularity analysis. Nonlinearity, 19(6), 1295-1312.
  • Gordoa, P., Joshi, N., Pickering, A. (2006). Second and fourth Painlevé hierarchies and Jimbo-Miwa linear problems. Journal of Mathematical Physics, 47(7), 073504-1-073504-16.
  • Gordoa, P., Joshi, N., Pickering, A. (2005). Backlund transformations for fourth Painleve hierarchies. Journal of Differential Equations, 217(1), 124-153. [More Information]
  • Joshi, N., Lafortune, S. (2005). How to detect integrability in cellular automata. Journal of Physics A: Mathematical and General, 38(28), L499-L504.
  • Joshi, N., Kitaev, A. (2005). The Dirichlet boundary value problem for real solutions of the first Painleve equation on segments in non-positive semi-axis. Journal fuer die Reine und Angewandte Mathematik: Crelle's journal, 583, 29-86.
  • Joshi, N., Kajiwara, K., Mazzocco, M. (2004). Generating Function Associated With The Determinant Formula For The Solutions Of The Painlevé II Equation. Asterisque, 297(2004), 67-78.
  • Joshi, N., Nijhoff, F., Ormerod, C. (2004). Lax pairs for ultra-discrete Painlevé cellular automata. Journal of Physics A: Mathematical and General, 37(2004), L559-L565.
  • Joshi, N. (2004). The second Painleve hierarchy and the stationary KdV hierarchy. Publications of the Research Institute for Mathematical Sciences, 40(2004), 1039-1061.
  • Gordoa, P., Joshi, N., Pickering, A. (2003). A new technique in nonlinear singularity analysis. Publications of the Research Institute for Mathematical Sciences, 39, 435-449.
  • Maruno, K., Ohta, Y., Joshi, N. (2003). Exact localized solutions of quintic discrete nonlinear Schrodinger equation. Physics Letters. Section A: General, Atomic and Solid State Physics, 311(2-3), 214-220.
  • Joshi, N., Mazzocco, M. (2003). Existence and uniqueness of tri-tronquee solutions of the second Painlevé hierarchy. Nonlinearity, 16(2), 427-439.
  • Clarkson, P., HONE, A., Joshi, N. (2003). Hierarchies of difference equations and Backlund transformations. Journal Of Nonlinear Mathematical Physics, 10(Suppl 2), 13-26.
  • Joshi, N. (2003). Tritronquee solutions of perturbed first painleve equations. Theoretical and Mathematical Physics, 11, 1515-1519.
  • HONE, A., Joshi, N., Kitaev, A. (2002). An entire function defined by a nonlinear recurrence relation. Journal of The London Mathematical Society, 66(2), 377-387.
  • CRESSWELL, C., Joshi, N. (2002). Consistent composition of Backlund transformations produces confined maps. Letters in Mathematical Physics, 61(1), 1-14.
  • Gordoa, P., Joshi, N., Pickering, I. (2001). Mappings preserving locations of movable poles: II. The third and fifth Painlevé equations. Nonlinearity, 14(3), 567-582.
  • Gordoa, P., Joshi, N., Pickering, I. (2001). On a Generalized 2 + 1 Dspersive Water Wave Hierarchy. Publications of the Research Institute for Mathematical Sciences, 37, 327-347.
  • Joshi, N., Kitaev, A. (2001). On Boutroux's Tritronquée Solutions of the First Painlevé Equation. Studies in Applied Mathematics, 107(3), 253-291.
  • Joshi, N. (2001). Regularizing the KdV equation near a blow-up surface. Theoretical and Mathematical Physics, 127(3), 744-750.
  • Gordoa, P., Joshi, N., Pickering, I. (2001). Truncation-type methods and Bäcklund transformations for ordinary differential equations: the third and fifth Painlevé equations. Glasgow Mathematical Journal, 43(A), 23-32.

Edited Journals

  • Joshi, N., Noumi, M., Sakai, H., Viallet, C. (2013). Journal of Nonlinear Mathematical Physics. Journal Of Nonlinear Mathematical Physics, 20(Supplement 1).

Conferences

  • Brown, N., Bower, M., Skalicky, J., Wood, L., Donovan, D., Loch, B., Bloom, W., Joshi, N. (2010). A professional development framework for teaching in higher education. 33rd Higher Education Research and Development Society of Australasia International Conference: HERDSA 2010 Reshaping Higher Education, Milperra: Higher Education Research and Development Society of Australasia.
  • Joshi, N. (2005). Asymptotics for Extended Cellular Automata. Recent Trends in Exponential Asymptotics, Kyoto: RIMS.

2014

  • Howes, P., Joshi, N. (2014). Global Asymptotics of the Second Painlevé Equation in Okamoto’s Space. Constructive Approximation, 39(1), 11-41. [More Information]

2013

  • Joshi, N., Noumi, M., Sakai, H., Viallet, C. (2013). Journal of Nonlinear Mathematical Physics. Journal Of Nonlinear Mathematical Physics, 20(Supplement 1).
  • Atkinson, J., Joshi, N. (2013). Singular-Boundary Reductions of Type-Q ABS Equations. International Mathematics Research Notices, 7, 1451-1481. [More Information]

2012

  • Joshi, N., Shi, Y. (2012). Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem. II. Hypergeometric solutions. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 468(2146), 3247-3264. [More Information]
  • Atkinson, J., Joshi, N. (2012). The Schwarzian variable associated with discrete KdV-type equations. Nonlinearity, 25(6), 1851-1866. [More Information]

2011

  • Joshi, N., Shi, Y. (2011). Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions. Proceedings of the Royal Society of London. Mathematical, Physical and Engineering Sciences, 467, 3443-3468. [More Information]
  • Duistermaat, J., Joshi, N. (2011). Okamoto's Space for the First Painlevé Equation in Boutroux Coordinates. Archive for Rational Mechanics and Analysis, 202(3), 707-785. [More Information]
  • Wood, L., Vu, T., Bower, M., Brown, N., Skalicky, J., Donovan, D., Loch, B., Joshi, N., Bloom, W. (2011). Professional development for teaching in higher education. International Journal of Mathematical Education in Science and Technology, 42(7), 997-1009. [More Information]

2010

  • Brown, N., Bower, M., Skalicky, J., Wood, L., Donovan, D., Loch, B., Bloom, W., Joshi, N. (2010). A professional development framework for teaching in higher education. 33rd Higher Education Research and Development Society of Australasia International Conference: HERDSA 2010 Reshaping Higher Education, Milperra: Higher Education Research and Development Society of Australasia.
  • Butler, S., Joshi, N. (2010). An inverse scattering transform for the lattice potential KdV equation. Inverse Problems, 26(115012), 1-28. [More Information]
  • Kassotakis, P., Joshi, N. (2010). Integrable Non-QRT Mappings of the Plane. Letters in Mathematical Physics, 91(1), 71-81. [More Information]
  • Ramani, A., Grammaticos, B., Joshi, N. (2010). Second-degree discrete Painlevé equations conceal first-degree ones. Journal of Physics A: Mathematical and Theoretical, 43(17), 1-9. [More Information]

2009

  • Joshi, N., Spicer, P. (2009). Direct "Delay" Reductions of the Toda Hierarchy. Journal of the Physical Society of Japan, 78(9), 094006-1-094006-5. [More Information]
  • Joshi, N. (2009). Direct 'delay' reductions of the Toda equation. Journal of Physics A: Mathematical and Theoretical, 42(2), 1-8. [More Information]
  • Joshi, N., Morrison, T. (2009). Existence and uniqueness of Tronquee solutions of the fourth-order Jimbo-Miwa second Painleve equation. Proceedings of the American Mathematical Society, 137(6), 2005-2014.
  • Joshi, N., Lafortune, S., Ramani, A. (2009). Hirota bilinear formalism and ultra-discrete singularity analysis. Nonlinearity, 22(4), 871-887.
  • Joshi, N., Kitaev, A., Treharne, P. (2009). On the linearization of the first and second Painleve equations. Journal of Physics A: Mathematical and Theoretical, 42(5), 055208-1-055208-18.

2008

  • Joshi, N., Morrison, T. (2008). New exact solutions of spatially and temporally varying reaction-diffusion equations. Analysis and Applications, 6(4), 371-381.
  • Field, C., Joshi, N., Nijhoff, F. (2008). q-difference equations of kd V type and Chazy-type second degree difference equations. Journal of Physics A: Mathematical and Theoretical, 41, 332005-332018.

2007

  • Hay, M., Hietarinta, J., Joshi, N., Nijhoff, F. (2007). A Lax pair for a lattice modified KdV equation, reductions to q-Painlevé equations and associated Lax pairs. Journal of Physics A: Mathematical and Theoretical, 40(2), F61-F73. [More Information]
  • Joshi, N., Kitaev, A., Treharne, P. (2007). On the linearization of the Painleve' III-VI equations and reductions of the three-wave resonant system. Journal of Mathematical Physics, 48(10), 103512-1-103512-42.
  • Joshi, N., Ormerod, C. (2007). The general theory of linear difference equations over the max-plus semi-ring. Studies in Applied Mathematics, 118(1), 85-97.
  • Clarkson, P., Joshi, N., Mazzocco, M. (2007). The Lax Pair for the mKdV Hierarchy. Seminaires et Congres, 14(Theories Asymptotiques et Equations de Painleve), 53-64.

2006

  • Ramani, A., Joshi, N., Grammaticos, B., Tamizhmani, T. (2006). Deconstructing an integrable lattice equation. Journal of Physics A: Mathematical and General, 39(8), L145-L149.
  • Joshi, N., Grammaticos, B., Tamizhmani, T., Ramani, A. (2006). From integrable lattices to non-QRT mappings. Letters in Mathematical Physics, 78(1), 27-37. [More Information]
  • Joshi, N., Pickering, A. (2006). Generalized Halphen systems. Proceedings of the Royal Society of Edinburgh Section A (Mathematics), 136A (6), 1287-1301.
  • Joshi, N., Kajiwara, K., Mazzocco, M. (2006). Generating function associated with the Hankel determinant formula for the solutions of the Painlevé IV equation. Funkcialaj Ekvacioj, Serio Internacia, 49(3), 451-468.
  • Joshi, N., Lafortune, S. (2006). Integrable ultra-discrete equations and singularity analysis. Nonlinearity, 19(6), 1295-1312.
  • Gordoa, P., Joshi, N., Pickering, A. (2006). Second and fourth Painlevé hierarchies and Jimbo-Miwa linear problems. Journal of Mathematical Physics, 47(7), 073504-1-073504-16.

2005

  • Joshi, N. (2005). Asymptotics for Extended Cellular Automata. Recent Trends in Exponential Asymptotics, Kyoto: RIMS.
  • Gordoa, P., Joshi, N., Pickering, A. (2005). Backlund transformations for fourth Painleve hierarchies. Journal of Differential Equations, 217(1), 124-153. [More Information]
  • Joshi, N., Lafortune, S. (2005). How to detect integrability in cellular automata. Journal of Physics A: Mathematical and General, 38(28), L499-L504.
  • Joshi, N., Kitaev, A. (2005). The Dirichlet boundary value problem for real solutions of the first Painleve equation on segments in non-positive semi-axis. Journal fuer die Reine und Angewandte Mathematik: Crelle's journal, 583, 29-86.

2004

  • Kruskal, M., Joshi, N., Halburd, R. (2004). Analytical And Asymptotic Methods For Nonlinear Singularity Analysis: A Review And Extensions Of Tests For The Painlevé Property. In Basil Grammaticos, Yvette Kosmann-Schwarzbach, K. M. Tamizhmani (Eds.), Integrability of Nonlinear Systems, (pp. 175-208). Berlin: Springer.
  • Joshi, N., Kajiwara, K., Mazzocco, M. (2004). Generating Function Associated With The Determinant Formula For The Solutions Of The Painlevé II Equation. Asterisque, 297(2004), 67-78.
  • Joshi, N., Nijhoff, F., Ormerod, C. (2004). Lax pairs for ultra-discrete Painlevé cellular automata. Journal of Physics A: Mathematical and General, 37(2004), L559-L565.
  • Joshi, N. (2004). The second Painleve hierarchy and the stationary KdV hierarchy. Publications of the Research Institute for Mathematical Sciences, 40(2004), 1039-1061.

2003

  • Gordoa, P., Joshi, N., Pickering, A. (2003). A new technique in nonlinear singularity analysis. Publications of the Research Institute for Mathematical Sciences, 39, 435-449.
  • Maruno, K., Ohta, Y., Joshi, N. (2003). Exact localized solutions of quintic discrete nonlinear Schrodinger equation. Physics Letters. Section A: General, Atomic and Solid State Physics, 311(2-3), 214-220.
  • Joshi, N., Mazzocco, M. (2003). Existence and uniqueness of tri-tronquee solutions of the second Painlevé hierarchy. Nonlinearity, 16(2), 427-439.
  • Clarkson, P., HONE, A., Joshi, N. (2003). Hierarchies of difference equations and Backlund transformations. Journal Of Nonlinear Mathematical Physics, 10(Suppl 2), 13-26.
  • Joshi, N. (2003). Hunting mathematical butterflies. In Ball, Akhmediev (Eds.), Nonlinear Dynamics: from Lasers to Butterflies, (pp. 77-114). USA: World Scientific Publishing.
  • Joshi, N. (2003). Tritronquee solutions of perturbed first painleve equations. Theoretical and Mathematical Physics, 11, 1515-1519.

2002

  • HONE, A., Joshi, N., Kitaev, A. (2002). An entire function defined by a nonlinear recurrence relation. Journal of The London Mathematical Society, 66(2), 377-387.
  • CRESSWELL, C., Joshi, N. (2002). Consistent composition of Backlund transformations produces confined maps. Letters in Mathematical Physics, 61(1), 1-14.

2001

  • Gordoa, P., Joshi, N., Pickering, I. (2001). Mappings preserving locations of movable poles: II. The third and fifth Painlevé equations. Nonlinearity, 14(3), 567-582.
  • Gordoa, P., Joshi, N., Pickering, I. (2001). On a Generalized 2 + 1 Dspersive Water Wave Hierarchy. Publications of the Research Institute for Mathematical Sciences, 37, 327-347.
  • Joshi, N., Kitaev, A. (2001). On Boutroux's Tritronquée Solutions of the First Painlevé Equation. Studies in Applied Mathematics, 107(3), 253-291.
  • Joshi, N. (2001). Regularizing the KdV equation near a blow-up surface. Theoretical and Mathematical Physics, 127(3), 744-750.
  • Gordoa, P., Joshi, N., Pickering, I. (2001). Truncation-type methods and Bäcklund transformations for ordinary differential equations: the third and fifth Painlevé equations. Glasgow Mathematical Journal, 43(A), 23-32.

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