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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

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

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

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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

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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). Google Scholar

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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). Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[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. Google Scholar

[7]

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

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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]. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[40]

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

[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. Google Scholar

[42]

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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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