ABSTRACT
In this paper, using the algorithm due to Ablowitz et al. [Phys. Rev. Lett. 128, 184101 (2022); J. Phys. A: Math. Gen. 55, 384010 (2022)], we explore the anomalous dispersive relations, inverse scattering transform, and fractional N-soliton solutions of the integrable fractional higher-order nonlinear Schrödinger (fHONLS) equations, containing the fractional third-order NLS (fTONLS), fractional complex mKdV (fcmKdV), and fractional fourth-order nonlinear Schrödinger (fFONLS) equations, etc. The inverse scattering problem can be solved exactly by means of the matrix Riemann-Hilbert problem with simple poles. As a consequence, an explicit formula is found for the fractional N-soliton solutions of the fHONLS equations in the reflectionless case. In particular, we analyze the fractional one-, two-, and three-soliton solutions with anomalous dispersions of fTONLS and fcmKdV equations. The wave, group, and phase velocities of these envelope fractional one-soliton solutions are related to the power laws of their amplitudes. Moreover, we also deduce the formula for the fractional N-soliton solutions of all fHONLS equations and analyze some velocities of the one-soliton solution. These obtained fractional N-soliton solutions may be useful to explain the related super-dispersion transports of nonlinear waves in fractional nonlinear media.
ABSTRACT
OBJECTIVE: This study aimed to understand the prevalence rate, epidemiological characteristics and relevant factors of arthritis in Shanghai. METHODS: A sample of 7 575 residents aged 15 years and above was drawn from 6 communities under multiple stage cluster sampling. A household survey with questionnaire was carried out to differentiate both undiagnosed patients and those with definite arthritis. Those who had not been diagnosed before were asked to carry further clinical examinations by a rheumatologist. RESULTS: The prevalence rate of arthritis was 6.11%, including osteoarthritis (OA) 4.18%, rheumatoid arthritis (RA) 0.52%, gout 0.28%, ankylosing spondylitis (AS) 0.28%, rheumatic arthritis 0.49% and other types arthritis 0.82%. Arthritis was significantly related to cardiovascular disease, pulmonary disease and gastrointestinal disease. Age, female and obesity might serve as risk factors for arthritis. Physical labors and living in rural area might have protecting effects. CONCLUSION: Elderly and female seemed to be at high risk for arthritis. Weight control and more exercise should be encouraged to reduce the risks. For arthritis patients, treatment to other chronic diseases should not be ignored.
