Kanehisa Takasaki

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All published works
Action Title Year Authors
+ PDF Chat Generalized ILW hierarchy: solutions and limit to extended lattice GD hierarchy 2023 Kanehisa Takasaki
+ Extended lattice Gelfand–Dickey hierarchy 2022 Kanehisa Takasaki
+ Generalized ILW hierarchy: Solutions and limit to extended lattice GD hierarchy 2022 Kanehisa Takasaki
+ Dressing operators in equivariant Gromov–Witten theory of CP1 2021 Kanehisa Takasaki
+ PDF Chat Cubic Hodge integrals and integrable hierarchies of Volterra type 2021 Kanehisa Takasaki
+ Integrable structures of specialized hypergeometric tau functions 2020 Kanehisa Takasaki
+ PDF Chat Three-Partition Hodge Integrals and the Topological Vertex 2019 Toshio Nakatsu
Kanehisa Takasaki
+ Differential Fay identities and auxiliary linear problem of integrable hierarchies 2019 Kanehisa Takasaki
+ Integrable structure of melting crystal model with external potentials 2019 Toshio Nakatsu
Kanehisa Takasaki
+ 4D limit of melting crystal model and its integrable structure 2018 Kanehisa Takasaki
+ PDF Chat Hurwitz numbers and integrable hierarchy of Volterra type 2018 Kanehisa Takasaki
+ Initial Value Problem for the Toda Lattice Hierarchy 2018 Kanehisa Takasaki
+ PDF Chat Toda hierarchies and their applications 2018 Kanehisa Takasaki
+ PDF Chat Call for papers: special issue on fifty years of the Toda lattice 2017 Vladimir V. Bazhanov
Patrick Dorey
Kenji Kajiwara
Kanehisa Takasaki
+ Quantum curve and 4D limit of melting crystal model 2017 Kanehisa Takasaki
+ PDF Chat q-Difference Kac-Schwarz Operators in Topological String Theory 2017 Kanehisa Takasaki
Toshio Nakatsu
+ Pfaffian structures and certain solutions to BKP hierarchies II. Multiple integrals 2016 A. Yu. Orlov
Takahiro Shiota
Kanehisa Takasaki
+ PDF Chat Open string amplitudes of closed topological vertex 2015 Kanehisa Takasaki
Toshio Nakatsu
+ PDF Chat Orbifold melting crystal models and reductions of Toda hierarchy 2015 Kanehisa Takasaki
+ Remarks on partition functions of topological string theory on generalized conifolds (Algebraic Combinatorics related to Young diagram and statistical physics) 2014 Kanehisa Takasaki
+ PDF Chat Generalized Ablowitz–Ladik hierarchy in topological string theory 2014 Kanehisa Takasaki
+ PDF Chat Modified melting crystal model and Ablowitz-Ladik hierarchy 2014 Kanehisa Takasaki
+ PDF Chat A modified melting crystal model and the Ablowitz–Ladik hierarchy 2013 Kanehisa Takasaki
+ Remarks on partition functions of topological string theory on generalized conifolds 2013 Kanehisa Takasaki
+ PDF Chat Old and New Reductions of Dispersionless Toda Hierarchy 2012 Kanehisa Takasaki
+ PDF Chat An ħ-dependent formulation of the Kadomtsev-Petviashvili hierarchy 2012 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat An hbar-expansion of the Toda hierarchy: a recursive construction of solutions 2012 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat An $${\hbar}$$ -expansion of the Toda hierarchy 2012 Kanehisa Takasaki
Takashi Takebe
+ Pfaffian structures and certain solutions to BKP hierarchies I. Sums over partitions 2012 A. Yu. Orlov
Takahiro Shiota
Kanehisa Takasaki
+ Integrable structure of modified melting crystal model 2012 Kanehisa Takasaki
+ PDF Chat An $\hbar$-dependent formulation of the Kadomtsev - Petviashvili hierarchy 2012 Kanehisa Takasaki
Kanehisa Takasaki
Takashi Takebe
Takashi Takebe
+ Generalized string equations for double Hurwitz numbers 2011 Kanehisa Takasaki
+ PDF Chat Thermodynamic limit of random partitions and dispersionless Toda hierarchy 2011 Kanehisa Takasaki
Toshio Nakatsu
+ PDF Chat hbar-Dependent KP hierarchy 2011 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat Toda Tau Functions with Quantum Torus Symmetries 2011 Kanehisa Takasaki
+ Toda tau functions with quantum torus symmetries 2011 Kanehisa Takasaki
+ PDF Chat Two extensions of 1D Toda hierarchy 2010 Kanehisa Takasaki
+ PDF Chat KP AND TODA TAU FUNCTIONS IN BETHE ANSATZ 2010 Kanehisa Takasaki
+ PDF Chat Non-degenerate solutions of the universal Whitham hierarchy 2010 Kanehisa Takasaki
Takashi Takebe
Lee Peng Teo
+ PDF Chat Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy 2009 Kanehisa Takasaki
+ PDF Chat Integrable structure of melting crystal model with two <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>q</mml:mi></mml:math>-parameters 2009 Kanehisa Takasaki
+ hbar-expansion of KP hierarchy: Recursive construction of solutions 2009 Takashi Takebe
Kanehisa Takasaki
+ PDF Chat LĂśwner equations, Hirota equations and reductions of the universal Whitham hierarchy 2008 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat Extended 5d Seiberg–Witten theory and melting crystal 2008 Toshio Nakatsu
Yui Noma
Kanehisa Takasaki
+ PDF Chat Melting Crystal, Quantum Torus and Toda Hierarchy 2008 Toshio Nakatsu
Kanehisa Takasaki
+ PDF Chat INTEGRABLE STRUCTURE OF 5d$\mathcal{N} = 1$ SUPERSYMMETRIC YANG-MILLS AND MELTING CRYSTAL 2008 Toshio Nakatsu
Yui Noma
Kanehisa Takasaki
+ Integrable structure of melting crystal model with external potentials 2008 Toshio Nakatsu
Kanehisa Takasaki
+ PDF Chat Universal Whitham hierarchy, dispersionless Hirota equations and multicomponent KP hierarchy 2007 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat Hamiltonian Structure of PI Hierarchy 2007 Kanehisa Takasaki
+ PDF Chat Hamiltonian Structure of PI Hierarchy 2007 Kanehisa Takasaki
+ Differential Fay identities and auxiliary linear problem of integrable hiearchies 2007 Kanehisa Takasaki
+ PDF Chat LĂśwner Equations and Dispersionless Hierarchies 2006 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat Dispersionless Hirota Equations of Two-Component BKP Hierarchy 2006 Kanehisa Takasaki
+ Radial Lowner equation and dispersionless cmKP hierarchy 2006 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat Elliptic Spectral Parameter and Infinite-Dimensional Grassmann Variety 2006 Kanehisa Takasaki
+ PDF Chat Explicit solutions of the classical Calogero and Sutherland systems for any root system 2006 Ryu Sasaki
Kanehisa Takasaki
+ Dispersionless Hirota equations of two-component BKP hierarchy 2006 Kanehisa Takasaki
+ Radial Loewner equation and dispersionless cmKP hierarchy 2006 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat q-Analogue of Modified KP Hierarchy and its Quasi-Classical Limit 2005 Kanehisa Takasaki
+ PDF Chat Free fermion and Seiberg–Witten differential in random plane partitions 2005 Takashi Maeda
Toshio Nakatsu
Kanehisa Takasaki
Takeshi Tamakoshi
+ PDF Chat Five-Dimensional Supersymmetric Yang-Mills Theories and Random Plane Partitions 2005 Takashi Maeda
Toshio Nakatsu
Kanehisa Takasaki
Takeshi Tamakoshi
+ Tyurin parameters of commuting pairs and infinite dimensional Grassmann manifold 2005 Kanehisa Takasaki
+ Tyurin parameters of commuting pairs and infinite dimensional Grassmann manifold 2005 Kanehisa Takasaki
+ Loewner equations and dispersionless hierarchies 2005 Kanehisa Takasaki
Takashi Takebe
+ Integrable systems whose spectral curves are the graph of a function 2004 Kanehisa Takasaki
+ PDF Chat Landau–Lifshitz Hierarchy and Infinite-Dimensional Grassmann Variety 2004 Kanehisa Takasaki
+ PDF Chat Spectral Curve, Darboux Coordinates and Hamiltonian Structure of Periodic Dressing Chains 2003 Kanehisa Takasaki
+ PDF Chat Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero–Gaudin system 2003 Kanehisa Takasaki
+ AKNS hierarchy coupled with Tyurin parameters 2003 Kanehisa Takasaki
+ Tyurin parameters and elliptic analogue of nonlinear Schr\"odinger hierarchy 2003 Kanehisa Takasaki
+ Tyurin parameters and elliptic analogue of nonlinear SchrĂśdinger hierarchy 2003 Kanehisa Takasaki
+ Elliptic spectral parameter and infinite dimensional Grassmann variety 2003 Kanehisa Takasaki
+ PDF Chat An integrable system on the moduli space of rational functions and its variants 2002 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat Hierarchy of (2 + 1)-Dimensional Nonlinear SchrĂśdinger Equation, Self-Dual Yang-Mills Equation, and Toroidal Lie Algebras 2002 Saburo Kakei
Tomonori Ikeda
Kanehisa Takasaki
+ Integrable systems whose spectral curve is the graph of a function 2002 Kanehisa Takasaki
+ PDF Chat Quantum Inozemtsev model, quasi-exact solvability and 𝒩-fold supersymmetry 2001 Ryu Sasaki
Kanehisa Takasaki
+ PDF Chat Hyperelliptic integrable systems on K3 and rational surfaces 2001 Kanehisa Takasaki
+ Painleve-Calogero correspondence (Analysis of Painleve equations) 2001 Kanehisa Takasaki
+ PDF Chat Anti-self-dual Yang–Mills equations on noncommutative space–time 2001 Kanehisa Takasaki
+ PDF Chat Painlevé–Calogero correspondence revisited 2001 Kanehisa Takasaki
+ None 2001 Takeshi Ikeda
Kanehisa Takasaki
+ PDF Chat WHITHAM DEFORMATIONS OF SEIBERG–WITTEN CURVES FOR CLASSICAL GAUGE GROUPS 2000 Kanehisa Takasaki
+ Toroidal Lie algebras and Bogoyavlensky's 2+1-dimensional equation 2000 Tomonori Ikeda
Kanehisa Takasaki
+ None 2000 Kanehisa Takasaki
+ PDF Chat Elliptic Calogero–Moser systems and isomonodromic deformations 1999 Kanehisa Takasaki
+ PDF Chat Calogero-Moser Models. IV: Limits to Toda Theory 1999 S. Pratik Khastgir
Ryu Sasaki
Kanehisa Takasaki
+ PDF Chat Quasiclassical limit of KP hierarchy,W-symmetries, and free fermions 1999 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat INTEGRABLE HIERARCHIES AND CONTACT TERMS IN u-PLANE INTEGRALS OF TOPOLOGICALLY TWISTED SUPERSYMMETRIC GAUGE THEORIES 1999 Kanehisa Takasaki
+ PDF Chat Calogero-Moser Models. II 1999 J. Andrew Bordner
Ryu Sasaki
Kanehisa Takasaki
+ PDF Chat Whitham Deformations and Tau Functions in<i>N</i>= 2 Supersymmetric Gauge Theories 1999 Kanehisa Takasaki
+ PDF Chat Calogero-Moser Models II: Symmetries and Foldings 1998 Andrew J. Bordner
Ryu Sasaki
Kanehisa Takasaki
+ PDF Chat Dispersionless hierarchies, Hamilton-Jacobi theory and twistor correspondences 1998 Partha Guha
Kanehisa Takasaki
+ None 1998 Kanehisa Takasaki
+ Calogero-Moser Models II: Symmetries and Foldings 1998 Andrew J. Bordner
Ryu Sasaki
Kanehisa Takasaki
+ PDF Chat Spectral curves and Whitham equations in isomonodromic problems of Schlesinger type 1998 Kanehisa Takasaki
+ None 1998 Kanehisa Takasaki
+ Dual Isomonodromic Problems and Whitham Equations 1997 Kanehisa Takasaki
+ Spectral Curves and Whitham Equations in Isomonodromic Problems of Schlesinger Type 1997 Kanehisa Takasaki
+ Gaudin Model, KZ Equation, and Isomonodromic Problem on Torus 1997 Kanehisa Takasaki
+ PDF Chat ISOMONODROMIC DEFORMATIONS AND SUPERSYMMETRIC GAUGE THEORIES 1996 Kanehisa Takasaki
Toshio Nakatsu
+ PDF Chat Toda lattice hierarchy and generalized string equations 1996 Kanehisa Takasaki
+ PDF Chat WHITHAM-TODA HIERARCHY AND N=2 SUPERSYMMETRIC YANG-MILLS THEORY 1996 Toshio Nakatsu
Kanehisa Takasaki
+ Integrable System and $N=2$ Supersymmetric Yang-Mills Theory 1996 Toshio Nakatsu
Kanehisa Takasaki
+ PDF Chat INTEGRABLE HIERARCHIES AND DISPERSIONLESS LIMIT 1995 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat Symmetries and tau function of higher dimensional dispersionless integrable hierarchies 1995 Kanehisa Takasaki
+ PDF Chat Quantum and classical aspects of deformed c = 1 strings 1995 Toshio Nakatsu
Kanehisa Takasaki
S. Tsujimaru
+ PDF Chat Dispersionless Toda hierarchy and two-dimensional string theory 1995 Kanehisa Takasaki
+ PDF Chat Nonabelian KP hierarchy with Moyal algebraic coefficients 1994 Kanehisa Takasaki
+ Dressing operator approach to Moyal algebraic deformation of selfdual gravity 1994 Kanehisa Takasaki
+ PDF Chat Dressing operator approach to Moyal algebraic deformation of selfdual gravity 1994 Kanehisa Takasaki
+ PDF Chat Integrable hierarchy underlying topological Landau-Ginzburg models of D-type 1993 Kanehisa Takasaki
+ PDF Chat Quasi-classical limit of BKP hierarchy andW-infinity symmetries 1993 Kanehisa Takasaki
+ PDF Chat Quasi-classical limit of Toda hierarchy andW-infinity symmetries 1993 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat Volume-preserving diffeomorphisms in integrable deformations of self-dual gravity 1992 Kanehisa Takasaki
+ PDF Chat SDIFF(2) KP HIERARCHY 1992 Kanehisa Takasaki
Takashi Takebe
+ W algebra, twistor, and nonlinear integrable systems 1992 Kanehisa Takasaki
+ Quasi-classical limit of KP hierarchy, W-symmetries and free fermions 1992 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat SDiff(2) Toda equation — Hierarchy, Tau function, and symmetries 1991 Kanehisa Takasaki
Takashi Takebe
+ PDF Chat Structure and duality of D-modules related to KP hierarchy 1991 Toshikazu Miyajima
Atsushi Nakayashiki
Kanehisa Takasaki
+ Area-Preserving Diffeomorphisms and Nonlinear Integrable Systems 1991 Kanehisa Takasaki
+ Hidden symmetries of integrable systems in Yang-Mills theory and Kähler geometry 1991 Kanehisa Takasaki
+ Symmetries of hyper-Kähler (or Poisson gauge field) hierarchy 1990 Kanehisa Takasaki
+ PDF Chat Hierarchy structure in integrable systems of gauge fields and underlying Lie algebras 1990 Kanehisa Takasaki
+ Integrable Systems in Gauge Theory, Kahler Geometry and Super KP Hierarchy - Symmetries and Algebraic Point of View 1990 Kanehisa Takasaki
+ An infinite number of hidden variables in hyper-Kähler metrics 1989 Kanehisa Takasaki
+ Symmetries of the super KP hierarchy 1989 Kanehisa Takasaki
+ GEOMETRY OF UNIVERSAL GRASSMANN MANIFOLD FROM ALGEBRAIC POINT OF VIEW 1989 Kanehisa Takasaki
+ Issues of Multi-Dimensional Integrable Systems 1988 Kanehisa Takasaki
+ PDF Chat Aspects of Integrability In Self-Dual Einstein Metrics and Related Equations 1986 Kanehisa Takasaki
+ Differential Equations and Grassmann Manifolds : from Prof. Sato's lectures(Theta Functions and Related Topics) 1986 Kanehisa Takasaki
+ Conformally Self-Dual Metrics and Integrability(Hyperfunctions and Differential Equations) 1986 Kanehisa Takasaki
+ On Riemann-Hilbert transformations 1985 Kanehisa Takasaki
+ PDF Chat A new approach to the self-dual Yang-Mills equations 1984 Kanehisa Takasaki
+ PDF Chat A class of solutions to the self-dual Yang-Mills equations 1983 Kanehisa Takasaki
+ PDF Chat Singular cauchy problems for a class of weakly hyperbolic differential operators 1982 Kanehisa Takasaki
+ PDF Chat Singular Cauchy problems for a class of weakly hyperbolic differential operators 1981 Kanehisa Takasaki
Common Coauthors
Commonly Cited References
Action Title Year Authors # of times referenced
+ PDF Chat INTEGRABLE HIERARCHIES AND DISPERSIONLESS LIMIT 1995 Kanehisa Takasaki
Takashi Takebe
37
+ PDF Chat The dispersionless Lax equations and topological minimal models 1992 I. M. Krichever
26
+ TRANSFORMATION GROUPS FOR SOLITON EQUATIONS 1994 Etsurō Date
23
+ PDF Chat Loop groups and equations of KdV type 1985 Graeme Segal
George Wilson
23
+ PDF Chat Melting Crystal, Quantum Torus and Toda Hierarchy 2008 Toshio Nakatsu
Kanehisa Takasaki
20
+ PDF Chat Quantum Calabi-Yau and Classical Crystals 2007 AndreÄ­ Okounkov
Nicolai Reshetikhin
Cumrun Vafa
20
+ PDF Chat Solitons and Infinite Dimensional Lie Algebras 1983 Michio Jimbo
Tetsuji Miwa
19
+ Soliton Equations as Dynamical Systems on Infinite Dimensional Grassmann Manifold 1983 Mikio Sato
18
+ PDF Chat SDiff(2) Toda equation — Hierarchy, Tau function, and symmetries 1991 Kanehisa Takasaki
Takashi Takebe
17
+ Representation theoretical meaning of the initial value problem for the Toda lattice hierarchy: I 1991 Takashi Takebe
16
+ PDF Chat Five-Dimensional Supersymmetric Yang-Mills Theories and Random Plane Partitions 2005 Takashi Maeda
Toshio Nakatsu
Kanehisa Takasaki
Takeshi Tamakoshi
15
+ Additional symmetries for integrable equations and conformal algebra representation 1986 A. Yu. Orlov
E. I. Schulman
15
+ PDF Chat Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory 1994 Nathan Seiberg
Edward Witten
14
+ A method for solving the dispersionless KP equation and its exact solutions 1988 Yuji Kodama
14
+ PDF Chat The Topological Vertex 2004 Mina Aganagic
Albrecht Klemm
Marcos Mari�o
Cumrun Vafa
14
+ Initial Value Problem for the Toda Lattice Hierarchy 2018 Kanehisa Takasaki
14
+ Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram 2003 AndreÄ­ Okounkov
Nicolai Reshetikhin
14
+ PDF Chat SDIFF(2) KP HIERARCHY 1992 Kanehisa Takasaki
Takashi Takebe
12
+ PDF Chat Seiberg-Witten Theory and Random Partitions 2007 Nikita Nekrasov
AndreÄ­ Okounkov
12
+ An infinite hierarchy of conservation laws and nonlinear superposition principles for self-dual Einstein spaces 1985 Charles P. Boyer
Jerzy Plebański
12
+ PDF Chat Quasi-classical limit of Toda hierarchy andW-infinity symmetries 1993 Kanehisa Takasaki
Takashi Takebe
11
+ PDF Chat Toda lattice hierarchy and generalized string equations 1996 Kanehisa Takasaki
11
+ PDF Chat Toda equations for Hurwitz numbers 2000 AndreÄ­ Okounkov
11
+ Integrable structure of melting crystal model with external potentials 2019 Toshio Nakatsu
Kanehisa Takasaki
11
+ The τ‐function of the universal whitham hierarchy, matrix models and topological field theories 1994 I. M. Krichever
11
+ Transformation groups for soliton equations 1982 Etsurō Date
Michio Jimbo
Masaki Kashiwara
Tetsuji Miwa
11
+ PDF Chat Seiberg-Witten Prepotential from Instanton Counting 2003 Nikita Nekrasov
11
+ Gromov–Witten theory, Hurwitz theory, and completed cycles 2006 Andreĭ Okounkov
Rahul Pandharipande
10
+ PDF Chat Toda lattice hierarchy and the topological description of the c = 1 string theory 1994 Tohru Eguchi
Hiroaki Kanno
10
+ PDF Chat Quantum and classical aspects of deformed c = 1 strings 1995 Toshio Nakatsu
Kanehisa Takasaki
S. Tsujimaru
10
+ PDF Chat Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds 2010 Jim Bryan
Benjamin Young
10
+ PDF Chat Some solutions of complex Einstein equations 1975 Jerzy Plebański
9
+ PDF Chat Dual isomonodromic deformations and moment maps to loop algebras 1994 J. Harnad
9
+ None 2001 A. Zabrodin
9
+ PDF Chat A matrix integral solution to two-dimensionalW p-gravity 1992 Mark Adler
Pierre van Moerbeke
9
+ PDF Chat Extended Seiberg-Witten theory and integrable hierarchy 2007 A. Marshakov
Nikita Nekrasov
9
+ PDF Chat A method for solving the dispersionless KP hierarchy and its exact solutions. II 1989 Yuji Kodama
John Gibbons
9
+ Extended conformal symmetries in real heavens 1990 Q‐Han Park
9
+ PDF Chat A new approach to the self-dual Yang-Mills equations 1984 Kanehisa Takasaki
9
+ The structure of theW ∞ algebra 1990 Ioannis Bakas
9
+ Group Representations and Systems of Differential Equations 1984 9
+ PDF Chat Analytic Functions and Integrable Hierarchies–Characterization of Tau Functions 2003 Lee-Peng Teo
8
+ PDF Chat Integrable hierarchies and the mirror model of local <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi mathvariant="double-struck">C</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">P</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> 2011 Andrea Brini
Guido Carlet
Paolo Rossi
8
+ Symmetries of hyper-Kähler (or Poisson gauge field) hierarchy 1990 Kanehisa Takasaki
8
+ Stable bundles and integrable systems 1987 Nigel Hitchin
8
+ PDF Chat Darboux coordinates and Liouville-Arnold integration in loop algebras 1993 Malcolm R. Adams
J. Harnad
Jacques Hurtubise
8
+ PDF Chat The equivariant Gromov–Witten theory of P<sup>1</sup> 2006 Andreĭ Okounkov
Rahul Pandharipande
8
+ Transformation theory for anti-self-dual equations and the Riemann-Hilbert problem 1982 Kohei Ueno
Yoshimasa Nakamura
8
+ PDF Chat Dispersionless Toda hierarchy and two-dimensional string theory 1995 Kanehisa Takasaki
8
+ Complete integrability of the Kadomtsev-Petviashvili equation 1984 Motohico Mulase
8