Integration of a~non-linear Hirota type equation with additional terms

A. B. Khasanov ; R. Kh. Eshbekov ; T. G. Hasanov

Izvestiya. Mathematics , Tome 89 (2025) no. 1, pp. 196-219.

Résumé

In this paper, the inverse spectral problem method is used to integrate a Hirota type equation with additional terms in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of six times continuously differentiable periodic infinite-gap functions is proved. It is also shown that the Cauchy problem is solvable at all times for sufficiently smooth initial conditions.

Détail

Keywords: non-linear Hirota type equation with additional terms, Dirac operator, spectral data, system of Dubrovin equations, trace formulas.

Bibliographie

[1] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg–de Vries equation”, Phys. Rev. Lett., 19 (1967), 1095–1097 | DOI | Zbl

[2] L. D. Faddeev, “Properties of the $S$-matrix of the one-dimensional Schrödinger equation”, Amer. Math. Soc. Transl. Ser. 2, 65, Amer. Math. Soc., Providence, RI, 1967, 139–166 | DOI

[3] V. A. Marchenko, Sturm–Liouville operators and applications, Oper. Theory Adv. Appl., 22, Birkhäuser Verlag, Basel, 1986 | DOI | MR | Zbl

[4] B. M. Levitan, Inverse Sturm–Liouville problems, VSP, Zeist, 1987 | MR | Zbl

[5] P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves”, Comm. Pure Appl. Math., 21:5 (1968), 467–490 | DOI | MR | Zbl

[6] V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media”, Soviet Physics JETP, 34:1 (1972), 62–69

[7] M. Wadati, “The exact solution of the modified Korteweg–de Vries equation”, J. Phys. Soc. Japan, 32:6 (1972), 1681 | DOI

[8] R. Hirota, “Exact envelop-soliton solutions of a nonlinear wave equation”, J. Math. Phys., 14 (1973), 805–809 | DOI | MR | Zbl

[9] V. E. Zakharov, L. A. Takhtadzhyan, and L. D. Faddeev, “Complete description of solutions of the “sine-Gordon” equation”, Soviet Physics Dokl., 19:12 (1974), 824–826

[10] M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “Method for solving the sine-Gordon equation”, Phys. Rev. Lett., 30:25 (1973), 1262–1264 | DOI | MR

[11] K. Konno, W. Kameyama, and H. Sanuki, “Effect of weak dislocation potential on nonlinear wave propagation in anharmonic crystal”, J. Phys. Soc. Japan, 37:1 (1974), 171–176 | DOI

[12] Deng-yuan Chen, Da-jun Zhang, and Shu-fang Deng, “The novel multi-soliton solutions of the MKdV–sine Gordon equations”, J. Phys. Soc. Japan, 71:2 (2002), 658–659 | DOI | MR | Zbl

[13] A.-M. Wazwaz, “$N$-soliton solutions for the integrable modified KdV–sine-Gordon equation”, Phys. Scr., 89:6 (2014), 5–15 | DOI

[14] S. P. Popov, “Scattering of solitons by dislocations in the modified Korteweg de Vries–sine-Gordon equation”, Comput. Math. Math. Phys., 55:12 (2015), 2014–2024 | DOI | MR

[15] S. P. Popov, “Numerical analysis of soliton solutions of the modified Korteweg–de Vries–sine-Gordon equation”, Comput. Math. Math. Phys., 55:3 (2015), 437–446 | DOI | MR

[16] S. P. Popov, “Nonautonomous soliton solutions of the modified Korteweg–de Vries–sine-Gordon equation”, Comput. Math. Math. Phys., 56:11 (2016), 1929–1937 | DOI | MR

[17] Man Jia, Ji Lin, and Sen Yue Lou, “Soliton and breather molecules in few-cycle-pulse optical model”, Nonlinear Dyn., 100 (2020), 3745–3757 | DOI

[18] I. S. Frolov, “Inverse scattering problem for a Dirac system on the whole axis”, Soviet Math. Dokl., 13:1468–1472 (1972)

[19] L. P. Nizhnik and Fam Loi Vu, “The inverse scattering problem on a half-line with a non-selfconjugate potential matrix”, Ukr. Math. J., 26 (1975), 384–398 | DOI

[20] A. B. Khasanov and G. U. Urazboev, “On the sine-Gordon equation with a self-consistent source corresponding to multiple eigenvalues”, Differ. Equ., 43 (2007), 561–570 | DOI

[21] A. B. Khasanov and U. A. Khoitmetov, “On integration of Korteweg–de Vries equation in a class of rapidly decreasing complex-valued functions”, Russian Math. (Iz. VUZ), 62:3 (2018), 68–78 | DOI

[22] A. B. Khasanov and U. A. Khoitmetov, “Integration of the general loaded Korteweg–de Vries equation with an integral type source in the class of rapidly decreasing complex-valued functions”, Russian Math. (Iz. VUZ), 65:7 (2021), 43–57 | DOI

[23] A. B. Khasanov and U. A. Hoitmetov, “On integration of the loaded mKdV equation in the class of rapidly decreasing functions”, Bulletin of Irkutsk State University. Series Mathematics, 38 (2021), 19–35 | DOI | MR | Zbl

[24] A. R. Its and V. B. Matveev, “Schrödinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg–de Vries equation”, Theoret. and Math. Phys., 23:1 (1975), 343–355 | DOI

[25] B. A. Dubrovin and S. P. Novikov, “Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg–de Vries equation”, Soviet Physics JETP, 40:6 (1974), 1058–1063

[26] B. A. Dubrovin, “Periodic problems for the Korteweg–de Vries equation in the class of finite band potentials”, Funct. Anal. Appl., 9:3 (1975), 215–223 | DOI

[27] A. R. Its, “Inversion of hyperelliptic integrals, and integration of nonlinear differential equation”, Vestnik Leningrad. Univ., 7:2 (1976), 39–46 (Russian) | MR | Zbl

[28] A. R. Its and V. P. Kotljarov, “Explicit formulas for solutions og a nonlinear Schrödinger equation”, Dokl. Akad. Nauk Ukrain. SSR Ser. A , 11 (1976), 965–968 (Russian) | MR

[29] A. O. Smirnov, “Elliptic solutions of the nonlinear Schrödinger equation and the modified Korteweg–de Vries equation”, Russian Acad. Sci. Sb. Math., 82:2 (1995), 461–470 | DOI

[30] V. B. Matveev and A. O. Smirnov, “Solutions of the Ablowitz–Kaup–Newell–Segur hierarchy equations of the “rogue wave” type: a unified approach”, Theoret. and Math. Phys., 186:2 (2016), 156–182 | DOI

[31] V. B. Matveev and A. O. Smirnov, “Two-phase periodic solutions to the AKNS hierarchy equations”, J. Math. Sci. (N.Y.), 242:5 (2019), 722–741 | DOI

[32] V. B. Matveev and A. O. Smirnov, “Multiphase solutions of nonlocal symmetric reductions of equations of the AKNS hierarchy: general analysis and simplest examples”, Theoret. and Math. Phys., 204:3 (2020), 1154–1165 | DOI

[33] Yu. A. Mitropol'skii, N. N. Bogolyubov, Jr., A. K. Prikarpatskii, and V. G. Samoilenko, Integrable dynamic systems: spectral and differential-geometric aspects, Naukova Dumka, Kiev, 1987 (Russian) | MR | Zbl

[34] S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of solitons. The inverse scattering method, Contemp. Soviet Math., Consultants Bureau [Plenum], New York, 1984 | MR | Zbl

[35] F. Gesztesy and H. Holden, Soliton equations and their algebro-geometric solutions, v. 1, $(1+1)$-dimensional continuous models, Cambridge Univ. Press, Cambridge, 2003 | DOI | MR | Zbl

[36] 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., 41:3 (2005), 310–318 | DOI

[37] V. B. Matveev, “30 years of finite-gap integration theory”, Philos. Trans. Roy. Soc. A, 366:1867 (2008), 837–875 | DOI | MR | Zbl

[38] E. L. Ince, “Further investigations into the periodic Lame functions”, Proc. Roy. Soc. Edinburgh, 60 (1940), 83–99 | DOI | MR | Zbl

[39] N. I. Akhiezer, “A continuous analogue of orthogonal polynomials on a system of intervals”, Soviet Math. Dokl., 2 (1961), 1409–1412

[40] E. L. Ince, “A proof of the impossibility of the coexistence of two Mathien functions”, Proc. Cambridge Philos. Soc., 21 (1922), 117–120 | Zbl

[41] P. Djakov and B. S. Mityagin, “Instability zones of periodic 1-dimensional Schrödinger and Dirac operators”, Russian Math. Surveys, 61:4 (2006), 663–766 | DOI

[42] G. A. Mannonov and A. B. Khasanov, “Cauchy problem for the nonlinear Hirota equation in the class of periodic infinite-zone functions”, St. Petersburg Math. J., 34:5 (2023), 821–845 | DOI

[43] A. Khasanov, R. Eshbekov, and Kh. Normurodov, “Integration of a nonlinear Hirota type equation with finite density in the class of periodic functions”, Lobachevskii J. Math., 44:10 (2023), 4329–4347 | DOI | MR | Zbl

[44] A. B. Khasanov, K. N. Normurodov, and U. O. Khudaerov, “Integrating the modified Korteweg–de Vries–sine-Gordon equation in the class of periodic infinite-gap functions”, Theoret. and Math. Phys., 214:2 (2023), 170–182 | DOI

[45] A. B. Khasanov and U. O. Xudayorov, “Integration of the modified Korteweg–de Vries–Liouville equation in the class of periodic infinite-gap functions”, Math. Notes, 114:6 (2023), 1247–1259 | DOI

[46] A. B. Khasanov, Kh. N. Normurodov, and U. O. Khudayorov, “Cauchy problem for the nonlinear Liouville equation in the class of periodic infinite-gap functions”, Differ. Equ., 59:10 (2023), 1413–1426 | DOI

[47] U. B. Muminov and A. B. Khasanov, “Integration of a defocusing nonlinear Schrödinger equation with additional terms”, Theoret. and Math. Phys., 211:1 (2022), 514–531 | DOI

[48] U. B. Muminov and A. B. Khasanov, “The Cauchy problem for the defocusing nonlinear Schrödinger equation with a loaded term”, Siberian Adv. Math., 32:4 (2022), 277–298 | DOI

[49] P. G. Grinevich and I. A. Taimanov, “Spectral conservation laws for periodic nonlinear equations of the Melnikov type”, Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 224, Adv. Math. Sci., 61, Amer. Math. Soc., Providence, RI, 2008, 125–138 | DOI | MR | Zbl

[50] A. B. Hasanov and M. M. Hasanov, “Integration of the nonlinear Schrödinger equation with an additional term in the class of periodic functions”, Theoret. and Math. Phys., 199:1 (2019), 525–532 | DOI

[51] A. B. Khasanov and M. M. Matyakubov, “Integration of the nonlinear Korteweg–de Vries equation with an additional term”, Theoret. and Math. Phys., 203:2 (2020), 596–607 | DOI

[52] A. B. Khasanov and T. G. Khasanov, “Cauchy problem for the Korteweg–de Vries equation in the class of periodic infinite-gap functions”, J. Math. Sci. (N.Y.), 283:4 (2024), 674–689 | DOI

[53] A. B. Khasanov and T. Z. Allanazarova, “On the modified Korteweg–De-Vries equation with loaded term”, Ukrainian Math. J., 73:11 (2022), 1783–1809 | DOI | MR | Zbl

[54] A. V. Domrin, “Remarks on the local version of the inverse scattering method”, Proc. Steklov Inst. Math., 253 (2006), 37–50 | DOI

[55] P. Djakov and B. Mityagin, “Instability zones of a periodic 1D Dirac operator and smoothness of its potential”, Comm. Math. Phys., 259:1 (2005), 139–183 | DOI | MR | Zbl

[56] B. M. Levitan and I. S. Sargsjan, Sturm–Liouville and Dirac operators, Math. Appl. (Soviet Ser.), 59, Kluwer Acad. Publ., Dordrecht, 1991 | DOI | MR | Zbl

[57] T. V. Misjura, “Characterization of the spectra of the periodic and antiperiodic boundary value problems that are generated by the Dirac operator, I”, Teor. Funktsii Funktsional. Anal. i Prilozhen., 30, Izd. Khar'k. Univ., Khar'kov, 1978, 90–101 (Russian) | MR | Zbl

[58] A. B. Khasanov and A. B. Yakhshimuratov, “An analogue of G. Borg's inverse theorem for the Dirac operator”, Uzbek. Mat. Zh., 3 (2000), 40–46 (Russian) | MR | Zbl

[59] A. B. Khasanov and A. M. Ibragimov, “On an inverse problem for the Dirac operator with periodic potential”, Uzbek. Mat. Zh., 3-4 (2001), 48–55 (Russian) | MR

[60] S. Currie, T. T. Roth, and B. A. Watson, “Borg's periodicity theorems for first-order self-adjoint systems with complex potentials”, Proc. Edinb. Math. Soc. (2), 60:3 (2017), 615–633 | DOI | MR | Zbl

[61] E. Korotyaev and D. Mokeev, “Dubrovin equation for periodic Dirac operator on the half-line”, Appl. Anal., 101:1 (2022), 337–365 | DOI | MR | Zbl

[62] A. B. Khasanov and A. B. Yakhshimuratov, “Inverse problem on the half-line for the Sturm–Liouville operator with periodic potential”, Differ. Equ., 51:1 (2015), 23–32 | DOI

[63] I. V. Stankevič, “On an inverse problem of spectral analysis for Hill's equation”, Soviet Math. Dokl., 11 (1970), 582–586

[64] E. Trubowitz, “The inverse problem for periodic potentials”, Comm. Pure Appl. Math., 30:3 (1977), 321–337 | DOI | MR | Zbl

[65] G. Borg, “Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte”, Acta Math., 78 (1946), 1–96 | DOI | MR | Zbl

[66] A. B. Yakhshimuratov, “Integration of a higher-order nonlinear Schrödinger system with a self-consistent source in the class of periodic functions”, Theoret. and Math. Phys., 202:2 (2020), 137–149 | DOI

[67] A. A. Danielyan, B. M. Levitan, and A. B. Khasanov, “Asymptotic behavior of Weyel–Titchmarsh $m$-function in the case of the Dirac system”, Math. Notes, 50:2 (1991), 816–823 | DOI