June 2019, 11(2): 167-185. doi: 10.3934/jgm.2019009

Riemann-Hilbert problem, integrability and reductions

1. 

Department of Applied Mathematics, National Research Nuclear University MEPHI, 31 Kashirskoe Shosse, Moscow 115409, Russian Federation

2. 

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev Street, Sofia 1113, Bulgaria

3. 

School of Mathematical Sciences, Technological University Dublin - City Campus, Kevin Street, Dublin D08 NF82, Ireland

4. 

Faculty of Mathematics and Infromatics, Sofia University St. Kliment Ohridsky, 5 James Bourchier Blvd., Sofia 1164, Bulgaria

5. 

Institute for Advanced Physical Studies, New Bulgarian University, 21 Montevideo Street, Sofia 1618, Bulgaria

* Corresponding author: R. I. Ivanov

Received  April 2018 Revised  March 2019 Published  May 2019

The present paper is dedicated to integrable models with Mikhailov reduction groups $G_R \simeq \mathbb{D}_h.$ Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on the realization of the $G_R$-action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with $\mathbb{D}_h$ symmetries are presented.

Citation: Vladimir S. Gerdjikov, Rossen I. Ivanov, Aleksander A. Stefanov. Riemann-Hilbert problem, integrability and reductions. Journal of Geometric Mechanics, 2019, 11 (2) : 167-185. doi: 10.3934/jgm.2019009
References:
[1]

M. J. Ablowitz, B. Prinari and A. D. Trubach, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University press, London Mathematical Society Lecture Note Series, 2004.

[2]

M. Adler, On a trace functional for pseudo-differential operators and the symplectic structure of the Korteweg-deVries Equation, Invent. Math., 50 (1979), 219-248. doi: 10.1007/BF01410079.

[3]

N. C. BabalicR. Constantinescu and V. Gerdjikov, On the solutions of a family of Tzitzeica equations, J. Geom. Symm. Physics, 37 (2015), 1-24. doi: 10.7546/jgsp-37-2015-1-24.

[4]

N. C. BabalicR. Constantinescu and V. S. Gerdjikov, On Tzitzeica equation and spectral properties of related Lax operators, Balkan Journal of Geometry and Its Applications, 19 (2014), 11-22.

[5]

R. Beals and R. Coifman, Inverse Scattering and Evolution Equations, Commun. Pure Appl. Math., 38 (1985), 29-42. doi: 10.1002/cpa.3160380103.

[6]

G. Berkeley, A. V. Mikhailov and P. Xenitidis, Darboux transformations with tetrahedral reduction group and related integrable systems, Journal of Mathematical Physics, 57 (2016), 092701, 15pp, arXiv: 1603.03289 doi: 10.1063/1.4962803.

[7]

A. Borovik, $N$-soliton solutions of the Landau-Lifshitz equation, Pis'ma Zh. Eksp. Teor. Fiz., 28 (1978), 629-632.

[8]

R. T. Bury, Automorphic Lie Algebras, Corresponding Integrable Systems and Their Soliton Solutions, PhD thesis, University of Leeds, 2010.

[9]

R. T. Bury, A. V. Mikhailov and J. P. Wang, Wave fronts and cascades of soliton interactions in the periodic two dimensional Volterra system, Phys. D, 347 (2017), 21–41, arXiv: 1603.03106v1 [nlin.SI] doi: 10.1016/j.physd.2017.01.003.

[10]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661–1664; arXiv: patt-sol/9305002 doi: 10.1103/PhysRevLett.71.1661.

[11]

A. Constantin, V. S. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inv. Problems, 22 (2006), 2197–2207; arXiv: nlin/0603019v2 [nlin.SI]. doi: 10.1088/0266-5611/22/6/017.

[12]

A. ConstantinR. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis–Procesi equation, Nonlinearity, 23 (2010), 2559-2575. doi: 10.1088/0951-7715/23/10/012.

[13]

A. DegasperisD. Holm and A. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys., 133 (2002), 1461-1472. doi: 10.1023/A:1021186408422.

[14]

L. A. Dickey, Soliton Equations and Hamiltonian Systems, World scientific, 2003. doi: 10.1142/5108.

[15]

V. Drinfel'd and V. V. Sokolov, Lie algebras and equations of Korteweg - de Vries type, Sov. J. Math., 22 (1995), 25-86. doi: 10.1142/9789812798244_0002.

[16]

L. D. Faddeev and L. A. Takhtadjan, Hamiltonian Methods in the Theory of Solitons, Springer Verlag, Berlin, 1987. doi: 10.1007/978-3-540-69969-9.

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V. S. Gerdjikov, Algebraic and analytic aspects of $N$-wave type equations, Contemporary Mathematics, 301 (2002), 35-68. doi: 10.1090/conm/301/05158.

[18]

V. S. Gerdjikov, Riemann-Hilbert Problems with canonical normalization and families of commuting operators, Pliska Stud. Math. Bulgar., 21 (2012), 201–216; arXiv: 1204.2928v1 [nlin.SI].

[19]

V. S. Gerdjikov, Derivative nonlinear Schrödinger equations with $\mathbb{Z}_N $ and $\mathbb{D}_N $–reductions, Romanian Journal of Physics, 58 (2013), 573-582.

[20]

V. S. Gerdjikov, M. I. Ivanov, The quadratic bundle of general form and the nonlinear evolution equations. I. Expansions over the "squared" solutions are generalized Fourier transforms, Bulgarian J. Phys., 10 (1983), 13–26 (In Russian).

[21]

V. S. Gerdjikov and M. I. Ivanov, The quadratic bundle of general form and the nonlinear evolution equations. Ⅱ. Hierarchies of Hamiltonian structures, Bulgarian J. Phys., 10 (1983), 130–143 (In Russian).

[22]

V. S. GerdjikovG. G. GrahovskiA. V. Mikhailov and T. I. Valchev, On soliton interactions for the hierarchy of a generalised Heisenberg ferromagnetic model on SU(3)/S(U(1)$\times$ U(2)) symmetric space, Journal of Geometry and Symmetry in Physics, 25 (2012), 23-55. doi: 10.7546/jgsp-25-2012-23-55.

[23]

V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov and S. K. Varbev, MKdV-type of equations related to $B^{(1)}_{2}$ and $A^{(2)}_{4}$, in Nonlinear Mathematical Physics and Natural Hazards, (eds: Boyka Aneva, Mihaela Kouteva-Guentcheva), Springer Proceedings in Physics, 163 (2015), 59–69. ISBN: 978-3-319-14327-9 (Print) 978-3-319-14328-6 (Online). doi: 10.1007/978-3-319-14328-6_5.

[24]

V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov and S. K. Varbev, Soliton equations related to the affine Kac-Moody algebra $D^{(1)}_{4}$. Eur. Phys. J. Plus, 130 (2015), 106–123; arXiv: 1412.2383v1 [nlin.SI]. doi: 10.1140/epjp/i2015-15106-5.

[25]

V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov and S. K. Varbev, On mKdV equations related to the affine Kac-Moody algebra $A_{5}^{(2)}$, J. Geom. Sym. Phys., 39 (2015), 17–31, arXiv: 1512.01475 nlin: SI. doi: 10.7546/jgsp-39-2015-17-31.

[26]

V. Gerdjikov, G. Vilasi and A. Yanovski, Integrable Hamiltonian Hierarchies. Spectral and Geometric Methods, Lecture Notes in Physics, 748, Springer, Berlin - Heidelberg, 2008. doi: 10.1007/978-3-540-77054-1.

[27]

V. S. Gerdjikov and A. B. Yanovski, Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system, J. Math. Phys., 35 (1994), 3687-3725. doi: 10.1063/1.530441.

[28]

V. S. Gerdjikov and A. B. Yanovski, Riemann-Hilbert Problems, families of commuting operators and soliton equations, Journal of Physics: Conference Series, 482 (2014), 012017. doi: 10.1088/1742-6596/482/1/012017.

[29]

V. S. Gerdjikov and A. B. Yanovski, On soliton equations with $\mathbb{Z}_{ {h}}$ and $\mathbb{D}_{ {h}}$ reductions: conservation laws and generating operators, J. Geom. Symmetry Phys., 31 (2013), 57-92. doi: 10.7546/jgsp-31-2013-57-92.

[30]

V. S. Gerdjikov and A. B. Yanovski, CBC systems with Mikhailov reductions by Coxeter automorphism. Ⅰ. Spectral theory of the recursion operators, Studies in Applied Mathematics, 134 (2015), 145-180. doi: 10.1111/sapm.12065.

[31]

M. GürsesA. Karasu and V. V. Sokolov, On construction of recursion operators from Lax representation, Journal of Mathematical Physics, 40 (1999), 6473-6490. doi: 10.1063/1.533102.

[32]

J. Haberlin and T. Lyons, Solitons of shallow-water models from energy-dependent spectral problems, Eur. Phys. J. Plus, 133 (2018), 16, arXiv: 1705.04989 [math-ph] doi: 10.1140/epjp/i2018-11848-8.

[33] S. Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York-London, 1978.
[34]

D. D. Holm, Geometric Mechanics Part I: Dynamics and Symmetry, Imperial College Press: London, 2011. doi: 10.1142/p801.

[35]

D. D. Holm, Geometric Mechanics Part II: Rotating, Translating and Rolling, Imperial College Press: London, 2011. doi: 10.1142/p802.

[36]

D. Holm and R. Ivanov, Smooth and peaked solitons of the CH equation, J. Phys. A: Math. Theor., 43 (2010), 434003 (18pp). doi: 10.1088/1751-8113/43/43/434003.

[37]

D. Holm and R. Ivanov, Two-component CH system: Inverse scattering, peakons and geometry, Inverse Problems, 27 (2011), 045013, 19pp, arXiv: 1009.5374v1 [nlin.SI] doi: 10.1088/0266-5611/27/4/045013.

[38]

R. Ivanov, On the dressing method for the generalised Zakharov-Shabat system, Nuclear Physics B, 694 (2004), 509–524; math-ph/0402031. doi: 10.1016/j.nuclphysb.2004.06.039.

[39]

R. Ivanov and T. Lyons, Integrable models for shallow water with energy dependent spectral problems, Journal of Nonlinear Mathematical Physics, 19 (2012), 1240008 (17 pages). doi: 10.1142/S1402925112400086.

[40]

D. J. Kaup, The three-wave interaction - a nondispersive phenomenon, Stud. Appl. Math., 55 (1976), 9-44. doi: 10.1002/sapm19765519.

[41]

D. J. Kaup, On the inverse scattering problem for cubic eigenvalue problems of the class $\psi_{xxx}+6Q\psi_{x}+6R\psi = \lambda \psi $, Stud. Appl. Math., 62 (1980), 189-216. doi: 10.1002/sapm1980623189.

[42]

V. KnibbelerS. Lombardo and J. A. Sanders, Higher-Dimensional Automorphic Lie Algebras, Found. Comput. Math., 17 (2017), 987-1035. doi: 10.1007/s10208-016-9312-1.

[43]

S. Lombardo and A. V. Mikhailov, Reductions of integrable equations: Dihedral group, J. Phys. A, 37 (2004), 7727–7742; arXiv: nlin/0404013 [nlin.SI] doi: 10.1088/0305-4470/37/31/006.

[44]

S. Lombardo and A.V. Mikhailov, Reduction groups and Automorphic Lie Algebras, Communication in Mathematical Physics, 258 (2005), 179-202. doi: 10.1007/s00220-005-1334-5.

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S. Lombardo and J. Sanders, On the classification of automorphic Lie algebras, Communications in Mathematical Physics, 299 (2010), 793–824; arXiv: 0912.1697 [math.RA]. doi: 10.1007/s00220-010-1092-x.

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show all references

References:
[1]

M. J. Ablowitz, B. Prinari and A. D. Trubach, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University press, London Mathematical Society Lecture Note Series, 2004.

[2]

M. Adler, On a trace functional for pseudo-differential operators and the symplectic structure of the Korteweg-deVries Equation, Invent. Math., 50 (1979), 219-248. doi: 10.1007/BF01410079.

[3]

N. C. BabalicR. Constantinescu and V. Gerdjikov, On the solutions of a family of Tzitzeica equations, J. Geom. Symm. Physics, 37 (2015), 1-24. doi: 10.7546/jgsp-37-2015-1-24.

[4]

N. C. BabalicR. Constantinescu and V. S. Gerdjikov, On Tzitzeica equation and spectral properties of related Lax operators, Balkan Journal of Geometry and Its Applications, 19 (2014), 11-22.

[5]

R. Beals and R. Coifman, Inverse Scattering and Evolution Equations, Commun. Pure Appl. Math., 38 (1985), 29-42. doi: 10.1002/cpa.3160380103.

[6]

G. Berkeley, A. V. Mikhailov and P. Xenitidis, Darboux transformations with tetrahedral reduction group and related integrable systems, Journal of Mathematical Physics, 57 (2016), 092701, 15pp, arXiv: 1603.03289 doi: 10.1063/1.4962803.

[7]

A. Borovik, $N$-soliton solutions of the Landau-Lifshitz equation, Pis'ma Zh. Eksp. Teor. Fiz., 28 (1978), 629-632.

[8]

R. T. Bury, Automorphic Lie Algebras, Corresponding Integrable Systems and Their Soliton Solutions, PhD thesis, University of Leeds, 2010.

[9]

R. T. Bury, A. V. Mikhailov and J. P. Wang, Wave fronts and cascades of soliton interactions in the periodic two dimensional Volterra system, Phys. D, 347 (2017), 21–41, arXiv: 1603.03106v1 [nlin.SI] doi: 10.1016/j.physd.2017.01.003.

[10]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661–1664; arXiv: patt-sol/9305002 doi: 10.1103/PhysRevLett.71.1661.

[11]

A. Constantin, V. S. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inv. Problems, 22 (2006), 2197–2207; arXiv: nlin/0603019v2 [nlin.SI]. doi: 10.1088/0266-5611/22/6/017.

[12]

A. ConstantinR. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis–Procesi equation, Nonlinearity, 23 (2010), 2559-2575. doi: 10.1088/0951-7715/23/10/012.

[13]

A. DegasperisD. Holm and A. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys., 133 (2002), 1461-1472. doi: 10.1023/A:1021186408422.

[14]

L. A. Dickey, Soliton Equations and Hamiltonian Systems, World scientific, 2003. doi: 10.1142/5108.

[15]

V. Drinfel'd and V. V. Sokolov, Lie algebras and equations of Korteweg - de Vries type, Sov. J. Math., 22 (1995), 25-86. doi: 10.1142/9789812798244_0002.

[16]

L. D. Faddeev and L. A. Takhtadjan, Hamiltonian Methods in the Theory of Solitons, Springer Verlag, Berlin, 1987. doi: 10.1007/978-3-540-69969-9.

[17]

V. S. Gerdjikov, Algebraic and analytic aspects of $N$-wave type equations, Contemporary Mathematics, 301 (2002), 35-68. doi: 10.1090/conm/301/05158.

[18]

V. S. Gerdjikov, Riemann-Hilbert Problems with canonical normalization and families of commuting operators, Pliska Stud. Math. Bulgar., 21 (2012), 201–216; arXiv: 1204.2928v1 [nlin.SI].

[19]

V. S. Gerdjikov, Derivative nonlinear Schrödinger equations with $\mathbb{Z}_N $ and $\mathbb{D}_N $–reductions, Romanian Journal of Physics, 58 (2013), 573-582.

[20]

V. S. Gerdjikov, M. I. Ivanov, The quadratic bundle of general form and the nonlinear evolution equations. I. Expansions over the "squared" solutions are generalized Fourier transforms, Bulgarian J. Phys., 10 (1983), 13–26 (In Russian).

[21]

V. S. Gerdjikov and M. I. Ivanov, The quadratic bundle of general form and the nonlinear evolution equations. Ⅱ. Hierarchies of Hamiltonian structures, Bulgarian J. Phys., 10 (1983), 130–143 (In Russian).

[22]

V. S. GerdjikovG. G. GrahovskiA. V. Mikhailov and T. I. Valchev, On soliton interactions for the hierarchy of a generalised Heisenberg ferromagnetic model on SU(3)/S(U(1)$\times$ U(2)) symmetric space, Journal of Geometry and Symmetry in Physics, 25 (2012), 23-55. doi: 10.7546/jgsp-25-2012-23-55.

[23]

V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov and S. K. Varbev, MKdV-type of equations related to $B^{(1)}_{2}$ and $A^{(2)}_{4}$, in Nonlinear Mathematical Physics and Natural Hazards, (eds: Boyka Aneva, Mihaela Kouteva-Guentcheva), Springer Proceedings in Physics, 163 (2015), 59–69. ISBN: 978-3-319-14327-9 (Print) 978-3-319-14328-6 (Online). doi: 10.1007/978-3-319-14328-6_5.

[24]

V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov and S. K. Varbev, Soliton equations related to the affine Kac-Moody algebra $D^{(1)}_{4}$. Eur. Phys. J. Plus, 130 (2015), 106–123; arXiv: 1412.2383v1 [nlin.SI]. doi: 10.1140/epjp/i2015-15106-5.

[25]

V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov and S. K. Varbev, On mKdV equations related to the affine Kac-Moody algebra $A_{5}^{(2)}$, J. Geom. Sym. Phys., 39 (2015), 17–31, arXiv: 1512.01475 nlin: SI. doi: 10.7546/jgsp-39-2015-17-31.

[26]

V. Gerdjikov, G. Vilasi and A. Yanovski, Integrable Hamiltonian Hierarchies. Spectral and Geometric Methods, Lecture Notes in Physics, 748, Springer, Berlin - Heidelberg, 2008. doi: 10.1007/978-3-540-77054-1.

[27]

V. S. Gerdjikov and A. B. Yanovski, Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system, J. Math. Phys., 35 (1994), 3687-3725. doi: 10.1063/1.530441.

[28]

V. S. Gerdjikov and A. B. Yanovski, Riemann-Hilbert Problems, families of commuting operators and soliton equations, Journal of Physics: Conference Series, 482 (2014), 012017. doi: 10.1088/1742-6596/482/1/012017.

[29]

V. S. Gerdjikov and A. B. Yanovski, On soliton equations with $\mathbb{Z}_{ {h}}$ and $\mathbb{D}_{ {h}}$ reductions: conservation laws and generating operators, J. Geom. Symmetry Phys., 31 (2013), 57-92. doi: 10.7546/jgsp-31-2013-57-92.

[30]

V. S. Gerdjikov and A. B. Yanovski, CBC systems with Mikhailov reductions by Coxeter automorphism. Ⅰ. Spectral theory of the recursion operators, Studies in Applied Mathematics, 134 (2015), 145-180. doi: 10.1111/sapm.12065.

[31]

M. GürsesA. Karasu and V. V. Sokolov, On construction of recursion operators from Lax representation, Journal of Mathematical Physics, 40 (1999), 6473-6490. doi: 10.1063/1.533102.

[32]

J. Haberlin and T. Lyons, Solitons of shallow-water models from energy-dependent spectral problems, Eur. Phys. J. Plus, 133 (2018), 16, arXiv: 1705.04989 [math-ph] doi: 10.1140/epjp/i2018-11848-8.

[33] S. Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York-London, 1978.
[34]

D. D. Holm, Geometric Mechanics Part I: Dynamics and Symmetry, Imperial College Press: London, 2011. doi: 10.1142/p801.

[35]

D. D. Holm, Geometric Mechanics Part II: Rotating, Translating and Rolling, Imperial College Press: London, 2011. doi: 10.1142/p802.

[36]

D. Holm and R. Ivanov, Smooth and peaked solitons of the CH equation, J. Phys. A: Math. Theor., 43 (2010), 434003 (18pp). doi: 10.1088/1751-8113/43/43/434003.

[37]

D. Holm and R. Ivanov, Two-component CH system: Inverse scattering, peakons and geometry, Inverse Problems, 27 (2011), 045013, 19pp, arXiv: 1009.5374v1 [nlin.SI] doi: 10.1088/0266-5611/27/4/045013.

[38]

R. Ivanov, On the dressing method for the generalised Zakharov-Shabat system, Nuclear Physics B, 694 (2004), 509–524; math-ph/0402031. doi: 10.1016/j.nuclphysb.2004.06.039.

[39]

R. Ivanov and T. Lyons, Integrable models for shallow water with energy dependent spectral problems, Journal of Nonlinear Mathematical Physics, 19 (2012), 1240008 (17 pages). doi: 10.1142/S1402925112400086.

[40]

D. J. Kaup, The three-wave interaction - a nondispersive phenomenon, Stud. Appl. Math., 55 (1976), 9-44. doi: 10.1002/sapm19765519.

[41]

D. J. Kaup, On the inverse scattering problem for cubic eigenvalue problems of the class $\psi_{xxx}+6Q\psi_{x}+6R\psi = \lambda \psi $, Stud. Appl. Math., 62 (1980), 189-216. doi: 10.1002/sapm1980623189.

[42]

V. KnibbelerS. Lombardo and J. A. Sanders, Higher-Dimensional Automorphic Lie Algebras, Found. Comput. Math., 17 (2017), 987-1035. doi: 10.1007/s10208-016-9312-1.

[43]

S. Lombardo and A. V. Mikhailov, Reductions of integrable equations: Dihedral group, J. Phys. A, 37 (2004), 7727–7742; arXiv: nlin/0404013 [nlin.SI] doi: 10.1088/0305-4470/37/31/006.

[44]

S. Lombardo and A.V. Mikhailov, Reduction groups and Automorphic Lie Algebras, Communication in Mathematical Physics, 258 (2005), 179-202. doi: 10.1007/s00220-005-1334-5.

[45]

S. Lombardo and J. Sanders, On the classification of automorphic Lie algebras, Communications in Mathematical Physics, 299 (2010), 793–824; arXiv: 0912.1697 [math.RA]. doi: 10.1007/s00220-010-1092-x.

[46]

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Figure 1.  Contour of a RHP with $ \mathbb{Z}_3 $ symmetry
Figure 2.  Contour of the RHP $ \mathbb{D}_3 $ symmetry
Figure 3.  Contour of the RHP for $ \mathbb{D}_2 $ symmetry (upper panel) and for $ \mathbb{D}_4 $ symmetry (lower panel)
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