第六屆非線性數(shù)學物理國際會議暨全國第十三屆可積系統(tǒng)學術研討會1文稿an integrable nonlinear diffusion hierarchy with a common2d arbitrary function

1、 An Integrable nonlinear diffusion hierarchy with a common 2D arbitrary functionAppl.Math.Lett.50(2015)105SY Lou with LL Xue and QP Liu and 2015.8.23-Weifang 濰坊 Introduction (Levi, Anco)From hereditary operator to integrable hierarchiesAbitrary first order scalar PDEs are integrableAll 1D differenti

2、al scalar hereditary operators (HO)General integrable nonlinear diffusion hierarchySymmetries and Lax pairs of INDHDamped Cole-Hopf hierarchyPrimary branch solutions and special solutionsInfinitely many (nonlocal) conservation lawsDiscussionsIntroduction (symmetries and conservation laws) Modeling P

3、hysics (standard model in particle phys) Symmetries and symmetry breaking Symmetries and conservation laws (Ancos talk) From symmetries to exact solutions (all solutions?) From strong symmetries to symmetries Strong symmetry+hereditary Recursion operator From symmetries to integrable models From exa

4、ct solutions to (conditional) symmetries Classification of hereditary operators From hereditary operator to integrable hierarchies Fokas, Fuchsteiner, Stramp, Santini TuGZ,TianC,MaWX,LiuQP,QiaoZJ, ZhouRG,LouS Definition on HO (for arbitrary f and g): (*) From HO to integrable hierarchies (IHs) Speci

5、al IHs traditional ones ，negative IHs breaking soliton IHs Abitrary first order scalar PDEs are integrable (*) All the scalar functional solution of (*) in the form Unique result: for arbitrary F and n Abitrary first order scalar PDEs (*) are integrable for arbitrary F and n. (*) is C-integrable mod

6、el and its all solutions can be find by symmetry approach SY Lou and RX Yao, 2014, arXiv:1402.6938 v1 All 1D differential scalar hereditary operators (HO) (*) All 1D differential scalar operator of (*) : F and G are arbitrary func. of the indicated vars . K is determined by G. General integrable non

7、linear diffusion hierarchy 1+1 D 1+1 D integrable nonlinear diffusion equation Symmetries and Lax pairs of INDHModel: Lax Pair(s):Symmetries : Damped Cole-Hopf hierarchy (DCHH) Damped Cole-Hopf equation DCHH Special DCHH (SDCHH) by taking g=0 A special diffusion equation from SDCHH (n=1) (%) Conserv

8、ation laws and of SymmetriesSDCHE Local CL Nonlocal CLs Lie point symmetries Hodograph transformation and linearization of SDCHH Special nonlinear diffusion equation (#) Hodograph transformation Linearized hierarchy (n=2 for (#) Special symmetry reduction solutions of the special nonlinear diffusion

9、 equation Travelling wave solution (Lambert W func. solution) Bessel func. Solutions Discussions the only scalar hereditary function All 1st order scalar systems are C-integrable the only 1D scalar HOs All scalar PDEs with only 1D differential HO are C-integrable (yes for all differential HO?) classification of the hereditary matrices classifications of differential matrix o