Darboux Transformation And New Solutions Of A Variable-Coefficient Modified Korteweg-de Vries Equation

The protagonist of this work is a variable-coefcient modified Korteweg-de Vries(vc-mKdV) equation, which plays an important role in the researches of Bose-Einsteincondensates and fluid dynamics.The n-fold Darboux transformation of the vc-mKdV equation is constructed in theform of determinant based on the technique used to solve Ablowitz-Kaup-Newell-Segursystem. By virtue of this, a series of new soliton solutions of the vc-mKdV equation areobtained, and the characteristics of these solutions are analysed in detail. Particularly,what roles the variable coefcients play in the solitonic propagation are illustrated witha group of graphs. And a classification of the two-soliton solutions is demonstrated,i.e. the signs of two spectral parameters determine whether the two-soliton is bright-bright-soliton, bright-dark-soliton or dark-dark-soliton. Here, a group of pictures arealso given for the further statement. Meanwhile, we compare the Darboux method inthe present work with the mapping method in [36], and simply discuss their respectiveadvantages.Many learned men have investigated the positon solutions of the nonlinear evolu-tion equations since Matveev put forward the term “positon”. In this paper, based onthe soliton solutions, we get the positon and soliton-positon solutions of the vc-mKdVequation using the Taylor series expansion. It’s worth noting that these positon solu-tions are nonsingular, which is diferent from the singular positon given by Matveevand others.