On modified Korteweg – de Vries equation with a loaded term
Abstract
UDC 517.957
In this paper, the method of the inverse spectral problem is applied to finding a solution to the Cauchy problem for the modified Korteweg–de Vries equation (mKdV) in the class of periodic infinite-gap functions.
A simple derivation of the Dubrovin system of differential equations is proposed.
The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five times continuously differentiable periodic infinite-gap functions is proved.
It is shown that the sum of a uniformly converging functional series constructed from the solutions of the infinite system of Dubrovin equations and the formulas for the first trace do indeed satisfy the mKdV equation. Moreover, it was proved that:
1) if the initial function is a real $ \pi $-periodic analytic function, then the solution of the Cauchy problem for the mKdV equation with a loaded term is also a real analytic function with respect to the variable $x;$
2) if the number $ \dfrac {\pi} {2} $ is the period (antiperiod) of the original function, then $ \dfrac {\pi} {2} $ is also the period (antiperiod) in the variable $x$ of the solution to the Cauchy problem for the mKdV equation with a loaded term.
References
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/10.1137/1019101 DOI: https://doi.org/10.1103/PhysRevLett.19.1095
L. D. Faddeev, Svojstva $S$-matry`czy odnomernogo uravneny`ya Shredy`ngera, Tr. Mat. y`n-ta AN SSSR, 73, 314 – 336 (1964).
V. A. Marchenko, Operatory Shturma – Lyuvyllia y ykh prylozhenyia, Nauk. dumka, Kyev (1977).
B. M. Levytan, Obratnye zadachy Shturma – Lyuvyllia, Nauka, Moskva (1984).
P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. 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, Tochnaia teoryia dvumernoi samofokusyrovky v odnomernoi avtomoduliatsyy voln v nelyneinykh sredakh, Zhurn. eksperym. y teor. fyzyky, 61, № 1, 118 – 134 (1971).
M. Wadati, The exact solution of the modified Korteweg – de Vries equation, J. Phys. Soc. Jаpаn, 32, № 6, 44 – 47 (1972), https://doi.org/10.1143/JPSJ.34.1289 DOI: https://doi.org/10.1143/JPSJ.32.1681
V. E. Zakharov, L. A. Takhtadzhian, L. D. Faddeev, Polnoe opysanye reshenyi „sin-Gordon” uravnenyia, Dokl. AN SSSR, 219, № 6, 1334 – 1337 (1974).
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). DOI: https://doi.org/10.1103/PhysRevLett.30.1262
Y. S. Frolov, Obratnaia zadacha dlia systemы Dyraka na vsei osy, Dokl. AN SSSR, 207, № 1, 44 – 47 (1972).
L. P. Nyzhnyk, Fam Loi Vu, Obratnaia zadacha rasseianyia na poluosy s nesamosopriazhennoi potentsyalnoi matrytsei, Ukr. mat. zhurn., 26, № 4, 469 – 485 (1974).
L. A. Takhtadzhian, L. D. Faddeev, Hamyltonov podkhod v teoryy solytonov, Nauka, Moskva (1986).
V. K. Mel’nikov, Integration of the nonlinear Schr¨odinger equation with a source, Inverse Problems, 8, no. 1, 133 – 147 (1992), http://stacks.iop.org/0266-5611/8/133 DOI: https://doi.org/10.1088/0266-5611/8/1/009
V. K. Mel’nikov, Integration of the nonlinear Schr¨odinger equation with a self-consistent source, Commun. Math. Phys., 137, 359 – 381 (1991). DOI: https://doi.org/10.1007/BF02431884
Y. Shao, Y. Zeng, The solutions of the NLS equations with self-consistent sources, J. Phys. A: Math. and Gen., 38, 2441 – 2467 (2005), https://doi.org/10.1088/0305-4470/38/11/008 DOI: https://doi.org/10.1088/0305-4470/38/11/008
Y. Zeng, W. Ma, R. Lin, Integration of the soliton hierarchy with self-consistent sources, J. Math. Phys., 41, № 8, 5453 – 5489 (2000), https://doi.org/10.1063/1.533420 DOI: https://doi.org/10.1063/1.533420
A. B. Khasanov, H. U. Urazboev, Ob uravnenyy sine-Hordon s samosohlasovannym ystochnykom, sootvetstvuiushchym kratnym sobstvennym znachenyiam, Dyfferents. uravnenyia, 43, № 4, 544 – 552 (2007). DOI: https://doi.org/10.1134/S0012266107040143
A. B. Khasanov, H. U. Urazboev, Ob yntehryrovanyy uravnenyia sine-Hordon s samosohlasovannym ystochnykom yntehralnoho typa v sluchae kratnykh sobstvennykh znachenyi, Yzv. vuzov. Matematyka, № 3, 55 – 66 (2009).
A. B. Khasanov, H. U. Urazboev, Ob yntehryrovanyy uravnenyia sine-Hordon s samosohlasovannym ystochnykom, Mat. trudy, 11, № 1, 153 – 166 (2008).
A. B. Khasanov, U. A. Khoytmetov, Ob yntehryrovanyy uravnenyia Korteveha – de Fryza v klasse bystroubyvaiushchykh kompleksnoznachnykh funktsyi, Yzv. vuzov. Matematyka,№ 3, 79 – 90 (2018).
A. B. Khasanov, U. A. Khoytmetov, Yntehryrovanye obshcheho nahruzhennoho uravnenyia Korteveha – de Fryza s yntehralnym ystochnykom v klasse bystroubyvaiushchykh kompleksnoznachnykh funktsyi, Yzv. vuzov. Matematyka, № 7, 52 – 66 (2021).
A. R. Yts, V. B. Matveev, Operatory Shredynhera s konechnozonnym spektrom y $N$-solytonnye reshenyia uravnenyia Korteveha – de Fryza, Teor. y mat. fyzyka, 23, № 1, 51 – 68 (1975).
B. A. Dubrovyn, S. P. Novykov, Peryodycheskyi y uslovno peryodycheskyi analohy mnohosolytonnykh reshenyi uravnenyia Korteveha – de Fryza, Zhurn. eksperym. y teor. fyzyky, 67, № 12, 2131 – 2143 (1974).
P. D. Lax, Periodic solutions of the KdV equations, Commun. Pure and Appl. Math., 28, 141 – 188 (1975), https://doi.org/10.1002/cpa.3160280105 DOI: https://doi.org/10.1002/cpa.3160280105
A. R. Yts, V. P. Kotliarov, Yavnye formuly dlia reshenyi nelyneinoho uravnenyia Shredynhera, Dokl. AN USSR. Ser. A, № 11, 965 – 968 (1976).
A. O. Smyrnov, Ellyptycheskye reshenyia nelyneinoho uravnenyia Shredynhera y modyfytsyrovannoho uravnenyia Korteveha – de Fryza, Mat. sb., 185, № 8, 103 – 114 (1994).
Yu. A. Mytropolskyi, N. N. Boholiubov (ml.), A. K. Prykarpatskyi, V. H. Samoilenko, Yntehryruemye dynamycheskye systemy: spektralnye y dyfferentsyalno-heometrycheskye aspekty, Nauk. dumka, Kyev (1987).
V. E. Zakharov, S. V. Manakov, S. P. Novykov, L. P. Pytaevskyi, Teoryia solytonov: metod obratnoi zadachy, Nauka, Moskva (1980).
L. A. Takhtadzhian, L. D. Faddeev, Hamyltonov podkhod v teoryy solytonov, Nauka, Moskva (1986).
E. L. Ince, Ordinary differential equations, Dover, New York (1956).
P. B. Dzhakov, B. S. Mytiahyn, Zonyneustoichyvosty odnomernykh peryodycheskykh operatorov Shredynhera y Dyraka, Uspekhy mat. nauk, 61, № 4, 77 – 182 (2006). DOI: https://doi.org/10.4213/rm2121
B. A. Dubrovyn, Peryodycheskaia zadacha dlia uravnenyia Korteveha – de Fryza v klasse konechnozonnykh potentsyalov, Funktsyon. analyz y eho pryl., 9, vyp. 3, 41 – 51 (1975).
P. G. Grinevich, 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, AMS, Providence, RI, 125 – 138 (2008), https://doi.org/10.1090/trans2/224/05 DOI: https://doi.org/10.1090/trans2/224/05
A. B. Khasanov, A. B. Yakhshymuratov, Ob uravnenyy Korteveha – de Fryza s samosohlasovannym ystochnykom v klasse peryodycheskykh funktsyi, Teor. y mat. fyzyka,164, № 2, 214 – 221 (2010). DOI: https://doi.org/10.4213/tmf6535
A. B. Khasanov, M. M. Matiakubov, Yntehryrovanye nelyneinoho uravnenyia Korteveha – de Fryza s dopolnytelnym chlenom, Teor. y mat. fyzyka, 203, № 2, 192 – 204 (2020). DOI: https://doi.org/10.4213/tmf9693
H. McKean, E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presense of infinitely many branch-points, 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, Amer. Math. Soc., 84, № 9, 1052 – 1085 (1978), https://doi.org/10.1090/S0002-9904-1978-14542-X DOI: https://doi.org/10.1090/S0002-9904-1978-14542-X
M. U. Schmidt, Integrable systems and Riemann surfaces of infinite genus, Mem. Amer. Math. Soc., 122, № 581, (1996), https://doi.org/10.1090/memo/0581 DOI: https://doi.org/10.1090/memo/0581
P. 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
A. V. Domryn, Meromorfnoe prodolzhenye reshenyi solytonnykh uravnenyi, Yzv. RAN. Ser. mat., 74, vyp.3, 23 – 44 (2010). DOI: https://doi.org/10.4213/im2784
A. B. Khasanov, M. M. Khasanov, Yntehryrovanye nelyneinoho uravnenyia Shredynhera s dopolnytelnыm chlenom v klasse peryodycheskykh funktsyi, Teor. y mat. fyzyka, 199, № 1, 60 – 68 (2019). DOI: https://doi.org/10.4213/tmf9581
B. M. Levitan, I. S. Sargsyan, , Sturm-Liouville and Dirac operators (Russian), Nauka, Moscow, (1988).
A. B. Khasanov, A. M. Ybrahymov, Ob obratnoi zadache dlia operatora Dyraka s peryodycheskym potentsyalom, Uzb. mat. zhurn., № 3-4, 48 – 55 (2001).
A. B. Khasanov, A. B. Yakhshymuratov, Analoh obratnoi teoremy H. Borha dlia operatora Dyraka, Uzb. mat. zhurn., № 3-4, 40 – 46 (2000).
T. V. Mysiura, Kharakterystyka spektrov peryodycheskoi y antyperyodycheskoi kraevykh zadach, porozhdaemykh operatsyei Dyraka, I, II, Teoryia funktsyi, funktsyon. analyz y ykh pryl., vyp. 30, 90 – 101 (1978); vyp.31, 102 – 109 (1979).
L. V. Stankevych, Ob odnoi zadache spektralnoho analyza dlia uravnenyia Khylla, Dokl. AN SSSR, 192, № 1, 34 – 37 (1970).
E. Trubowitz, The inverse problem for periodic potentials, Commun. Pure and Appl. Math., 30, 321 – 337 (1977), https://doi.org/10.1002/cpa.3160300305 DOI: https://doi.org/10.1002/cpa.3160300305
G. Borg, Eine Umkehrung der Sturm – Liouvilleschen Eigenwertaufgabe, Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math., 78, 1 – 96 (1946), https://doi.org/10.1007/BF02421600 DOI: https://doi.org/10.1007/BF02421600
S. Currie, T. Roth, B. Watson, Borg’s periodicity theorems for first-order self-adjoint systems with complex potentials, Proc. Edinburgh Math. Soc., 60, № 3, 615 – 633 (2017), https://doi.org/10.1017/S0013091516000389 DOI: https://doi.org/10.1017/S0013091516000389
N. Y. Akhyezer, Kontynualnyi analoh ortohonalnykh mnohochlenov na systeme yntervalov, Dokl. AN SSSR, 141, № 2, 262 – 266 (1961).
H. Flashka, 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
H. Hochstadt, A generalization of Borg’s inverse theorem for Hill’s equation, J. Math. Anal. and Appl., 102, no. 2, 599 – 605 (1984), https://doi.org/10.1016/0022-247X(84)90195-1 DOI: https://doi.org/10.1016/0022-247X(84)90195-1
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