| Accompanied with the rapid development of the information science technology, symbolic computation, which is the new branch of the artificial intelligence, has been extensively applied in certain branches of scientific researches, and it has been an assis-tant tool for the study of science and technology. Certain interest has been focused on the nonlinear science, which is one of the fields of current advanced scientific researches, and the soliton theory is one of the three main branches of the nonlinear science. Based on symbolic computation and Darboux transformation (DT), the dissertation is to in-vestigate the integrable properties, analytic solutions and physical applications of some nonlinear evolution equations which have physical backgrounds, some algebraic algo-rithms that can be performed with symbolic computation are also proposed. Types of equations investigated in the dissertation are mainly the variable-coefficient, coupled, higher-dimensional and higher-order nonlinear evolution equations.The work of this dissertation includes the following aspects:(1) Comparing with the equations with constant coefficients, the corresponding ones with variable-coefficients can describe the physical mechanism of nonlinear phenomena in physical situations with inhomogeneities media or nonuniformities of boundaries, and be considered to be more practical. To describe various complex nonlinear phenomena in real situation, some equations with time and space variable-coefficients from certain physical settings have been proposed. Due to its relevance to fundamental aspects of physics and technology, coherent atom optics is the subject of much current interest. For that purpose, the lower dimensional condensates have been the subject of active studies. Certain dynamics of the Bose-Einstein condensates (BECs) have been claimed to be governed by the one-dimensional Gross-Pitaevskii (GP) equation for a time-dependent trap. In this chapter, by virtue of the Painleve analysis and symbolic computation, we will derive the integrable condition for the GP equation with the time-dependent scat-tering length in the presence ot a confining or expulsive time-dependent trap. Lax pair for the equation will be directly obtained via the Ablowitz-Kaup-Newell-Segur (AKNS) scheme under the integrable condition. Bright one-soliton-like solution of the GP equa-tion will be presented via the auto-Backlund transformation and some analytic solutions with variable amplitudes will be obtained by the ansatz method. In addition, an infinite number of conservation laws will be derived. Those results could be of some values for the studies on the BECs.(2) Nowadays,(2+1)-dimensional nonlinear evolution equations in soliton theory and mathematical physics are the subject of active studies. Due to the limitation of the dimension, the pure (1+1)-dimensional systems can not account for some ob-served features. In realistic situations, the higher dimensional systems may provide more complex models. As a typical example in (2+1)-dimensional systems, the Kadomtsev-Petviashvili (KP) system has been derived from many physical applications in the Bose-Einstein condensation, plasma physics and nonlinear optics, etc. In this chapter, a coupled KP system is investigated with symbolic computation, some researches have proposed that the system could be decomposed into the first two members of the (1+1)-dimensional AKNS hierarchy, and based on that decomposition, we will obtain three kinds of DTs of its reduced equations. Moreover, the multi-soliton-like solutions of the coupled KP system will be derived. Finally some figures will be plotted to discuss the propagation features of the soliton solutions. Those solutions could be of some values for the studies in the context of random matrix theory.(3) In this chapter, by virtue of the DT and symbolic computation, the Kundu-Eckhaus (KE) equation with the cubic and quintic nonlinear terms and KE equation with variable coefficients will be investigated. The KE equation appears in the nonlin-ear optics, quantum field theory and weakly nonlinear dispersive matter waves. The KE equation with variable coefficients, which is an extension of the KE one, possesses the soliton solutions with more parametric freedom, therefore its solutions can be expected to model more complex situations in reality than the KE counterpart. Through DT, one-, two-and three-soli ton solutions of the KE equation will be presented in the form of modulus; Some physical quantities such as the amplitude, width, initial phase and energy of soliton solutions will be obtained; The asymptotic behavior of the two-soli ton solution will be analyzed, and shows that the collision is elastic; Figures will be plotted to discuss the propagation features of the soliton solutions:(ⅰ) Elastic collisions of the two solitons, the head-on and overtaking;(ii) Periodic attraction and repulsion of the bounded states of the two solitons;(iii) Two types of the dynamic characters of the three solitons:Three solitons intersect at a point at the moment of the collision and the soli-tons keep their shapes except for the phase shifts; Two solitons keep the bounded state and one another soliton retains its own shape invariant before and after the collision. Through the extension of the KE equation, the KE equation with variable coefficients will also be investigated in this chapter by virtue of the DT and symbolic computation. Lax pair of the equation will be obtained, and the corresponding DT will be constructed. One-, two-and three-soliton solutions will be presented, and the asymptotic behaviors of the two-soliton solution will be analyzed. Figures will be plotted to discuss the propaga-tion features of the soliton solutions:(i) As the one-soli ton solution, for different choices of the nonlinear dispersion, we obtain different soliton structures:When the nonlinear dispersion is constant, the soliton propagates with an invariant energy, amplitude and uniform velocity; When the nonlinear dispersion is variable, we present five different types of soliton structures with the varying velocities;(ii) The head-on collision, over-taking collision, parabolic solitons, periodic oscillation and the bounded states of the two solitons are analyzed;(iii) Four different dynamic characters of the three solitons are presented:Three solitons collide at a point, keeping their shapes unchanged except for the phase shifts before and after the collision; Two solitons keep the bounded state and the third one retains its own shape invariant before and after the collision; Three parabolic solitons; Collisions between one parabolic soliton and a two-soliton-bounded structure.(4) In this chapter, the quintic generalization of the coupled cubic nonlinear Schrodinger equations from twin-core nonlinear optical fibers and waveguides will be studied, which describe the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. Lax pair of the equations will be obtained and the corresponding DT will be constructed. Moreover, one-, two-and three-soliton solutions will be presented in the form of modulus. Dynamic features of solitons will be graphically discussed:(i) Head-on and overtaking elastic collisions of the two solitons;(ii) Periodic attraction and repulsion of the bounded states of two solitons;(iii) Energy-exchanging collisions of the three solitons. Finally an infinite number of conservation laws of the extension equation will also be presented. |
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