Study Of Soliton Theory

In this paper, we discuss mainly three aspects about the Hirota's direct method, perturbation, symmetries and similarity solutions of nonlinear partial differential equations, and they read as follows:1. Firstly the significant characters of the bilinear operator are studied, then we investigate the characters of the nonlinear PDE directly obtained by the logarithmic transformation for the KdV-type bilinear equation. According to these characters, the polynomial transformations with nonempty linear term between the original polynomial equation with nonempty linear term and its KdV-type bilinear equation are obtained in details and also are confirmed with some examples. Some necessary conditions for the given polynomial nonlinear PDE with nonempty linear term, which owns corresponding polynomial transformation with nonempty linear term, are found. If the given nonlinear PDE owns its Kdv-type bilinear form, according to these necessary conditions, we can find the relationships between linear operators of the original equation and the bilinear operators of its bilinear form. And the bilinear form of the given original equation can be obtained directly in virtue of the relationships. These results make us straightforth understanding of the KdV-type nonlinear PDE, which help us improve directly some results appearing in literature.2,As for the 1~st-order perturbed nonlinear PDE, we obtain the conditional equality that the coefficient of the 1~st-order term of the Taylor series expansion aroundε=0 for any asymptotical analytical solution of the perturbed PDE with perturbing parameterεmust be admitted. By making use of the conditional equality, we may obtain some transformations, which directly map the analytical solutions of the given unperturbed PDE to the asymptotical analytical solutions of the corresponding perturbed one. The Lie-Baclund symmetries is introduced in order to obtaining more transformations. Hence, we can directly create more transformations in virtue of known Lie-Baclund symmetries and recursion operators of corresponding unperturbed equation. The perturbed Burgers equation and the perturbed Korteweg-de Vries(KdV) equation are used as examples. As for the higher-order cases, we only give the main results with a simple example. We should notice that not any group theoretical knowledge is introduced in the deduce process for obtaining these theoretical results and transformations. In addition, the N-solitons solutions of the perturbed KdV equations are investigated by Hirota's direct method approach and the 1-soliton diagram are illustrated.3. By a known transformation, the Broer-Kaup(BK) equations are combined to a single one. After the classical Lie symmetry analysis and similarity reductions are performed for the single one, new similar solutions are obtained. From the special kind of its similarity reductions, we find the relationship between the BK equations and the famous Burgers equation and the heat equation by observation. By making use of these relationship, more similarity solution can be created directly by abundant known solutions of the Burgers equation and the heat equation.