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

By Wenxiu Ma, Yuncheng You. 2004

534 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b46793e747bc653fe235f6320109eae2d33f37a2

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Citations in the corpus

The inverse scattering transform and soliton solutions of a combined modified Korteweg–de Vries equation (2019, methodology): “Many other approaches to soliton solutions are available in the field of integrable equations, among which are the Hirota direct method [15], the generalized bilinear technique [26], the Wronskian technique [9,32] and the Darboux transformation [33,36].”
Lump and lump-soliton solutions to the $$(2+1)$$(2+1)-dimensional Ito equation (2018, methodology): “Hirota bilinear forms are one of the integrablility characteristics of nonlinear partial differential equations [1] and associated bilinear equations can be solved by the Wronskian technique [2,3].”
Interaction solutions to Hirota-Satsuma-Ito equation in (2 + 1)-dimensions (2019, background): “, [36]) between two different kinds of solutions to integrable equations.”
VARIATION OF PARAMETERS METHOD FOR SOLVING SIXTH-ORDER BOUNDARY VALUE PROBLEMS (2009, methodology): “The variation of parameters method [12, 13, 18-22, 26, 27] provides the general solution of equation (1) as follows”
Long-Time Asymptotics of a Three-Component Coupled mKdV System (2019, methodology): “direct method [59], the generalized bilinear technique [60], the Wronskian technique [61,62] and the 198”
Riemann–Hilbert problems andN-soliton solutions for a coupled mKdV system (2018, methodology): “There are many other approaches to soliton solutions in the field of integrable systems, which include the bilinear method [31,32], the Wronskian technique [33,34] and the Darboux transformation [35].”
A second Wronskian formulation of the Boussinesq equation (2009, methodology): “Therefore, in order to constructWronskian solutions as in the case of the KdV equation [32], we need to solve the following two representative systems: φxxx = λφ + h, φt = ± √ 3φxx, (3.”
A refined invariant subspace method and applications to evolution equations (2012, methodology): “f motion. Based on the classiﬁcation of elementary functions, there are only three kinds of explicit elementary exact solutions, classiﬁed as soliton-like, positon-like and complexiton-like solutions [1]-[4], to diﬀerential equations, both linear and nonlinear. Hirota direct method presents Wronskia”
Application of the Riemann–Hilbert approach to the multicomponent AKNS integrable hierarchies (2019, methodology): “available in the field of integrable equations, among which are the Hirota direct method [50], the generalized bilinear technique [51,52], the Wronskian technique [53,54] and the Darboux transformation [55].”