Higher-order corrections to the short-pulse equation

Using renormalization group techniques, we derive an extended short- pulse equation as approximation to a nonlinear wave equation. We investigate the new equation numerically and show that the new equation captures efficiently higher- order effects on pulse propagation in cubic nonlinear media. We illustrate our findings using one- and two-soliton solutions of the first-order short-pulse equation as initial conditions in the nonlinear wave equation

The Sasa-Satsuma higher order nonlinear Schrodinger equation and its bilinearization and multi-soliton solutions

Higher order and multicomponent generalizations of the nonlinear Schrodinger equation are important in various applications, e.g., in optics. One of these equations, the integrable Sasa-Satsuma equation, has particularly interesting soliton solutions. Unfortunately the construction of multi-soliton solutions to this equation presents difficulties due to its complicated bilinearization. We discuss briefly some previous attempts and then give the correct bilinearization based on the interpretation of the Sasa-Satsuma equation as a reduction of the three-component Kadomtsev-Petvishvili hierarchy. In the process we also get bilinearizations and multi-soliton formulae for a two component generalization of the Sasa-Satsuma equation (the Yajima-Oikawa-Tasgal-Potasek model), and for a (2+1)-dimensional generalization.Comment: 13 pages in RevTex, added reference

Local Isometric immersions of pseudo-spherical surfaces and evolution equations

The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat [3], is characterized by the property that to each solution of a differential equation, within the class, there corresponds a 2-dimensional Riemannian metric of curvature equal to $-1$. The class of differential equations describing pseudo-spherical surfaces carries close ties to the property of complete integrability, as manifested by the existence of infinite hierarchies of conservation laws and associated linear problems. As such, it contains many important known examples of integrable equations, like the sine-Gordon, Liouville and KdV equations. It also gives rise to many new families of integrable equations. The question we address in this paper concerns the local isometric immersion of pseudo-spherical surfaces in ${\bf E}^{3}$ from the perspective of the differential equations that give rise to the metrics. Indeed, a classical theorem in the differential geometry of surfaces states that any pseudo-spherical surface can be locally isometrically immersed in ${\bf E}^{3}$. In the case of the sine-Gordon equation, one can derive an expression for the second fundamental form of the immersion that depends only on a jet of finite order of the solution of the pde. A natural question is to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudo-spherical surfaces. In an earlier paper [11], we have shown that this property fails to hold for all other second order equations, except for those belonging to a very special class of evolution equations. In the present paper, we consider a class of evolution equations for $u(x,t)$ of order $k\geq 3$ describing pseudo-spherical surfaces. We show that whenever an isometric immersion in ${\bf E}^3$ exists, depending on a jet of finite order of $u$, then the coefficients of the second fundamental forms are functions of the independent variables $x$ and $t$ only.Comment: Fields Institute Communications, 2015, Hamiltonian PDEs and Applications, pp.N

Discrete Integrable Systems and Hodograph Transformations Arising from Motions of Discrete Plane Curves

We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam equation, the complex Dym equation, and the short pulse equation. They are related to the modified KdV or the sine-Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler-Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.Comment: 19 page

Morphologic characteristics of the circle of Willis in patients with migraine

Background: An incomplete circle of Willis (CW) and other abnormalities of CW may predispose to cerebral hypoperfusion which can be one of the factors leading to white matter abnormalities in patients with migraine. This study assessed the relationship between the morphologic characteristics of the CW and migraine.Methods: The study included 150 patients with migraine (age 18-56,12 men and 138 women, 21 patients with migraine with aura and 129 without aura) and 119 patients of controls without migraine who had no history of cerebral ischemia or infarcts on MRI (age 1 8 -6 2 ,4 0 men, 79 women). All patients underwent 3-dimensional time of flight magnetic resonance angiography of the CW. According to the vessel size seen on 3D TOF MR angiograms, the component vessels consisting of CW were regarded as normal, hypoplastic, or absent. The anterior part of the CW (APCW) was considered complete when the anterior communicating artery (ACoA) as well as the A1 segments of the anterior cerebral artery (АСА) were present on visual examination, incomplete when any of these vessels was missing and indeterminate when we observed a hypoplastic A1 segment of АСА. The posterior part of the circle of Willis (PPCW) was considered complete when both posterior (PCoA) and the P1 segments of the posterior cerebral artery (PCA) were present on visual examination, incomplete when one of these vessels was missing and indeterminate when we observed hypoplastic PCoA or P1 segment of PCA.Results: We found the following morphologic characteristics of the CW in patients with migraine: complete APCW -137, incomplete APCW - 0, indeterminate APCW -13, complete PPCW -101, incomplete PPCW — 4, indeterminate PPCW - 45; hypoplasia (n=36) or aplasia of the vertebral artery (n=6), duplication of АСА (n=3) or of A1 segment (n=3), hypoplasia of inferior (n=3) or superior cerebellar artery (n=2). Among these characteristics, only indeterminate patterns of posterior CW were significantly more common in migraineurs than in the control group (30% vs 19%; P < .05; odds ratio 1.79; 95% Cl: 1.0-3.1). No difference was found between migraineurs with and without aura.Введение. Разомкнутый Виллизиев круг и различные аномалии строения Виллизиева круга могут предрасполагать к церебральной гипоперфузии, которая может быть одним из факторов, приводящих к формированию белых очагов в веществе головного мозга у пациентов с мигренью. Целью данного исследования было изучить встречаемость различных аномалий Виллизиева круга у больных с мигренью и дать характеристику морфологического строения Виллизиева круга. Материалы и методы. В исследование включены 150 пациентов с мигренью (12 мужчин и 138 женщин) в возрасте от 18 до 56 лет. Из них 21 пациент с мигренью с аурой и 129 пациентов с мигренью без ауры. Мигрень была диагностирована согласно международным критериям International Headache Society criteria. В качестве контрольной группы использовались 119 пациентов без мигрени (40 мужчин и 79 женщин), у которых не было инсультов в анамнезе и инфарктов по МРТ. Их возраст варьировал от 18 до 62 лет. Всем больным с мигренями и пациентам контрольной группы выполнена магнитно-резонансная томография (МРТ) головного мозга и магнитно-резонансная ангиография (МР-АГ) сосудов головного мозга. Согласно размерам сосудов на МР-АГ составляющие Виллизиев круг сосуды были подразделены на нормальные, гиполастичные и отсутствующие. Передняя часть Виллизиева круга была определена как замкнутая, если обе передние соединительные артерии и А1 сегмент передней мозговой артерии (ПМА) присутствовали на МР-ангиограмме; при отсутствии одного из зтих сосудов - незамкнутой и при наличии гипоплазии сегмента А1 ПМА - промежуточной. Задняя часть Виллизиева круга была определена как замкнутая, если обе задние соединительные артерии (ЗСА) и Р1 сегмент задней мозговой артерии (ЗМА) присутствовали на МР-ангиограмме; при отсутствии одного из зтих сосудов - незамкнутой и при наличии гипоплазии ЗСА или сегмента Р1 ЗМА — промежуточной. Результаты. Мы выявили следующие морфологические характеристики Виллизиева круга у больных с мигренями: замкнутая передняя часть Виллизиева круга (n=137), промежуточный тип строения передней части Виллизиева круга (n=13), замкнутая задняя часть Виллизиева круга (n=101), незамкнутая задняя часть Виллизиева круга (n= 4), промежуточный тип строения задней части Виллизиева круга (n=45); гипоплазия позвоночной артерии (n=36), аплазия позвоночной артерии (n=6), удвоение ПМА (n=3), удвоение А1 сегмента ПМА (n=3), гипоплазия нижней мозжечковой артерии (n=3), гипоплазия верхней мозжечковой артерии (n=2). Среди зтих вариантов строения Виллизиева круга только незамкнутая задняя часть Виллизиева круга и промежуточный тип строения задней части Виллизиева круга встречались значительно чаще у пациентов с мигренью с аурой по сравнению с контрольной группой (33% против 19%; Р < 0,05; отношение шансов: 2.0; 95% доверительный интервал: 1,1 -3,5). Не было найдено различий в строении Виллизиева круга у больных с мигренью с аурой и без ауры. Вывод: Наши результаты показали, что промежуточный тип строения задней части Виллизиева круга представляет характерную черту строения Виллизиева круга у исследуемых больных с мигренью

Large scale analytic calculations in quantum field theories

We present a survey on the mathematical structure of zero- and single scale quantities and the associated calculation methods and function spaces in higher order perturbative calculations in relativistic renormalizable quantum field theories.Comment: 25 pages Latex, 1 style fil

Perturbation theory for nearly integrable multi-component nonlinear PDEs

The Riemann-Hilbert problem associated with the integrable PDE is used as a nonlinear transformation of the nearly integrable PDE to the spectral space. The temporal evolution of the spectral data is derived with account for arbitrary perturbations and is given in the form of exact equations, which generate the sequence of approximate ODEs in successive orders with respect to the perturbation. For vector nearly integrable PDEs, embracing the vector NLS and complex modified KdV equations, the main result is formulated in a theorem. For a single vector soliton the evolution equations for the soliton parameters and first-order radiation are given in explicit formComment: Submitted to Journal of Mathematical Physics (References are corrected

Generalizations of the short pulse equation

We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation

Geographical Distribution, Incidence, Malignancies, and Outcome of 136 Eastern Slavic Patients With Nijmegen Breakage Syndrome and NBN Founder Variant c.657_661del5

Nijmegen breakage syndrome (NBS) is a DNA repair disorder characterized by combined immunodeficiency and a high predisposition to lymphoid malignancies. The majority of NBS patients are identified with a homozygous five base pair deletion in the Nibrin (NBN) gene (c.657_661del5, p.K219fsX19) with a founder effect observed in Caucasian European populations, especially of Slavic origin. We present here an analysis of a cohort of 136 NBS patients of Eastern Slav origin across Belarus, Ukraine, Russia, and Latvia with a focus on understanding the geographic distribution, incidence of malignancy, and treatment outcomes of this cohort. Our analysis shows that Belarus had the highest prevalence of NBS (2.3 per 1,000,000), followed by Ukraine (1.3 per 1,000,000), and Russia (0.7 per 1,000,000). Of note, the highest concentration of NBS cases was observed in the western regions of Belarus and Ukraine, where NBS prevalence exceeds 20 cases per 1,000,000 people, suggesting the presence of an “Eastern Slavic NBS hot spot.” The median age at diagnosis of this cohort ranged from 4 to 5 years, and delay in diagnosis was more pervasive in smaller cities and rural regions. A total of 62 (45%) patients developed malignancies, more commonly in males than females (55.2 vs. 34.2%; p=0.017). In 27 patients, NBS was diagnosed following the onset of malignancies (mean age: 8 years). Malignancies were mostly of lymphoid origin and predominantly non-Hodgkin lymphoma (NHL) (n=42, 68%); 38% of patients had diffuse large B-cell lymphoma. The 20-year overall survival rate of patients with malignancy was 24%. However, females with cancer experienced poorer event-free survival rates than males (16.6% vs. 46.8%, p=0.036). Of 136 NBS patients, 13 underwent hematopoietic stem cell transplantation (HSCT) with an overall survival of 3.5 years following treatment (range: 1 to 14 years). Indications for HSCT included malignancy (n=7) and immunodeficiency (n=6). Overall, 9% of patients in this cohort reached adulthood. Adult survivors reported diminished quality of life with significant physical and cognitive impairments. Our study highlights the need to improve timely diagnosis and clinical management of NBS among Eastern Slavs. Genetic counseling and screening should be offered to individuals with a family history of NBS, especially in hot spot regions. © Copyright © 2021 Sharapova, Pashchenko, Bondarenko, Vakhlyarskaya, Prokofjeva, Fedorova, Savchak, Mareika, Valiev, Popa, Tuzankina, Vlasova, Sakovich, Polyakova, Rumiantseva, Naumchik, Kulyova, Aleshkevich, Golovataya, Minakovskaya, Belevtsev, Latysheva, Latysheva, Beznoshchenko, Akopyan, Makukh, Kozlova, Varabyou, Ballow, Ong, Walter, Kondratenko, Kostyuchenko and Aleinikova.We thank all doctors for clinical help for patients. We also appreciate the support of patient and their parents for agreeing to take part in this study. TP thanks Sergey?Nikulshin, Marika Grutupa, and Zanna Kovalova. We thank Joseph Dasso for editing this manuscript, primarily for proper English