Integration of a nonlinear sine-Gordon–Liouville-type equation in the class of periodic infinite-gap functions

  • A. B. Khasanov Samarkand State University, Uzbekistan https://orcid.org/0000-0003-2571-5179
  • Kh. N. Normurodov Samarkand State University, Uzbekistan
  • T. G. Khasanov Urgench State University, Uzbekistan
Keywords: sine-Gordon type equation, Liouville equation, Dirac operator, spectral data, system of Dubrovin differential equations, trace formulas.

Abstract

UDC 517.9

The method of inverse spectral problem  is used to integrate a nonlinear sine-Gordon–Liouville-type equation in the class of periodic infinite-gap functions. The evolution of the spectral data for the periodic Dirac operator  is introduced in which the coefficient of the Dirac operator is a solution of a nonlinear sine-Gordon–Liouville-type  equation. The solvability of the Cauchy problemc is proved for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin differential equations and the first-trace formula satisfies the sine-Gordon–Liouville-type equation.

References

A. V. Zhiber, R. D. Murtazina, I. T. Habibullin, A. B. Shabat, Harakteristicheskie kol'ca Li i nelineinye integriruemye uravnenija, Moscow, Izhevsk (2012).

C. Gardner, I. Green, M. Kruskal, R. Miura, A method for solving the Korteweg–de Vries equation, Phys. Rev. Lett., 19, 1095–1098 (1967); https://doi.org/10.1103/PhysRevLett.19.1095. DOI: https://doi.org/10.1103/PhysRevLett.19.1095

L. D. Faddeev, Properties of the $S$-matrix of the one-dimensional Schrödinger equation, Tr. Mat. Inst. Steklova, 73, 314–336 (1964); https://www.mathnet.ru/links/f792a8563be600a674d64326eca0f1d1/tm1633.pdf.

V. A. Marchenko, Sturm–Liouville operators and their applications, Naukova Dumka, Kiev (1977).

B. M. Levitan, Inverse Sturm–Liouville problems, Nauka, Moscow (1984).

P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure and Appl. Math., 21, 467–490 (1968); https://doi.org/10.1002/cpa.3160210503. DOI: https://doi.org/10.1002/cpa.3160210503

V. E. Zakharov, A. B. Shabat, Tochnaya teoriya dvumernoi samofokusirovki v odnomernoi avtomodulyatsii voln v nelineinykh sredakh, JETF, 61, 118–134 (1971); https://doi.org/10.31857/S0044466922060072.

M. Wadati, The exact solution of the modified Korteweg–de Vries equation, J. Phys. Soc. Jap., 32, №6, 44–47 (1972); https://doi.org/10.1142/S0129055X14300064. DOI: https://doi.org/10.1143/JPSJ.32.1681

R. Hirota, Exact envelop-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14, 805–809 (1973); https://doi.org/10.1063/1.1666399. DOI: https://doi.org/10.1063/1.1666399

V. E. Zakharov, L. A. Takhtadjyan, L. D. Faddeev, Polnoe opisanie reshenii ``Sin-Gordon'' uravnenija, Dokl. AN SSSR, 219, № 6, 1334–1337 (1974); https://www.mathnet.ru/rus/dan38752.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, Method for solving the sine-Gordon equation, Phys. Rev. Lett., 30, №25, 1262–1264 (1973); https://doi.org/10.1103/PhysRevLett.30.1262. DOI: https://doi.org/10.1103/PhysRevLett.30.1262

K. Konno, W. Kameyama, H. Sanuki, Effect of weak dislocation potential on nonlinear wave propagation in anharmonic cristal, J. Phys. Soc. Jap., 37, №1, 171–176 (1974); https://doi.org/10.1143/JPSJ. 37.171. DOI: https://doi.org/10.1143/JPSJ.37.171

Deng-yuan Chen, Da-jun Zhang, Shu-fang Deng, The Novel multi-soliton solutions of the MKdV–Sine Gordon equations, J. Phys. Soc. Jap., 71, 658–659 (2002); http://dx.doi.org/10.1143/JPSJ. 71.658. DOI: https://doi.org/10.1143/JPSJ.71.658

Abdul-Majid Wazwaz, $N$-soliton solutions for the integrable modified KdV–sine-Gordon equation, Phys. Scr., 89, №6, Article 065805 (2014); http://dx.doi.org/10.1088/0031-8949/89/6/065805. DOI: https://doi.org/10.1088/0031-8949/89/6/065805

S. P. Popov, Scattering of solitons by dislocations in the modified Korteweg–de Vries–sine-Gordon equation, Comput. Math. and Math. Phys., 55, №12, 2014–2024 (2015); http://dx.doi.org/10.1134/S0965542515120143. DOI: https://doi.org/10.1134/S0965542515120143

S. P. Popov, Numerical analysis of soliton solutions of the modified Korteweg–de Vries–sine-Gordon equation, Comput. Math. and Math. Phys., 55, №3, 437–446 (2015); https://doi.org/10.1134/S0965542515030136. DOI: https://doi.org/10.1134/S0965542515030136

S. P. Popov, Nonautonomous soliton solutions of the modified Korteweg–de Vries–sine-Gordon equation, Comput. Math. and Math. Phys., 56, №11, 1929–1937 (2016); https://doi.org/10.1134/S0965542516110105. DOI: https://doi.org/10.1134/S0965542516110105

Man Jia, Ji Lin, Sen Yue Lou, Soliton and breather molecules in few-cycle-pulse optical model, Nonlinear Dynam., 100, 3745–3757 (2020); https://doi.org/10.1007/s11071-020-05695-3. DOI: https://doi.org/10.1007/s11071-020-05695-3

I. S. Frolov, Obratnaja zadacha rassejanija dlja sistemy Diraka na vsei osi, Dokl. AN SSSR, 207, №1, 44–47 (1972).

L. P. Nijnik, Lou Vu Fam, Inverse scattering problem on a semiaxis with a non-self-adjoint potential matrix, Ukr. Math. J., 26, 469–485 (1974).

L. A. Takhtadjyan, L. D. Faddeev, Hamiltonian approach in the theory of solitons, Nauka, Moscow (1984); https://www.mathnet.ru/links/a3e594a10fd314a3d0584e710a86749d/aa156.pdf.

A. B. Khasanov, Obratnaya zadacha teorii rasseyaniya dlya sistemy dvux nesamosopryajennykh differentsialnykh uravnenii pervogo poryadka, Dokl. AN SSSR, 277, 559–562 (1984).

A. B. Khasanov, G. U. Urazboev, On the sine-Gordon equation with a self-consistent source corresponding to multiple eigenvalues, Different. Equat., 43, №4, 561–570 (2007); http://dx.doi.org/10.1134/S0012266107040143. DOI: https://doi.org/10.1134/S0012266107040143

A. B. Khasanov, G. U. Urazboev, On the sine-Gordon equation with a self-consistent source of the integral type, Sib. Adv. Math., 19, №1, 13–23 (2009); http://dx.doi.org/10.3103/S1055134409010027. DOI: https://doi.org/10.3103/S1055134409010027

A. B. Hasanov, U. A. Hoitmetov, On integration of the loaded Korteweg–de Vries equation in the class of rapidly decreasing functions, Proc. Inst. Math. and Mech., 47, №2, 250–261 (2021). DOI: https://doi.org/10.30546/2409-4994.47.2.250

A. B. Khasanov, U. A. Hoitmetov, 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, 43–57 (2021); https://doi.org/10.3103/S1066369X21070069. DOI: https://doi.org/10.3103/S1066369X21070069

A. B. Khasanov, U. A. Hoitmetov, On integration of the loaded mKdV equation in the class of rapidly decreasing functions, Bull. Irkutsk State Univ., Ser. Math., 38, 19–35 (2021); https://doi.org/10.26516/1997-7670.2021.38.19. DOI: https://doi.org/10.26516/1997-7670.2021.38.19

A. R. Its, V. B. Matveev, Operatory Shredingera s konechnozonnym spektrom i $N$-solitonnye resheniya uravneniya Kortevega–de Friza, TMP, 23, 51–68 (1975).

B. A. Dubrovin, S. P. Novikov, Periodicheskie i uslovno periodicheskie analogi mnogosolitonnykh reshenii uravneniya Kortevega–de Friza, JETF, 67, 2131–2143 (1974).

A. R. Its, Inversion of hyperelliptic integrals, and integration of nonlinear differential equations, Vestn. Leningrad. Univ. Mat. Mekh. Astronom., 7, 39–46 (1976).

A. R. Its, V. P. Kotlyarov, Explicit formulas for solutions of a nonlinear Schrödinger equation, Dokl. Akad. Nauk Ukr. SSR Ser. A, 11, 965–968 (1976); https://doi.org/10.1070/SM1995v082n02ABEH003575.

A. O. Smirnov, Elliptic solutions of the nonlinear Schrödinger equation and the modified Korteweg– de Vries equation, Sb. Math., 185, 103–114 (1994); https://doi.org/10.1070/SM1995v082n02ABEH003575. DOI: https://doi.org/10.1070/SM1995v082n02ABEH003575

V. B. Matveev, A. O. Smirnov, Solutions of the Ablowitz–Kaup–Newell–Segur hierarchy equations of the rogue wave type: a unified approach, Theor. and Math. Phys., 186, 156–182 (2016); https://doi.org/10.1134/S0040577916020033. DOI: https://doi.org/10.1134/S0040577916020033

V. B. Matveev, A. O. Smirnov, Two-phase periodic solutions to the AKNS hierarchy equations, J. Math. Sci., 242, 722–741 (2019); https://doi.org/10.1007/s10958-019-04510-8. DOI: https://doi.org/10.1007/s10958-019-04510-8

V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskiy, Soliton theory: inverse problem method, Nauka, Moscow (1980).

Yu. A. Mitropolskiy, N. N. Bogolyubov (ml.), A. K. Prikarpatskiy, V. G. Samoylenko, Integriruemye dinamicheskie sistemy: spektralnye i differentsialno-geometricheskie aspekty, Naukova Dumka, Kiev (1987).

V. B. Matveev, 30 years of finite-gap integration theory, Philos. Trans. Roy. Soc. A, 366, 837–875 (2018); https://doi.org/10.1098/rsta.2007.2055. DOI: https://doi.org/10.1098/rsta.2007.2055

A. B. Khasanov, A. B. Yakhshimuratov, Inverse problem on the half-line for the Sturm–Liouville operator with periodic potential, Different. Equat., 51, №1, 23–32 (2015); http://dx.doi.org/10.1134/S0012266115010036. DOI: https://doi.org/10.1134/S0012266115010036

B. M. Levitan, A. B. Khasanov, Estimation of the Cauchy function for finite-zone nonperiodic potentials, Funct. Anal. and Appl., 26, 91–98 (1992); https://doi.org/10.1007/BF01075268. DOI: https://doi.org/10.1007/BF01075268

E. L. Ince, A proof of the impossibility of the coexistence of two Mathieu functions, Proc. Cambridge Phil. Soc., 21, 117–120 (1922).

P. B. Djakov, B. S. Mityagin, Instability zones of periodic 1-dimensional Schrödinger and Dirac operators, Russian Math. Surveys, 61, 77–182 (2006); https://doi.org/10.1070/RM2006v061n04ABEH004343. DOI: https://doi.org/10.1070/RM2006v061n04ABEH004343

G. A. Mannonov, A. B. Khasanov, The Cauchy problem for the nonlinear Hirota equation in the class of periodic infinite-gap functions, Algebra and Anal., 34, 139–172 (2022); https://www.mathnet.ru/rus/aa1833. DOI: https://doi.org/10.1090/spmj/1780

A. B. Khasanov, Kh. N. Normurodov, U. O. Khudoyorov, Integrating the modified Korteweg–de Vries–sine-Gordon equation in the class of periodic infinite-gap functions, TMP, 214, 170–182 (2023); https://doi.org/10.1134/ S0040577923020022. DOI: https://doi.org/10.1134/S0040577923020022

A. B. Hasanov, M. M. Hasanov, Integration of the nonlinear Schrödinger equation with an additional term in the class of periodic functions, TMP, 199, №1, 525–532 (2019); https://doi.org/10.1134/S0040577919040044. DOI: https://doi.org/10.1134/S0040577919040044

A. B. Khasanov, M. M. Matyakubov, Integration of the nonlinear Korteweg–de Vries equation with an additional term, TMP, 203, №2, 596–607 (2020); https://doi.org/10.1134/S0040577920050037. DOI: https://doi.org/10.1134/S0040577920050037

A. B. Khasanov, T. G. Khasanov, The Cauchy problem for the Korteweg–de Vries equation in the class of periodic infinite-gap functions, Zap. Nauchn. Sem. POMI, 506, 258–279 (2021); https://www.mathnet.ru/rus/znsl7154.

A. B. Khasanov, T. G. Khasanov, Integration of a nonlinear Korteweg–de Vries equation with a loaded term and a source, J. Appl. and Ind. Math., 16, №2, 227–239 (2022); https://doi.org/10.33048/SIBJIM.2021.25.209. DOI: https://doi.org/10.1134/S1990478922020053

A. V. Domrin, Remarks on the local version of the inverse scattering method, Proc. Steklov Inst. Math., 253, 37–50 (2006); https://doi.org/10.1134/S0081543806020040. DOI: https://doi.org/10.1134/S0081543806020040

A. B. Khasanov, A. B. Yakhshimuratov, The almost-periodicity of infinite-gap potensials of the Dirac operator, Dokl. Math., 54, №2, 767–769 (1996).

U. B. Muminov, A. B. Khasanov, Integration of a defocusing nonlinear Schrödinger equation with additional terms, Theoret. and Math. Phys., 211, №1, 514–531 (2022); https://doi.org/10.1134/S0040577922040067. DOI: https://doi.org/10.1134/S0040577922040067

U. B. Muminov, A. B. Khasanov, The Cauchy problem for the defocusing nonlinear Schrödinger equation with a loaded term, Sib. Adv. Math., 32, №4, 277–298 (2022); http://dx.doi.org/10.1134/S1055134422040046. DOI: https://doi.org/10.1134/S1055134422040046

A. B. Khasanov, T. Zh. Allanazarova, On the modified Korteweg–de-Vries equation with loaded term, Ukr. Math. J., 73, 1783–1809 (2022); http://dx.doi.org/10.1007/s11253-022-02030-4. DOI: https://doi.org/10.1007/s11253-022-02030-4

B. A. Babazhanov, A. B. Khasanov, Integration of equation of Toda periodic chain kind, Ufa Math. J., 9, №2, 17–24 (2017); http://dx.doi.org/10.13108/2017-9-2-17. DOI: https://doi.org/10.13108/2017-9-2-17

H. McKean, E. Trubowitz, Hill's operator and hyperelliptic function theory in the presence of infinitely many branchpoints, Commun. Pure and Appl. Math., 29, 143–226 (1976); https://doi.org/10.1002/CPA.3160290203. DOI: https://doi.org/10.1002/cpa.3160290203

H. McKean, E. Trubowitz, Hill's surfaces and their theta functions, Bull. Amer. Math. Soc., 84, 1052–1085 (1978). DOI: https://doi.org/10.1090/S0002-9904-1978-14542-X

P. D. Lax, Almost periodic solutions of the KdV equation, SIAM

Rev., 18, №3, 351–375 (1976); https://doi.org/ 10.1137/1018074. DOI: https://doi.org/10.1137/1018074

B. M. Levitan, Pochti periodichnost' beskonechno-zonnykh potencialov, Izv. AN SSSR, Ser. Mat., 45, №2, 291–320 (1981); https://doi.org/10.1070/IM1982v018n02ABEH001388. DOI: https://doi.org/10.1070/IM1982v018n02ABEH001388

A. A. Danielyan, B. M. Levitan, A. B. Khasanov, Asymptotic behavior of Weyl–Titchmarsh $m$-function in the case of the Dirac system, Math. Notes, 50, №2, 816–823 (1991); https://doi.org/10.1007/BF01157568. DOI: https://doi.org/10.1007/BF01157568

T. V. Misjura, Harakteristika spektrov periodicheskoi i antiperiodicheskoi kraevykh zadach, porozhdaemykh operaciei Diraka I, Teorija funkcii, funkcional'nyi analiz i ikh prilozhenija, 30, Vysshaja Shkola, Khar'kov, 90–101 (1978); Kharakteristika spektrov periodicheskoi i antiperiodicheskoi kraevykh zadach, porozhdaemykh operaciei Diraka II, 31, 102–109 (1979).

A. B. Khasanov, A. M. Ibragimov, Ob obratnoi zadache dlja operatora Diraka s periodicheskim potentsialom, Uzb. Mat. Zh., № 3-4, 48–55 (2001).

A. B. Khasanov, A. B. Yahshimuratov, Analog obratnoi teoremy G. Borga dlja operatora Diraka, Uzb. Mat. Zh., № 3-4, 40–46 (2000).

S. Currie, T. Roth, B. Watson, Borg's periodicity theorems for first-order selfadjoint systems with complex potentials, Proc. Edinb. Math. Soc., 60, №3, 615–633 (2017); https://doi.org/10.1017/S0013091516000389. DOI: https://doi.org/10.1017/S0013091516000389

D. Battig, B. Grebert, J. C. Guillot, T. Kappeler, Folation of phase space for the cubic non-linear Schrödinger equation, Compos. Math., 85, №2, 163–199 (1993); http://dx.doi.org/10.5167/uzh-22665.

B. Grebert, J. C. Guillot, Gap of one dimensional periodic AKNS systems, Forum Math., 5, №5, 459–504 (1993); http://dx.doi.org/10.1515/form.1993.5.459. DOI: https://doi.org/10.1515/form.1993.5.459

E. Korotayev, Inverse problem and estimates for periodic Zakharov–Shabat systems, J. reine and angew. Math., 583, 87–115 (2005); https://doi.org/10.1515/crll.2005.2005.583.87. DOI: https://doi.org/10.1515/crll.2005.2005.583.87

E. Korotayev, D. Mokeev, Dubrovin equation for periodic Dirac operator on the half-line, Appl. Anal., 101, №1, 1–29 (2020); https://doi.org/10.1080/00036811.2020.1742882. DOI: https://doi.org/10.1080/00036811.2020.1742882

E. Trubowitz, The inverse problem for periodic potential, Commun. Pure and Appl. Math., 30, 321–337 (1977); https://doi.org/10.1002/cpa.3160300305. DOI: https://doi.org/10.1002/cpa.3160300305

I. V. Stankevich, Ob odnoi obratnoi zadache spektral'nogo analiza dlja uravnenija Hilla, Dokl. AN SSSR, 192, №1, 34–37 (1970); https://www.mathnet.ru/rus/dan35384.

N. I. Ahiezer, Kontinual'nyi analog ortogonal'nykh mnogochlenov na sisteme intervalov, Dokl. AN SSSR, 144, №2, 262–266 (1961); https://www.mathnet.ru/rus/dan25745.

H. Flaschka, On the inverse problem for Hill's operator, Arch. Ration. Mech. and Anal., 59, №4, 293–309 (1975); https://doi.org/10.1007/BF00250422. DOI: https://doi.org/10.1007/BF00250422

Published
04.09.2024
How to Cite
KhasanovA. B., NormurodovK. N., and KhasanovT. G. “Integration of a Nonlinear Sine-Gordon–Liouville-Type Equation in the Class of Periodic Infinite-Gap Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, no. 8, Sept. 2024, pp. 1217 -34, doi:10.3842/umzh.v76i8.7610.
Section
Research articles