Classical and Quantum Integrability in Laplacian Growth

Main Article Content

Eldad Bettelheim


We provide a review of Laplacian growth geared at making a link between this problem and quantum integrability. The purpose is to put to use the link between quantum integrability and conformal feld theory in order to provide a theory for the fractal structure of Laplacian growth clusters. The paper only provides the framework for this conjectural connection, while leaving the realization of this program for later work.

Article Details

How to Cite
BETTELHEIM, Eldad. Classical and Quantum Integrability in Laplacian Growth. Quarterly Physics Review, [S.l.], v. 3, n. 2, july 2017. ISSN 2572-701X. Available at: <>. Date accessed: 05 dec. 2022.


[1] A. Levermann and I. Procaccia. Algorithm for parallel Laplacian growth by iterated conformal maps. Phys. Rev. E., 69(3):031401, 2004.

[2] O. Praud and H. L. Swinney. Fractal dimension and unscreened angles measured for radial viscous fingering. Phys. Rev. E, 72(1):011406, 2005.

[3] T. C. Halsey, B. Duplantier, and K. Honda. Multifractal Dimensions and Their Fluctuations in Diffusion-Limited Aggregation. Physical Review Letters, 1997.

[4] B. Duplantier. Conformally Invariant Fractals and Potential Theory. Physical Review Letters, 2000.

[5] B. Duplantier. Conformal Fractal Geometry and Boundary Quantum Gravity. math-ph/0303034, 2003.

[6] M. Mineev-Weinstein, P. B. Wiegmann, and A. Zabrodin. Integrable structure of interface dynamics.
Phys. Rev. Lett., 84:5106–5109, 2000.

[7] I. Krichever, M. Mineev-Weinstein, P. Wiegmann, and A. Zabrodin. Laplacian growth and
Whitham equations of soliton theory. Physica D Nonlinear Phenomena, 198:1–28, 2004.

[8] P. B. Wiegmann and A. Zabrodin. Conformal Maps and Integrable Hierarchies. Communications
in Mathematical Physics, 213:523–538, 2000.

[9] S. Richardson. Hele-shaw flows with free boundary produced by the injection of fluid into a
narrow channel. Journal of Fluid Mechanics, 56:609, 1972.

[10] R. Teodorescu, P. Wiegmann, and A. Zabrodin. Unstable Fingering Patterns of Hele-Shaw Flows
as a Dispersionless Limit of the Kortweg de Vries Hierarchy. Phys. Rev. Lett., 95(4):044502,

[11] V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov. Integrable structure of conformal
field theory, quantum KdV theory and Thermodynamic Bethe Ansatz. Communications in
Mathematical Physics, 1996.

[12] V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov. Integrable Structure of Conformal
Field Theory II. Q-operator and DDV equation. Communications in Mathematical Physics,
190:247–278, 1997.

[13] V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov. Integrable Structure of Conformal
Field Theory III. The Yang-Baxter Relation. Communications in Mathematical Physics,
200:297–324, 1999.

[14] V. A. Fateev and S. L. Lukyanov. Poisson-Lie Groups and Classical W-Algebras. International
Journal of Modern Physics A, 7:853–876, 1992.

[15] V. A. Fateev and S. L. Lukyanov. Vertex Operators and Representations of Quantum Universal
Enveloping Algebras. International Journal of Modern Physics A, 7:1325–1359, 1992.

[16] A. N. Varchenko and P. I. Etingof. Why the boundary of a round drop becomes a curve of order four. American mathematical society, Providence, Rhode Island, 1991.

[17] B Gustafsson. Acta Aapplicandae Mathematicae, 1:209–240, 1983.

[18] K. Ueno and K. Takasaki. Toda lattice hierarchy. In K. Okamoto, editor, Group representations
and systems of differential equations, Amsterdam, 1984. North-Holland.

[19] A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov, and A. Orlov. Matrix models of 2-D
gravity and Toda theory. Nucl. Phys., B357:565–618, 1991.

[20] H. F. Baker. Abelian Functions. Cambridge University Press, Cambridge, UK, 1897.

[21] I. M. Krichever. Integration of nonlinear equations by the methods of algebraic geometry. Func.
Anal. Appl., 11:12–26, 1977.

[22] B. A. Dubrovin. Theta functions and non-linear equations. Russian Math. Surveys, 36:2:11–92,

[23] R. Carroll. Remarks on the Whitham equations. arXiv:solv-int/9511009, 1995.

[24] G. B. Whitham. Linear and Nonlinear Waves. Wiley, New York, 1974.

[25] G. B. Whitham. Nonlinear dispersive waves. SIAM Journal Appl. Math, 14(4):956–958, 1966.

[26] G. B. Whitham. A general approach to linear and non-linear dispersive waves using a Lagrangian.
J. Fluid. Mech., 22:273–283, 1965.

[27] F. Fucito, A. Gamba, M. Martellini, and O. Ragnisco. Non-Linear WKB Analysis of the String
Equation. International Journal of Modern Physics B, 6:2123–2147, 1992.

[28] I. M. Krichever. Method of averaging two-dimensional integrable equations. Functional Analysis
and its Applications, 22(3):200–213, 1988.

[29] A. Das. Integrable Models. World Scientific, Singapore, 1989.

[30] I. M. Krichever. Spectral theory of two-dimensional periodic operators. Russian Math. Surveys,
44:2:145–225, 1989.

[31] H. Flaschka, M. G. Forest, and D. W. McLaughlin. Multiphase averaging and the inverse spectral
solution of KdV. Comm. Pure. Appl. Math., 33:739–784, 1980.

[32] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura. Method for Solving the KortewegdeVries
Equation. Physical Review Letters, 1967.

[33] H. Flaschka and D. W. McLaughlin. Canonically Conjugate Variables for the Korteweg-de Vries
Equation and the Toda Lattice with Periodic Boundary Conditions. Progress of Theoretical
Physics, 1976.

[34] Ludwig Faddeev and Leon Takhtajan. Hamiltonian methods in the theory of solitons. Springer
Science & Business Media, 2007.

[35] E. K. Sklyanin. J. Soviet Math., 31:3417–3431, 1985.

[36] R. Sasaki and I. Yamanaka. Field theoretical construction of an infinite set of quantum commuting operators related with soliton equations. Communications in Mathematical Physics, 108:691–704, 1987.

[37] A. B. Zamolodchikov. On the thermodynamic Bethe ansatz equations for reflectionless ADE
scattering theories. Physics Letters B, 253:391–394, 1991.

[38] A. B. Zamolodchikov. Thermodynamic Bethe ansatz in relativistic models: Scaling 3-state potts
and Lee-Yang models. Nuclear Physics B, 342:695–720, 1990.

[39] A. B. Zamolodchikov. Integrals of Motion in Scaling 3-STATE Potts Model Field Theory.
International Journal of Modern Physics A, 3:743–750, 1988.

[40] Feodor A Smirnov. Form factors in completely integrable models of quantum field theory. volume
14 of Advanced Series in Mathematical Physics. World Scientific, 1992.

[41] D. Bernard and A. Leclair. Residual quantum symmetries of the Restricted sine-Gordon theories.
Nuclear Physics B, 1990.

[42] F. A. Smirnov. Reductions of the sine-Gordon model as a perturbation of minimal models of
conformal field theory. Nuclear Physics B, 337:156–180, June 1990.

[43] O. Babelon, D. Bernard, and F. A. Smirnov. Quantization of Solitons and the Restricted SineGordon
Model. Communications in Mathematical Physics, 1996.

[44] O. Babelon, D. Bernard, and F. A. Smirnov. Null-Vectors in Integrable Field Theory. Communications
in Mathematical Physics, 186:601–648, 1997.

[45] O. Babelon, D. Bernard, and F. A. Smirnov. Form factors, KdV and Deformed Hyperelliptic
Curves. Nuclear Physics B Proceedings Supplements, 1997.

[46] F. A. Smirnov. Quasi-classical Study of Form Factors in Finite Volume. arXiv:hep-th/9802132,

[47] Harry Kesten. Hitting probabilities of random walks on Zd
. Stochastic Processes and their
Applications, 25:165–184, 1987.

[48] T. A. Witten and L. M. Sander. Diffusion limited aggregation. Phys. Rev., B27:5686, 1983.

[49] Hershel M Farkas and Irwin Kra. Riemann surfaces. Springer, 1992.

[50] Gerald Teschl. Jacobi operators and completely integrable nonlinear lattices, chapter A. Number
72. American Mathematical Soc., 2000.