American Institute of Mathematical Sciences

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July  2012, 11(4): 1439-1452. doi: 10.3934/cpaa.2012.11.1439

On the integrability of KdV hierarchy with self-consistent sources

Vladimir S. Gerdjikov 1, , Georgi Grahovski 2, and  Rossen Ivanov 3,

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

Received  April 2011 Published  January 2012

Non-holonomic deformations of integrable equations of the KdV hierarchy are studied by using the expansions over the so-called ``squared solutions'' (squared eigenfunctions). Such deformations are equivalent to perturbed models with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV equation. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV hierarchy with generic self-consistent sources are analyzed. This allows one to formulate a set of conditions on the perturbation terms that preserve the integrability. The perturbation corrections to the scattering data and to the corresponding action-angle variables are studied. The analysis shows that although many nontrivial solutions of KdV equations with generic self-consistent sources can be obtained by the Inverse Scattering Transform (IST), there are solutions that, in principle, can not be obtained via IST. Examples are considered showing the complete integrability of KdV6 with perturbations that preserve the eigenvalues time-independent. In another type of examples the soliton solutions of the perturbed equations are presented where the perturbed eigenvalue depends explicitly on time. Such equations, however in general, are not completely integrable.
Keywords: soliton perturbations, Inverse scattering method, KdV6 equation., self-consistent sources, KdV hierarchy.
Mathematics Subject Classification: Primary: 37K15, 37K40, 37K55; Secondary: 35P10, 35P25, 35P3.
Citation: Vladimir S. Gerdjikov, Georgi Grahovski, Rossen Ivanov. On the integrability of KdV hierarchy with self-consistent sources. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1439-1452. doi: 10.3934/cpaa.2012.11.1439
References:
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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.  Google Scholar

[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.  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 (E-print: patt-sol/9305002). doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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G. Eilenberger, "Solitons: Mathematical Methods for Physicists," Springer Series in Solid-State Sciences. vol. 19, Springer-Verlag, Berlin, (1981).  Google Scholar

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L. D. Faddeev and L. A. Takhtajan, Poisson structure for the KdV equation, Lett. MAth. Phys., 10 (1985), 183-188. doi: 10.1007/BF00398156.  Google Scholar

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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.  Google Scholar

[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.  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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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

V. I. Karpman and E. M. Maslov, Perturbation theory for solitons, Soviet Phys. JETP, 46 (1977), 537-559.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

B. A. Kupershmidt, KdV6: an integrable system, Phys. Lett. A, 372 (2008), 2634-2639. doi: :10.1016/j.physleta.2007.12.019.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

J. Leon, Spectral transform and solitons for generalized coupled Bloch systems, J. Math. Phys., 29 (1988), 2012-2019. doi: 10.1063/1.527859.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

A. C. Newell, In "Solitons" (R. K. Bullough and P. J. Caudrey eds.), Springer Verlag, 1980.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

Jing Ping Wang, Extension of integrable equations, J. Phys. A: Math. Theor., 42 (2009), 362204. doi: 10.1088/1751-8113/42/36/362004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[50]

V. Zakharov and L. Faddeev, Korteweg-de Vries equation is a completely integrable Hamiltonian system, Func. Anal. Appl., 5 (1971), 280-287 (English).  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

G. Eilenberger, "Solitons: Mathematical Methods for Physicists," Springer Series in Solid-State Sciences. vol. 19, Springer-Verlag, Berlin, (1981).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

V. I. Karpman and E. M. Maslov, Perturbation theory for solitons, Soviet Phys. JETP, 46 (1977), 537-559.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

B. A. Kupershmidt, KdV6: an integrable system, Phys. Lett. A, 372 (2008), 2634-2639. doi: :10.1016/j.physleta.2007.12.019.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

J. Leon, Spectral transform and solitons for generalized coupled Bloch systems, J. Math. Phys., 29 (1988), 2012-2019. doi: 10.1063/1.527859.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

A. C. Newell, In "Solitons" (R. K. Bullough and P. J. Caudrey eds.), Springer Verlag, 1980.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

Jing Ping Wang, Extension of integrable equations, J. Phys. A: Math. Theor., 42 (2009), 362204. doi: 10.1088/1751-8113/42/36/362004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[50]

V. Zakharov and L. Faddeev, Korteweg-de Vries equation is a completely integrable Hamiltonian system, Func. Anal. Appl., 5 (1971), 280-287 (English).  Google Scholar

[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.  Google Scholar

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