Asymptotic analysis on bright solitons and breather solutions of a generalized higher-order nonlinear Schrödinger equation in an optical fiber or a planar waveguide

We study a generalized higher-order nonlinear Schr¨odinger equation in an optical fiber or a planar waveguide.We obtain the Lax pair and N-fold Darboux transformation(DT)with N being a positive integer.Based on Lax pair obtained by us,we derive the infinitely-many conservation laws.We give the bright one-,two-,and N-soliton solutions,and the first-,second-,and Nth-order breather solutions based on the N-fold DT.We conclude that the velocities of the bright solitons are influenced by the distributed gain function,g(z),and variable coefficients in equation,h1(z),p1(z),r1(z),and s1(z)via the asymptotic analysis,where z represents the propagation variable or spatial coordinate.We also graphically observe that:the velocities of the first-and second-order breathers will be affected by h1(z),p1(z),r1(z),and s1(z),and the background wave depends on g(z).

Rogue waves of a(3+1)-dimensional BKP equation

We investigate certain rogue waves of a(3+1)-dimensional BKP equation via the Kadomtsev-Petviashili hierarchy reduction method.We obtain semi-rational solutions in the determinant form,which contain two special interactions:(i)one lump develops from a kink soliton and then fuses into the other kink one;(ii)a line rogue wave arises from the segment between two kink solitons and then disappears quickly.We find that such a lump or line rogue wave only survives in a short time and localizes in both space and time,which performs like a rogue wave.Furthermore,the higher-order semi-rational solutions describing the interaction between two lumps(one line rogue wave)and three kink solitons are presented.

Lax pair and vector semi-rational nonautonomous rogue waves for a coupled time-dependent coefficient fourth-order nonlinear Schrodinger system in an inhomogeneous optical fiber

Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion(GVD) and fourth-order dispersion(FOD) coefficients are the constants, we exhibit the first-and second-order vector semirational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first-and second-order periodic vector semi-rational rogue waves, first-and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.

Rogue Waves for a (2+1)-Dimensional Coupled Nonlinear Schr?dinger System with Variable Coefficients in a Graded-Index Waveguide

Studied in this paper is a(2+1)-dimensional coupled nonlinear Schr?dinger system with variable coefficients,which describes the propagation of an optical beam inside the two-dimensional graded-index waveguide amplifier with the polarization effects. According to the similarity transformation, we derive the type-Ⅰ and type-Ⅱ rogue-wave solutions. We graphically present two types of the rouge wave and discuss the influence of the diffraction parameter on the rogue waves.When the diffraction parameters are exponentially-growing-periodic, exponential, linear and quadratic parameters, we obtain the periodic rogue wave and composite rogue waves respectively.

The structure of the compound eyes and phototaxis in two phenotypes of the bean pest Callosobruchus maculatus(Coleoptera:Bruchinae)

Callosobruchus maculatus(Fabricius,1775)(Coleoptera:Bruchinae)is a destructive agricultural pest that is harmful to beans worldwide and an important quarantine pest in China.It was divided into two phenotypes based on polyphenism:normal and flight forms.In this study,we first compared the morphological structures of the compound eyes of the two forms.According to the results of scanning electron microscopy(SEM),transmission electron microscopy(TEM),microcomputed tomography(micro-CT),and computer three-dimensional reconstruction,there are no differences in the structures of the compound eyes between the normal and flight forms except for the number of ommatidia.From the internal structure,the compound eyes have a biconvex cornea with open rhabdom and acone eye,crystalline cone directly connected with rhabdom,and no clear zone.It is a kind of apposition eye.Ommatidia facets range in shape from quadrilateral to hexagonal and some irregular shapes.On electroretinograms(ERGs),the normal and flight forms showed different spectral sensitivities:the normal form had the strongest response to ultraviolet light,whereas the flight form had the strongest response to white light.Behavioral assays revealed that the normal and flight forms showed completely opposite phototaxis behaviors;the flight form exhibited positive phototaxis,whereas the normal form exhibited negative phototaxis.This study not only enriches our knowledge on coleopteran compound eyes but also provides a foundation for in-depth research on the photoreceptor mechanisms of compound eyes,which may be useful in pest control management.