Geodesic
The Cauchy problem for the defocusing nonlinear Schr\"odinger equation with a loaded term
Matematičeskie trudy, Tome 25 (2022) no. 1, pp. 102-133.

Voir la notice de l'article provenant de la source Math-Net.Ru

The method of inverse spectral problems is applied for integrating the defocusing nonlinear Scrödinger (DNS) equation with loaded terms in the class of infinite-gap periodic functions. We describe the evolution of the spectral data for a periodic Dirac operator whose coefficient is a solution to the DNS equation with loaded terms. We prove the following assertions. (1) It the initial function is real-valued, $\pi$-periodic, and analytic then the solution of the Cauchy problem for the DNS equation with loaded terms is a real-valued analytic function in $x$. (2) If $\pi/2$ is the period (or antiperiod) of the initial function then $\pi/2$ is the period (antiperiod) of the solution of the Cauchy problem problem with respect to $x$. 
@article{MT_2022_25_1_a4,
     author = {U. B. Muminov and A. B. Khasanov},
     title = {The {Cauchy} problem for the defocusing nonlinear {Schr\"odinger} equation with a loaded term},
     journal = {Matemati\v{c}eskie trudy},
     pages = {102--133},
     publisher = {mathdoc},
     volume = {25},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2022_25_1_a4/}
}
TY  - JOUR
AU  - U. B. Muminov
AU  - A. B. Khasanov
TI  - The Cauchy problem for the defocusing nonlinear Schr\"odinger equation with a loaded term
JO  - Matematičeskie trudy
PY  - 2022
SP  - 102
EP  - 133
VL  - 25
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2022_25_1_a4/
LA  - ru
ID  - MT_2022_25_1_a4
ER  - 
%0 Journal Article
%A U. B. Muminov
%A A. B. Khasanov
%T The Cauchy problem for the defocusing nonlinear Schr\"odinger equation with a loaded term
%J Matematičeskie trudy
%D 2022
%P 102-133
%V 25
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2022_25_1_a4/
%G ru
%F MT_2022_25_1_a4
U. B. Muminov; A. B. Khasanov. The Cauchy problem for the defocusing nonlinear Schr\"odinger equation with a loaded term. Matematičeskie trudy, Tome 25 (2022) no. 1, pp. 102-133. http://geodesic.mathdoc.fr/item/MT_2022_25_1_a4/

[1] Bikbaev R. F., “Vremennaya asimptotika resheniya nelineinogo uravneniya Shredingera s granichnymi usloviyami tipa «stupenki»”, TMF, 81:1 (1989), 3–11 | MR | Zbl

[2] Bikbaev R. F., Sharipov R. A., “Asimptotika pri $t\to\infty$ resheniya zadachi Koshi dlya uravneniya Kortevega-de Friza v klasse potentsialov s konechnozonnym povedeniem pri $x\to\pm\infty$”, TMF, 78:3 (1989), 345–356 | MR | Zbl

[3] Dzhakov P. B., Mityagin B. S., “Zony neustoichivosti odnomernykh periodicheskikh operatorov Shrëdingera i Diraka”, UMN, 61:4(370) (2006), 77–182 | DOI | MR | Zbl

[4] Domrin A. V., “Zamechaniya o lokalnom variante metoda obratnoi zadachi rasseyaniya”, Tr. MIRAN, 253, 2006, 46–60 | Zbl

[5] Domrin A. V., “O veschestvenno-analiticheskikh resheniyakh nelineinogo uravneniya Shrëdingera”, Tr. MMO, 75, no. 2, 2014, 205–218 | Zbl

[6] Dubrovin B. A., Novikov S. P., “Periodicheskii i uslovno periodicheskii analogi mnogosolitonnykh reshenii uravneniya Kortevega-de Friza”, ZhETF, 67:12 (1974), 2131–2143

[7] Zakharov V. E., Manakov S. V., Novikov S. P., Pitaevskii L. P., Teoriya solitonov: metod obratnoi zadachi, Nauka, M., 1980

[8] Zakharov V. E., Takhtadzhyan L. A., Faddeev L. D., “Polnoe opisanie reshenii «Sin-Gordon» uravneniya”, Dokl. AN SSSR, 219:6 (1974), 1334–1337 | Zbl

[9] Zakharov V. E., Shabat A. B., “Tochnaya teoriya dvumernoi samofokusirovki v odnomernoi avtomodulyatsii voln v nelineinykh sredakh”, ZhETF, 61:1 (1971), 118–134 | MR

[10] Its A. R., “Obraschenie giperellipticheskikh integralov i integrirovanie nelineinykh differentsialnykh uravnenii”, Vestn. Leningr. un-ta. Ser. Matem. Mekhan. Astron., 7:2 (1976), 39–46 | Zbl

[11] Its A. R., Kotlyarov V. P., “Yavnye formuly dlya reshenii nelineinogo uravneniya Shredingera”, Dokl. AN USSR. Ser. A, 11 (1976), 965–968

[12] Its A. R., Matveev V. B., “Operatory Shredingera s konechnozonnym spektrom i $N$-solitonnye resheniya uravneniya Kortevega-de Friza”, TMF, 23:1 (1975), 51–68 | MR

[13] Kozhanov A. I., “Nelineinye nagruzhennye uravneniya i obratnye zadachi”, Zh. vychisl. matem. i matem. fiz., 44 (2004), 694–716 | Zbl

[14] Levitan B. M., Obratnye zadachi Shturma — Liuvillya, Nauka, M., 1984 | MR

[15] Levitan B. M., Sargsyan I. S., Operatory Shturma — Liuvillya i Diraka, Nauka, M., 1988 | MR

[16] Levitan B. M., Khasanov A. B., “Otsenka funktsii Koshi v sluchae konechnozonnykh neperiodicheskikh potentsialov”, Funkts. analiz i ego pril., 26:2 (1992), 18–28 | MR | Zbl

[17] Marchenko V. A., Operatory Shturma — Liuvillya i ikh prilozheniya, Naukova dumka, Kiev, 1977 | MR

[18] Misyura T. V., “Kharakteristika spektrov periodicheskikh i antiperiodicheskikh kraevykh zadach, porozhdaemykh operatsiei Diraka. I”, Teoriya funktsii, funktsionalnyi analiz i ikh prilozheniya, 30, Vischa shkola, Kharkov, 1978, 90–101 | MR

[19] Mitrapolskii Yu. A., Bogolyubov N. N (ml.), Prikarpatskii A. K., Samoilenko V. G., Integriruemye dinamicheskie sistemy: spektralnye i differentsialno-geometricheskie aspekty, Naukova dumka, Kiev, 1987

[20] Smirnov A. O., “Ellipticheskie resheniya nelineinogo uravneniya Shredingera i modifitsirovannogo uravneniya Kortevega-de Friza”, Matem. sb., 185:8 (1994), 103–114 | Zbl

[21] Smirnov A. O., “Ellipticheskie po $t$ resheniya nelineinogo uravneniya Shredingera”, TMF, 107:2 (1996), 188–200 | DOI | MR | Zbl

[22] Stankevich I. V., “Ob odnoi zadache spektralnogo analiza dlya uravneniya Khilla”, Dokl. AN SSSR, 192:1 (1970), 34–37 | Zbl

[23] Takhtadzhyan L. A., Faddeev L. D., Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR

[24] Faddeev L. D., “Svoistva $S$-matritsy odnomernogo uravneniya Shredingera”, Tr. MIAN SSSR, 73, 1964, 314–336 | Zbl

[25] Frolov I. S., “Obratnaya zadacha rasseyaniya dlya sistemy Diraka na vsei osi”, Dokl. AN SSSR, 207:1 (1972), 44–47 | Zbl

[26] Khasanov A. B., “Obratnaya zadacha teorii rasseyaniya dlya sistemy dvukh nesamosopryazhennykh differentsialnykh uravnenii pervogo poryadka”, Dokl. AN SSSR, 277:3 (1984), 559–562 | MR | Zbl

[27] Khasanov A. B., “Obratnaya zadacha rasseyaniya dlya vozmuschennogo konechnozonnogo operatora Shturma — Liuvillya”, Dokl. AN SSSR, 318:5 (1991), 1095–1098

[28] Khasanov A. B., Ibragimov A. M., “Ob obratnoi zadache dlya operatora Diraka s periodicheskim potentsialom”, Uzb. matem. zh., 2001, no. 3, 4, 48–55

[29] Khasanov A. B., Matyakubov M. M., “Integrirovanie nelineinogo uravneniya Kortevega-de Friza s dopolnitelnym chlenom”, TMF, 203:2 (2020), 192–204 | DOI | MR | Zbl

[30] Khasanov A. B., Urazboev G. U., “Ob uravnenii sine-Gordon s samosoglasovannym istochnikom, sootvetstvuyuschem kratnym sobstvennym znacheniyam”, Differents. uravneniya, 43:4 (2007), 544–552 | MR | Zbl

[31] Khasanov A. B., Urazboev G. U., “Ob integrirovanii uravneniya sine-Gordon s samosoglasovannym istochnikom integralnogo tipa v sluchae kratnykh sobstvennykh znachenii”, Izv. vuzov. Matem., 2009, no. 3, 55–66 | MR | Zbl

[32] Khasanov A. B., Khoitmetov U. A., “Ob integrirovanii uravneniya Kortevega-de Friza v klasse bystroubyvayuschikh kompleksnoznachnykh funktsii”, Izv. vuzov. Matem., 2018, no. 3, 79–90 | Zbl

[33] Khasanov A. B., Yakhshimuratov A. B., “Analog obratnoi teoremy G. Borga dlya operatora Diraka”, Uzb. matem. zh., 2000, no. 3, 4, 40–46 | Zbl

[34] Khasanov A. B., Yakhshimuratov A. B., “Ob uravnenii Kortevega-de Friza s samosoglasovannym istochnikom v klasse periodicheskikh funktsii”, TMF, 164:2 (2010), 214–221 | DOI | Zbl

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

[36] Babazhanov B. A. and Khasanov A. B., “Inverse problem for a quadratic pencil of Sturm–Liouville operators with finite-gap periodic potential on the half-line”, Differ. Equ., 43:6 (2007), 737–744 | DOI | MR | Zbl

[37] Borg G., “Eine Umkehrung der Sturm-Liouvillschen Eigenwertaufgable. Bestimmung der Differentialgleichung durch die Eigenwete”, Acta Math., 78 (1946), 1–96 | DOI | MR | Zbl

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

[39] Gardner C. S., Greene J. M., Kruskal M. D., and Miura R. M., “A method for solving the Korteweg-de Vries equation”, Phys. Rev. Lett., 19 (1967), 1095–1098 | DOI | MR

[40] Grinevich P. G. and Taimanov I. A., “Spectral conservation laws for periodic nonlinear equations of the Melnikov type”, Geometry, Topology, and Mathematical Physics, S. P. Novikov's Seminar: 2006–2007, American Mathematical Society Translations: Series 2, 224, 2006 | DOI | MR

[41] Ince E. L., Ordinary Differential Equations, Dover, New York, 1956 | MR

[42] Khasanov A. B. and Urazboev G. U., “On the sine-Gordon equation with a self-consistent source”, Siberian Adv. Math., 19:1 (2009), 13–23 | DOI | MR

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

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

[45] Mamedov K. A., “Integration of mKdV equation with a self-consistent source in the class of finite density functions in the case of moving eigenvalues”, Russian Math., 64:10 (2020), 66–78 | DOI | MR | Zbl

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

[47] Wadati M., “The exact solution of the modified Korteweg-de Vries equation”, J. Phys. Soc. Japan, 32:6 (1972), 44–47 | DOI | MR