Pub Date : 2024-04-24DOI: 10.26907/0021-3446-2024-4-80-88
Sh. T. Ishmukhametov, B. Mubarakov, R. Rubtsova, E. V. Oleynikova
The Baillie PSW hypothesis was formulated in 1980 and was named after the authors R. Baillie, C. Pomerance, J. Selfridge and S. Wagstaff. The hypothesis is related to the problem of the existence of odd numbers n equiv pm 2 (mod 5), which are both Fermat and Lucas-pseudoprimes (in short, FL-pseudoprimes). A Fermat pseudoprime to base a is a composite number n satisfying the condition an - 1 equiv 1 (mod n). Base a is chosen to be equal to 2. A Lucas pseudoprime is a composite n satisfying Fn - e(n) equiv 0 (mod n), where n(e) is the Legendre symbol e(n) = bigl( n 5 bigr) , Fm the mth term of the Fibonacci series. According to Baillie’s PSW conjecture, there are no FL-pseudoprimes. If the hypothesis is true, the combined primality test checking Fermat and Lucas conditions for odd numbers not divisible by 5 gives the correct answer for all numbers of the form n equiv pm 2 (mod 5), which generates a new deterministic polynomial primality test detecting the primality of 60 percent of all odd numbers in just two checks. In this work, we continue the study of FL-pseudoprimes, started in our article "On a combined primality test" published in the journal "Izvestia VUZov.Matematika" No. 12 in 2022. We have established new restrictions on probable FL-pseudoprimes and described new algorithms for checking FL-primality, and with the help of them we proved the absence of such numbers up to the boundary B = 1021, which is more than 30 times larger than the previously known boundary 264 found by J. Gilchrist in 2013. An inaccuracy in the formulation of theorem 4 in the mentioned article has also been corrected.
Baillie PSW 假设提出于 1980 年，以作者 R. Baillie、C. Pomerance、J. Selfridge 和 S. Wagstaff 的名字命名。该假说与存在奇数 n equiv pm 2 (mod 5) 的问题有关，这些奇数既是费马假素数，又是卢卡斯假素数（简称 FL 假素数）。以 a 为底数的费马假素数是满足 an - 1 equiv 1 (mod n) 条件的合数 n。卢卡斯伪素数是满足 Fn - e(n) equiv 0 (mod n) 条件的合数 n，其中 n(e) 是勒让德符号 e(n) = bigl( n 5 bigr) ，Fm 是斐波那契数列的第 m 项。根据贝利的 PSW 猜想，不存在 FL 伪素数。如果假设成立，那么对不能被 5 整除的奇数进行费马条件和卢卡斯条件的联合原始性检验，就能对所有 n equiv pm 2 (mod 5) 形式的数给出正确答案，这就产生了一种新的确定性多项式原始性检验，只需两次检验就能检测出 60% 的奇数的原始性。在这项工作中，我们将继续研究 FL 伪素数，我们的文章《论组合式初等性检验》于 2022 年发表在《Izvestia VUZov.Matematika》杂志第 12 期上。我们建立了对可能的 FL 伪素数的新限制，并描述了检查 FL 原始性的新算法，在这些算法的帮助下，我们证明了在 B = 1021 边界以内不存在这样的数，这比 J. Gilchrist 在 2013 年发现的已知边界 264 大 30 多倍。我们还纠正了上述文章中定理 4 表述的不准确之处。
{"title":"On the Baillie PSW-conjecture","authors":"Sh. T. Ishmukhametov, B. Mubarakov, R. Rubtsova, E. V. Oleynikova","doi":"10.26907/0021-3446-2024-4-80-88","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-4-80-88","url":null,"abstract":"The Baillie PSW hypothesis was formulated in 1980 and was named after the authors R. Baillie, C. Pomerance, J. Selfridge and S. Wagstaff. The hypothesis is related to the problem of the existence of odd numbers n equiv pm 2 (mod 5), which are both Fermat and Lucas-pseudoprimes (in short, FL-pseudoprimes). A Fermat pseudoprime to base a is a composite number n satisfying the condition an - 1 equiv 1 (mod n). Base a is chosen to be equal to 2. A Lucas pseudoprime is a composite n satisfying Fn - e(n) equiv 0 (mod n), where n(e) is the Legendre symbol e(n) = bigl( n 5 bigr) , Fm the mth term of the Fibonacci series. According to Baillie’s PSW conjecture, there are no FL-pseudoprimes. If the hypothesis is true, the combined primality test checking Fermat and Lucas conditions for odd numbers not divisible by 5 gives the correct answer for all numbers of the form n equiv pm 2 (mod 5), which generates a new deterministic polynomial primality test detecting the primality of 60 percent of all odd numbers in just two checks. In this work, we continue the study of FL-pseudoprimes, started in our article \"On a combined primality test\" published in the journal \"Izvestia VUZov.Matematika\" No. 12 in 2022. We have established new restrictions on probable FL-pseudoprimes and described new algorithms for checking FL-primality, and with the help of them we proved the absence of such numbers up to the boundary B = 1021, which is more than 30 times larger than the previously known boundary 264 found by J. Gilchrist in 2013. An inaccuracy in the formulation of theorem 4 in the mentioned article has also been corrected.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"33 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140661274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-24DOI: 10.26907/0021-3446-2024-4-39-46
N. A. Rather, A. Bhat, M. Shafi
For the polynomial P(z) = n sum j=0 cjzj of degree n having all its zeros in | z| leq k, k geq 1, V. Jain in “On the derivative of a polynomial”, Bull. Math. Soc. Sci. Math. Roumanie Tome 59, 339–347 (2016) proved that max | z| =1 | P prime (z)| geq n biggl( | c0| + | cn| kn+1 | c0| (1 + kn+1) + | cn| (kn+1 + k2n) biggr) max | z| =1 | P(z)| . In this paper we strengthen the above inequality and other related results for the polynomials of degree n geq 2.
对于多项式 P(z) = n sum j=0 cjzj of degree n，其所有零点都在|z| leq k, k geq 1，V. Jain 在 "论多项式的导数"，Bull.Math.Soc.Roumanie Tome 59, 339-347 (2016) 证明了 max | z| =1 | P prime (z)| geq n biggl( | c0| + | cn| kn+1 | c0| (1 + kn+1) + | cn| (kn+1 + k2n) biggr) max | z| =1 | P(z)| 。本文将对 n geq 2 度的多项式加强上述不等式及其它相关结果。
{"title":"Sharpening of Tur´an-type inequality for polynomials","authors":"N. A. Rather, A. Bhat, M. Shafi","doi":"10.26907/0021-3446-2024-4-39-46","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-4-39-46","url":null,"abstract":"For the polynomial P(z) = n sum j=0 cjzj of degree n having all its zeros in | z| leq k, k geq 1, V. Jain in “On the derivative of a polynomial”, Bull. Math. Soc. Sci. Math. Roumanie Tome 59, 339–347 (2016) proved that max | z| =1 | P prime (z)| geq n biggl( | c0| + | cn| kn+1 | c0| (1 + kn+1) + | cn| (kn+1 + k2n) biggr) max | z| =1 | P(z)| . In this paper we strengthen the above inequality and other related results for the polynomials of degree n geq 2.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"27 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140661844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-24DOI: 10.26907/0021-3446-2024-4-47-66
A. K. Aydamir Khazretovich Stash
The research topic of this work is at the junction of the theory of Lyapunov exponents and oscillation theory. In this paper, we study the spectra (i.e., the sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential equations with coefficients continuous on the positive semi-axis. In the first part of the paper, we build a third order linear differential equation with the following property: the spectra of all upper and lower strong and weak exponents of oscillation of strict and non-strict signs, zeros, roots and hyper roots contain a countable set of different essential values, both metrically and topologically. Moreover, all these values are implemented on the same sequence of solutions of the constructed equation, that is, for each solution from this sequence, all of the oscillation exponents coincide with each other. In the construction of the indicated equation and in the proof of the required results, we used analytical methods of the qualitative theory of differential equations and methods from the theory of perturbations of solutions of linear differential equations, in particular, the author’s technique for controlling the fundamental system of solutions of such equations in one special case. In the second part of the paper, the existence of a third order linear differential equation with continuum spectra of the oscillation exponents is established, wherein the spectra of all oscillation exponents fill the same segment of the number axis with predetermined arbitrary positive incommensurable ends. It turned out that for each solution of the constructed differential equation, all of the oscillation exponents coincide with each other. The obtained results are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential equations.
{"title":"On infinite spectra of oscillation exponents of third order linear differential equations","authors":"A. K. Aydamir Khazretovich Stash","doi":"10.26907/0021-3446-2024-4-47-66","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-4-47-66","url":null,"abstract":"The research topic of this work is at the junction of the theory of Lyapunov exponents and oscillation theory. In this paper, we study the spectra (i.e., the sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential equations with coefficients continuous on the positive semi-axis. In the first part of the paper, we build a third order linear differential equation with the following property: the spectra of all upper and lower strong and weak exponents of oscillation of strict and non-strict signs, zeros, roots and hyper roots contain a countable set of different essential values, both metrically and topologically. Moreover, all these values are implemented on the same sequence of solutions of the constructed equation, that is, for each solution from this sequence, all of the oscillation exponents coincide with each other. In the construction of the indicated equation and in the proof of the required results, we used analytical methods of the qualitative theory of differential equations and methods from the theory of perturbations of solutions of linear differential equations, in particular, the author’s technique for controlling the fundamental system of solutions of such equations in one special case. In the second part of the paper, the existence of a third order linear differential equation with continuum spectra of the oscillation exponents is established, wherein the spectra of all oscillation exponents fill the same segment of the number axis with predetermined arbitrary positive incommensurable ends. It turned out that for each solution of the constructed differential equation, all of the oscillation exponents coincide with each other. The obtained results are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential equations.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"64 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140663993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-24DOI: 10.26907/0021-3446-2024-4-89-93
V. Sekorin
We investigate the decidability of first-order logic extensions. For example, it is established in A. S. Zolotov’s works that a logic with a unary transitive closure operator for the one successor theory is decidable. We show that in a similar case, a logic with a unary partial fixed point operator is undecidable. For this purpose, we reduce the halting problem for the counter machine to the problem of truth of the underlying formula. This reduction uses only one unary non-nested partial fixed operator that is applied to a universal or existential formula.
我们研究一阶逻辑扩展的可判定性。例如，佐洛托夫（A. S. Zolotov）在其著作中指出，具有一元传递闭包算子的一阶理论逻辑是可判定的。我们证明，在类似情况下，具有一元部分定点算子的逻辑是不可判定的。为此，我们将计数器的停止问题简化为底层公式的真值问题。这种还原只使用一个一元非嵌套部分定点算子，它适用于一个普遍式或存在式。
{"title":"On undecidability of unary non-nested PFP-operators for one successor function theory","authors":"V. Sekorin","doi":"10.26907/0021-3446-2024-4-89-93","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-4-89-93","url":null,"abstract":"We investigate the decidability of first-order logic extensions. For example, it is established in A. S. Zolotov’s works that a logic with a unary transitive closure operator for the one successor theory is decidable. We show that in a similar case, a logic with a unary partial fixed point operator is undecidable. For this purpose, we reduce the halting problem for the counter machine to the problem of truth of the underlying formula. This reduction uses only one unary non-nested partial fixed operator that is applied to a universal or existential formula.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"2 23","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140664702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-22DOI: 10.26907/0021-3446-2024-4-15-19
S. Demir
Let phi in S with int phi (x) dx = 1, and define phi t(x) = 1 tn phi Bigl( x t Bigr) , and denote the function family { phi tast f(x)} t>0 by Phi ast f(x). Let scrJ be a subset of BbbR (or more generally an ordered index set), and suppose that there exists a constant C1 such that sum tin scrJ | ^phi t(x)| 2 < C1 for all x in BbbR n. Then i) There exists a constant C2 > 0 such that | V2(Phi ast f)| Lp leq C2| f| Hp, n n + 1 < p leq 1 for all f in Hp(BbbR n), n n + 1 < p leq 1. ii) The lambda -jump operator Nlambda (Phi ast f) satisfies | lambda [Nlambda (Phi ast f)]1/2| Lp leq C3| f| Hp, n n + 1 < p leq 1, uniformly in lambda > 0 for some constant C3 > 0.
让 phi in S with int phi (x) dx = 1，定义 phi t(x) = 1 tn phi Bigl( x t Bigr) ，并用 Phi ast f(x) 表示函数族 { phi tast f(x)} t>0.让 scrJ 是 BbbR 的一个子集（或者更一般地说是一个有序索引集），假设存在一个常数 C1，使得 sum tin scrJ | ^phi t(x)| 2 < C1 for all x in BbbR n。Then i) Thereists a constant C2 > 0 such that | V2(Phi ast f)| Lp leq C2| f| Hp, n n + 1 < p leq 1 for all fin Hp(BbbR n), n n + 1 < p leq 1.ii) The lambda -jump operator Nlambda (Phi ast f) satisfies | lambda [Nlambda (Phi ast f)]1/2| Lp leq C3| f| Hp, n n + 1 < p leq 1, uniformly in lambda > 0 for some constant C3 > 0.
{"title":"Variation and λ-jump inequalities on Hp spaces","authors":"S. Demir","doi":"10.26907/0021-3446-2024-4-15-19","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-4-15-19","url":null,"abstract":"Let phi in S with int phi (x) dx = 1, and define phi t(x) = 1 tn phi Bigl( x t Bigr) , and denote the function family { phi tast f(x)} t>0 by Phi ast f(x). Let scrJ be a subset of BbbR (or more generally an ordered index set), and suppose that there exists a constant C1 such that sum tin scrJ | ^phi t(x)| 2 < C1 for all x in BbbR n. Then i) There exists a constant C2 > 0 such that | V2(Phi ast f)| Lp leq C2| f| Hp, n n + 1 < p leq 1 for all f in Hp(BbbR n), n n + 1 < p leq 1. ii) The lambda -jump operator Nlambda (Phi ast f) satisfies | lambda [Nlambda (Phi ast f)]1/2| Lp leq C3| f| Hp, n n + 1 < p leq 1, uniformly in lambda > 0 for some constant C3 > 0.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"92 14","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140676793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-22DOI: 10.26907/0021-3446-2024-4-31-38
M. I. Muminov, U. R. Shadiev
We consider a family of bounded self-adjoint matrix operators (generalized Friedrichs models) acting on the direct sum of one-particle and two-particle subspaces of the Fock space. Conditions for the existence of eigenvalues of the matrix operators are obtained.
{"title":"On the existence of an eigenvalue of the generalized Friedrichs model","authors":"M. I. Muminov, U. R. Shadiev","doi":"10.26907/0021-3446-2024-4-31-38","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-4-31-38","url":null,"abstract":"We consider a family of bounded self-adjoint matrix operators (generalized Friedrichs models) acting on the direct sum of one-particle and two-particle subspaces of the Fock space. Conditions for the existence of eigenvalues of the matrix operators are obtained.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"5 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140674452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-22DOI: 10.26907/0021-3446-2024-4-20-30
V. E. Kruglov
With the help of the formula for the general solution of a difference equation with constant coefficients, it is shown that the set of solutions to this equation contains classical solutions of the type kmλk. We present necessary and sufficient conditions on the coefficients of the equation and the initial parameters under which such solutions are obtained.
{"title":"Conditions for the existence of power solutions to a linear difference equation with constant coefficients","authors":"V. E. Kruglov","doi":"10.26907/0021-3446-2024-4-20-30","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-4-20-30","url":null,"abstract":"With the help of the formula for the general solution of a difference equation with constant coefficients, it is shown that the set of solutions to this equation contains classical solutions of the type kmλk. We present necessary and sufficient conditions on the coefficients of the equation and the initial parameters under which such solutions are obtained.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"21 7","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140674072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-22DOI: 10.26907/0021-3446-2024-4-3-14
V. Gorbatsevich
The article considers decompositions of nilpotent Lie algebras and nilpotent Lie groups, and connections between them. Also, descriptions of irreducible transitive actions of nilpotent Lie groups on the plane and on three-dimensional space are given.
{"title":"On decompositions and transitive actions of nilpotent Lie groups","authors":"V. Gorbatsevich","doi":"10.26907/0021-3446-2024-4-3-14","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-4-3-14","url":null,"abstract":"The article considers decompositions of nilpotent Lie algebras and nilpotent Lie groups, and connections between them. Also, descriptions of irreducible transitive actions of nilpotent Lie groups on the plane and on three-dimensional space are given.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"58 23","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140675809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-09DOI: 10.26907/0021-3446-2024-3-91-96
T. Rasulov, D. E. Ismoilova
{"title":"Spectral relations for a matrix model in fermionic Fock space","authors":"T. Rasulov, D. E. Ismoilova","doi":"10.26907/0021-3446-2024-3-91-96","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-3-91-96","url":null,"abstract":"","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140724247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-09DOI: 10.26907/0021-3446-2024-3-70-83
A. B. Khasanov, Khasun Normurodov
In this paper, the inverse spectral problem method is used to integrate a nonlinear sine-Gordon type equation with an additional term 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 three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula satisfies sine-Gordon-type equations with an additional term.
{"title":"Integration of a sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions","authors":"A. B. Khasanov, Khasun Normurodov","doi":"10.26907/0021-3446-2024-3-70-83","DOIUrl":"https://doi.org/10.26907/0021-3446-2024-3-70-83","url":null,"abstract":"In this paper, the inverse spectral problem method is used to integrate a nonlinear sine-Gordon type equation with an additional term 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 three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula satisfies sine-Gordon-type equations with an additional term.","PeriodicalId":507800,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika","volume":"105 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140725766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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