Symbolic Computation On The Soliton And Rogue-wave Solutions For The Nonlinear Equations In Optical Fibers And Other Fields

Nonlinear evolution equations (NLEEs) can be used to describe the nonlinear phenomena in such fields as the optical fibers, Heisenberg fer-romagnets, fluids and plasmas. There exist some rational solutions for the NLEEs, such as the soliton and rogue-wave solutions. Because of the balance between the dispersion and nonlinearity, soliton appears. As one of the solutions localized in both time and space, Peregrine soliton can be viewed as the mathematical description of the rogue wave. With the symbolic computation, the solitons and rogue waves in the optical fibers and other fields are analytical studied based on some nonlinear Schrodinger(NLS)-type equations. The main research contents of this dissertation are as follows:(1) The (2+1)-dimensional coupled NLS equations with variable co-efficients are investigated. Firstly, via the rational dependent variable transformation, the equations are transformed to the bilinear forms, then bright one- and two-soliton solutions for the equations are derived. Based on the soliton solutions obtained, propagation of the one soliton and col-lision between the two solitons are shown and analyzed with graphics.(2) Investigated is the fourth-order variable-coefficient NLS equation.Under the integrable conditions, dark one- and two-soliton solutions for this equation are derived with the Hirota bilinear method. Based on the soliton solutions obtained, propagation of the one soliton and collision between the two solitons are displayed in some figures, and effects of some coefficients on the propagation of the one soliton and collision between the two solitons are analyzed.(3) Under investigation is the Kundu-Eckhaus equation with vari-able coefficients. Firstly, with the Lax pair, we construct the generalized Darboux transformation for the equation based on the Darboux trans-formation. Then, the first- and second-order rogue-wave solutions are respectively derived. Finally, effects of the nonlinear dispersion on the first- and second-order rogue waves are graphically discussed.(4) The generalized nonautonomous nonlinear equation is studied.(a) Under the integrable conditions, bilinear forms for this equation are derived via the rational dependent variable transformation, then bright one- and two-soliton solutions are obtained. Influences of the coefficients on the propagation of the one soliton and collision between the two soli-tons are graphically investigated. Besides, with the split-step Fourier method, we study the stability of the solitons with respect to the finite initial perturbations. (b) Under the integrable conditions, the general-ized Darboux transformation for this equation is constructed based on the Darboux transformation. Then, the first- and second-order rogue-wave solutions are derived, and the properties of the rogue waves are graphically discussed.(5) The discrete Ablowitz-Ladik equation with constant coefficients and the one with variable coefficients are respectively studied. Firstly,dark-soliton solutions for the discrete Ablowitz-Ladik equation with con-stant coefficients and bright-soliton solutions for the discrete Ablowitz-Ladik equation with variable coefficients are derived with the Hirota bi-linear method. Then, propagation of the one soliton and collision between the two solitons are graphically discussed.(6) Investigation is carried out on the coupled cubic-quintic NLS equations. Firstly, via the Hirota bilinear method, bright-bright soliton solutions for the equations are derived. Then, some types of the soliton collisions: Bound states, head-on and overtaking collisions between the two solitons are observed in figures.(7) Investigated is the nonlinear system with variable coefficients.Based on the Darboux transformation, we construct the generalized Dar-boux transformation for the system. Then, the first- and second-order rogue-wave solutions are obtained, respectively. Finally, influences of the coefficients on the first- and second-order rogue waves are analyzed graph-ically.