References
[1] B.A. Babadzhanov, A.B. Khasanov, and A.B. Yakhshimuratov, On the inverse problem for a quadratic pencil of Sturm–Liouville operators with periodic potential, Differ. Equ. 41 (2005), 310–318. CrossRef

[2] B.A. Babadzhanov and A.B. Khasanov, Inverse problem for a quadratic pencil of Shturm–Liouville operators with finite-gap pereodic potential on the half-line, Differ. Equ. 43 (2007), 723–730. CrossRef

[3] B.A. Babajanov, M. Fechkan, and G.U. Urazbaev, On the periodic Toda lattice with self-consistent source, Commun. Nonlinear Sci. Numer. Simul. 22 (2015), 379–388. CrossRef

[4] B.A. Babajanov, M. Fechkan, and G.U. Urazbaev, On the periodic Toda lattice hierarchy with an integral source, Comm. Nonlinear Sci. Numer. Simul. 52 (2017), 110–123. CrossRef

[5] B.A. Babajanov and A.B. Khasanov, Periodic Toda chain with an integral source, Theoret. Math. Phys. 184 (2015), 1114–1128. CrossRef

[6] A. Cabada and A. Yakhshimuratov, The system of Kaup equations with a selfconsistent source in the class of periodic functions, Zh. Mat. Piz., Anal., Geom. 9 (2013), 287–303.

[7] C. Claude, J. Leon, and A. Latifi, Nonlinear resonant scattering and plasma instability: an integrable model, J. Math. Phys. 32 (1991), 3321–3330. CrossRef

[8] B.A. Dubrovin, The periodic problem for the Korteweg-de Vries equation in the class of finite-gap potentials, Anal. Funct. Its Appl. 9 (1975), No. 3, 41–51. CrossRef

[9] B.A. Dubrovin and S.P. Novikov, Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg-de Vries equation, J. Exp. Theor. Phys. 40 (1975), No. 6, 1058–1063

[10] P.G. Grinevich and I.A. Taimanov, Spectral conservation laws for periodic nonlinear equations of the Melnikov type Amer. Math. Soc. Transl. Ser. 2 224 (2008), 125–138. CrossRef

[11] G.Sh. Guseinov, On a quadratic pencil of Sturm–Liouville operators with periodic coefficients, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3 (1984), No. 2, 14–21.

[12] G.Sh. Guseinov, Asymptotic formulas for solutions and eigenvalues of quadratic pencil of Sturm–Liouville equations, Preprint No. 113, Inst. Phys. Akad. Nauk Azerb. SSR, Baku, 1984 (Russian).

[13] G.Sh. Guseinov, Spectrum and eigenfunction expansions of a quadratic pencil of Sturm–Liouville operators with periodic coefficients, Spectral Theory of Operators and its Applications, No. 6, Elm, Baku, 1985, 56–97.

[14] G.Sh. Guseinov, On spectral analysis of a quadratic pencil of Sturm–Liouville operators, Soviet Math. Dokl. 32 (1985), No. 3, 1859–1862.

[15] G.Sh. Guseinov, Inverse problems for a quadratic pencil of Sturm–Liouville operators on a finite interval, Spectral Theory of Operators and its Applications, No. 7, Elm, Baku, 1986, 51–101.

[16] B. Hu and T. Xia, The Binary Nonlinearization of the Super Integrable System and Its Self-Consistent Sources, Int. J. Nonlinear Sci. Numer. Simul. 18 (2017), No. 3-4, 281–297. CrossRef

[17] B. Hu and T. Xia, A Riemann–Hilbert approach to the initial-boundary value problem for Kundu–Eckhaus equation on the half line, Complex Var. Elliptic. 64 (2019), 2019–2039. CrossRef

[18] A.R. Its, Inversion of hyperelliptic integrals and integration of nonlinear differential equations, Vestn. Lening. Univ. Mat. 9 (1981), 121–129.

[19] M. Jaulent and I. Miodek, Nonlinear Evolution Equation Associated with EnergyDependent Schrodinger Potentials, Lett. Math. Phys. 1 (1976), No. 3, 243–250. CrossRef

[20] D.J. Kaup, A higher-order water-wave equation and the method for solving it, Prog. Theor. Phys. 54 (1975), 396–408. CrossRef

[21] A.B. Khasanov and A.B. Yakhshimuratov, The Korteweg-de Vries equation with a self-consistent source in the class of periodic functions, Theor. Math. Phys. 164 (2010), 1008–1015. CrossRef

[22] P.D. Lax, Periodic solutions of the KdV equations, Nonlinear Wave Motion, Lecture in Appl. Math., 15, Amer Math. Soc., Providence, RI, 1974, 85-96.

[23] J. Leon and A. Latifi, Solution of an initial-boundary value problem for coupled nonlinear waves, J. Phys. A: Math. Gen. 23 (1990), 1385–1403. CrossRef

[24] Q. Li, D.J. Zhang, and D.Y. Chen, Solving the hierarchy of the nonisospectral KdV equation with self-consistent sources via the inverse scattering transform, J. Phys. A Math. Theor. 41 (2008), 355209. CrossRef

[25] R.L. Lin, Y.B. Zeng, and W.X. Ma, Solving the KdV hierarchy with self-consistent sources by inverse scattering method Physics, J. Phys. A Math. Theor. 291 (2001), 287–298. CrossRef

[26] V.B. Matveev and M.I. Yavor, Solutions presque périodiques et a N -solitons de l’équation hydrodynamique nonlinéaire de Kaup, Ann. Inst. Henri Poincaré, Sect. A 31 (1979), 25–41.

[27] V.K. Melnikov, A direct method for deriving a multisoliton solution for the problem of interaction of waves on the x, y plane, Commun. Math. Phys. 112 (1987), 639– 652. CrossRef

[28] V.K. Melnikov, Integration method of the Korteweg-de Vries equation with a selfconsistent source, Phys. Lett. A 133 (1988), 493–506. CrossRef

[29] V.K. Melnikov, Integration of the Korteweg-de Vries equation with a source. Inverse Problems, 6 (1990), 233–246. CrossRef

[30] V.K. Melnikov, Integration of the nonlinear Schroedinger equation with a selfconsistent source, Commun. Math. Phys. 137 (1991), 359–381. CrossRef

[31] Yu.A. Mitropol’skii, N.N. Bogolyubov (Jr.), A.K. Prikarpatskii, and V.G. Samoilenko, Integrable Dynamical Systems: Spectral and Differential Geometric Aspects, Naukova Dumka, Kiev, 1987 (Russian).

[32] A.O. Smirnov, Real finite-gap regular solutions of the Kaup-Boussinesq equation, Theor. Math. Phys. 66 (1986), 19–31. CrossRef

[33] A.O. Smirnov, A matrix analogue of Appell’s theorem and reductions of multidimensional Riemann theta-functions Math. USSR-Sb. 61 (1988), 379–388. CrossRef

[34] V.S. Shchesnovich and E.V. Doktorov, Modified Manakov system with selfconsistent source, Phys. Lett. A 213 (1996), 23–31. CrossRef

[35] G. Urazboev and A. Babadjanova, On the integration of the matrix modified Korteweg-de Vries equation with self-consistent source, Tamkang J. Math. 50 (2019), No. 3, 281–291. CrossRef

[36] A.B. Yakhshimuratov, The nonlinear Schrodinger equation with a self-consistent source in the class of periodic functions, Math. Phys., Anal. and Geom. 14 (2011), 153–169. CrossRef

[37] A.B. Yakhshimuratov and B.A. Babajanov, Integration of equation of Kaup system kind with a self-consistent source in the class of periodic functions, Ufa Math. J. 12 (2020), No. 1, 104–114. CrossRef

[38] R. Yamilov, Symmetries as integrability criteria for differential difference equations, J. Phys. A: Math. Gen. 39 (2006), R541–R623. CrossRef

[39] Y. Zeng, W.X. Ma, and Runliang Lin, Integration of the soliton hierarchy with self-consistent sources, J. Math. Phys. 41 (2000), 5453–5488. CrossRef

[40] Y.B. Zeng, W.X. Ma, and Y. J. Shao, Two binary Darboux transformations for the KdV hierarchy with self-consistent sources, J. Math. Phys. 42 (2001), 2113–2128. CrossRef

[41] Y.B. Zeng, Y.J. Shao, and W.M. Xue, Negaton and positon solutions of the soliton equation with self-consistent sources, J. Phys. A: Math. Gen. 36 (2003), 5035–5043. CrossRef

[42] D.J. Zhang and D.Y. Chen, The N -soliton solutions of the sine-Gordon equation with self-consistent sources, Physics A 321 (2003), 467–481. CrossRef