Investigation On The Soliton And Rogue-wave Phenomena In Nonlinear Optics And Other Fields Via Mathematical Methods

Nonlinear evolution equations can be used to describe such nonlinear phenomena as solitons and rogue waves in nonlinear optics,molecular biology,fluid mechanics and other fields.In this dissertation,based on some nonlinear evolution equations in nonlinear optics,molecular biology and fluid mechanics,we obtain soliton,rogue wave and breather solutions via the Darboux transformation and Hirota method.Moreover,we discuss and analyse the properties of the rogue waves,solitons and breathers,and the interactions between rogue waves,solitons and breathers.In what follows,we give out the main contents of this dissertation:In chapter 1,we briefly introduce the background of nonlinear science and the research progress of nonlinear waves represented by solitons and rogue waves.Some mathematical methods used in this dissertation to study the soliton and rogue-wave phenomena are also introduced,as well as the main work and arrangement of this dis-sertation.In chapter 2,studied are the vector rational and semi-rational rogue waves in a non-Kerr medium,through the coupled cubic-quintic nonlinear Schr(?)dinger system,which describes the effects of quintic nonlinearity on the ultrashort optical pulse propagation in the medium.Applying the gauge transformations,we derive the Nth-order Darboux transformtion and Nth-order vector rational and semi-rational rogue wave solutions,where N is a positive integer.Based on such solutions,we present three types of the second-order rogue waves with the triangle structure,and the third-order vector rogue waves with the merged,triangle and pentagon structures in each component.Moreover,we show the first-and second-order vector semi-rational rogue waves which display the coexistence of the rogue waves and the breathers.In chapter 3,we study a higher-order coupled nonlinear Schr(?)dinger system,which describes the simultaneous propagation of two ultrashort optical pulses in an optical fiber.We construct the dark-bright one-,two-soliton solutions,Nth-order dark-bright semi-rational soliton solutions for the mixed interactions,and the Nth-order breather solutions for the focusing interactions.Based on such solutions,we graphically discuss the dark-bright one-,two-solitons,dark-bright semi-rational solitons and the shapes of the breathers.In chapter 4,a coupled fourth-order nonlinear Schr(?)dinger system,which describes the propagation of ultrashort optical pulses in a birefringent optical fiber,is studied.The N th-order Darboux transformation and corresponding vector breather solutions are constructed.Based on such solutions,we graphically show four different types of breathers.Moreover,we find that the range of the breather along the time axis decreases,and the angle between the breather and the time axis increases as the value of |?|decreases,where ? represents the strength of higher-order linear and nonlinear effcts.In chapter 5,investigation is madeon the coupled variable-coefficient fourth-order nonlinear Schr(?)dinger equations,which describe the simultaneous propagation of optical pulses in an inhomogeneous optical fiber.Via the generalized Darboux transformation,the first-and second-order rogue wave solutions are constructed.Based on such solutions,effects of the group velocity dispersion coefficient and the fourth-order dispersion coefficient on the rogue waves are graphically analyzed.The firstorder rogue waves with the ey e-shaped distribution,the interactions between the first-order rogue waves with solitons,and the second-order rogue waves with one highest peak and with the triangular structure are displayed.When the value of the group velocity dispersion coefficient or the fourth-order dispersion increases,range of the first-order rogue wave increases.Composite rogue waves are obtained,where the temporal separation between two adjacent rogue waves can be changed if we adjust the group velocity dispersion coefficient and fourth-order dispersion coefficient.Periodic rogue waves are presented.Periods of such rogue waves decrease with the periods of the group-velocity dispersion and fourth-order dispersion coefficient decreasing.In chapter 6,we investigate the vector multi-rogue waves for the three-coupled fourth-order nonlinear Schr(?)dinger equations,which describe the dynamics of an alpha helical protein with the nearest and next-nearest neighbour interactions and interspine coupling.Via the Darboux-dressing transformation,the vector multi-rogue-wave solutions are derived.Based on such solutions,we present the single vector rogue wave,vector rogue wave pair and triple vector rogue wave graphically.We show that a four-petaled rogue wave with two humps and two valleys appears in two components,while the other component appears an eye-shaped rogue wave.Existing time of the rogue wave decreases with the strength of higher-order linear and nonlinear effects.We also obtain the separated and interacting vector rogue wave pairs,as well as the triple vector rogue waves.Moreover,we verify the baseband modulation instability through the linear stability analysis.In chapter 7,investigation is made on a Kadomtsev-Petviashvili-based system,which can be seen in fluid dynamics and plasma physics.Based on the Hirota method,bilinear form and Backlund transformation are derived.N-soliton solutions in terms of the Wronskian are constructed,and it can be verified that the N-soliton solutions in terms of the Wronskian satisfy the bilinear form and Backlund transformation.Via the N-soliton solutions in terms of the Wronskian,we graphically obtain the kink-dark-like solitons and parallel solitons,which keep their shapes and velocities unchanged during the propagation.In chapter 8,we summarize the work of this dissertation,and point out some shortcomings.Besides,we will give out the future research work.