The dispersionless 2D Toda equation: dressing, Cauchy problem, longtime behavior, implicit solutions and wave breaking
aa r X i v : . [ n li n . S I] N ov The dispersionless 2D Toda equation: dressing,Cauchy problem, longtime behaviour,implicit solutions and wave breaking
S. V. Manakov , § and P. M. Santini , § Landau Institute for Theoretical Physics, Moscow, Russia Dipartimento di Fisica, Universit`a di Roma ”La Sapienza”, andIstituto Nazionale di Fisica Nucleare, Sezione di Roma 1Piazz.le Aldo Moro 2, I-00185 Roma, Italy § e-mail: [email protected], [email protected] September 16, 2018
Abstract
We have recently solved the inverse spectral problem for one-parameter families of vector fields, and used this result to constructthe formal solution of the Cauchy problem for a class of integrable non-linear partial differential equations in multidimensions, including thesecond heavenly equation of Plebanski and the dispersionless Kadomt-sev - Petviashvili (dKP) equation, arising as commutation of vectorfields. In this paper we make use of the above theory i) to constructthe nonlinear Riemann-Hilbert dressing for the so-called two dimen-sional dispersionless Toda equation ( exp ( ϕ )) tt = ϕ ζ ζ , elucidatingthe spectral mechanism responsible for wave breaking; ii) we presentthe formal solution of the Cauchy problem for the wave form of it:( exp ( ϕ )) tt = ϕ xx + ϕ yy ; iii) we obtain the longtime behaviour of thesolutions of such a Cauchy problem, showing that it is essentiallydescribed by the longtime breaking formulae of the dKP solutions,confirming the expected universal character of the dKP equation asprototype model in the description of the gradient catastrophe of two-dimensional waves; iv) we finally characterize a class of spectral dataallowing one to linearize the RH problem, corresponding to a class ofimplicit solutions of the PDE. Introduction
It was observed long ago [1] that the commutation of multidimensional vectorfields can generate integrable nonlinear partial differential equations (PDEs)in arbitrary dimensions. Some of these equations are dispersionless (or quasi-classical) limits of integrable PDEs, having the dispersionless Kadomtsev- Petviashvili (dKP) equation u xt + u yy + ( uu x ) x = 0 [2],[3] as universalprototype example, they arise in various problems of Mathematical Physicsand are intensively studied in the recent literature (see, f.i., [4] - [18]). Inparticular, an elegant integration scheme applicable, in general, to nonlinearPDEs associated with Hamiltonian vector fields, was presented in [7] and anonlinear ¯ ∂ - dressing was developed in [13]. Special classes of nontrivialsolutions were also derived (see, f.i., [12], [15]).The Inverse Spectral Transform (IST) for 1-parameter families of mul-tidimensional vector fields has been developed in [19] (see also [20]). Thistheory, introducing interesting novelties with respect to the classical IST forsoliton equations [21, 22], has allowed one to construct the formal solutionof the Cauchy problems for the second heavenly equation [23] in [19] and forthe novel system of PDEs u xt + u yy + ( uu x ) x + v x u xy − v y u xx = 0 ,v xt + v yy + uv xx + v x v xy − v y v xx = 0 , (1)in [24]. The Cauchy problem for the v = 0 reduction of (1), the dKP equation,was also presented in [24], while the Cauchy problem for the u = 0 reductionof (1), an integrable system introduced in [16], was given in [25]. This ISTand its associated nonlinear Riemann - Hilbert (RH) dressing turn out to beefficient tools to study several properties of the solution space, such as: i)the characterization of a distinguished class of spectral data for which theassociated nonlinear RH problem is linearized and solved, corresponding to aclass of implicit solutions of the PDE (as it was done for the dKP equation in[26] and for the Dunajski generalization [27] of the second heavenly equationin [28]); ii) the construction of the longtime behaviour of the solutions of theCauchy problem [26]; iii) the possibility to establish whether or not the lackof dispersive terms in the nonlinear PDE causes the breaking of localizedinitial profiles and, if yes, to investigate in a surprisingly explicit way theanalytic aspects of such a wave breaking (as it was recently done for the(2+1)-dimensional dKP model in [26]).2n this paper we make use of this theory to study another distinguishedmodel arising as the commutation of vector fields, the so-called 2 dimensionaldispersionless Toda (2ddT) equation φ ζ ζ = (cid:0) e φ t (cid:1) t , φ = φ ( ζ , ζ , t ) , (2)or ϕ ζ ζ = ( e ϕ ) tt , ϕ = φ t , (3)also called Boyer-Finley [29] equation or SU( ∞ ) Toda equation [30], thenatural continuous limit of the 2 dimensional Toda lattice [34, 35] φ nζ ζ = c (cid:0) e φ n +1 − φ n − e φ n − φ n − (cid:1) , φ = φ n ( ζ , ζ ) . (4)The 2ddT equation was probably first derived in [31] as an exact reductionof the second heavenly equation; then in [32] as a distinguished example ofan integrable system in multidimensions. Some of its integrability propertieshave been investigated in [33] and the integration method presented in [7]is applicable to it. Both elliptic and hyperbolic versions of (2) are relevant,describing, for instance, integrable H -spaces (heavens) [29, 36] and integrableEinstein - Weyl geometries [37]-[38],[30]. String equations solutions [9] of itare relevant in the ideal Hele-Shaw problem [39, 40, 41, 42, 43].The integrability of (2) follows from the fact that (2) is the condition ofcommutation [ ˆ L , ˆ L ] = 0 for the following pair of one-parameter families ofvector fields [8]: ˆ L = ∂ ζ + λv∂ t + (cid:16) − λv t + φ ζ t (cid:17) λ∂ λ , ˆ L = ∂ ζ + λ − v∂ t + (cid:16) λ − v t − φ ζ t (cid:17) λ∂ λ , (5)where v = e φt (6)and λ ∈ C is the spectral parameter, implying the existence of commoneigenfunctions of both vector fields; i.e., the existence of the Lax pair: ψ ζ = − λvψ t + (cid:16) λv t − φ ζ t (cid:17) λψ λ ,ψ ζ = − λ − vψ t − (cid:16) λ − v t − φ ζ t (cid:17) λψ λ . (7)3quations (7) and (2) can be written in the following ”Hamiltonian” form[8]: ψ ζ + {H , ψ } ( λ,t ) = 0 , ψ ζ + {H , ψ } ( λ,t ) = 0 , H ζ − H ζ − {H , H } ( λ,t ) = 0 , (8)where H = λv − φ ζ , H = − λ − v + φ ζ (9)and { f, g } ( λ,t ) := λ ( f λ g t − f t g λ ) . (10)If, in particular, ζ = z = x + iy , ζ = ¯ z = x − iy , x, y ∈ R , (11)equation (2) becomes the following nonlinear wave equation φ xx + φ yy = (cid:0) e φ t (cid:1) t , (12)or ϕ xx + ϕ yy = ( e ϕ ) tt . (13)In addition, if | λ | = 1 and φ ∈ R , equations (9) give the “real” Hamiltonianformulation [32] ψ x + { H , ψ } ( θ,t ) = 0 , ψ y + { H , ψ } ( θ,t ) = 0 ,H y − H x − { H , H } ( θ,t ) = 0 , (14)of (7) and (12), for the real Hamiltonians H , H : H = i ( H + H ) = sin θe φt − φ y ,H = ( H − H ) = − cos θe φt + φ x (15)and the Poisson bracket { f, g } ( θ,t ) := f θ g t − f t g θ , (16)having introduced the parametrization λ = e − iθ , θ ∈ R . (17)The paper is organized as follows. In § § § § In this section we introduce the vector nonlinear RH problem enabling oneto construct large classes of solutions of the Lax pair (7) and of the 2ddTequations (2) and (12).
Proposition . Consider the following vector nonlinear RH problem ξ + j ( λ ) = ξ − j ( λ ) + R j ( ξ − ( λ ) + ν ( λ ) , ξ − ( λ ) + ν ( λ )) , λ ∈ Γ , j = 1 , λ plane, where ~R ( ~s ) =( R ( s , s ) , R ( s , s )) T are given differentiable spectral data depending onthe second argument s through exp ( is ) and satisfying the constraint {R ( s , s ) , R ( s , s ) } ( s ,s ) = 1 , R j ( s , s ) := s j + R j ( s , s ) , j = 1 , , (19)with { f, g } ( s ,s ) := f s g s − f s g s ; (20)where ν j , j = 1 , ~ν = (cid:18) ν ν (cid:19) = (cid:18) ( ζ λ + ζ λ − ) v − t − ζ φ ζ i ln λ + i φ t (cid:19) (21)5nd ~ξ + = ( ξ +1 , ξ +2 ) T and ~ξ − = ( ξ − , ξ − ) T are the unknown vector solutionsof the RH problem (18), analytic rispectively inside and outside the contourΓ and such that ~ξ − → ~ λ → ∞ . Then, assuming that the above RHproblem and its linearized form are uniquely solvable, we have the followingresults.1) If lim λ →∞ ( iλξ − ) = φ ζ e − φt ,iξ +2 (0) = φ t , (22)it follows that ~π ± = ~ξ ± + ~ν are common eigenfunctions of ˆ L , : ˆ L ~π ± =ˆ L ~π ± = ~ { π ± , π ± } ( λ,t ) = λ ( π ± λ π ± t − π ± t π ± λ ) = i (23)and the potentials φ ζ , φ t , reconstructed through (22), solve the 2ddT equa-tion (2).2) In addition, if the variables ζ , ζ are specified as in (11), if the RH datasatisfy the additional reality constraint ~ R (cid:16) ~ R ( ~s ) (cid:17) = ~s, ∀ ~s ∈ C (24)and Γ is the unit circle, then the eigenfunctions satisfy the following symme-try relation: ~π − ( λ ) = ~π + (1 / ¯ λ ) (25)and φ ∈ R . Remark 1
The RH problem (18) can obviously be formulated directly interms of the eigenfunctions ~π ± as follows: π + j ( λ ) = R j ( π − ( λ ) , π − ( λ )) = π − j ( λ ) + R j ( π − ( λ ) , π − ( λ )) , λ ∈ Γ , j = 1 , , (26)with the normalization ~π − ( λ ) = ~ν ( λ ) + O ( λ − ) , | λ | >> . (27) Remark 2
The dependence of ~R on s through exp ( is ) ensures that thesolutions ~ξ ± of the RH problem do not exhibit the ln λ singularity containedin the normalization (21). It follows that the eigenfunctions π ± contain the6n λ singularity only as an additive singularity, while π ± do not exhibit suchsingularity. Remark 3
Before adding the closure conditions (22), the solutions ~ξ ± of theRH problem depend, via the normalization (21), on the undefined fields φ t and φ ζ , through the combination ( t + ζ φ ζ ); then the two closure condi-tions (22) must be viewed as a nonlinear system of two algebraic equationsfor φ t and φ ζ defining implicitly the solution φ ζ , φ t of the 2ddT equation.Therefore, as in the dKP case [26], we expect that this spectral features beresponsible for the wave breaking of localized initial data evolving accordingto the nonlinear wave equation (3). Details on how two-dimensional wavesevolving according to the 2ddT equation break will be presented elsewhere.An alternative closure, perhaps useful in the reality reduction case describedin part 2) of the above Proposition, is given by the equations − i ξ +1 (0)2 = I m ( t + zφ z ) ,iξ +2 (0) = φ t . (28)With this closure, indeed, we obtain a system of algebraic equations involving t + zφ z , its imaginary part and φ t . Remark 4
The symmetry relations (25) are a distinguished example of thefollowing symmetry of the common eigenfunctions of the Lax pair (7), when ζ = z, ζ = ¯ z as in (11) and φ ∈ R : if ψ ( λ ) is a solution of (7), then ψ (1 / ¯ λ ) is a solution too . Remark 5
For part 2) of the above Proposition, when the contour Γ isthe unit circle, the RH problem is characterized by the following system ofnonlinear integral equations for ξ ± j ( λ ) , | λ | = 1 (having parametrized λ as in(17)): ξ ± j ( e − iθ ) − π π R dθ ′ − (1 ∓ ǫ ) e i ( θ ′− θ ) R j (cid:16) ( ze − iθ ′ + ¯ ze iθ ′ ) v − t − zφ z + ξ − ( e − iθ ′ ) ,θ ′ + i φ t + ξ − ( e − iθ ′ ) (cid:17) = 0 , j = 1 , , (29)and the closure conditions (22) read φ z e − φt = πi π R dθe − iθ R (cid:16) ( ze − iθ + ¯ ze iθ ) v − t − zφ z + ξ − ( e − iθ ) ,θ + i φ t + ξ − ( e − iθ ) (cid:17) = 0 , (30)7 t = − πi π R dθR (cid:16) ( ze − iθ + ¯ ze iθ ) v − t − zφ z + ξ − ( e − iθ ) , θ + i φ t + ξ − ( e − iθ ) (cid:17) = 0 . (31) Proof . For part 1), we first apply the operators ˆ L , to the RH problem (26),obtaining the linearized RH problem ˆ L j ~π + = A ˆ L j ~π − , where A is the Jacobianmatrix of the transformation (26): A ij = ∂ R i /∂s j , i, j = 1 ,
2. Since, due tothe normalization (27), ˆ L j ~π − → ~ λ → ∞ , it follows that, by uniqueness, ~π ± are common eigenfunctions of the vector fields ˆ L , : ˆ L , ~π ± = ~ φ t , φ z are solutions of the 2ddT equation (2). Then theeigenfunctions exhibit the following asymptotics: π − = λzv − t − zφ z + λ − (¯ zv + a − v − ) + O ( λ − ) , | λ | >> ,π +1 = λ − ¯ zv − t − ¯ zφ ¯ z + λ ( zv + a + v − ) + O ( λ ) , | λ | << ,π − = i ln λ + i φ t − iλ − φ z v − + O ( λ − ) , | λ | >> ,π +2 = i ln λ − i φ t + iλφ ¯ z v − + O ( λ ) , | λ | << , (32)where a − ¯ z = − a + z = ( ze φ t ) z − (¯ ze φ t ) ¯ z . (33)implying the closure conditions (22),(28). Since, from (26), { π +1 , π +2 } ( t,λ ) = {R , R } ~π − { π − , π − } ( t,λ ) , λ ∈ Γ, equation (19) implies that { π +1 , π +2 } ( t,λ ) = { π − , π − } ( t,λ ) , λ ∈ Γ; i.e., the Poisson brackets of the ± eigenfunctions areanalytic in the whole complex λ plane. Since { π − , π − } → i as λ → ∞ , itfollows that { π +1 , π +2 } = { π − , π − } = i . For part 2), applying ~ R ( · ) to thecomplex conjugate of the RH problem (26) and using the reality condition(24) it follows that ~π + ( λ ) = ~π − ( λ ) , | λ | = 1. By the Schwartz reflectionprinciple, it follows the symmetry (25) and, using the Lax pair (7), the realitycondition φ ∈ R . ✷ In this section we present the formal solution of the Cauchy problem for thewave form of the 2ddT equation: (cid:0) e φ t (cid:1) t = φ xx + φ yy , x, y ∈ R , t > , φ ( x, y, t ) ∈ R ,φ ( x, y,
0) = A ( x, y ) , φ t ( x, y,
0) = B ( x, y ) . (34)8here the assigned initial conditions A ( x, y ) , B ( x, y ) are localized in the ( x, y )plane for x + y → ∞ . To do it, we use the IST for vector fields developedin [19, 20, 24, 25].In this respect, we recall two basic facts: since the Lax pair of the 2ddTis made of vector fields, i) the space of eigenfunctions is a ring (if f and f are eigenfunctions, any differentiable function F ( f , f ) is an eigenfunction);ii) since the vector fields are also Hamiltonian, the space of eigenfunctions isalso a Lie algebra, whose Lie bracket is the Poisson bracket (10) (if f and f are eigenfunctions, also { f , f } ( λ,t ) is an eigenfunction).Multiplying the first and second equations of (7) (with ζ = z, ζ = ¯ z asin (11)) by λ − and λ respectively, then adding and subtracting the resultingequations, one obtains the equivalent and more convenient Lax pair:ˆ L ψ := λψ ¯ z − λ − ψ z − (cid:16) − v t + λ φ ¯ zt + λ − φ zt (cid:17) λψ λ = 0 , (35)ˆ L ψ := ψ t + v − ( λψ ¯ z + λ − ψ z ) − v − ( λφ ¯ zt − λ − φ zt ) λψ λ = 0 , (36)where the first equation must be viewed as the spectral problem (in which v t shall be replaced, in the direct problem, by ( φ z ¯ z / exp ( − φ t / t -evolution of the eigenfunction. Eigenfunctions and spectral data . Now we introduce the Jost and an-alytic eigenfunctions for the spectral problem (35). Since the associatedundressed operator: λ∂ ¯ z − λ − ∂ z coincides with the undressed operator ofthe spectral problem for the (2 + 1)-dimensional self-dual Yang-Mills equa-tion [44], the construction of the Jost and analytic Green’s functions is takenfrom there.We define Jost eigenfunctions of the spectral problem (35) on the unitcircle of the complex λ plane, using the parametrization (17). Introducingthe convenient real variables ξ, η, θ ′ as follows: ξ = cos θ x + sin θ y,η = − sin θ x + cos θ y,θ ′ = θ, (37)the Lax pair (35),(36) becomesˆ L ψ := ψ η − [ − ( φ ξξ + φ ηη ) v − + φ ξt ] ( ηψ ξ − ξψ η + ψ θ ′ ) = 0 , (38)ˆ L ψ := ψ t + v − ψ ξ − ( v − ) η ( ηψ ξ − ξψ η + ψ θ ′ ) = 0 . (39)9 convenient basis of Jost eigenfunctions are the solutions f and f of equa-tion (38) satisfying the boundary conditions ~f ( ξ, η, θ ′ ) := (cid:18) f ( ξ, η, θ ′ ) f ( ξ, η, θ ′ ) (cid:19) → (cid:18) ξθ ′ (cid:19) , as η → −∞ ; (40)they are characterized by the linear integral equation ~f = (cid:18) ξθ ′ (cid:19) + η R −∞ dη ′ [ − ( φ ξξ + φ η ′ η ′ ) v − + φ ξt ] ( η ′ ~f ξ − ξ ~f η ′ + ~f θ ′ ) . (41)It follows that f ( ξ, η, θ ) and f ( ξ, η, θ ) − θ are 2 π -periodic in θ .The η → ∞ limit of ~f defines the scattering vector ~σ ( ξ, θ ) = ( σ ( ξ, θ ) , σ ( ξ, θ )) T as follows ~f ( ξ, η, θ ) → ~ S ( ξ, θ ) = (cid:18) ξθ (cid:19) + ~σ ( ξ, θ ) , as η → ∞ ; (42)namely: ~σ ( ξ, θ ) = 12 Z R dη (cid:2) − ( φ ξξ + φ ηη ) v − + φ ξt (cid:3) ( η ~f ξ − ξ ~f η + ~f θ ) . (43)Also the scattering vector is 2 π -periodic in θ : ~σ ( ξ, θ + 2 π ) = ~σ ( ξ, θ ); i.e., itsdependence on the second argument θ is through exp ( iθ ).The analytic eigenfunctions of (35) are defined instead via the integralequations: ~ψ ± ( z, ¯ z, λ ) = (cid:18) ψ ± ( z, ¯ z, λ ) ψ ± ( z, ¯ z, λ ) (cid:19) = (cid:18) λz + λ − ¯ zi ln λ (cid:19) + i R C dz ′ ∧ d ¯ z ′ G ± ( z − z ′ , ¯ z − ¯ z ′ , λ ) h − φ z ¯ z v − + λ φ ¯ z ′ t + λ − φ z ′ t i λ ~ψ ± λ ( z ′ , ¯ z ′ , λ ) , (44)where G ± are the analytic Green’s functions G ± ( z, ¯ z, λ ) = ∓ π λz + λ − ¯ z ) , sgn(1 − | λ | ) = ± λG ± ¯ z − λ − G ± z = δ ( z ), reducing, on the unit circle | λ | = 1, to G ± ( z, ¯ z, λ ) = ∓ π ξ ∓ iǫη , < ǫ << . (46)10ince G + and G − are analytic respectively inside and outside the unitcircle of the complex λ plane, then ( ψ +1 , ψ +2 ) and ( ψ − , ψ − ) are also analytic,respectively, inside and outside the unit circle of the complex λ plane, af-ter subtracting their singular parts, given respectively by ( λ − ¯ zv, i ln λ ) and( λzv, i ln λ ), as it can be seen by solving the integral equations (44) by iter-ation or from the following λ - asymptotics: ψ − = λzv − zφ z + λ − (¯ zv + a − v − ) + O ( λ − ) , | λ | >> ,ψ +1 = λ − ¯ zv − ¯ zφ ¯ z + λ ( zv + a + v − ) + O ( λ ) , | λ | << ,ψ − = i ln λ + i φ t − iλ − φ z v − + O ( λ − ) , | λ | >> ,ψ +2 = i ln λ − i φ t + iλφ ¯ z v − + O ( λ ) , | λ | << , (47)where a ± are defined in (33).In addition, equations (46) imply the limits G + ( z − z ′ , ¯ z − ¯ z ′ , λ ) → − π ξ − ξ ′ ± iε , as η → ∓∞ ,G − ( z − z ′ , ¯ z − ¯ z ′ , λ ) → π ξ − ξ ′ ∓ iε , as η → ∓∞ . (48)Therefore, on the unit circle | λ | = 1, the η → −∞ limit of ( ψ +1 , ψ +2 ) and( ψ − , ψ − ) are analytic respectively in the upper and lower parts of the com-plex ξ plane, while the η → ∞ limit of ( ψ +1 , ψ +2 ) and ( ψ − , ψ − ) are analyticrespectively in the lower and upper parts of the complex ξ plane. This mech-anism, first observed in [44], plays an important role in the IST for vectorfields (see [19],[20],[24],[25]).Since the Jost eigenfunctions ~f = ( f , f ) T are a good basis in the spaceof eigenfunctions of the spectral problem (38) for | λ | = 1, one can expressthe analytic eigenfunctions in terms of them through the following formulae,valid for | λ | = 1: ~ψ ± = ~ K ± ( ~f ) = ~f + ~χ ± ( f , f ) , (49)defining the spectral data ~χ ± as differentiable functions of two arguments.In the η → −∞ limit, equations (49) reduce tolim η →−∞ ~ψ ± − (cid:18) ξθ (cid:19) = ~χ ± ( ξ, θ ) , (50)implying that i) ~χ + ( ξ, θ ) and ~χ − ( ξ, θ ) are analytic in the first variable ξ respectively in the upper and lower half parts of the complex ξ plane, and ii) ~χ ± ( ξ, θ ) are 2 π -periodic in θ : ~χ ± ( ξ, θ + 2 π ) = ~χ ± ( ξ, θ ) (their dependence onthe second argument θ is through exp ( iθ )).11t η → ∞ , equations (49) reduce tolim η →∞ ~ψ ± − (cid:18) ξθ (cid:19) = ~σ + ~χ ± ( ξ + σ , θ + σ ) . (51)Applying the operator R R dξ π R dθ π e − i ( ωξ + nθ ) · , n ∈ Z to equations (51) andusing the above established analiticity properties in ξ and the 2 π -periodicityin θ , we obtain the following linear integral equations connecting the (Fouriertransforms of the) scattering data ~σ to the (Fourier transforms of the) spectraldata ~χ ± :˜ ~χ ± ( ω, n ) + H ( ± ω ) (cid:18) ˜ ~σ ( ω, n ) + R R dω ′ ∞ P n ′ = −∞ ˜ ~χ ± ( ω ′ , n ′ ) Q ( ω ′ , n ′ , ω, n ) (cid:19) = ~ , (52)where H is the Heaviside step function and Q ( ω ′ , n ′ , ω, n ) = R R dξ π π R dθ π e i ( ξ ( ω ′ − ω )+( n ′ − n ) θ ) (cid:0) e i ( ω ′ σ ( ξ,θ )+ n ′ σ ( ξ,θ )) − (cid:1) , ˜ ~χ ± ( ω, n ) = R R dξ π R dθ π e − i ( ωξ + nθ ) ~χ + ( ξ, θ ) , ˜ ~σ ( ω, n ) = R R dξ π R dθ π e − i ( ωξ + nθ ) ~σ + ( ξ, θ ) . (53)At last, eliminating, from equations (49), the Jost eigenfunctions ~f , oneobtains, through algebraic manipulation, the following vector nonlinear RHproblem on the unit circle of the complex λ plane: ψ +1 = R ( ψ − , ψ − ) = ψ − + R ( ψ − , ψ − ) , | λ | = 1 ,ψ +2 = R ( ψ − , ψ − ) = ψ − + R ( ψ − , ψ − ) . (54)We remark that the 2 π -periodicity properties of the scattering data ~χ ± ( ξ, θ )in the variable θ imply that the dependence of ~R on the second argument s is also through exp ( is ), to guaranty that the ln λ singularity is just anadditive one for ψ ± , and is absent for ψ ± .Recapitulating, in the direct problem, at t = 0, we go from the initialconditions φ, φ t of the 2ddT equation to the initial scattering vector ~σ ( ξ, θ );from it we construct, through the linear integral equations (52), the scat-tering data ~χ ± ( ξ, θ ) and, through algebraic manipulation, the RH spectral12ata ~R ( ~s ) = ( R ( s , s ) , R ( s , s )). In the inverse problem, one gives theRH spectral data ~R ( ~s ) and reconstructs the vector solutions ~ψ ± of the RHproblem (54), defined by the normalization: ~ψ − = ( zλ + ¯ zλ − ) e φt − zφ z i ln λ + i φ t ! + ~O ( λ − ) , | λ | >> . (55)At last, the closure conditionslim λ →∞ λ ( iψ − + ln λ ) = φ z e − φt , lim λ → ( iψ +2 + ln λ ) = φ t , (56)consequences of the asymptotics (47), allow one to reconstruct the solution ofthe 2ddT equation through the solution of a system of two algebraic equationsfor φ t and φ z . Time evolution of the spectral data . To construct the t -evolution ofthe spectral data we observe that ~f and ~ψ ± , eigenfunctions of the spectralproblem (35): ˆ L ~f = ˆ L ~ψ ± = ~
0, are solutions of the following equationsinvolving the second Lax operator ˆ L ~f = ˆ L ~ψ ± = (1 , T , implying thefollowing elementary time evolutions of the data: ~σ ( ξ, θ, t ) = ~σ ( ξ − t, θ, , ~χ ± ( ξ, θ, t ) = ~χ ± ( ξ − t, θ, ,~R ( ξ, θ, t ) = ~R ( ξ − t, θ, . (57)In addition, it follows that the common Jost eigenfunctions ~J and the com-mon analytic eigenfunctions ~π ± of the Lax pair (35),(36) are obtained from ~f and ~ψ ± simply as follows: ~J := ~f − t (1 , T ,~π ± := ~ψ ± − t (1 , T . (58)It is easy to verify that the analytic eigenfunctions ~π ± , the RH data ~R ( ~s )and the associated RH problem of this section coincide with those appearingin the dressing construction of § Hamiltonian constraints on the data . The Hamiltonian character of the2ddT dynamics implies the following formulae for the Poisson brackets of therelevant eigenfunctions: { J , J } ( λ,t ) = { π ± , π ± } ( λ,t ) = i (59)13hich, in turn, imply that the transformations ~s → ~ K ± ( ~s ) and ~s → ~ R ( ~s ) arecanonical: {K ± , K ± } ( s ,s ) = {R , R } ( s ,s ) = 1 . (60)To prove (59), one first shows that J := { J , J } ( λ,t ) → i as η → −∞ , π − := { π − , π − } ( λ,t ) → i as λ → ∞ , π +3 := { π +1 , π +2 } ( λ,t ) → i as λ →
0. Sincethe vector fields are Hamiltonian, J , π ± are also common eigenfunctions,and equations (59) hold, by uniqueness. Equations (60) are consequences of(59) and of the relations ~π ± = ~ K ± (cid:16) ~J (cid:17) , ~π + = ~ R (cid:0) ~π − (cid:1) . (61) Reality constraints . The definition (58b) and the condition φ ∈ R implythe symmetry relations ~π − ( λ ) = ~π + (1 / ¯ λ ); (62)consequently, from (18), the reality constraint (24) on the RH data holdstrue. Small field limit and Radon Transform . As for the IST of the heavenly[20] and dKP [24] equations, in the small field limit | φ | , | φ t | <<
1, the directand inverse spectral transforms presented in this section reduce to the directand inverse Radon transform [45]. Indeed, the mapping from the initial data { A ( x, y ) , B ( x, y ) } to the scattering vector ~σ reduces to the direct Radontransform: ~σ ( ξ, θ ) ∼ R R (cid:18) η (cid:19) h − ( ∂ ξ + ∂ η ) A ( x ( ξ, η, θ ) , y ( ξ, η, θ ))+ ∂ ξ B ( x ( ξ, η, θ ) , y ( ξ, η, θ )) i dη,x ( ξ, η, θ ) = ξ cos θ − η sin θ, y ( ξ, η, θ ) = ξ sin θ + η cos θ, (63)while the spectral data ~χ ± and ~R are constructed from ~σ as follows: ~χ ± ( ξ, θ ) ∼ − ˆ P ± ξ ~σ ( ξ, θ ) , ~R ( ξ, θ ) ∼ − i ˆ H ξ ~σ ( ξ, θ ) , (64)where ˆ P ± ξ and ˆ H ξ are rispectively the ( ± ) analyticity projectors and theHilbert transform in the variable ξ :ˆ P ± ξ g ( ξ ) := ± πi Z R dξ ′ ξ ′ − ( ξ ± i g ( ξ ′ ) , ˆ H ξ g ( ξ ) := 1 π P Z R dξ ′ ξ − ξ ′ g ( ξ ′ ) . (65)14t last, the first of the closure conditions (56) of the inverse problem reducesto the inverse Radon transform φ t ( x, y, t ) ∼ − πi π R dθR ( ξ − t, θ ) ∼ − π π R dθ P R R dξ ′ ξ ′ − ( ξ − t ) σ ( ξ ′ , θ ) ,ξ = x cos θ + y sin θ, (66)that can be shown to be equivalent to the well-known Poisson formula φ ( x, y, t ) = ∂ t R R dx ′ dy ′ π L ( x − x ′ , y − y ′ , t ) A ( x ′ , y ′ ) + R R dx ′ dy ′ π L ( x − x ′ ,y − y ′ , t ) B ( x ′ , y ′ ) , (67)where L ( x, y, t ) := H ( t − x − y ) p t − x − y (68)and H ( · ) is the Heaviside step function, describing the solution of the Cauchyproblem φ tt = φ xx + φ yy , x, y ∈ R , t > , φ ( x, y, t ) ∈ R ,φ ( x, y,
0) = A ( x, y ) , φ t ( x, y,
0) = B ( x, y ) . (69)for the linear wave equation in 2+1 dimensions. In this section we show, as it was done in the dKP case [26], that the spectralmechanism causing the breaking of a localized initial condition evolving ac-cording to the 2ddT equation is present also in the longtime regime. We willactually show that the longtime breaking of the 2ddT solutions is essentiallydescribed by the longtime breaking formulae of the dKP solutions found in[26]; this is an important confirmation of the expected universal characterof the dKP equation as prototype model in the description of the gradientcatastrophe of two-dimensional waves .We remark that it is clearly meaningful to study the longtime behaviourof the solutions of the 2ddT equation only if no breaking takes place before,at finite time. In this section we assume that the initial condition be small,then the nonlinearity becomes important only in the longtime regime and nobreaking takes place before. 15otivated by the longtime behaviour of the solutions of the linear waveequation u tt = u xx + u yy , localized, with amplitude O ( t − ), in the region p x + y − t = O (1), we study the longtime behaviour of the solutions ofthe 2ddT equation in the space-time region z = t + r e iα , α, r ∈ R , α = O (1) , t >> , (70)implying that r = p x + y − t, α = arctan yx . (71)Substituting (70) into the integral equations (29) and keeping in mind that,in the longtime regime, φ t is small, so that, f. i., v ∼ φ t / φ t /
8, weobtain ξ ± j ( λ ) − π π R dθ ′ − (1 ∓ ǫ ) e i ( θ ′− θ ) R j (cid:16) − t sin (cid:0) θ ′ − α (cid:1) + r cos( θ ′ − α )+ t + r cos( θ ′ − α ) φ t (1 + φ t ) − zφ z + ξ − ( e − iθ ′ ) , θ ′ + ξ − ( e − iθ ′ ) (cid:17) ∼ , j = 1 , . (72)Since the main contribution to these integrals occurs when sin(( θ ′ − α ) / ∼ θ ′ = α − µ ′ / √ t , obtaining ξ ± j ( λ ) − π √ t R R dµ ′ − (1 ∓ ǫ ) e i ( α − θ − µ ′√ t ) R j (cid:16) − µ ′ + X + ξ − (cid:18) e − i ( α − µ ′√ t ) (cid:19) ,α + ξ − (cid:18) e − i ( α − µ ′√ t ) (cid:19) (cid:17) ∼ , j = 1 , , (73)where X := r + t + r φ t + t φ t − zφ z . (74)If | θ − α | >> t − / , equations (73) imply that ξ ± j ( λ ) = O ( t − / ): ξ ± j ( λ ) ∼ π √ t ( − (1 ∓ ǫ ) e i ( α − θ ) ) R R dµ ′ R j (cid:16) − µ ′ + X + ξ − (cid:18) e − i ( α − µ ′√ t ) (cid:19) ,α + ξ − (cid:18) e − i ( α − µ ′√ t ) (cid:19) (cid:17) , j = 1 , . (75)If, instead, θ − α = − µt − / , | µ | = O (1), then ξ ± j ( λ ) = O (1): ξ ± j (cid:16) e − i ( α − µ √ t ) (cid:17) ∼ πi R R dµ ′ µ ′ − ( µ ± iǫ ) R j (cid:16) − µ ′ + X + ξ − (cid:18) e − i ( α − µ ′√ t ) (cid:19) ,α + ξ − (cid:18) e − i ( α − µ ′√ t ) (cid:19) (cid:17) , j = 1 , . (76)16herefore it is not possible to neglect, in the above integral equations, ξ − j , j =1 , R j , j = 1 ,
2; it follows that these integral equationsremain nonlinear even in the longtime regime.At last, using equations (75), the asymptotic form of the closure condi-tions read, for t >> φ t ∼ − πi √ t R R dµ ′ R (cid:16) − µ ′ + X + ξ − (cid:18) e − i ( α − µ ′√ t ) (cid:19) ,α + ξ − (cid:18) e − i ( α − µ ′√ t ) (cid:19) (cid:17) , (77) φ z = − e − iα φ t (1 + φ t ) (1 + O ( t − )) . (78)Comparing (77) and (78), and using (70), we infer that zφ z ∼ − t + r φ t (1+ φ t ).Using this asymptotic relation in (77), we finally obtain the following result.In the space-time region z = t + r e iα , α, r ∈ R , t >> ,X := r + ( t + r ) φ t + t φ t = p x + y − t + p x + y φ t + t φ t = O (1) , (79)the longtime t >> expφ t ) t = φ xx + φ yy is described by the following implicit (scalar) equation: φ t = √ t F (cid:16)p x + y − t + p x + y φ t + t φ t , arctan yx (cid:17) + o (cid:16) √ t (cid:17) , (80)where F is given by F ( X, α ) = − πi R R dµ ′ R (cid:16) − µ ′ + X + a ( µ ′ ; X, α ) , α + a ( µ ′ ; X, α ) (cid:17) (81)and a j ( µ ; X, α ) , j = 1 , a j ( µ ; X, α ) = πi R R dµ ′ µ ′ − ( µ − iǫ ) R j (cid:16) − µ ′ + X + a ( µ ′ ; X, α ) , α + a ( µ ′ ; X, α ) (cid:17) , j = 1 , . (82)Outside the asymptotic region (79) the solution decays faster.We first remark that, since φ t = O ( t − ), the condition X = O (1) impliesthat r = p x + y − t = O ( √ t ); it follows that, in the longtime regime t >>
17, the solution of the Cauchy problem for the 2ddT equation is concentrated,with amplitude O ( t − ), in the asymptotic region p x + y − t = O ( √ t ). Wealso remark that the asymptotic solution (80)-(82) is connected to the initialconditions of the Chauchy problem through the direct problem presented inthe previous section. In this section, in analogy with the results of [26],[28], we construct a class ofexplicit solutions of the vector nonlinear RH problem (18) and, correspond-ingly, a class of implicit solutions of the 2ddT equation parametrized by anarbitrary real spectral function of one variable.Suppose that the two components of the RH spectral data ~R in (18) aregiven by: R j ( s , s ) = ( − j +1 if (cid:0) e s + s (cid:1) , j = 1 , , (83)in terms of the single real spectral function f of a single argument, dependingon s and s only through their sum.Then the RH problem (26) becomes π +1 = π − + if (cid:16) e π − + π − (cid:17) , | λ | = 1 ,π +2 = π − − if (cid:16) e π − + π − (cid:17) (84)and the following properties hold.i) The reality and Hamiltonian constraints (24) and (19) are satisfied.ii) π +1 + π +2 = π − + π − . Consequently, using the analyticity properties of theeigenfunctions, it follows that the functions ∆ + and ∆ − , defined by∆ ± := π ± + π ± − ( zλ + ¯ zλ − ) v − i ln λ, (85)are analytic respectively inside and outside the unit circle of the λ -plane andsatisfy the equation ∆ + = ∆ − ; therefore they are equal to a constant in λ .Evaluating such a constant at λ = 0 and at λ → ∞ , we obtain the followingequalities ∆ + = ∆ − = − t − ¯ zφ ¯ z − i φ t − t − zφ z + i φ t . (86)18his implies thati) the solutions of the 2ddT equation generated by the above RH problemsatisfy the linear (2 + 1)-dimensional PDE φ t = i (¯ zφ ¯ z − zφ z ) (87)and, substituting in (12) the expression of φ t in terms of φ z , φ ¯ z given in (87),one obtains the following nonlinear two dimensional constraint: i (¯ z∂ ¯ z − z∂ z ) (cid:0) e i (¯ zφ ¯ z − zφ z ) (cid:1) = φ z ¯ z (88)on the solutions of 2ddT constructed by the above RH problem.ii) π +1 + π +2 = π − + π − is the following explicit and elementary function of λ : w ( λ ) := π +1 + π +2 = π − + π − = ( zλ + ¯ zλ − ) e − φt + i ln λ − t − zφ z + i φ t . (89)iii) Since, from (89), π − + π − = w ( λ ) is an explicit function of λ , the vectornonlinear RH problem (84) decouples into two scalar, linear RH problems: π +1 = π − + if (cid:0) e w ( λ ) (cid:1) ,π +2 = π − − if (cid:0) e w ( λ ) (cid:1) , (90)whose explicit solutions are given by ξ ± j ( λ ) = ( − j +1 12 πi H | λ | =1 dλ ′ λ ′ − (1 ∓ ǫ ) e argλ f (cid:0) e w ( λ ′ ) (cid:1) , j = 1 , , (91)where ξ ± j = π ± j − ν j , and the closure conditions (22) read φ z e − φt = − πi I | λ | =1 dλf (cid:0) e w ( λ ) (cid:1) , φ t = 12 πi I | λ | =1 dλλ f (cid:0) e w ( λ ) (cid:1) . (92)Although the RH problem (90) is linear, since w ( λ ) in (89) depends on theunknowns φ t , φ z , the closure conditions (92) are a nonlinear algebraic systemof two equations for the two unknowns φ t , φ z , defining implicitly a class ofsolutions of the 2ddT equation parametrized by the arbitrary real spectralfunction f ( · ) of a single variable. Acknowledgements . This research has been supported by the RFBRgrants 07-01-00446, 06-01-90840, and 06-01-92053, by the bilateral agree-ment between the Consortium Einstein and the RFBR, and by the bilateralagreement between the University of Roma “La Sapienza” and the LandauInstitute for Theoretical Physics of the Russian Academy of Sciences.19 eferences [1] V. E. Zakharov and A. B. Shabat, Functional Anal. Appl. , 166-174(1979).[2] R. Timman, “Unsteady motion in transonic flow”, Symposium Transson-icum, Aachen 1962. Ed. K. Oswatitsch, Springer 394-401.[3] E. A. Zobolotskaya and R. V. Kokhlov, “Quasi - plane waves in thenonlinear acoustics of confined beams”, Sov. Phys. Acoust. , n. 1 (1969)35-40.[4] Y. Kodama and J. Gibbons, “Integrability of the dispersionless KP hi-erarchy”, Proc. 4th Workshop on Nonlinear and Turbulent Processes inPhysics, World Scientific, Singapore 1990.[5] B. Kupershmidt, J. Phys. A: Math. Gen. , 871 (1990).[6] V. E. Zakharov, “Dispersionless limit of integrable systems in 2+1 dimen-sions”, in Singular Limits of Dispersive Waves , edited by N.M.Ercolaniet al., Plenum Press, New York, 1994.[7] I. M. Krichever, “The τ -function of the universal Witham hierarchy, ma-trix models and topological field theories”, Comm. Pure Appl. Math. ,437-475 (1994).[8] K. Takasaki and T. Takebe, arXiv:hep-th/9112042 .[9] K. Takasaki and T. Takebe, Rev. Math. Phys. , 743 (1995).[10] M. Dunajski and L. J. Mason, “Hyper-K¨ahler hierachies and theirtwistor theory”, Comm. Math. Phys. , 641-672 (2000). “Twistor the-ory of hyper-K¨ahler metrics with hidden symmetries”, J. Math. Phys., , 3430-3454 (2003).[11] M. Dunajski, L. J. Mason and P. Tod, “Einstein-Weyl geometry, thedKP equation and twistor theory”, J. Geom. Phys. ∂ -approach for the dispersionless KP hierarchy”, J.Phys. A: Math. Gen. (2003) 6457-6472.[16] M. V. Pavlov: “Integrable hydrodynamic chains”, J. Math. Phys. (2003) 4134-4156.[17] E. V. Ferapontov and K. R. Khusnutdinova: “On integrability of (2+1)-dimensional quasilinear systems”, Comm. Math. Phys. (2004) 187-206.[18] B. Konopelchenko and F. Magri, “Dispersionless integrable equa-tions as coisotropic deformations. Extensions and reductions”.arXiv:nlin/0608010.[19] S. V. Manakov and P. M. Santini: “Inverse scattering problem for vectorfields and the Cauchy problem for the heavenly equation”, Physics LettersA (2006) 613-619. http://arXiv:nlin.SI/0604017.[20] S. V. Manakov and P. M. Santini: “Inverse scattering problem for vectorfields and the heavenly equation”; http://arXiv:nlin.SI/0512043.[21] V. E. Zakharov, S. V. Manakov, S. P. Novikov and L. P. Pitaevsky, Theory of solitons , Plenum Press, New York, 1984.2122] M. J. Ablowitz and P. A. Clarkson,
Solitons, nonlinear evolution equa-tions and Inverse Scattering , London Math. Society Lecture Note Series,vol. 194, Cambridge University Press, Cambridge (1991).[23] J. F. Plebanski, “Some solutions of complex Einstein equations”, J.Math. Phys. , 2395-2402 (1975).[24] S. V. Manakov and P. M. Santini: “The Cauchy problem on the planefor the dispersionless Kadomtsev-Petviashvili equation”; JETP Letters, , No 10, 462-466 (2006). http://arXiv:nlin.SI/0604016.[25] S. V. Manakov and P. M. Santini: Theor. Math. Phys. (1), 1004-1011 (2007).[26] S. V. Manakov and P. M. Santini: “On the solutions of the dKP equa-tion: the nonlinear Riemann-Hilbert problem, longtime behaviour, im-plicit solutions and wave breaking”; J. Phys. A: Math. Theor. (2008)055204 (23pp).[27] M. Dunajski, Proc. Royal Soc. A , 1205.[28] L. Bogdanov, V. Dryuma and S. V. Manakov: “Dunajski generaliza-tion of the second heavenly equation: dressing method and the hier-archy”; J. Phys. A: Math. Theor. 40 14383-14393, doi: 10.1088/1751-8113/40/48/005.[29] C. Boyer and J. D. Finley, “Killing vectors in self-dual, Euclidean Ein-stein spaces”, J. Math. Phys. (1982), 1126-1128.[30] R. S. Ward, ”Einstein-Weyl spaces and SU( ∞ ) Toda fields”, Class.Quantum Grav. (1990) L95-L98.[31] J. D. Finley and J. F. Plebanski: “The classification of all K spacesadmitting a Killing vector”, J. Math. Phys. , 1938 (1979).[32] V. E. Zakharov: “Integrable systems in multidimensional spaces”, Lec-ture Notes in Physics, Springer-Verlag, Berlin (1982), 190-216.[33] M. V. Saveliev, Commun. Math. Phys. (1989), 283. M. V. Saveliev,Teoreticheskaya i Matematicheskaya Fisika, No. 3, (1992) 457-465.2234] G. Darboux,