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On the integrability of KdV hierarchy with self-consistent sources
1. | Institute for Nuclear Research and Nuclear Energy, Bulgarian academy of sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria |
2. | School of Electronic Engineering, Dublin City University, Glasnevin, Dublin 9 |
3. | School of Mathematical Science, Dublin Institute of Technology, Kevin Street, Dublin 8 |
References:
[1] |
V. A. Arkad'ev, A. K. Pogrebkov and M. K. Polivanov, Expansions with respect to squares, symplectic and Poisson structures associated with the Sturm-Liouville problem. I, Theor. Math. Phys., 72 (1987), 909-920. |
[2] |
V. A. Arkad'ev, A. K. Pogrebkov and M. K. Polivanov, Expansions with respect to squares, symplectic and Poisson structures associated with the Sturm-Liouville problem. II, Theor. Math. Phys., 75 (1988), 448-460.
doi: 10.1007/BF01017483. |
[3] |
G. Borg, Eine Umkehrung der Sturm-Liouvilleschen eigenwertaufgabe. Bestimmung der differentialgleichung durch die eigenwerte, Acta Math., 78 (1946), 1-96, (German).
doi: 10.1007/BF02421600. |
[4] |
F. Calogero, A. Degasperis, "Spectral Transform and Solitons Vol 1. Tools to Solve and Investigate Nonlinear Evolution Equations," Studies in Mathematics and its Applications 13 (Lecture Notes in Computer Science vol 144) Amsterdam: North-Holland (1982), p. 516. |
[5] |
R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664 (E-print: patt-sol/9305002).
doi: 10.1103/PhysRevLett.71.1661. |
[6] |
C. Claude, A. Latifi and J. P. Leon, Nonlinear resonant scattering and plasma instability: an integrable model, J. Math Phys., 32 (1991), 3321-3330.
doi: 10.1063/1.529443. |
[7] |
A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inv. Problems, 22 (2006), 2197-2207 (E-print: nlin/0603019).
doi: 10.1088/0266-5611/22/6/017. |
[8] |
A. Constantin, V. Gerdjikov and R. Ivanov, Generalized Fourier transform for the Camassa-Holm hierarchy, Inverse Problems, 23 (2007), 1565-1597 (E-print: arXiv:0707.2048).
doi: 10.1088/0266-5611/23/4/012. |
[9] |
A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi Equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186 (E-print: arXiv:0709.0905).
doi: 10.1007/s00205-008-0128-2. |
[10] |
G. Eilenberger, "Solitons: Mathematical Methods for Physicists," Springer Series in Solid-State Sciences. vol. 19, Springer-Verlag, Berlin, (1981). |
[11] |
L. D. Faddeev and L. A. Takhtajan, Poisson structure for the KdV equation, Lett. MAth. Phys., 10 (1985), 183-188.
doi: 10.1007/BF00398156. |
[12] |
V. S. Gerdjikov, Generalised Fourier transforms for the soliton equations. Gauge-covariant formulation, Inv. Problems, 2 (1986), 51-74.
doi: 10.1088/0266-5611/2/1/005. |
[13] |
V. S. Gerdjikov, The generalized Zakharov-Shabat system and the soliton perturbations, Theoret. and Math. Phys., 99 (1994), 593-598.
doi: 10.1007/BF01016144. |
[14] |
V. S. Gerdjikov and M. I. Ivanov, Expansions over the "squared" solutions and the inhomogeneous nonlinear Schrödinger equation, Inv. Problems, 8 (1992), 831-847.
doi: 10.1088/0266-5611/8/6/004. |
[15] |
V. S. Gerdjikov and E. Kh. Khristov, Evolution equations solvable by the inverse-scattering method. I. Spectral theory, Bulgarian J. Phys., 7 (1980), 28-41. (In Russian). On evolution equations solvable by the inverse scattering method. II. Hamiltonian structure and Bäcklund transformations Bulgarian J. Phys., 7 (1980), 119-133 (In Russian). |
[16] |
V. S. Gerdjikov, G. Vilasi and A. B. Yanovski, "Integrable Hamiltonian Hierarchies. Spectral and Geometric Methods," Lecture Notes in Physics, 748. Springer-Verlag, Berlin, 2008.
doi: 10.1007/978-3-540-77054-1. |
[17] |
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. |
[18] |
G. G. Grahovski and R. I. Ivanov, Generalised Fourier transform and perturbations to soliton equations, Discr. Cont. Dyn. Syst. B, 12 (2009), 579-595 (E-print: arXiv:0907.2062).
doi: 10.3934/dcdsb.2009.12.579. |
[19] |
P. Guha, Nonholonomic deformation of generalized KdV-type equations, J. Phys. A: Math. Theor., 42 (2009), 345201.
doi: 10.1088/1751-8113/42/34/345201. |
[20] |
I. Iliev, E. Khristov and K. Kirchev, "Spectral Methods in Soliton Equations," Pitman Monographs and Surveys in Pure and Appl. Math., 73, Pitman, London, 1994. |
[21] |
R. I. Ivanov, Water waves and integrability, Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci., 365 (2007), 2267-2280 (E-print: arXiv:0707.1839).
doi: 10.1098/rsta.2007.2007. |
[22] |
R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid. Mech., 457 (2002), 63-82.
doi: 10.1017/S0022112001007224. |
[23] |
R. S. Johnson, On solutions of the Camassa-Holm equation, Proc. Roy. Soc. London A, 459 (2003), 1687-1708.
doi: 10.1016/S0169-5983(03)00036-4. |
[24] |
A. Karasu-Kalkantli, A. Karasu, A. Sakovich, S. Sakovich and R. Turhan, A new integrable generalization of the KdV equation, J. Math. Phys., 49 (2008), 073516.
doi: 10.1063/1.2953474. |
[25] |
V. I. Karpman and E. M. Maslov, Perturbation theory for solitons, Soviet Phys. JETP, 46 (1977), 537-559. |
[26] |
D. J. Kaup, A perturbation expansion for the Zakharov-Shabat inverse scattering transform, SIAM J. Appl. Math., 31 (1976), 121-133.
doi: 10.1137/0131013. |
[27] |
D. J. Kaup, Closure of the squared Zakharov-Shabat eigenstates, J. Math. Anal. Appl., 54 (1976), 849-864.
doi: 10.1016/0022-247X(76)90201-8. |
[28] |
D. J. Kaup, In "Significance of Nonlinearity in the Natural Science" (Kursunoglu, A. Perlmutter and L. F. Scott eds.), Plenum Press, p. 97, 1977. |
[29] |
D. J. Kaup and A. C. Newell, Solitons as particles, oscillators, and in slowly changing media: A singular perturbation theory, Proc. Roy. Soc., A361 (1978), 413-446.
doi: 10.1098/rspa.1978.0110. |
[30] |
K. P. Kirchev and E. Kh. Hristov, Expansions connected with the products of the solutions of two regular Sturm-Liouville problems, Sibirsk. Mat. Zh., 21 (1980), 98-109 (Russian). |
[31] |
A. Kundu, Exact accelerating soliton in nonholonomic deformation of the KdV equation with two-fold integrable hierarchy, J. Phys. A: Math. Theor., 41 (2008), 495201.
doi: 10.1088/1751-8113/41/49/495201. |
[32] |
A. Kundu, R. Sahadevan and L. Nalinidevi, Nonholonomic deformation of KdV and mKdV equations and their symmetries, hierarchies and integrability, J. Phys. A: Math. Theor., 42 (2009), 115213.
doi: 10.1088/1751-8113/42/11/115213. |
[33] |
A. Kundu, Nonlinearizing linear equations to integrable systems including new hierarchies with nonholonomic deformations, J. Math. Phys., 50 (2009), 102702.
doi: 10.1063/1.3204081. |
[34] |
A. Kundu, Two-fold integrable hierarchy of nonholonomic deformation of the derivative nonlinear Schröinger and the Lenells-Fokas equation, J. Math. Phys., 51 (2010), 022901.
doi: 10.1063/1.3276447. |
[35] |
B. A. Kupershmidt, KdV6: an integrable system, Phys. Lett. A, 372 (2008), 2634-2639.
doi: :10.1016/j.physleta.2007.12.019. |
[36] |
J. P. Leon, General evolution of the spectral transform from the $\partial$-approach, Phys. Lett, 123A (1987), 65-70.
doi: 10.1016/0375-9601(87)90657-8. |
[37] |
J. P. Leon and A. Latifi, Solution of an initial-boundary value problem for coupled nonlinear waves, J. Phys. A: Math. Gen., 23 (1990), 1385-1403.
doi: 10.1088/0305-4470/23/8/013. |
[38] |
J. P. Leon, Nonlinear evolutions with singular dispersion laws and forced systems, Phys. Lett, 144A (1990), 444-452.
doi: 10.1016/0375-9601(90)90512-M. |
[39] |
J. Leon, Spectral transform and solitons for generalized coupled Bloch systems, J. Math. Phys., 29 (1988), 2012-2019.
doi: 10.1063/1.527859. |
[40] |
V. K. Melnikov, Integration of the Korteweg-de Vries eqtion with source, Inverse Probl., 6 (1990), 233-246.
doi: 10.1088/0266-5611/6/2/007. |
[41] |
V. K. Melnikov, Creation and annihilation of solitons in the system described by the Kortweg-de Vries equation with a self-consistent source, Inverse Probl., 6 (1990), 809-823.
doi: 10.1088/0266-5611/6/5/010. |
[42] |
A. C. Newell, In "Solitons" (R. K. Bullough and P. J. Caudrey eds.), Springer Verlag, 1980. |
[43] |
S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, "Theory of Solitons: the Inverse Scattering Method," Plenum, New York, 1984. |
[44] |
A. Ramani, B. Grammaticos and R. Willox, Bilinearization and solutions of the KdV6 equation, Anal. Appl., 6 (2008), 401-412.
doi: 10.1142/S0219530508001249. |
[45] |
T. Valchev, On the Kaup-Kupershmidt equation. Completeness relations for the squared solutions, in "Nineth International Conference on Geometry, Integrability and Quantization, June 8-13 2007, Varna, Bulgaria" (eds. I. Mladenov and M. de Leon), SOFTEX, Sofia (2008), 308-319. |
[46] |
Jing Ping Wang, Extension of integrable equations, J. Phys. A: Math. Theor., 42 (2009), 362204.
doi: 10.1088/1751-8113/42/36/362004. |
[47] |
A-M. Wazwaz, The integrable KdV6 equations: Multiple soliton solutions and multiple singular soliton solutions, Applied Mathematics and Computation, 204 (2008), 963-972.
doi: 10.1016/j.amc.2008.08.007. |
[48] |
Y. Q. Yao and Y. B. Zeng, Integrable Rosochatius deformations of higher-order constrained flows and the soliton hierachy with self-consistent source, J. Phys. A: Math. Theor., 41 (2008), 295205.
doi: 10.1088/1751-8113/41/29/295205. |
[49] |
Y. Q. Yao and Y. B. Zeng, The bi-Hamiltonian structure and new solutions of KdV6 equation, Lett. Math. Phys., 86 (2008), 193-208.
doi: 10.1007/s11005-008-0281-4. |
[50] |
V. Zakharov and L. Faddeev, Korteweg-de Vries equation is a completely integrable Hamiltonian system, Func. Anal. Appl., 5 (1971), 280-287 (English). |
[51] |
Y. B. Zeng, Y. J. Shao and W. Xue, Negaton and positon solutions of the soliton equation with sself-consistent sources, J. Phys. A Math. Gen., 36 (2003), 5035-5043.
doi: 10.1088/0305-4470/36/18/308. |
show all references
References:
[1] |
V. A. Arkad'ev, A. K. Pogrebkov and M. K. Polivanov, Expansions with respect to squares, symplectic and Poisson structures associated with the Sturm-Liouville problem. I, Theor. Math. Phys., 72 (1987), 909-920. |
[2] |
V. A. Arkad'ev, A. K. Pogrebkov and M. K. Polivanov, Expansions with respect to squares, symplectic and Poisson structures associated with the Sturm-Liouville problem. II, Theor. Math. Phys., 75 (1988), 448-460.
doi: 10.1007/BF01017483. |
[3] |
G. Borg, Eine Umkehrung der Sturm-Liouvilleschen eigenwertaufgabe. Bestimmung der differentialgleichung durch die eigenwerte, Acta Math., 78 (1946), 1-96, (German).
doi: 10.1007/BF02421600. |
[4] |
F. Calogero, A. Degasperis, "Spectral Transform and Solitons Vol 1. Tools to Solve and Investigate Nonlinear Evolution Equations," Studies in Mathematics and its Applications 13 (Lecture Notes in Computer Science vol 144) Amsterdam: North-Holland (1982), p. 516. |
[5] |
R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664 (E-print: patt-sol/9305002).
doi: 10.1103/PhysRevLett.71.1661. |
[6] |
C. Claude, A. Latifi and J. P. Leon, Nonlinear resonant scattering and plasma instability: an integrable model, J. Math Phys., 32 (1991), 3321-3330.
doi: 10.1063/1.529443. |
[7] |
A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inv. Problems, 22 (2006), 2197-2207 (E-print: nlin/0603019).
doi: 10.1088/0266-5611/22/6/017. |
[8] |
A. Constantin, V. Gerdjikov and R. Ivanov, Generalized Fourier transform for the Camassa-Holm hierarchy, Inverse Problems, 23 (2007), 1565-1597 (E-print: arXiv:0707.2048).
doi: 10.1088/0266-5611/23/4/012. |
[9] |
A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi Equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186 (E-print: arXiv:0709.0905).
doi: 10.1007/s00205-008-0128-2. |
[10] |
G. Eilenberger, "Solitons: Mathematical Methods for Physicists," Springer Series in Solid-State Sciences. vol. 19, Springer-Verlag, Berlin, (1981). |
[11] |
L. D. Faddeev and L. A. Takhtajan, Poisson structure for the KdV equation, Lett. MAth. Phys., 10 (1985), 183-188.
doi: 10.1007/BF00398156. |
[12] |
V. S. Gerdjikov, Generalised Fourier transforms for the soliton equations. Gauge-covariant formulation, Inv. Problems, 2 (1986), 51-74.
doi: 10.1088/0266-5611/2/1/005. |
[13] |
V. S. Gerdjikov, The generalized Zakharov-Shabat system and the soliton perturbations, Theoret. and Math. Phys., 99 (1994), 593-598.
doi: 10.1007/BF01016144. |
[14] |
V. S. Gerdjikov and M. I. Ivanov, Expansions over the "squared" solutions and the inhomogeneous nonlinear Schrödinger equation, Inv. Problems, 8 (1992), 831-847.
doi: 10.1088/0266-5611/8/6/004. |
[15] |
V. S. Gerdjikov and E. Kh. Khristov, Evolution equations solvable by the inverse-scattering method. I. Spectral theory, Bulgarian J. Phys., 7 (1980), 28-41. (In Russian). On evolution equations solvable by the inverse scattering method. II. Hamiltonian structure and Bäcklund transformations Bulgarian J. Phys., 7 (1980), 119-133 (In Russian). |
[16] |
V. S. Gerdjikov, G. Vilasi and A. B. Yanovski, "Integrable Hamiltonian Hierarchies. Spectral and Geometric Methods," Lecture Notes in Physics, 748. Springer-Verlag, Berlin, 2008.
doi: 10.1007/978-3-540-77054-1. |
[17] |
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. |
[18] |
G. G. Grahovski and R. I. Ivanov, Generalised Fourier transform and perturbations to soliton equations, Discr. Cont. Dyn. Syst. B, 12 (2009), 579-595 (E-print: arXiv:0907.2062).
doi: 10.3934/dcdsb.2009.12.579. |
[19] |
P. Guha, Nonholonomic deformation of generalized KdV-type equations, J. Phys. A: Math. Theor., 42 (2009), 345201.
doi: 10.1088/1751-8113/42/34/345201. |
[20] |
I. Iliev, E. Khristov and K. Kirchev, "Spectral Methods in Soliton Equations," Pitman Monographs and Surveys in Pure and Appl. Math., 73, Pitman, London, 1994. |
[21] |
R. I. Ivanov, Water waves and integrability, Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci., 365 (2007), 2267-2280 (E-print: arXiv:0707.1839).
doi: 10.1098/rsta.2007.2007. |
[22] |
R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid. Mech., 457 (2002), 63-82.
doi: 10.1017/S0022112001007224. |
[23] |
R. S. Johnson, On solutions of the Camassa-Holm equation, Proc. Roy. Soc. London A, 459 (2003), 1687-1708.
doi: 10.1016/S0169-5983(03)00036-4. |
[24] |
A. Karasu-Kalkantli, A. Karasu, A. Sakovich, S. Sakovich and R. Turhan, A new integrable generalization of the KdV equation, J. Math. Phys., 49 (2008), 073516.
doi: 10.1063/1.2953474. |
[25] |
V. I. Karpman and E. M. Maslov, Perturbation theory for solitons, Soviet Phys. JETP, 46 (1977), 537-559. |
[26] |
D. J. Kaup, A perturbation expansion for the Zakharov-Shabat inverse scattering transform, SIAM J. Appl. Math., 31 (1976), 121-133.
doi: 10.1137/0131013. |
[27] |
D. J. Kaup, Closure of the squared Zakharov-Shabat eigenstates, J. Math. Anal. Appl., 54 (1976), 849-864.
doi: 10.1016/0022-247X(76)90201-8. |
[28] |
D. J. Kaup, In "Significance of Nonlinearity in the Natural Science" (Kursunoglu, A. Perlmutter and L. F. Scott eds.), Plenum Press, p. 97, 1977. |
[29] |
D. J. Kaup and A. C. Newell, Solitons as particles, oscillators, and in slowly changing media: A singular perturbation theory, Proc. Roy. Soc., A361 (1978), 413-446.
doi: 10.1098/rspa.1978.0110. |
[30] |
K. P. Kirchev and E. Kh. Hristov, Expansions connected with the products of the solutions of two regular Sturm-Liouville problems, Sibirsk. Mat. Zh., 21 (1980), 98-109 (Russian). |
[31] |
A. Kundu, Exact accelerating soliton in nonholonomic deformation of the KdV equation with two-fold integrable hierarchy, J. Phys. A: Math. Theor., 41 (2008), 495201.
doi: 10.1088/1751-8113/41/49/495201. |
[32] |
A. Kundu, R. Sahadevan and L. Nalinidevi, Nonholonomic deformation of KdV and mKdV equations and their symmetries, hierarchies and integrability, J. Phys. A: Math. Theor., 42 (2009), 115213.
doi: 10.1088/1751-8113/42/11/115213. |
[33] |
A. Kundu, Nonlinearizing linear equations to integrable systems including new hierarchies with nonholonomic deformations, J. Math. Phys., 50 (2009), 102702.
doi: 10.1063/1.3204081. |
[34] |
A. Kundu, Two-fold integrable hierarchy of nonholonomic deformation of the derivative nonlinear Schröinger and the Lenells-Fokas equation, J. Math. Phys., 51 (2010), 022901.
doi: 10.1063/1.3276447. |
[35] |
B. A. Kupershmidt, KdV6: an integrable system, Phys. Lett. A, 372 (2008), 2634-2639.
doi: :10.1016/j.physleta.2007.12.019. |
[36] |
J. P. Leon, General evolution of the spectral transform from the $\partial$-approach, Phys. Lett, 123A (1987), 65-70.
doi: 10.1016/0375-9601(87)90657-8. |
[37] |
J. P. Leon and A. Latifi, Solution of an initial-boundary value problem for coupled nonlinear waves, J. Phys. A: Math. Gen., 23 (1990), 1385-1403.
doi: 10.1088/0305-4470/23/8/013. |
[38] |
J. P. Leon, Nonlinear evolutions with singular dispersion laws and forced systems, Phys. Lett, 144A (1990), 444-452.
doi: 10.1016/0375-9601(90)90512-M. |
[39] |
J. Leon, Spectral transform and solitons for generalized coupled Bloch systems, J. Math. Phys., 29 (1988), 2012-2019.
doi: 10.1063/1.527859. |
[40] |
V. K. Melnikov, Integration of the Korteweg-de Vries eqtion with source, Inverse Probl., 6 (1990), 233-246.
doi: 10.1088/0266-5611/6/2/007. |
[41] |
V. K. Melnikov, Creation and annihilation of solitons in the system described by the Kortweg-de Vries equation with a self-consistent source, Inverse Probl., 6 (1990), 809-823.
doi: 10.1088/0266-5611/6/5/010. |
[42] |
A. C. Newell, In "Solitons" (R. K. Bullough and P. J. Caudrey eds.), Springer Verlag, 1980. |
[43] |
S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, "Theory of Solitons: the Inverse Scattering Method," Plenum, New York, 1984. |
[44] |
A. Ramani, B. Grammaticos and R. Willox, Bilinearization and solutions of the KdV6 equation, Anal. Appl., 6 (2008), 401-412.
doi: 10.1142/S0219530508001249. |
[45] |
T. Valchev, On the Kaup-Kupershmidt equation. Completeness relations for the squared solutions, in "Nineth International Conference on Geometry, Integrability and Quantization, June 8-13 2007, Varna, Bulgaria" (eds. I. Mladenov and M. de Leon), SOFTEX, Sofia (2008), 308-319. |
[46] |
Jing Ping Wang, Extension of integrable equations, J. Phys. A: Math. Theor., 42 (2009), 362204.
doi: 10.1088/1751-8113/42/36/362004. |
[47] |
A-M. Wazwaz, The integrable KdV6 equations: Multiple soliton solutions and multiple singular soliton solutions, Applied Mathematics and Computation, 204 (2008), 963-972.
doi: 10.1016/j.amc.2008.08.007. |
[48] |
Y. Q. Yao and Y. B. Zeng, Integrable Rosochatius deformations of higher-order constrained flows and the soliton hierachy with self-consistent source, J. Phys. A: Math. Theor., 41 (2008), 295205.
doi: 10.1088/1751-8113/41/29/295205. |
[49] |
Y. Q. Yao and Y. B. Zeng, The bi-Hamiltonian structure and new solutions of KdV6 equation, Lett. Math. Phys., 86 (2008), 193-208.
doi: 10.1007/s11005-008-0281-4. |
[50] |
V. Zakharov and L. Faddeev, Korteweg-de Vries equation is a completely integrable Hamiltonian system, Func. Anal. Appl., 5 (1971), 280-287 (English). |
[51] |
Y. B. Zeng, Y. J. Shao and W. Xue, Negaton and positon solutions of the soliton equation with sself-consistent sources, J. Phys. A Math. Gen., 36 (2003), 5035-5043.
doi: 10.1088/0305-4470/36/18/308. |
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