The Cauchy problem for the Pavlov equation with large data Derchyi Wu Institute of Mathematics, Academia Sinica, Taipei, Taiwan 5 January 26, 2015 1 0 Abstract 2 n The Pavlov equation is one of the simplest integrable systems of a vector fields arising from various problems of mathematical physics J and differential geometry which are intensively studied in recent liter- 3 ature. Inthisreport,solvinganonlinearRiemann-Hilbertproblemvia 2 a Newtonian iterationscheme, we complete the inverse scattering the- ory and prove a short time unique solvability of the Cauchy problem ] I of the Pavlov equation with large initial data. S . n i l n 1 Introduction [ 1 v Integrable dPDEs (dispersionless partial differential equations), including 0 the dispersionless Kadomtsev-Petviashvili equation [11], [25], [28], the first 8 and second heavenly equations of Plebanski [24], the dispersionless 2D Toda 7 5 (or Boyer-Finley) equation [3], [8], and the Pavlov equation [6], [7], [23], are 0 defined by a commutation [L,M] = 0 of pairs of one-parameter families of . 1 vector fields. They arise in various problems of mathematical physics and 0 are intensively studied recently. 5 Due to lack of dispersion, integrable dPDEs may or may not exhibit 1 : a gradient catastrophe at finite time. Since the Lax operators are vector v i fields, the Fourier transform theory used in soliton theory for proving the X existence of eigenfunctions fails and the inverse problem is intrinsically non- ar linear for integrable dPDEs, unlike the ∂¯-problem formulated for general soliton equations [1], [2]. At last, no explicit regular localized solutions, like solitons or lumps, exist for integrable dPDEs. Therefore, it is important to solve the inverse scattering problem for integrable dPDEs. A formal inverse scattering theory has been recently constructed, including i) to solve their Cauchy problem, ii) obtain the longtime behavior of solutions, iii) costruct distinguished classes of exact implicit solutions, iv) establish if, due to the lack of dispersion, the nonlinearity of the PDE is “strong enough” to cause the gradient catastrophe of localized multidimensional disturbances, and v) to study analytically the breaking mechanism [12]-[22]. 1 The Pavlov equation, v +v +v v −v v = 0, xt yy x xy y xx (1.1) v = v(x,y,t) ∈ R, x,y,t ∈R, arising in the study of integrable hydrodynamic chains [23], and in differen- tial geometry as a particular example of Einstein - Weyl metric [6], is the simplest integrable dPDE available in the literature [23], [7], [6]. It was first derived in [5] as a conformal symmetry of the second heavenly equation. In our previous work [10], we solve the forward problem via a Beltrami-type equation, a first order PDE, and a shifted Riemann-Hilbert problem and the inverse problem by a nonlinear integral equation under a small data constraint. More precisely, we justify the complex eigenfunction Φ(x,z,λ), defined by ∂ Φ(x,y,λ)+(λ+v )∂ Φ(x,y,λ) = 0, x, y ∈ R, λ ∈ C± y x x (1.2) Φ(x,y,λ)−(x−λy) → 0, |x|, |y| → ∞, is link to the real eigenfunction ϕ(x,y,λ), defined by ∂ ϕ+(λ+v )∂ ϕ =0, x, y ∈ R, λ ∈ R y x x (1.3) ϕ−(x−λy)→ 0, y → −∞ by the equation Φ−(x,y,λ) = ϕ(x,y,λ)+χ−(ϕ(x,y,λ),λ) (1.4) where χ−(ξ,λ) satisfies a shifted Riemann-Hilbert problem σ(ξ,λ)+χ+(ξ+σ(ξ,λ),λ)−χ−(ξ,λ) = 0, ξ ∈ R, (1.5) σ(ξ,λ) = lim (ϕ(ξ,y,λ)−ξ). y→∞ Therefore, the inverse scattering theory reduces to deriving uniform ξ-, λ- asymptotic estimates of χ(ξ,λ) in the direct problem, and solving the asso- ciated Riemann-Hilbert problem via the nonlinear integral equation 1 χ−(ψ(x,y,t,ζ),ζ) ψ(x,y,t,λ)+χ−(ψ(x,y,t,λ),λ) = x−λy−λ2t− I dζ R π ζ −λ ZR (1.6) in the inverse problem. However, for technical reasons, we have to impose two strong assumptions, the compact supportcondition and small data con- straint on the initial data v(x,y,0), to make the above resolution scheme possible [10]. 2 In this report, we aim to solving a large data problem. The difficulties are to find an effective scheme to solve the nonlinear inverse problem with large data and to establish a Fredholm alternative for the linearization. To do it, we transform the conjunction formula (1.4) to Φ+(x,y,λ) = Φ−(x,y,λ)+R(Φ−(x,y,λ),λ). (1.7) and the inverse problem (1.6) to the nonlinear Riemann-Hilbert problem Ψ+(x,y,t,λ) = Ψ−(x,y,t,λ)+R(Ψ−,λ), λ ∈ R, ∂ Ψ(x,y,t,λ) = 0, λ ∈ C±, λ (1.8) Ψ(x,y,t,λ)−(x−λy−λ2t)→ 0, |λ|→ ∞, Ψ(x,y,0,λ) = Φ(x,y,λ). [15]-[19]. Furthermore, we adopt a Newtonian iteration approach to study (1.8). The linearization of the Newtonian process turns out to be a non homogeneous Riemann-Hilbert problem of which a Fredholm theory has been well investigated [9]. Consequently, an index zero condition on the linearization1+∂ R,vitalforaFredholmtheory,andadeformationproperty ζ of R(Φ−(x,y,λ),λ), needed in the Newtonian iteration scheme, should be justified to make the Newtonian iteration approach feasible. The dispersion term x−λy−λ2t causes the estimates, necessary for the above Newtonian iteration approach, growing inevitably unbounded when |y| → ∞ or t gets larger. Precisely, the obstruction to the global solvability is the non existence of R(ω− +x−λy −λ2t,λ) when t gets larger. Hence onlyalocalsolvability ofthenonlinearRiemann-Hilbertproblemisachieved in general. Moreover, without compact support constraints, the quadratic dispersion term destroys the L1(R,dλ) property of ∂R/∂t as t > 0. So only a local solvability of the Lax pair of the Pavlov equation holds. To derive a local solvability of the Cauchy problem of the Pavlov equation, we still need to impose the compact support condition [10]. Thecontentsofthepaperareasfollows. InSection2and3,removingthe compactsupportconditionontheinitialdatav(x,y)andusingtheLp(R,dξ) -Hilberttransformtheory, wederiveλ-asymptotics of therealeigenfunction ϕ(x,y,λ) and the solution to the shifted Riemann-Hilbert problem χ(ξ,λ) via uniform Lp(R,dξ) estimates on them. In Section 4, we derive the con- junction formula (1.7) between complex eigenfunctions Φ±(x,y,λ) and de- fine R(ζ,λ) as the spectral data. An index zero condition on 1+∂ R and ζ a deformation property of R(Φ−(x,y,λ),λ) are justified as well. In Sec- tion 5, via a Newtonian iteration method and using the Lp(R,dλ) - Hilbert transform theory, we solve the short time unique existence of the nonlinear 3 Riemann-Hilbert problem (1.8) with prescribed initial data. The short time unique existence of the Lax pair and of the Cauchy problem of the Pavlov equation (1.1) are established in Secion 6. Acknowledgments. The author would like to thank S. Manakov and P. Santini for their pioneer contribution in the IST of dispersionless PDEs and S. X. Chen of Fudan university for helpful discussion in the Newtonian iteration method. The author feel very grateful to P. Grinevich for many brilliant inputs in discussion. The author was partially supported by NSC 103-2115-M-001 -003 -2. 2 The forward problem I: the real eigenfunctions The forward problem of the Pavlov equation with compactly supported ini- tial datahas beensolved in [10]. Removing thecompact supportrestriction, we prove theexistence of realeigenfunction ϕ(x,y,λ) andderive uniformes- timates of scattering data σ(ξ,λ) in this section. Throughout this report, S = {f :R2 → R| f(x,y) is Schwartz in x, y}, Lp(R,dλ) = {f : R → C| |f|Lp(R,dλ) = ( |f(λ)|pdλ)1p <∞}, ZR Hp(R,dλ) = {f : R → C| f, ∂ f ∈ Lp(R,dλ)}. λ Consider, for each fixed λ ∈ R, ∂ ϕ +(λ+v )∂ ϕ = 0, for x, y ∈ R, (2.1) y ± x x ± ϕ (x,y,λ)−ξ → 0, as y → ±∞, (2.2) ± where ξ = x− λy. The solvability and uniqueness of the boundary value problem of the first order partial differential equation (2.1), (2.2) is shown by solving the ordinary differential equation [10] dx = λ+v (x,y), x = x(y;x ,y ,λ), x(y ;x ,y ,λ) = x , (2.3) x 0 0 0 0 0 0 dy or, equivalently, dh = v (h+λy,y), dy x h = h(y;ξ ,y ,λ) = x(y;x ,y ,λ)−λy, (2.4) 0 0 0 0 h(y ;ξ ,y ,λ) = x −λy = ξ . 0 0 0 0 0 0 4 Hence y′ h(y′;x−λy,y,λ) = ξ+ v h(y′′;x−λy,y,λ)+λy′′,y′′ dy′′,(2.5) x Zy ϕ (x,y,λ) = h(±∞;x−(cid:0)λy,y,λ) (cid:1) (2.6) ± ±∞ = ξ+ v h(y′′;x−λy,y,λ)+λy′′,y′′ dy′′, x Zy (cid:0) (cid:1) and |∂β∂α∂µ h(y′;ξ,y,λ)−ξ |≤ C . (2.7) y′ λ ξ µ,α,β (cid:0) (cid:1) The real eigenfunction of the Pavlov equation is defined by ϕ(x,y,λ) = ϕ (x,y,λ) (2.8) − and the scattering data σ(ξ,λ) is defined as ϕ (x,y,λ) = ϕ (x,y,λ)+σ(ϕ (x,y,λ),λ), (2.9) + − − where σ(ξ,λ) =h(∞;ξ,−∞,λ)−ξ ∞ (2.10) = v h(y′′;ξ,−∞,λ)+λy′′,y′′ dy′′ x Z−∞ (cid:0) (cid:1) Lemma 2.1. Suppose v ∈ S. Then 0 < C <1+∂ σ(ξ,λ) < C . (2.11) 1 ξ 2 Proof. Comparing (2.5) and (2.10), to prove (2.11), it suffices to show that 0 < c ≤ ∂ h(y′;x−λy,y,λ) ≤ c (2.12) 1 x 2 for two constants c and c . Taking derivatives on (2.4), one obtains 1 2 ∂ ∂h(y′;x−λy,y,λ) ∂h(y′;x−λy,y,λ) = v (h+λy′,y′) , (2.13) ∂y′ ∂x xx ∂x ∂h(y;x−λy,y,λ) = 1. (2.14) ∂x Therefore, − |u (h+λy′′,y′′)|dy′′ ≤ log ∂h(y′;x−λy,y,λ) ≤ |u (h+λy′′,y′′)|dy′′ R xx ∂x R xx R R 5 and e−RR|uxx(h+λy′′,y′′)|dy′′ ≤ ∂h(y′;x−λy,y,λ) ≤ eRR|uxx(h+λy′′,y′′)|dy′′ (2.15) ∂x So 0 < C < ∂ h(y′,ξ,y,λ) < C , (2.16) 1 ξ 2 by the Schwartz condition and (2.12) is proved. Proposition 1. Suppose v ∈ S and p ≥ 1. Then the scattering data σ satisfies C |∂λν∂ξµσ(ξ,λ)|L∞ ≤ 1+|λ|2+µ+ν, (2.17) C |∂ν∂µσ(ξ,λ)| ≤ . (2.18) λ ξ Lp(dξ) 2+µ+ν−1 1+|λ| p Proof. The method in the proof of Proposition 3.2 in [10] can be applied to the non-compact case as well. Indeed, let y′ x = h(y′;ξ,−∞,λ)+λy′ = ξ +λy′+ v dy′′. (2.19) x Z−∞ For λ ≫ 1, from the implicit function theorem, there exist y′ = H(ξ,x,λ) and H (ξ,x,λ) so that 1 −ξ+x H (ξ,x,λ) 1 H(ξ,x,λ) = + . (2.20) λ λ2 So H (ξ,∞,λ) 1 σ(ξ,λ) = − . (2.21) λ From (2.19), one has ∂y′ 1 = , (2.22) ∂x λ+v (x,H(ξ,x,λ)) x or, equivalently ∂H1(ξ,x,λ) = − vx(x,−ξλ+x + Hλ21) , (2.23) ∂x 1+v (x,−ξ+x + H1)/λ x λ λ2 H (ξ,−∞,λ) = 0. (2.24) 1 6 Define 1 ξ ξˆ 1 λˆ = , ξˆ= , Hˆ (ξˆ,x,λˆ)= H ( ,x, ). λ λ 1 1 λˆ λˆ So (2.23) and (2.24) turn into ∂Hˆ (ξˆ,x,λˆ) v (x,−ξˆ+λˆx+λˆ2Hˆ (ξˆ,x,λˆ)) 1 x 1 = − , (2.25) ∂x 1+λˆv (x,−ξˆ+λˆx+λˆ2Hˆ (ξˆ,x,λˆ)) x 1 Hˆ (ξˆ,−∞,λˆ) = 0. (2.26) 1 For |λˆ| < 1 , the right hand side of (2.25) is smooth in ξˆ, λˆ. 2max|vx(x,y)| Expanding it at λˆ = 0 and using the boundary condition (2.26), we obtain Hˆ (ξˆ,∞,λˆ) 1 ∞ v (x,−ξˆ+λˆx+λˆ2Hˆ (ξˆ,x,λˆ)) x 1 = − dx Z−∞ 1+λˆvx(x,−ξˆ+λˆx+λˆ2Hˆ1(ξˆ,x,λˆ)) (2.27) ∞ = − v (x,−ξˆ)dx+O(λˆ) x Z−∞ =O(λˆ) by the mean value theorem, fundamental theorem of calculus, and v ∈ G. From the Hadamard’s lemma it follows σ(ξˆ/λˆ,1/λˆ) Hˆ (ξˆ,∞,λˆ) 1 = − λˆ2 λˆ is regular in ξˆ, λˆ for |λˆ|< 1 . Consequently, by 2max|vx(x,y)| ∂ = −λˆ2∂ −λˆξˆ∂ , ∂ = ∂ˆ , (2.28) λ λˆ ξˆ ξ ξˆ we obtain C |∂λν∂ξµσ(ξ,λ)|L∞ ≤ 1+|λ|2+µ+ν, (2.29) if λ ≫ 1. By analogy, one can show (2.29) if λ ≪ −1. Thus (2.17) is proved. Moreover, (2.18) follows from (2.27), (2.28), and the Minkowski inequality. 7 3 The forward problem II: the complex eigenfunc- tions With the positivity property (2.11), one has the unique solvability of the shifted Riemann-Hilbert problem [9] σ(ξ,λ)+χ+(ξ +σ(ξ,λ),λ)−χ−(ξ,λ) = 0, ξ ∈ R, ∂ξ¯χ = 0, ξ ∈ C±, (3.1) χ → 0, |ξ|→ ∞, ξ ∈ C. Applying (2.17), (2.18), and the boundedness of the Hilbert transform, we show that Proposition 2. If v ∈ S and p > 1, then the shifted Riemann-Hilbert problem (3.1) admits a unique bounded solution χ satisfying C |∂λν∂ξµχ−(ξ,λ)|L∞ ≤ 2+µ+ν−1 , ∀ξ ∈ R, ∀λ∈ R, (3.2) 1+|λ| p C |∂λν∂ξµχ(ξ,λ)|L∞ ≤ 2+µ+ν−1 , ∀ξ ∈ C±, ∀λ∈ R. (3.3) 1+|λ| p Proof. In[21],theuniquesolvabilityoftheshiftedRiemann-Hilbertproblem (3.1) has been justified by converting it to the following linear equation 1 χ−(ξ,λ)− f(ξ,ξ′,λ)χ−(ξ′,λ)dξ′ +g(ξ,λ) = 0, (3.4) 2πi ZR where f(ξ,ξ′,λ) = ∂ξ′s(ξ′,λ) − 1 , s(ξ′,λ)−s(ξ,λ) ξ′−ξ g(ξ,λ) = −1σ(ξ,λ)+ 1 ∂ξ′s(ξ′,λ) σ(ξ′,λ)dξ′, (3.5) 2 2πi s(ξ′,λ)−s(ξ,λ) ZR s(ξ,λ) = ξ+σ(ξ,λ). Hence ityields toshowingtheuniformestimate (3.2)withoutthecompactly supported condition. Taking derivatives of (3.4) and applying Proposition 1, we obtain ∂ν∂µχ−(ξ,λ) (3.6) λ ξ = 1 ∂λν ∂ξ +∂ξ′ µ[k(ξ,ξ′,λ)χ−(ξ′,λ)]dξ′ 2πi ξ′−ξ ZR (cid:0) (cid:1) 8 − 1 ∂λν ∂ξ +∂ξ′ µ[k(ξ,ξ′,λ)σ(ξ′,λ)]dξ′ 2πi ξ′−ξ ZR (cid:0) (cid:1) + 1 ∂λν∂ξµ′σ(ξ′,λ)dξ′+ 1∂ν∂µσ 2πi ξ′−ξ 2 λ ξ ZR where k(ξ,ξ′,λ) = s(ξ,λ)−s(ξ′,λ)−∂ξ′s(ξ′,λ)(ξ −ξ′). (3.7) s(ξ,λ)−s(ξ′,λ) Applying Proposition 1, Hadamard’s lemma, and an induction argument, one can derive C |∂λν∂ξµ1∂ξµ′2k(ξ,ξ′,λ)|L∞ ≤ 1+|λ|µ1+µ2+ν, (3.8) and then C |∂λν ∂ξ +∂ξ′ µkχ−−k∂λν∂ξµ′χ−|Lp(R,dξ) ≤ 2+µ+ν−1 , 1+|λ| p (cid:0) (cid:1) (3.9) C |∂λν ∂ξ +∂ξ′ µkσ|Lp(R,dξ) ≤ 2+µ+ν−1 . 1+|λ| p (cid:0) (cid:1) Plugging (3.9) into (3.6), we obtain C |∂ν∂µχ−−K ∂ν∂µχ− | ≤ , λ ξ λ ξ Lp(R,dξ) 2+µ+ν−1 1+|λ| p (cid:16) (cid:17) where 1 Kψ(ξ,λ) = f(ξ,ξ′,λ)ψ(ξ′,λ)dξ′. 2πi ZR Hence the boundedness of the Hilbert transform on Lp for p > 1, C |∂ν∂µχ−(ξ,λ)| ≤ , |λ| ≫ 1. (3.10) λ ξ Lp(R,dξ) 2+µ+ν−1 1+|λ| p So C |∂λν∂ξµχ−(ξ,λ)|L∞ ≤ 2+µ+ν−1 , |λ| ≫ 1, (3.11) 1+|λ| p from the Sobolev imbedding theorem. Finally (3.2) is achieved for ∀λ ∈ R by continuity. 9 Theorem 1. If v ∈ S, then for fixed x, y ∈ R, there exists a unique con- tinuous Φ(x,y,λ) such that Φ(x,y,λ) is holomorphic for λ ∈ C± and has continuous limits on both sides of λ ∈ R satisfying Φ−(x,y,λ) = ϕ(x,y,λ)+χ−(ϕ(x,y,λ),λ), (3.12) Φ+(x,y,λ) = Φ−(x,y,λ). (3.13) Moreover, Φ(x,y,λ) is a complex eigenfunction, i.e. for ∀λ ∈ C± fixed, (∂ +(λ+v ))∂ Φ= 0, for ∀x, y ∈R, (3.14) y x x Φ(x,y,λ)−(x−λy) → 0, as x, y → ∞, and Φ(x,y,λ) = Φ(x,y,λ¯), (3.15) C |∂λν∂xk∂yh Φ±−x+λy |L∞ ≤ 2+ν+k−1. (3.16) 1+|λ| p (cid:0) (cid:1) Proof. We refer to Theorem 3.1 in [10] for a detailed proof of this theorem exceptfor(3.12). Notethatthewholeprooftheredoesnotrequireanysmall data assumption nor a compact support condition. Moreover, theconjunctionformula(3.12)forcomplexandrealeigenfunc- tions has been justified by changes of variables in the proof of Theorem 3.2 in [10] under a compact support restriction. However, the same proof can be refined and holds for v(x,y) ∈ G as well. Indeed, denote λ = λ +iλ , R I the first change of variables F :(x,y) → (x ,y ) is defined by 1 1 1 x = lim h(y′;x−λ y,y,λ ) = ϕ (x,y,λ ), y < 0 1 R R − R y′→−∞  x1 = lim h(y′;x−λRy,y,λR) = ϕ+(x,y,λR), y > 0 (3.17)  y′→+∞   y = y 1  In the new variables, L = ∂ +(λ+v )∂ = ∂ +iλ κ(x ,y )∂ , (3.18) y x x y1 I 1 1 x1 where ∂ϕ ± κ(x1,y1) = ∂x (x,y,λR)|(x,y)=F1−1(x1,y1), y 6= 0. (3.19) By (2.15), there exists a pair of positive constants C , C such that: 1 2 0< C ≤ κ(x ,y )≤ C . (3.20) 1 1 1 2 10