Hierarchies and Hamiltonian structures of the Nonlinear Schrödinger family using geometric and spectral techniques

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April 2019, 24(4): 1677-1695. doi: 10.3934/dcdsb.2018287

Hierarchies and Hamiltonian structures of the Nonlinear Schrödinger family using geometric and spectral techniques

Partha Guha ^1,2, and Indranil Mukherjee ^3,

1.	S.N. Bose National Centre for Basic Sciences, JD Block, Sector Ⅲ, Salt Lake, Kolkata - 700106, India
2.	Instituto de Física de São Carlos; IFSC/USP, Universidade de São Paulo Caixa Postal 369, CEP 13560-970, São Carlos-SP, Brazil
3.	School of Management and Sciences, Maulana Abul Kalam Azad University of Technology, West Bengal, BF 142, Sector I, Salt Lake, Kolkata-700064, India

Received March 2017 Revised February 2018 Published April 2019 Early access August 2018

This paper explores the class of equations of the Non-linear Schrödinger (NLS) type by employing both geometrical and spectral analysis methods. The work is developed in three stages. First, the geometrical method (AKS theorem) is used to derive different equations of the Non-linear Schrödinger (NLS) and Derivative Non-linear Schrödinger (DNLS) families. Second, the spectral technique (Tu method) is applied to obtain the hierarchies of equations belonging to these types. Third, the trace identity along with other techniques is used to obtain the corresponding Hamiltonian structures. It is found that the spectral method provides a simple algorithmic procedure to obtain the hierarchy as well as the Hamiltonian structure. Finally, the connection between the two formalisms is discussed and it is pointed out how application of these two techniques in unison can facilitate the understanding of integrable systems. In concurrence with Tu's method, Gesztesy and Holden also formulated a method of derivation of the trace formulas for integrable nonlinear evolution equations, this method is based on a contour-integration technique.

Keywords: Adler-Kostant-Symes theorem, Nonlinear Schrödinger equation, loop algebra, bihamiltonian system, Gesztesy-Holden methods, Tu methodology, trace identity.

Mathematics Subject Classification: 35Q55, 37K10, 37K30.

Citation: Partha Guha, Indranil Mukherjee. Hierarchies and Hamiltonian structures of the Nonlinear Schrödinger family using geometric and spectral techniques. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1677-1695. doi: 10.3934/dcdsb.2018287

References:

[1]	M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, (Cambridge: Cambridge University Press, 1991). doi: 10.1017/CBO9780511623998.
[2]	M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, The inverse scattering transform - Fourier analysis for nonlinear problems, Stud. Appl. Math., 53 (1974), 249-315. doi: 10.1002/sapm1974534249.
[3]	M. Adler, On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg de Vries type equations, Invent. Math., 50 (1979), 219-248. doi: 10.1007/BF01410079.
[4]	M. Adler and P. van Moerbeke, Completely integrable systems, Euclidean Lie algebras and curves, Adv. Math., 38 (1980), 267-317. doi: 10.1016/0001-8708(80)90007-9.
[5]	C. Athorne and A. Fordy, Generalised KdV and mKdV equations associated with symmetric spaces, J. Phys. A, 20 (1987), 1377-1386. doi: 10.1088/0305-4470/20/6/021.
[6]	C. Athorne and A. Fordy, Integrable equations in (2 + 1) diemensions associated with symmetric and homogeneous spaces, J. Math. Phys., 28 (1987), 2018-2024. doi: 10.1063/1.527463.
[7]	F. Calogero and W. Eckhaus, Nonlinear evolution equations, rescalings, model PDES and their integrability: I, Inverse Problems, 3 (1987), 229-262. doi: 10.1088/0266-5611/3/2/008.
[8]	H. H. Chen, Y. C. Lee and C. S. Liu, Integrability of nonlinear Hamiltonian systems by inverse scattering method. Special issue on solitons in physics, Phys. Scripta, 20 (1979), 490-492. doi: 10.1088/0031-8949/20/3-4/026.
[9]	W. Eckhaus, The long-time behaviour for perturbed wave-equations and related problems, Trends in Applications of Pure Mathematics to Mechanics (Bad Honnef, 1985), Lecture Notes in Phys., Springer, Berlin, 249 (1986), 168-194. doi: 10.1007/BFb0016391.
[10]	L. Faddeev and L. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-540-69969-9.
[11]	G. Falqui, Separation of variables for Lax systems: A bihamiltonian point of view, Fourth Italian-Latin American Conference on Applied and Industrial Mathematics (Havana, 2001), 393-403, Inst. Cybern. Math. Phys., Havana, 2001.
[12]	G. Falqui, F. Magri and M. Pedroni, Soliton Equations, Bi-Hamiltonian Manifolds and Integrability, 21° Coloquio Brasileiro de Matematica. [21st Brazilian Mathematics Colloquium] Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1997.
[13]	G. Falqui, F. Magri and M. Pedroni, A bihamiltonian approach to separation of variables in mechanics, Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years after NEED '79 (Gallipoli, 1999), World Science, River Edge, NJ, 2000,258-266.
[14]	H. Flaschka, A. C. Newell and T. Ratiu, Kac-Moody Lie algebras and soliton equations. Ⅱ. Lax equations associated with A₁⁽¹⁾, Phys. D, 9 (1983), 300-323. doi: 10.1016/0167-2789(83)90274-9.
[15]	A. P. Fordy and P. P. Kulish, Nonlinear Schrödinger equations and simple Lie algebras, Comm. Math. Phys., 89 (1983), 427-443.
[16]	I. M. Gelfand and I. Zakharevich, Webs, Veronese curves, and bihamiltonian systems, J. of Func. Anal., 99 (1991), 150-178. doi: 10.1016/0022-1236(91)90057-C.
[17]	I. M. Gelfand and I. Zakharevich, On the local geometry of bihamiltonian structures, The Gelfand Mathematical Seminar, 1990-1992 (Boston), Birkhäuser, 1993, 51-112.
[18]	I. M. Gelfand and I. Zakharevich, Webs, Lenard schemes, and the local geometry of biHamiltonian Toda and Lax structures, Selecta Math. (N.S.), 6 (2000), 131-183. doi: 10.1007/PL00001387.
[19]	V. S. Gerdjikov and M. I. Ivanov, The quadratic bundle of general form and the nonlinear evolution equations: Ⅱ. Hierarchies of Hamiltonian structures, Bulg. J. Phys., 10 (1983), 130-143.
[20]	F. Gesztesy and H. Holden, Soliton equations and their algebro-geometric solutions Volume Ⅰ : (1+1)Dimensional Continuous Models, Cambridge University Press, 2003. doi: 10.1017/CBO9780511546723.
[21]	F. Gesztesy and H. Holden, A combined sine-Gordon and modified Korteweg-deVries hierarchy and its algebro-geometric solutions, Weikard, R and Weinstein, G (eds.), Differential Equations and Mathematical Physics (Studies in Advanced Mathematics, Providence, . RI, and Boston, MA: Amer. Math. Soc and International Press, 16 (2000), 133-173.
[22]	F. Gesztesy and H. Holden, Trace formulas and conservation laws for nonlinear evolution equations, Rev.Math.Phys., 6 (1994), 51-95. doi: 10.1142/S0129055X94000055.
[23]	F. Gesztesy and H. Holden, On new trace formulae for Schrödinger operators, Acta Appl. Math., 39 (1995), 315-333. doi: 10.1007/BF00994640.
[24]	F. Gesztesy and H. Holden, Dubrovin equations and integrable systems on hyperelliptic curves, Math. Scand., 91 (2002), 91-126. doi: 10.7146/math.scand.a-14381.
[25]	F. Gesztesy, R. Ratnaseelan and G. Teschl, The KdV hierarchy and associated trace formulas, Gohberg, I, Lancaster, P and Shivakumar, P. N. (eds.), Recent Developments in Operator Theory and its applications (Operator Theory: Advances and Applications, Basel: Birkhauser, 87 (1996), 125-163.
[26]	F. Gesztesy and R. Weikard, A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy, Acta Math., 181 (1998), 63-108. doi: 10.1007/BF02392748.
[27]	F. Gesztesy and R. Weikard, Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies - an analytic approach, Bull. Amer. Math. Soc. (N.S), 35 (1998), 271-317. doi: 10.1090/S0273-0979-98-00765-4.
[28]	P. Guha, On Commuting flows of AKS hierarchy and twistor correspondence, Journal of Geom. Phys., 20 (1996), 207-217. doi: 10.1016/0393-0440(95)00056-9.
[29]	P. Guha, Adler-Kostant-Symes construction, bi-Hamiltonian manifolds, and KdV equations, J. Math. Phys., 38 (1997), 5167-5182. doi: 10.1063/1.531935.
[30]	P. Guha, AKS hierarchy and bi-Hamiltonian geometry of Gelfand-Zakharevich type, J. Math. Phys., 45 (2004), 2864-2884. doi: 10.1063/1.1756698.
[31]	D. J. Kaup and A. C. Newell, An exact solution for a derivative nonlinear Schrodinger equation, J. Math. Phys., 19 (1978), 798-801. doi: 10.1063/1.523737.
[32]	B. Kostant, Quantization and Representation Theory, in Representation Theory of Lie Groups, Lond. Math. Soc. Lect. Note, 34, edited by M. F. Atiyah.
[33]	A. Kundu, Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger type equations, J. Math. Phys., 25 (1984), 3433-3438. doi: 10.1063/1.526113.
[34]	A. Kundu, Exact solutions to higher-order nonlinear equations through gauge transformation, Physica D, 25 (1987), 399-406. doi: 10.1016/0167-2789(87)90113-8.
[35]	J. Lenells, Exactly solvable model for nonlinear pulse propogation in optical fibers, Stud. Appl.Maths., 123 (2009), 215-232. doi: 10.1111/j.1467-9590.2009.00454.x.
[36]	J. Lenells and A. S. Fokas, On a novel integrable generalization of the nonlinear Schrodinger equation, Nonlinearity, 22 (2009), 11-27. doi: 10.1088/0951-7715/22/1/002.
[37]	F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19 (1978), 1156-1162. doi: 10.1063/1.523777.
[38]	A. C. Newell, Solitons in Mathematics and Physics, (Society for Industrial and Applied Mathematics, 1985). doi: 10.1137/1.9781611970227.
[39]	T. Ratiu, The C. Neumann problem as a complete integrable system on an adjoint orbit, Trans. Amer. Math. Soc., 264 (1981), 321-329. doi: 10.1090/S0002-9947-1981-0603766-3.
[40]	A. G. Reiman and M. A. Semenov-Tian-Sanskii, Reduction of Hamiltonian systems, affine Lie algebras and Lax equations Ⅰ, Invent. Math., 54 (1979), 81-100. doi: 10.1007/BF01391179.
[41]	A. G. Reiman and M. A. Semenov-Tian-Sanskii, Soviet Math. Dokl., Current Algebras and Nonlinear Partial Differential Equations, 21 (1980), 630-634.
[42]	C.-L. Terng, Soliton equations and differential geometry, J. Differential Geometry, 45 (1997), 407-445. doi: 10.4310/jdg/1214459804.
[43]	G. Tu, The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys., 30 (1989), 330-338. doi: 10.1063/1.528449.
[44]	G. Tu, A trace identity and its applications to the theory of discrete integrable systems, J. Phys. A: Math. Gen., 23 (1990), 3903-3922. doi: 10.1088/0305-4470/23/17/020.
[45]	G. Tu, A new hierarchy of integrable systems and its Hamiltonian structure, Science in China (Series A), 32 (1989), 142-153.
[46]	G. Tu, On Liouville integrability of zero-curvature equations and the Yang hierarchy, J. Phys. A: Math. Gen., 22 (1989), 2375-2392. doi: 10.1088/0305-4470/22/13/031.

show all references

References:

[1]	M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, (Cambridge: Cambridge University Press, 1991). doi: 10.1017/CBO9780511623998.
[2]	M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, The inverse scattering transform - Fourier analysis for nonlinear problems, Stud. Appl. Math., 53 (1974), 249-315. doi: 10.1002/sapm1974534249.
[3]	M. Adler, On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg de Vries type equations, Invent. Math., 50 (1979), 219-248. doi: 10.1007/BF01410079.
[4]	M. Adler and P. van Moerbeke, Completely integrable systems, Euclidean Lie algebras and curves, Adv. Math., 38 (1980), 267-317. doi: 10.1016/0001-8708(80)90007-9.
[5]	C. Athorne and A. Fordy, Generalised KdV and mKdV equations associated with symmetric spaces, J. Phys. A, 20 (1987), 1377-1386. doi: 10.1088/0305-4470/20/6/021.
[6]	C. Athorne and A. Fordy, Integrable equations in (2 + 1) diemensions associated with symmetric and homogeneous spaces, J. Math. Phys., 28 (1987), 2018-2024. doi: 10.1063/1.527463.
[7]	F. Calogero and W. Eckhaus, Nonlinear evolution equations, rescalings, model PDES and their integrability: I, Inverse Problems, 3 (1987), 229-262. doi: 10.1088/0266-5611/3/2/008.
[8]	H. H. Chen, Y. C. Lee and C. S. Liu, Integrability of nonlinear Hamiltonian systems by inverse scattering method. Special issue on solitons in physics, Phys. Scripta, 20 (1979), 490-492. doi: 10.1088/0031-8949/20/3-4/026.
[9]	W. Eckhaus, The long-time behaviour for perturbed wave-equations and related problems, Trends in Applications of Pure Mathematics to Mechanics (Bad Honnef, 1985), Lecture Notes in Phys., Springer, Berlin, 249 (1986), 168-194. doi: 10.1007/BFb0016391.
[10]	L. Faddeev and L. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-540-69969-9.
[11]	G. Falqui, Separation of variables for Lax systems: A bihamiltonian point of view, Fourth Italian-Latin American Conference on Applied and Industrial Mathematics (Havana, 2001), 393-403, Inst. Cybern. Math. Phys., Havana, 2001.
[12]	G. Falqui, F. Magri and M. Pedroni, Soliton Equations, Bi-Hamiltonian Manifolds and Integrability, 21° Coloquio Brasileiro de Matematica. [21st Brazilian Mathematics Colloquium] Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1997.
[13]	G. Falqui, F. Magri and M. Pedroni, A bihamiltonian approach to separation of variables in mechanics, Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years after NEED '79 (Gallipoli, 1999), World Science, River Edge, NJ, 2000,258-266.
[14]	H. Flaschka, A. C. Newell and T. Ratiu, Kac-Moody Lie algebras and soliton equations. Ⅱ. Lax equations associated with A₁⁽¹⁾, Phys. D, 9 (1983), 300-323. doi: 10.1016/0167-2789(83)90274-9.
[15]	A. P. Fordy and P. P. Kulish, Nonlinear Schrödinger equations and simple Lie algebras, Comm. Math. Phys., 89 (1983), 427-443.
[16]	I. M. Gelfand and I. Zakharevich, Webs, Veronese curves, and bihamiltonian systems, J. of Func. Anal., 99 (1991), 150-178. doi: 10.1016/0022-1236(91)90057-C.
[17]	I. M. Gelfand and I. Zakharevich, On the local geometry of bihamiltonian structures, The Gelfand Mathematical Seminar, 1990-1992 (Boston), Birkhäuser, 1993, 51-112.
[18]	I. M. Gelfand and I. Zakharevich, Webs, Lenard schemes, and the local geometry of biHamiltonian Toda and Lax structures, Selecta Math. (N.S.), 6 (2000), 131-183. doi: 10.1007/PL00001387.
[19]	V. S. Gerdjikov and M. I. Ivanov, The quadratic bundle of general form and the nonlinear evolution equations: Ⅱ. Hierarchies of Hamiltonian structures, Bulg. J. Phys., 10 (1983), 130-143.
[20]	F. Gesztesy and H. Holden, Soliton equations and their algebro-geometric solutions Volume Ⅰ : (1+1)Dimensional Continuous Models, Cambridge University Press, 2003. doi: 10.1017/CBO9780511546723.
[21]	F. Gesztesy and H. Holden, A combined sine-Gordon and modified Korteweg-deVries hierarchy and its algebro-geometric solutions, Weikard, R and Weinstein, G (eds.), Differential Equations and Mathematical Physics (Studies in Advanced Mathematics, Providence, . RI, and Boston, MA: Amer. Math. Soc and International Press, 16 (2000), 133-173.
[22]	F. Gesztesy and H. Holden, Trace formulas and conservation laws for nonlinear evolution equations, Rev.Math.Phys., 6 (1994), 51-95. doi: 10.1142/S0129055X94000055.
[23]	F. Gesztesy and H. Holden, On new trace formulae for Schrödinger operators, Acta Appl. Math., 39 (1995), 315-333. doi: 10.1007/BF00994640.
[24]	F. Gesztesy and H. Holden, Dubrovin equations and integrable systems on hyperelliptic curves, Math. Scand., 91 (2002), 91-126. doi: 10.7146/math.scand.a-14381.
[25]	F. Gesztesy, R. Ratnaseelan and G. Teschl, The KdV hierarchy and associated trace formulas, Gohberg, I, Lancaster, P and Shivakumar, P. N. (eds.), Recent Developments in Operator Theory and its applications (Operator Theory: Advances and Applications, Basel: Birkhauser, 87 (1996), 125-163.
[26]	F. Gesztesy and R. Weikard, A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy, Acta Math., 181 (1998), 63-108. doi: 10.1007/BF02392748.
[27]	F. Gesztesy and R. Weikard, Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies - an analytic approach, Bull. Amer. Math. Soc. (N.S), 35 (1998), 271-317. doi: 10.1090/S0273-0979-98-00765-4.
[28]	P. Guha, On Commuting flows of AKS hierarchy and twistor correspondence, Journal of Geom. Phys., 20 (1996), 207-217. doi: 10.1016/0393-0440(95)00056-9.
[29]	P. Guha, Adler-Kostant-Symes construction, bi-Hamiltonian manifolds, and KdV equations, J. Math. Phys., 38 (1997), 5167-5182. doi: 10.1063/1.531935.
[30]	P. Guha, AKS hierarchy and bi-Hamiltonian geometry of Gelfand-Zakharevich type, J. Math. Phys., 45 (2004), 2864-2884. doi: 10.1063/1.1756698.
[31]	D. J. Kaup and A. C. Newell, An exact solution for a derivative nonlinear Schrodinger equation, J. Math. Phys., 19 (1978), 798-801. doi: 10.1063/1.523737.
[32]	B. Kostant, Quantization and Representation Theory, in Representation Theory of Lie Groups, Lond. Math. Soc. Lect. Note, 34, edited by M. F. Atiyah.
[33]	A. Kundu, Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger type equations, J. Math. Phys., 25 (1984), 3433-3438. doi: 10.1063/1.526113.
[34]	A. Kundu, Exact solutions to higher-order nonlinear equations through gauge transformation, Physica D, 25 (1987), 399-406. doi: 10.1016/0167-2789(87)90113-8.
[35]	J. Lenells, Exactly solvable model for nonlinear pulse propogation in optical fibers, Stud. Appl.Maths., 123 (2009), 215-232. doi: 10.1111/j.1467-9590.2009.00454.x.
[36]	J. Lenells and A. S. Fokas, On a novel integrable generalization of the nonlinear Schrodinger equation, Nonlinearity, 22 (2009), 11-27. doi: 10.1088/0951-7715/22/1/002.
[37]	F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19 (1978), 1156-1162. doi: 10.1063/1.523777.
[38]	A. C. Newell, Solitons in Mathematics and Physics, (Society for Industrial and Applied Mathematics, 1985). doi: 10.1137/1.9781611970227.
[39]	T. Ratiu, The C. Neumann problem as a complete integrable system on an adjoint orbit, Trans. Amer. Math. Soc., 264 (1981), 321-329. doi: 10.1090/S0002-9947-1981-0603766-3.
[40]	A. G. Reiman and M. A. Semenov-Tian-Sanskii, Reduction of Hamiltonian systems, affine Lie algebras and Lax equations Ⅰ, Invent. Math., 54 (1979), 81-100. doi: 10.1007/BF01391179.
[41]	A. G. Reiman and M. A. Semenov-Tian-Sanskii, Soviet Math. Dokl., Current Algebras and Nonlinear Partial Differential Equations, 21 (1980), 630-634.
[42]	C.-L. Terng, Soliton equations and differential geometry, J. Differential Geometry, 45 (1997), 407-445. doi: 10.4310/jdg/1214459804.
[43]	G. Tu, The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys., 30 (1989), 330-338. doi: 10.1063/1.528449.
[44]	G. Tu, A trace identity and its applications to the theory of discrete integrable systems, J. Phys. A: Math. Gen., 23 (1990), 3903-3922. doi: 10.1088/0305-4470/23/17/020.
[45]	G. Tu, A new hierarchy of integrable systems and its Hamiltonian structure, Science in China (Series A), 32 (1989), 142-153.
[46]	G. Tu, On Liouville integrability of zero-curvature equations and the Yang hierarchy, J. Phys. A: Math. Gen., 22 (1989), 2375-2392. doi: 10.1088/0305-4470/22/13/031.

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