3 research outputs found
Solutions to the complex Korteweg-de Vries equation: Blow-up solutions and non-singular solutions
In the paper two kinds of solutions are derived for the complex Korteweg-de
Vries equation, including blow-up solutions and non-singular solutions. We
derive blow-up solutions from known 1-soliton solution and a double-pole
solution. There is a complex Miura transformation between the complex
Korteweg-de Vries equation and a modified Korteweg-de Vries equation. Using the
transformation, solitons, breathers and rational solutions to the complex
Korteweg-de Vries equation are obtained from those of the modified Korteweg-de
Vries equation. Dynamics of the obtained solutions are illustrated.Comment: 12 figure
Complex-valued Burgers and KdV-Burgers equations
Spatially periodic complex-valued solutions of the Burgers and KdV-Burgers
equations are studied in this paper. It is shown that for any sufficiently
large time T, there exists an explicit initial data such that its corresponding
solution of the Burgers equation blows up at T. In addition, the global
convergence and regularity of series solutions is established for initial data
satisfying mild conditions