Quantum mechanics

Korteweg–de Vries equation

Statistics

Variation of parameters

Differential equation

Applied mathematics

Bilinear form

Physics

Nonlinear system

Mathematics

Bilinear interpolation

Wronskian

Mathematical analysis

Partial differential equation

Transactions of the American Mathematical Society

A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.

Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions