# Classical and Quantum Integrability in Laplacian Growth

## Main Article Content

## Abstract

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: <https://esmed.org/MRA/qpr/article/view/1200>. Date accessed: 05 dec. 2022.
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## References

[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,

2005.

[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,

1981.

[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,

1998.

[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.

[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,

2005.

[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,

1981.

[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,

1998.

[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.