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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

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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

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KP equation

(ut − uux − uxxx)x = uyy

Perturbative symmetry approach

(ut−εuux − uxxx)x = uyy

Dispersive deformation

  • ut − uux−ε2uxxx
  • x = uyy

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)

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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

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The method of hydrodynamic reductions

Applies to quasilinear equations

A(u)ux + B(u)uy + C(u)ut = 0

Consists of seeking N-phase solutions

u = u(R1, ..., RN)

The phases Ri(x, y, t) are required to satisfy a pair of commuting equations

Ri

y = µi(R)Ri x,

Ri

t = λi(R)Ri x

Commutativity conditions:

∂jµi µj−µi = ∂jλi λj−λi

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

  • f one variable.

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Example of dKP

(ut − uux)x = uyy

First order (hydrodynamic) form:

ut − uux = wy, uy = wx N-phase solutions: u = u(R1, ..., RN), w = w(R1, ..., RN) where Ri

y = µi(R)Ri x,

Ri

t = λi(R)Ri x

Then

∂iw = µi∂iu, λi = u + (µi)2

Equations for u(R) and µi(R) (Gibbons-Tsarev system):

∂jµi = ∂ju µj − µi , ∂i∂ju = 2 ∂iu∂ju (µj − µi)2

In involution! General solution depends on N arbitrary functions of one variable.

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Systems of hydrodynamic type in (2+1)D

ut + A(u)ux + B(u)uy = 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

N i

jk = V p j ∂upV i k − V p k ∂upV i j − V i p(∂ujV p k − ∂ukV p j )

Haantjes tensor

Hi

jk = N i prV p j V r k − N p jrV i pV r k − N p rkV i pV r j + N p jkV i r V r p

General case: Integrability =

⇒ H(V ) = 0 where V = (A + kE)−1(B + lE).

Hamiltonian case: Integrability ⇐

⇒ H(V ) = 0.

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Hydrodynamic chains

ut + V (u)ux = 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

u1

t = f(u1, u2)x,

u2

t = g(u1, u2, u3)x,

u3

t = h(u1, u2, u3, u4)x, ...

Hamiltonian chains

ut =

  • B d

dx + d dxBt ∂h ∂u, h(u1, u2, u3)

Generic case: h = (u3 + P(u1, u2))1/3 where P is a cubic polynomial.

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Equations of the dispersionless Hirota type

F(uxx, uxy, uyy, uxt, uyt, utt) = 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).

euxx + euyy = eutt utt = uxy uxt + 1 6η(uxx)u2

xt

where η solves the Chazy equation.

21-dimensional moduli space, equivalence group Sp(6, R)

Geometry: hypersurfaces in the Lagrangian Grassmannian

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Second order quasilinear PDEs

f11uxx + f22uyy + f33utt + 2f12uxy + 2f13uxt + 2f23uyt = 0 fij depend on the first order derivatives ux, uy, ut 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).

uxx + uyy − eututt = 0 α℘′(ux) − ℘′(uy) ℘(ux)℘(uy) uxy+β ℘′(ut) − ℘′(ux) ℘(ux)℘(ut) uxt+γ ℘′(uy) − ℘′(ut) ℘(uy)℘(ut) uyt = 0 20-dimensional moduli space, equivalence group SL(4, R)

Geometry: conformal structures in projective space

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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.

  • ut − uux−ε2uxxx
  • x = uyy

Look for deformed N-phase solutions in the form

u = u(R1, ..., RN)+ε2(. . . ) + ε4(. . . ) + . . .

where

Ri

y = µi(R)Ri x+ε2(. . . ) + ε4(. . . ) + . . .

Ri

t = λi(R)Ri x+ε2(. . . ) + ε4(. . . ) + . . .

Here (. . . ) are required to be polynomial and homogeneous in the derivatives of

  • Ri. Recall that λi = u + (µi)2, and µi, u satisfy the Gibbons-Tsarev system.

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Deformations of one-phase reductions of dKP

  • ut − uux−ε2uxxx
  • x = uyy

Deformed one-phase reductions (modulo the Miura group can assume u = R):

Ry =µRx +ε2

  • µ′Rxx + 1

2(µ′′ − (µ′)3)R2

x

  • x

+ O(ε4) Rt =(µ2 + R)Rx +ε2 (2µµ′ + 1)Rxx + (µµ′′ − µ(µ′)3 + (µ′)2/2)R2

x

  • x + O(ε4)

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.

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Generalized KP equation

ut − uux+ε(A1uxx + A2u2

x) + ε2(B1uxxx + B2uxuxx + B3u3 x) = wy

wx = uy

Require that all one-phase reductions can be deformed as

u = R, w = w(R)+ε2(. . . ) + ε4(. . . ) + . . .

where

Ry =µRx+ε2(. . . ) + ε4(. . . ) + . . . Rt =(µ2 + R)Rx+ε2(. . . ) + ε4(. . . ) + . . . w′ = µ. This gives A1 = A2 = B2 = B3 = 0, B1=const, = ⇒ KP

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Classification result: scalar third order (2+1)D soliton equations with simplest nonlocalities

ut = ϕux + ψuy + ηwy+ǫ(...) + ǫ2(...), wx = uy

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 Dyu is the nonlocality, no other non-local terms are

allowed.

  • Classify integrable dispersionless systems of the form

ut = ϕux + ψuy + ηwy, wx = uy

  • Add dispersive corrections which inherit all hydrodynamic reductions (sufficient

to consider 1-component reductions only)

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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.

ϕuu = −ϕ2

w + ψuϕw − 2ψwϕu

η , ϕuw = ηwϕu η , ϕww = ηwϕw η ψuu = −ϕwψw + ψuψw − 2ϕwηu + 2ηwϕu η , ψuw = ηwψu η , ψww = ηwψw η ηuu = −ηw (ϕw − ψu) η , ηuw = ηwηu η , ηww = η2

w

η

In involution, straightforward to solve: η = 1, η = u, η = ewh(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.

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Non-uniqueness of dispersive corrections: VN and mVN equations

Veselov-Novikov equation

ut = (uw)y+ǫ2uyyy, wx = uy

modified Veselov-Novikov equation

ut = (uw)y+ǫ2

  • uyy − 3

4 u2

y

u

  • y

, wx = uy

Dispersionless limit

ut = (uw)y, wx = uy

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BKP and CKP equations

ut−5(u2+w)ux−5uwx+5wy+ε2(uuxxx + wxxx + uxxx) − ε4 25uxxxxx = 0, wx = uy

and

ut−5(u2+w)ux−5uwx+5wy+ε2(uuxxx + wxxx + 5 2uxxx) − ε4 25uxxxxx = 0, wx = uy

Dispersionless limit

ut − 5(u2 + w)ux − 5uwx + 5wy, wx = uy

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New examples

ut = (βw + β2u2)ux − 3βuuy + wy+ǫ2[B3(u) − 2β2B2(u)ux]

here B = 2β2uDx − 2βDy.

ut = 4 27γ2u3ux + (w + γu2)uy + uwy+ǫ2[B3(u) − 1 3γuxB2(u)]

here B = 1

3γuDx + Dy.

ut = δ u3 ux − 2wuy + uwy−ǫ2 u 1 u

  • xxx

For δ = 0 it reduces to the Harry Dym equation.

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