Classification of Integrable Evolution Equations of the Form u_t=u_xxx+f(t,x,u,u_x,u_xx)

1996/05/17 by Ayşe Hümeyra Bilge, Bilge, Ayse Humeyra
Chemistry · Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Molecular spectroscopy and chirality #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.solv-int/9605004

openalex publication_date 1996/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain the classification of integrable equations of the form u_t=u_xxx+f(t,x,u,u_x,u_xx) using the formal symmetry method of Mikhailov et al [A.V.Mikhailov, A.B.Shabat and V.V.Sokolov, in \it What is Integrability edited by V.E. Zakharov (Springer-Verlag, Berlin 1991)]. We show that all such equations can be transformed to an integrable equation of the form v_t=v_xxx+f(v,v_x,v_xx) using transformations Φ(x,t,u,v,u_x,v_x)=0, and the u_xx dependence can be eliminated except for two equations.

Related

• Miura Transformations for Integrable Evolution Equations of the Form u_t=u_xxx+f(t,x,u,u_x,u_xx)
• The symmetry approach to integrability: recent advances
• Integrability conditions for Boussinesq type systems
• From fully-nonlinear to semilinear evolution equations: two symmetry-integrable examples
• Symmetry approach to integrability and non-associative algebraic structures