SOLITON EQUATIONS IN 2+1 DIMENSIONS: DEFORMATIONS OF DISPERSIONLESS - PowerPoint PPT Presentation

SOLITON EQUATIONS IN 2+1 DIMENSIONS: DEFORMATIONS OF DISPERSIONLESS LIMITS Eugene Ferapontov Department of Mathematical Sciences, Loughborough University, UK E.V.Ferapontov@lboro.ac.uk Collaboration: A Moro, V Novikov Enigma Workshop ‘Geometry and Integrability’ Obergurgl, 13-20 December 2008 1

KP equation ( u t − uu x − u xxx ) x = u yy Perturbative symmetry approach ( u t − εuu x − u xxx ) x = u yy Dispersive deformation u t − uu x − ε 2 u xxx � � x = u yy Program of classification of (2+1)D integrable systems: • Classify (2+1)D dispersionless systems which may (potentially) arise as dispersionless limits of integrable soliton equations (method of hydrodynamic reductions) • Understand how to add dispersive corrections (deformation of hydrodynamic reductions) 2

Plan: • The method of hydrodynamic reductions. Example of dKP • Classification of (2+1)D dispersionless integrable systems – Systems of hydrodynamic type – Hydrodynamic chains – Equations of the dispersionless Hirota type – Second order quasilinear PDEs • Dispersive deformations of dispersionless integrable systems • Classification of third order (2+1)D soliton equations with ‘simplest’ nonlocalities 3

The method of hydrodynamic reductions Applies to quasilinear equations A ( u ) u x + B ( u ) u y + C ( u ) u t = 0 Consists of seeking N-phase solutions u = u ( R 1 , ..., R N ) The phases R i ( x, y, t ) are required to satisfy a pair of commuting equations R i y = µ i ( R ) R i R i t = λ i ( R ) R i x , x ∂ j µ i ∂ j λ i µ j − µ i = Commutativity conditions: λ j − λ i Definition A quasilinear system is said to be integrable if, for any number of phases N, it possesses infinitely many N-phase solutions parametrized by 2N arbitrary functions of one variable. 4

Example of dKP ( u t − uu x ) x = u yy First order (hydrodynamic) form: u t − uu x = w y , u y = w x N -phase solutions: u = u ( R 1 , ..., R N ) , w = w ( R 1 , ..., R N ) where R i y = µ i ( R ) R i R i t = λ i ( R ) R i x , x Then λ i = u + ( µ i ) 2 ∂ i w = µ i ∂ i u, Equations for u ( R ) and µ i ( R ) (Gibbons-Tsarev system): ∂ j u ∂ i u∂ j u ∂ j µ i = µ j − µ i , ∂ i ∂ j u = 2 ( µ j − µ i ) 2 In involution! General solution depends on N arbitrary functions of one variable. 5

Systems of hydrodynamic type in (2+1)D u t + A ( u ) u x + B ( u ) u y = 0 E. V. Ferapontov and K. R. Khusnutdinova, Double waves in multi-dimensional systems of hydrodynamic type: the necessary condition for integrability, Proc. Royal Soc. A 462 (2006) 1197-1219. E.V. Ferapontov, A. Moro and V.V. Sokolov, Hamiltonian systems of hydrodynamic type in 2+1 dimensions, Comm. Math. Phys. 285 , no. 1 (2009) 31–65. Nijenhuis tensor jk = V p k − V p p ( ∂ u j V p k − ∂ u k V p N i j ∂ u p V i k ∂ u p V i j − V i j ) Haantjes tensor pr V p k − N p k − N p j + N p H i jk = N i j V r jr V i p V r rk V i p V r jk V i r V r p ⇒ H ( V ) = 0 where V = ( A + kE ) − 1 ( B + lE ) . General case: Integrability = ⇒ H ( V ) = 0 . Hamiltonian case: Integrability ⇐ 6

Hydrodynamic chains u t + V ( u ) u x = 0 E. V. Ferapontov and D. G. Marshall, Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor, Math. Ann. 339 , no. 1 (2007) 61-99. Haantjes tensor H ( V ) well-defined! Conservative chains u 1 t = f ( u 1 , u 2 ) x , u 2 t = g ( u 1 , u 2 , u 3 ) x , u 3 t = h ( u 1 , u 2 , u 3 , u 4 ) x , ... Hamiltonian chains � ∂h � B d dx + d dxB t h ( u 1 , u 2 , u 3 ) u t = ∂ u , Generic case: h = ( u 3 + P ( u 1 , u 2 )) 1 / 3 where P is a cubic polynomial. 7

Equations of the dispersionless Hirota type F ( u xx , u xy , u yy , u xt , u yt , u tt ) = 0 E.V. Ferapontov, L. Hadjikos and K.R. Khusnutdinova, Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian, arXiv: 0705.1774 (2007). e u xx + e u yy = e u tt u tt = u xy + 1 6 η ( u xx ) u 2 xt u xt where η solves the Chazy equation. 21 -dimensional moduli space, equivalence group Sp (6 , R ) Geometry: hypersurfaces in the Lagrangian Grassmannian 8

Second order quasilinear PDEs f 11 u xx + f 22 u yy + f 33 u tt + 2 f 12 u xy + 2 f 13 u xt + 2 f 23 u yt = 0 f ij depend on the first order derivatives u x , u y , u t only. P . A. Burovskii, E. V. Ferapontov and S. P . Tsarev, Second order quasilinear PDEs and conformal structures in projective space; arXiv:0802.2626v1, (2008). u xx + u yy − e u t u tt = 0 α℘ ′ ( u x ) − ℘ ′ ( u y ) u xy + β ℘ ′ ( u t ) − ℘ ′ ( u x ) u xt + γ ℘ ′ ( u y ) − ℘ ′ ( u t ) u yt = 0 ℘ ( u x ) ℘ ( u y ) ℘ ( u x ) ℘ ( u t ) ℘ ( u y ) ℘ ( u t ) 20 -dimensional moduli space, equivalence group SL (4 , R ) Geometry: conformal structures in projective space 9

Dispersive deformations of dispersionless integrable systems E. V. Ferapontov and A. Moro, Dispersive deformations of hydrodynamic reductions of 2D dispersionless integrable systems, J. Phys. A: Math. Theor. 42 (2009) 035211, 15pp. � u t − uu x − ε 2 u xxx � x = u yy Look for deformed N-phase solutions in the form u = u ( R 1 , ..., R N )+ ε 2 ( . . . ) + ε 4 ( . . . ) + . . . where R i y = µ i ( R ) R i x + ε 2 ( . . . ) + ε 4 ( . . . ) + . . . R i t = λ i ( R ) R i x + ε 2 ( . . . ) + ε 4 ( . . . ) + . . . Here ( . . . ) are required to be polynomial and homogeneous in the derivatives of R i . Recall that λ i = u + ( µ i ) 2 , and µ i , u satisfy the Gibbons-Tsarev system. 10

Deformations of one-phase reductions of dKP � u t − uu x − ε 2 u xxx � x = u yy Deformed one-phase reductions (modulo the Miura group can assume u = R ): R y = µR x � � µ ′ R xx + 1 2( µ ′′ − ( µ ′ ) 3 ) R 2 + ε 2 + O ( ε 4 ) x x R t =( µ 2 + R ) R x (2 µµ ′ + 1) R xx + ( µµ ′′ − µ ( µ ′ ) 3 + ( µ ′ ) 2 / 2) R 2 + ε 2 � � x + O ( ε 4 ) x Conjecture For any integrable system in (2+1)D, all hydrodynamic reductions of its dispersionless limit can be deformed into reductions of the dispersive counterpart. 11

Generalized KP equation u t − uu x + ε ( A 1 u xx + A 2 u 2 x ) + ε 2 ( B 1 u xxx + B 2 u x u xx + B 3 u 3 x ) = w y w x = u y Require that all one-phase reductions can be deformed as w = w ( R )+ ε 2 ( . . . ) + ε 4 ( . . . ) + . . . u = R, where R y = µR x + ε 2 ( . . . ) + ε 4 ( . . . ) + . . . R t =( µ 2 + R ) R x + ε 2 ( . . . ) + ε 4 ( . . . ) + . . . w ′ = µ . This gives A 1 = A 2 = B 2 = B 3 = 0 , B 1 =const, = ⇒ KP 12

Classification result: scalar third order (2+1)D soliton equations with simplest nonlocalities u t = ϕu x + ψu y + ηw y + ǫ ( ... ) + ǫ 2 ( ... ) , w x = u y here ϕ, ψ, η are functions of v and w , and ( . . . ) denote terms which are polynomial in the derivatives of v and w with respect to x and y of orders 2 and 3 , respectively. Here w = D − 1 x D y u is the nonlocality, no other non-local terms are allowed. • Classify integrable dispersionless systems of the form u t = ϕu x + ψu y + ηw y , w x = u y • Add dispersive corrections which inherit all hydrodynamic reductions (sufficient to consider 1-component reductions only) 13

Classification of integrable dispersionless limits Integrability conditions, based on E.V. Ferapontov and K.R. Khusnutdinova, The characterization of 2-component (2+1)-dimensional integrable systems of hydrodynamic type, J. Phys. A: Math. Gen. 37 , no. 8 (2004) 2949–2963. ϕ uu = − ϕ 2 w + ψ u ϕ w − 2 ψ w ϕ u ϕ uw = η w ϕ u ϕ ww = η w ϕ w , , η η η ψ uu = − ϕ w ψ w + ψ u ψ w − 2 ϕ w η u + 2 η w ϕ u , ψ uw = η w ψ u , ψ ww = η w ψ w η η η η ww = η 2 η uu = − η w ( ϕ w − ψ u ) η uw = η w η u w , , η η η In involution, straightforward to solve: η = 1 , η = u, η = e w h ( u ) Conjecture For any ϕ, ψ, η one can reconstruct (non-uniquely) dispersive corrections which inherit all hydrodynamic reductions. Infinite series in ǫ are required in general. 14

Non-uniqueness of dispersive corrections: VN and mVN equations Veselov-Novikov equation u t = ( uw ) y + ǫ 2 u yyy , w x = u y modified Veselov-Novikov equation � � u 2 u yy − 3 y u t = ( uw ) y + ǫ 2 , w x = u y 4 u y Dispersionless limit u t = ( uw ) y , w x = u y 15

BKP and CKP equations u t − 5( u 2 + w ) u x − 5 uw x +5 w y + ε 2 ( uu xxx + w xxx + u xxx ) − ε 4 25 u xxxxx = 0 , w x = u y and 2 u xxx ) − ε 4 u t − 5( u 2 + w ) u x − 5 uw x +5 w y + ε 2 ( uu xxx + w xxx + 5 25 u xxxxx = 0 , w x = u y Dispersionless limit u t − 5( u 2 + w ) u x − 5 uw x + 5 w y , w x = u y 16

New examples u t = ( βw + β 2 u 2 ) u x − 3 βuu y + w y + ǫ 2 [ B 3 ( u ) − 2 β 2 B 2 ( u ) u x ] here B = 2 β 2 uD x − 2 βD y . u t = 4 27 γ 2 u 3 u x + ( w + γu 2 ) u y + uw y + ǫ 2 [ B 3 ( u ) − 1 3 γu x B 2 ( u )] here B = 1 3 γuD x + D y . � 1 u 3 u x − 2 wu y + uw y − ǫ 2 u t = δ � u u xxx For δ = 0 it reduces to the Harry Dym equation. 17