Dynamics of fundamental and double-pole breathers and solitons for a nonlinear Schrodinger equation with sextic operator under non-zero boundary conditions

We study the dynamics of fundamental and double-pole breathers and solitons for the focusing and defocusing nonlinear Schrodinger equation with the sextic operator under non-zero boundary conditions. Our analysis mainly focuses onthe dynamical properties of simple- and double-pole solutions. Firstly, through verification, we find that solutions undernon-zero boundary conditions can be transformed into solutions under zero boundary conditions, whether in simple-pole ordouble-pole cases. For the focusing case, in the investigation of simple-pole solutions, temporal periodic breather and thespatial-temporal periodic breather are obtained by modulating parameters. Additionally, in the case of multi-pole solitons,we analyze parallel-state solitons, bound-state solitons, and intersecting solitons, providing a brief analysis of their interactions.In the double-pole case, we observe that the two solitons undergo two interactions, resulting in a distinctive “triangle”crest. Furthermore, for the defocusing case, we briefly consider two situations of simple-pole solutions, obtaining one andtwo dark solitons.

Riemann–Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation

We investigate the inverse scattering transform for the Schrödinger-type equation under zero boundary conditions with the Riemann–Hilbert(RH)approach.In the direct scattering process,the properties are given,such as Jost solutions,asymptotic behaviors,analyticity,the symmetries of the Jost solutions and the corresponding spectral matrix.In the inverse scattering process,the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem.Then,the reconstruction formula of potential and trace formula are also derived correspondingly.Thus,N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases.Furthermore,we present a single double-pole solution by taking some parameters,and it is analyzed in detail.

一类具有忆阻器的Lorenz型混沌系统余维二分岔及无穷远处动力学分析

基于经典的Lorenz系统,通过反馈控制的方式得到了一类具有忆阻器的三维混沌系统,对该系统分别从局部高余维分岔及无穷远全局动力学行为这两个方面进行了研究.首先,基于平均理论,对原点平衡点处的zero-Hopf分岔行为进行了分析;其次,基于中心流形理论,对原点平衡点处的double-zero分岔进行了分析;最后,根据Poincaré紧致化方法,对该系统在无穷远处的动力学行为进行了研究.

Effect of dietary protein sources and storage temperatures on egg internal quality of stored shell eggs

This study was conducted to evaluate the effects of various protein sources(soybean meal, SBM; cottonseed protein, CSP; double-zero rapeseed meal, DRM) on the internal quality(Haugh unit, yolk index,albumen pH,yolk hardness and yolk springiness) of eggs when stored at either 4 or 28℃ for 28 d. A total of 288 laying hens(32 wk of age) were randomly allotted to 6 treatment groups(4 replicates per treatment) and fed diets containing SBM, CSP, or DRM individually or in combination with equal crude protein content(SBM-CSP, SBM-DRM, and CSP-DRM) as the protein ingredient(s). A 6 x 2 factorial arrangement was employed with dietary types and storage temperatures(4 and 28℃) as the main effects. After 12 wk of diet feeding, a total of 216 eggs was collected for egg internal quality determination.The results showed as follows: 1) lower egg quality was observed in the DRM group compared with the other groups when stored at 4 and 28℃ for 28 d(P < 0.05), while there was no difference in egg internal quality among the other groups. 2) The CSP diet resulted in higher yolk hardness compared with the other diets when eggs were stored at 4℃ for 28 d(P < 0.05). Lower Haugh unit was observed in the DRM and SBM-DRM groups compared with the other groups when eggs were stored for 28 d at 4℃(P < 0.05).3) Yolk breakage occurred in the DRM group and eggs could not be analyzed for egg internal quality when stored at 28℃ for 28 d. The overall results indicated that CSP or DRM as the sole dietary protein source for laying hens may adversely affect the internal quality of stored eggs as compared with the SBM diet, and half replacement of CSP combined with SBM may maintain similar egg quality to SBM diet alone for eggs stored under refrigerated conditions.