The dual modified Korteweg-de Vries Fokas Qiao equation: Geometry and local analysis

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1 The dual modified Korteweg-de Vries Fokas Qiao equation: Geometry and local analysis Piotr Michał Bies, Przemysław Górka, and Enrique G. eyes Citation: J. Math. Phys. 53, ); doi: / View online: View Table of Contents: Published by the American Institute of Physics. elated Articles On the use of Fourier averages to compute the global isochrons of quasi)periodic dynamics Chaos, ) Matrix representation of the time operator J. Math. Phys. 53, ) The rogue wave and breather solution of the Gerdjikov-Ivanov equation J. Math. Phys. 53, ) Special singularity function for continuous part of the spectral data in the associated eigenvalue problem for nonlinear equations J. Math. Phys. 53, ) Liouvillian quasi-normal modes of Kerr-Newman black holes J. Math. Phys. 53, ) Additional information on J. Math. Phys. Journal Homepage: Journal Information: Top downloads: Information for Authors:

2 JOUNAL OF MATHEMATICAL PHYSICS 53, ) The dual modified Korteweg-de Vries Fokas Qiao equation: Geometry and local analysis Piotr Michał Bies, 1 Przemysław Górka, 1,a) and Enrique G. eyes,b) 1 Department of Mathematics and Information Sciences, Warsaw University of Technology, Pl. Politechniki 1, Warsaw, Poland Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307 Correo, Santiago, Chile eceived 0 April 01; accepted June 01; published online 3 July 01) We study a bi-hamiltonian equation with cubic nonlinearity shown to appear in the theory of water waves by Fokas, derived by Qiao using the two-dimensional Euler equation, and also known to arise as the dual of the modified Korteweg-de Vries equation thanks to work by Fokas, Fuchssteiner, Olver, and osenau. We present a quadratic pseudo-potential, we compute infinite sequences of local and nonlocal conservation laws, and we construct an infinite-dimensional Lie algebra of symmetries which contains a semi-direct sum of the sl, )-loop algebra and the centerless Virasoro algebra. As an application we prove a theorem on the existence of smooth solutions, and we construct some explicit examples. Moreover, we consider the Cauchy problem and we prove existence and uniqueness of weak solutions in the Sobolev space H q+ ), q > 1/. C 01 American Institute of Physics. [ I. INTODUCTION The equation m t + u u x )m x + u x m = 0, m = u u xx, 1) was found independently by Fokas, 9 by Fuchssteiner, 11 and by Olver and osenau 4 using algorithms designed to generate new bi-hamiltonian systems starting from bi-hamiltonian equations which were already known. In turn, the papers 9, 11, 4 were at least partially inspired by much earlier work by Fokas and Fuchssteiner, 10, 1 ). The basic observation appearing in efs. 10 and 1 then systematized and generalized in efs. 9, 11, and 4 was that the rearrangement of the pair of compatible hamiltonian operators of a given bi-hamiltonian equation would give rise to new bihamiltonian equations. When this method is applied to the Korteweg-de Vries KdV) equation, the new equation which arises is the well-known Camassa-Holm equation, a model for water waves which supports peakon traveling waves and admits wave breaking. 6 8, 0, When the method is applied to the modified Korteweg-de Vries mkdv) equation, the new equation which arises is Eq. 1). See Eq. 3.9) in ef. 9,Eq.5)inef.4, and Eq. 3.6) in ef. 11). Quite remarkably, in ef. 9 Fokas observed that 1) arises in the study of water waves, and in ef. 7 Qiao sketched a derivation of Eq. 1) starting from the two-dimensional Euler equation. Qiao noted the bi-hamiltonian character of 1), and he also observed that it has an sl, )-valued Lax pair see ef. 7, the latter paper, 18 and also ef. 30 for an earlier work on Lax pairs for dual equations). Qiao s work is particularly interesting for us because he presented in ef. 7 a careful analysis of traveling wave solutions of 1), and therefore provided motivation for our own work. For example, we note that Qiao found in ef. 7 W -shaped solitary wave solutions to Eq. 1), and that these solutions are not present in Camassa-Holm theory. 0 a) P.Gorka@mini.pw.edu.pl. b) ereyes@fermat.usach.cl /01/537)/073710/19/$ , C 01 American Institute of Physics

3 Bies, Górka, and eyes J. Math. Phys. 53, ) Qiao continued his research of Eq. 1)inef.8. Complementing his work, it was observed in ef. 18 that there exists a transformation from Qiao s equation to a negative flow of the modified KdV hierarchy. In this paper, we study Eq. 1) using the geometric and analytic approaches we considered in our previous papers 13, 14 on the Camassa-Holm and Hunter-Saxton equations, 15 respectively remarkably, the HS equation also appears as a dual equation, see ef. 4). It turns out that indeed we can investigate Eq. 1) using geometric means and we can also prove local well-posedness of the Cauchy problem but, for instance, while the Lie-algebraic structure of symmetries of 1) is similar to the one found in ef. 17 for the Camassa-Holm equation, the symmetries themselves are quite different from the ones in ef. 17. As we explain below, this fact makes their integration a non-trivial task. We organize our work as follows: In Sec. II we exploit the fact that Eq. 1) admits an sl, )- valued Lax pair. First, we use it to give a natural geometric interpretation to the variable m appearing in 1) in terms of foliations of pseudo-spherical surfaces naturally associated to 1). Second, we recall that the existence of an sl, )-valued Lax pair implies the existence of a quadratic pseudo-potential α see for instance ef. 6). This pseudo-potential is crucial for the computation of conservation laws using power series, as in classical Korteweg-de Vries or Camassa-Holm theory, and also for our symmetry analysis. We construct a sequence of nonlocal conservation laws and, using the gauge freedom inherent to the Lax pair formulation of partial differential equations, we show that in fact Eq. 1) admits an infinite number of non-trivial local conservation laws. Then we concentrate on nonlocal symmetries. The theory of local symmetries is of course classical, 3 but nonlocal symmetries are still a challenging topic of research. The great freedom with which we can define nonlocal objects makes the construction of nonlocal symmetries a non-trivial task, certainly different from the well-known construction of local symmetries which is basically an algorithmic task, as explained in efs. 1 and 3 and as implemented for instance in the recent symbolic computation package GeM ef. 5)). We find a nonlocal symmetry L 6 depending on the pseudo-potential α and we use it to construct a whole infinite-dimensional Lie algebra of nonlocal symmetries. This Lie algebra contains a semi-direct sum of the loop algebra over sl, ) and the centerless Virasoro algebra. We then concentrate on the symmetry L 6 just mentioned, by reasons explained in Subsections II B and II C. This symmetry is a first order generalized symmetry of certain augmented system to be introduced in Subsection II B) and so we can ask for its flow. 3 In contradistinction with the nonlocal symmetries appearing in our previous studies of the Camassa-Holm and Hunter- Saxton equations, 13, 14, 16, 17 finding the flow of L 6 is equivalent to solving a non-symmetric first order hyperbolic system of partial differential equations. We prove a rigorous theorem on existence of solutions for this system, basically by applying some results appearing in ef. 3, and we also integrate it explicitly in some non-trivial cases. In particular, in Subsection II C we present some explicit solutions different to the ones appearing in Qiao s paper. 7 In Sec. III we investigate the Cauchy problem for Eq. 1). This section builds on our previous research on the modified Camassa-Holm and Hunter-Saxton equations 13, 14 which, in turn, was influenced by the classical work of Li and Olver on the local well-posedness of the Camassa-Holm equation i.e., the dual KdV equation) in the Sobolev space H s ), s > 3/). We use regularization techniques to find aprioriestimates, and we show existence and uniqueness of weak solutions for initial data in the Sobolev space H q+ )forq 1/, ). II. GEOMETY AND INTEGABILITY We begin by recalling the Lax pair formulation of Eq. 1) following. 7 Equation m = u u xx, m t + u u x )m x + u x m = 0, ) is the integrability condition of the linear system x = X, t = T in which ) 1/ 1/) m λ X =, 3) 1/)m λ 1/

4 Bies, Górka, and eyes J. Math. Phys. 53, ) and λ + 1/) u u x T = ) λ 1 u u x ) 1/) mλu u x ) ) λ 1 u + u x ) + 1/) mλu u x ) λ 1/) u u x ). 4) As it is well-known, 4, 6 the existence of an sl, )-valued Lax pair such as the one determined by 3) and 4), is equivalent to the fact that, as explained below, Eq. ) describes pseudo-spherical surfaces: generic) solutions ux, t) toeq.) determine iemannian metrics of Gaussian curvature K = 1onthe domain of ux, t). We can be more specific in this case because of the special form the matrix X has. Indeed, motivated by efs. 3 and 33, we can show that 3) and 4) allow us to construct foliations on the pseudo-spherical surfaces determined by ). First of all, we define one-forms ω 1, ω, and ω 1 as ω 1 = dx + Pdt, ω = Q + dt, ω 1 = m λ dx + 1 Q ) dt, 5) in which P = λ + u u x, Q = λ 1 u u x ) m λu u x ), = λ 1 u + u x ) + mλu u x ). Then, it is easy to see that the structure equations d ω 1 = ω 1 ω, d ω = ω 1 ω 1, d ω 1 = K ω 1 ω with K = 1whenever u and m satisfy Eq. ). Whenever ω 1 ω 0, we can consider the coframe {ω 1, ω }, and the iemannian metric g = ω 1 ω 1 + ω ω on an open subset U of the x, t)-plane. The first two equations above tell us that ω 1 is the Levi-Civita connection one-form corresponding to the coframe {ω 1, ω }, and the last equation says that the Gaussian curvature of U, g)isk = 1. Now, the equation ω = 0 6) defines a foliation on U. The moving frame dual to the coframe {ω 1, ω } is e 1 = x, e = Q + t + P ). x The vector field e 1 is tangent to the leaves of the foliation 6), and if we write ω 1 = p ω 1 + q ω, the invariants of 6) are functions of p, q, and their covariant derivatives. A short computation yields ω 1 = mλω 1 + Pmλ + Q ω, Q + so that p = mλ, and q = Pmλ + Q = u xx, Q + u x in which we have used m = u u xx.nowweset = ) 1 x 1, an operator which will also appear in Sec. III, seeeq.78). The equation for q becomes q = p p. 7) p x Introducing covariant derivatives 1 f and f of functions f via df = 1 f)ω 1 + f)ω we find 1 p = p x, p = Q + p t + p x P), and therefore we can write 7) asq = p + p in which now = 1 1. This equation, or 1 p condition 7), characterize the foliations determined by solutions to Eq. ). Finally, we note that it follows from the identity ω 1 = p ω 1 + q ω and Gauss equation that the invariants p and q satisfy p = 1 q + p + q 1. 8)

5 Bies, Górka, and eyes J. Math. Phys. 53, ) This equation is precisely Eq. ), if written in terms of x, t, u. Thus, we conclude that Eq. ) not only can be written in a geometrically invariant fashion, but also that m not u!) has a direct geometric interpretation. A. A construction of conservation laws First of all, we prove that Eq. ) admits a quadratic pseudo-potential. The lemma below follows from the geometric observation that ) describes pseudo-spherical surfaces, see ef. 6, but it can be also checked directly. Lemma.1: Equation ) admits a quadratic pseudo-potential α. More precisely, the Pfaffian system α x = α mλ α + mλ, 9) α t = mλu x + u x λ + u λ + mλu) α + 4 λ u + u ) x α u x λ + mλu mλu x + u λ, 10) is completely integrable for α whenever m and u satisfy ). Lemma.1 allows us to construct a sequence of local and nonlocal) conservation laws for Eq. ) by using a simple recurrence relation. It is also important because, as we will show in Lemma., it is at the base of our construction of an infinite dimensional Lie algebra of nonlocal) symmetries for ). The following lemma can be also checked via a straightforward computation. Its geometric origin is explained in ef. 6: Lemma.: The system of equations δ x = α m λ 1, 11) δ t = m λu x u x λ u λ m λu) α + λ + u u x, 1) is compatible whenever ux, t) and mx, t) solve Eq. ) and αx, t) solves the system 9) and 10). Thus, we conclude that the equation α m λ 1) = t x m λu x u x λ u λ m λu) α + λ + u u x is a one-parameter family of conservation laws for Eq. 1). Expansion of α as a power series yields a sequence of conservation laws, as claimed. A consistent expansion for α is α = α n λ n. 14) This expansion yields the following equations for the coefficients α n : n=0 ) 13) α 0,x = α 0, 15) α 1,x = α m, 16) α n,x = α n + 1 n 1 m α k α n k, n. 17) k=0

6 Bies, Górka, and eyes J. Math. Phys. 53, ) We set α 0 = 0 and we easily find α 1 = 1 u u x). Thus, we conclude from 13) that mα 1 dx is a conserved density for Eq. ). We have, mα dx = 1 u u x )u u xx )dx = 1 uu u xx )dx up to total x-derivatives. This conserved density is the first Hamiltonian density appearing in Qiao s paper. 7 See also efs. 11 and 4). Further conserved densities found using 14) are nonlocal or zero. For example, the recursion relation 17) impliesthatα = 0, and that the function α 3 satisfies 1 8 u u xx)u u x ) α 3 α 3,x = 0. 18) We can easily check that α 3 is nonlocal. If α 3 = Fx, u, u x ), replacing into 18) and taking derivatives with respect to u xx we obtain the equation Solving for Fu, u x ) we get 1 8 u x uu x 1 8 u u x Fx, u, u x ) = 0. Fu, u x ) = 1 4 u3 x uu x 1 8 u u x + F1x, u), and replacing back into 18) we find that in fact the function F1x, u) does not exist. Now, we are free to modify the linear problem determined by 3) and 4) by application of gauge transformations. We use this freedom to prove that Eq. ) admits an infinite number of local conservation laws: consider the matrix ) 1/) 1/)i A =, 1 i apply the gauge transformation X AXA 1 + A x A 1, T AT A 1 + A t A 1, and set λ = iη. We obtain that Eq. ) is the integrability condition of the linear problem x = X g, t = T g in which ) 1/) m η 1/4 X g =, 19) 1 1/) m η and 1/η)u 1/) m η u + 1/) m η u x 1/η ) 1/4)u + 1/4)u x u ) x/η) T g = /η u + u x + u. 0) x/η 1/η)u + 1/) m η u 1/) m η u x We note that the matrices X g and T g appear in Schiff s paper 30 up to unessential coefficients. Instead of Lemma.1, we obtain that the equations γ x = γ + γ mη + 1/4 1) γ t = /ηu x + /η u + u x )γ /ηu + mηu mηu x )γ 1/4u 1/4u x + 1/η)u x 1/η ) ) are compatible whenever u and m are solutions to Eq. 1). Also, instead of Lemma., we now have the parameter-dependent potential equations ˆδ x = γ + m η, 3) ˆδ t = 4/η u x 4/η + u u x )γ + /η u m ηu + m η u x. 4)

7 Bies, Górka, and eyes J. Math. Phys. 53, ) Now we take γ = n=1 γ nη n and substitute in the equations for γ x. We obtain γ 1 = 1 4m, γ = 1 m γ 1,x, γ n+1 = 1 m γ n 1 n,x + γ k γ n k ), n. k=1 Substitution of the functions γ n in the equation for ˆδ x yields a sequence of local conserved densities for Eq. ). The odd functions γ n + 1 yield non-trivial conserved densities. Indeed, it can be checked by induction that γ n + 1 always includes a non-zero power of 1/m. The existence of this sequence of local conservation laws forces on us the question whether Eq. ) admits wave breaking or not. For instance, it is known in KdV theory that its infinitely many local conservation laws can be used to demonstrate the boundedness of the Sobolev norms of its solutions independently of time, and therefore they are important for proving global well-posedness see ef. and the review ef. 31). On the other hand, the Camassa-Holm CH) and Hunter-Saxton HS) equations have infinitely many local conservation laws as well see ef. 5 for CH and ef. 14 and references therein for HS), but these conservation laws cannot prevent the existence of smooth solutions which develop singularities in finite time. 8,, 34 We conjecture that this is what happens in the case of Eq. ). B. A Lie algebra of nonlocal symmetries In this subsection we consider nonlocal) symmetries of Eq. ). We recall the relevant definitions, further details are in efs. 13,14,16, and 6 and references therein. We assume that the reader is familiar with the theory of local symmetries as it appears in efs. 1 and 3. Definition.1: Let N be a non-zero integer. An N-dimensional covering π of a system of) partial differential equations) a = 0, a = 1,...,k, is a pair π = {γ b : b = 1,...,N}; {X ib : b = 1,...,N; i = 1,...,n} ) 5) of variables γ b and smooth functions X ib depending on x i, u α, γ b and a finite number of partial derivatives of u α, such that the equations γ b x i = X ib 6) are compatible whenever u α x i ) is a solution to a = 0. We usually write π = γ b ; X ib ) instead of 5). The variables γ b are new dependent variables, the nonlocal variables of the theory. Equation 6) relates them to the original variables u α. Definition.: Let a = 0, a = 1,...,k, be a system of partial differential equations, and let π = γ b ; X ib ) be a covering of a = 0. A nonlocal π-symmetry of a = 0 is a generalized symmetry X = i ξ i x i + α φ α u α + b ϕ b 7) γ b of the augmented system a = 0, γ b x i = X ib. 8)

8 Bies, Górka, and eyes J. Math. Phys. 53, ) Equation 6) allows us to assume that the functions ξ i, φ α, ϕ b of 7) depend at most on variables x i, u α, γ b, and a finite number of partial derivatives of u α. Definition. is essentially in ef. 1, pp ]. Thus, in order to find nonlocal π-symmetries, we can proceed as in the local case considered for instance in P. Olver s treatise: 3 we need to check the conditions see efs. 1 and 3, p. 90]) γ b ) pr X a ) = 0, and pr X X x i ib = 0, 9) in which X is a generalized vector field as in 7), and prx is the infinite prolongation of X, pr X = ξ i + φ α x i J u α + ϕ b J i α,j J γ b, b,j J where the sum runs over all multi-indices J = j 1,...,j k ), 0 j 1,...,j k n, k 0, and the functions φ α J, ϕb J are given by φ α J = D J φ α i ξ i u α i ) + i ξ i u α Ji, and ϕb J = D J ϕ b i ξ i γ b i ) + i ξ i γ b Ji. As explained in ef. 3, p. 91], it is enough to consider evolutionary vector fields X of the form m X = G α u + N H b α γ. 30) b α=1 If X is a nonlocal π-symmetry, we say following ef. 1 that G = G 1,...,G m )istheπ-shadow of the nonlocal symmetry 30). A fundamental result appearing in ef. 1 states that if G is a π-shadow, and we allow N the number of new dependent variables) to be infinite, it is always possible to find a further covering ˆπ and a bona fide nonlocal ˆπ-symmetry X of the form X = m α=1 Gα + N u α b=1 H b. γ b In many interesting cases it is not necessary to use an infinite number of new dependent variables in 13, 14, 16, 5, 6 order to reconstruct a nonlocal symmetry from a shadow see ef. 1, the later papers and references therein) and therefore the theory really is of interest for applications. As we will see momentarily, Equation ) is a further example of this fact. From the general theory of generalized symmetries see efs. 1 and 3, Chapter 5]) we conclude that nonlocal symmetries can be used to generate solutions: Lemma.3: If u α 0 xi ) and γ b 0 xi ) are solutions to the augmented system 8), the solution to the Cauchy problem u α τ = Gα, b=1 γ b τ = H b, u α x i, 0) = u α 0 xi ), γ b x i, 0) = γ b 0 xi ), is a one-parameter family of solutions to the augmented system 8). In particular, nonlocal π- symmetries send solutions to the system a = 0 to solutions of the same system. Now we construct a covering of Eq. ) and we compute nonlocal symmetries. The most difficult part of this construction is to discover a shadow of a nonlocal symmetry. Once a shadow is known, the construction of a covering π and the calculation of π-symmetries) follows more or less standard lines. Proposition.1: Consider the pseudo-potential α and the potential δ determined by Eqs. 3), 4), and 1), respectively. Then, the pair G u, G m ), in which G u = α 1) e δ, G m = λ e δ α m x + α m λ λ m ), 31) is a shadow for the Qiao equation 1).

9 Bies, Górka, and eyes J. Math. Phys. 53, ) The proof of this proposition is a very long but straightforward computation: it amounts to checking that the vector field X = G u u + G m m satisfies prx m + u u xx) = 0 and pr Xm t + u u x )m x + u x m ) = 0 on solutions of Eq. ), or equivalently, that the pair G u, G m ) satisfies the formal linearization of ) on solutions. Now, once we know a shadow for Eq. ), we need to investigate how the functions α and δ evolve as u changes along the flow of the vector field G u u + G m m. In other words, if we set u τ = G u and m τ = G m, we need to compute α τ and δ τ. It is not difficult to find α τ taking τ-derivatives of 3). We find α τ = m λ α 3 m λ α + α λ. e δ On the other hand, it is not possible to compute δ τ in terms of α, δ, and derivatives of) u. Indeed, we obtain from 11) δ xτ = δ τ ) x = λ α m) τ, and replacing the expressions we already have for m τ and α τ we find that the expression α m) τ appearing in the last equation is not a total x-derivative. The only way to resolve what is seemingly a contradiction is to define δ τ = β for a new nonlocal variable β, and to note that β must be determined by the equations β x = δ τ ) x = λ α m) τ, [ β t = δ τ ) t = δ t ) τ = m λu x u x λ u ] λ m λu) α + λ + u u x. τ These two equations are compatible. Now we make the crucial observation that the variation of β induced by u τ and m τ can be expressed in terms of u, m, their x-derivatives, and the nonlocal variables α, δ, β. Its explicit expression is rather involved and it will be given below. At this point we wish to stress the fact that these observations imply that now we have a covering π for Eq. ) with nonlocal variables α, δ, and β, and also a bona fide nonlocal π-symmetry, the vector G u u + G m m + α τ α + β δ + β τ β, which we just determined. We summarize the construction of π in the following proposition: Proposition.: Equation m = u u xx, m t + u u x )m x + u x m = 0 3) admits a three-dimensional covering π with nonlocal variables α, δ, and β determined by the following compatible equations: α x = α mλ α + mλ, 33) α t = mλu x + u x λ + u λ + mλu) α + 4 λ u + u ) x α u x λ + mλu mλu x + u λ, 34) δ x = α m λ 1, 35) δ t = m λu x u x λ u λ m λu) α + λ + u u x, 36)

10 Bies, Górka, and eyes J. Math. Phys. 53, ) and β x = λ e δ α λm α m + αm ), 37) β t = e δ [ u x m + 4 u x m m x u x + m x u mu ) λ 4 m ] α +e δ [ u x m + u m ) λ u x m + 4 mu) λ + 4 λ 1] α +e δ [ u + u x ]. 38) Equipped with this proposition we can write down the symmetry we have constructed by hand. However, we can do better: we can classify nonlocal symmetries. The following result was obtained with the help of the MAPLE packages VESSIOT see ef. 1) and GeM see ef. 5): Theorem.1: We consider the covering of Eq. 3) determined by Eqs. 33) 38) and the equations λ x = 0, λ t = 0. 39) The first order nonlocal -symmetries of Eq. 3) are linear combinations of L 1 = x, 40) L = t, 41) [ L 3 Aλ)) = Aλ) t t + m m + u u β β λ ], λ 4) L 4 = β, 43) L 5 = δ β β, 44) L 6 = 1 + α )exp δ) u + λ exp δ) λm + α m λ αm x ) m λαα mλ + mλ α)exp δ) α + β δ + exp δ)λ 3 α 3 m + 6exp δ)λ 4 α m + 4exp δ)λ 3 α 3 m x 1 ) β β. We have included λ as a new dependent variable because if the variables m, u, α, δ, β are subject to scaling, so must be λ. Otherwise the augmented system 3) 38) would not admit scaling symmetries. Also, we note that the vector fields L 1 L 3 project onto classical symmetries of Eq. 3), while the vector fields L 4 L 6 are genuine nonlocal symmetries. Of these, L 4 and L 5 are vertical symmetries; their flows do not modify the original dependent variables m and u. Thus, the most interesting symmetry from the point of view of generating solutions is 45). Corollary.1: The commutator table of the symmetries L 1 L 6 of Eq. 3) is: L 1 L L 3 Aλ)) L 4 L 5 L 6 L 1 L AL L 3 Bλ)) L L 3 λab BA )) L 4 BL 6 L 4 AL 4 L 4 L 5 L 5 L 4 L 6 L 6 AL 6 L 5 L 6 45)

11 Bies, Górka, and eyes J. Math. Phys. 53, ) As it happens in the cases of the Camassa-Holm and Hunter-Saxton equations see the previous papers 14, 17 ), we can extract from the infinite-dimensional Lie algebra spanned by L 1 L 6, a semi-direct sum of the loop algebra over sl, ) and the centerless Virasoro algebra. Proposition.3: Let us define the vector fields Tn 1 = λn L 5, Tn = λn L 4, Tn 3 = λn L 6, and W n = L 3 λ n ), in which n Z. Then, the following commutation relations hold: [Tm 1, T n ] = T m+n, [T m 1, T n 3 ] = T m+n 3, [T m, T n 3 ] = T m+n 1, 46) [W m, W n ] = m n)w m+n, 47) [T 1 m, W n] = mt 1 m+n, [T m, W n] = m 1)T m+n, [T 3 m, W n] = m + 1)T 3 m+n. 48) C. Integrating nonlocal symmetries As explained in Subsection IIB, the most interesting nonlocal symmetry for generating solutions to Eq. 3) is the vector field L 6 given by 45). The system of equations we need to solve in order to find the flow of L 6 is u τ = 1 + α ) e δ, 49) m τ = λ e δ λm + α m λ αm x ), 50) α τ = λαα mλ + mλ α) e δ, 51) δ = β, 5) τ β τ = e δ λ 3 α 3 m + 6 λ 4 α m + 4 λ 3 α 3 ) 1 m x β, 53) and we supplement 49) 53) with initial conditions ux, t, 0) = u 0, mx, t, 0) = m 0, αx, t, 0) = α 0, δx, t, 0) = δ 0, βx, t, 0) = β 0, 54) in which u 0 x, t), m 0 x, t), α 0 x, t), δ 0 x, t), and β 0 x, t) are particular solutions to 3) 38). We note that the system 49) 53) is not a symmetric hyperbolic system, in contradistinction with the systems appearing in the Camassa-Holm and Hunter-Saxton cases. 5,17,14 Nonetheless, we can prove a general theorem which yields solutions to Eq. 3) using standard techniques from the theory of nonlinear hyperbolic systems see for instance ef. 3): Theorem.: Take M = S 1 or M =, and assume that the initial functions mx, t, 0) = m 0 x, t), 55) γ x, t, 0) = γ 0 x, t), δx, t, 0) = δ 0 x, t), and βx, t, 0) = β 0 x, t), 56) belong to the Sobolev space H k M), with k > 3/. Then, the system 50) 53) with initial conditions 55) and 56) possesses solutions mx, t, τ), γ x, t, τ), δx, t, τ), and βx, t, τ), on an interval I about τ = 0, belonging to L I, H k M)) LipI, H k 1 M)).

12 Bies, Górka, and eyes J. Math. Phys. 53, ) Furthermore, if the initial data are smooth, then so are the local in τ) solutions mx, t, τ), γ x, t, τ), δx, t, τ), and βx, t, τ). Proof: We write the system 50) 53) using matrices. We set v = m,α,δ,β) t, and we define αλe δ λe δ λm + α m λ) B = , b = λαe δ α mλ + mλ α) β, 4α 3 λ 3 e δ e δ λ 3 α 3 m + 6λ 4 α m ) 1 β so that 50) 53) is equivalent to B x v = b. The matrix A 0 given by 1 + 4α 4 λ 4 e δ 0 0 α λ e δ A 0 = α λ e δ is a symmetrizer for the system B x v = b, that is, A 0 B is a symmetric matrix. Now the result follows from ef. 3, Chapter 16]. This theorem, together with standard symmetry theory, 3 immediately implies the existence of 1-parameter local groups of smooth solutions to Qiao s equation: Corollary.: Take M = S 1 or M =, and assume that the functions u 0 x, t), m 0 x, t), γ 0 x, t), δ 0 x, t), and β 0 x, t), 57) are smooth functions on M which solve the system 3) 38). Then, there exist at least for τ in an interval about τ = 0) a family of smooth solutions ux, t, τ) to Qiao s equation such that ux, t, 0) = u 0 x, t) for all x M. In actual fact, we can solve explicitly the initial value problem 49) 54) in some cases. As in the last theorem, we only investigate 50) 54), since none of these equations depend on u.we begin with a change of variables ξ = τ, in which η is a function to be determined. We have We choose η so that m τ = m ξ + m η η τ η = ηx,τ), η τ = λe δ λm + α m δ m λ) λαe η η x. η = λαe δ, ηx,τ = 0) = x, 58) x and we apply this change of variables to 50) 53). For example, we have α τ = α ξ + α η η τ = α ξ + α ) δ η λαe = α α λαe δ η x ξ x, and we can simplify this last expression using 33). Proceeding in this way we obtain the following system of equations: m ξ = λe δ λm + α m λ), 59) α ξ = e δ λ mα 3 3λα + λ αm), 60)

13 Bies, Górka, and eyes J. Math. Phys. 53, ) δ ξ = e δ λ α m λα) + β, 61) β ξ = λ4 α m e δ 1 β. 6) We have therefore replaced the system of first order nonlinear partial differential equations 50) 53) for a system of essentially) ordinary differential equations which is to be supplemented with Eq. 58). We can solve 58) 6) explicitly. We obtain the following six sets of solutions we suppress explicit dependence of the solutions on x and t): xξ) = η, mξ) = 0, αξ) = 0, δξ) = δ 0, βξ) = 0, 63) xξ) = η; mξ) = 0, αξ) = 0, δξ) = ln β 0 ξ + ) xξ) = η, mξ) = 4 m 0 ξβ 0 + ), αξ) = 0, ξβ m 0 λ v 0 ξ δξ) = ln β 0 ξ + ) 4 + δ 0, βξ) = ξ + β 0, 64) + δ 0, βξ) = ξ + β 0, in which v 0 = exp δ 0 ); xξ)= { arctan 9v 0 λ +4β0 4β 0v 0 λ)ξ +8β 0 4v 0 λ) arctan 16β 0 8v 0 λ) } +η, 3v 0 λ 3v 0 λ mξ) = 3, αξ) = 1, 66) 4 λ δξ) = ln 4ξ β0 + 16ξβ 0 4β 0 ξ v 0 λ 8ξv 0 λ + 9ξ v0 λ δ 0, 16 9ξv0 βξ) = λ + 4ξβ0 4ξβ 0v 0 λ + 8β 0 ) 4ξ β0 + 16ξβ 0 4β 0 ξ v 0 λ 8ξv 0 λ + 9ξ v0 λ + 16, in which v 0 = exp δ 0 ); xξ)= { arctan 9v 0 λ +4β0 +4β 0v 0 λ)ξ +8β 0 +4v 0 λ) arctan 16β 0+8v 0 λ) } +η, 3v 0 λ 3v 0 λ mξ) = 3 ; αξ) = 1, 67) 4 λ δξ) = ln 4ξ β0 + 16ξβ 0 + 4β 0 ξ v 0 λ + 8ξv 0 λ + 9ξ v0 λ δ 0, 16 9ξv0 βξ) = λ + 4ξβ0 + 4ξβ 0v 0 λ + 8β 0 ) 4ξ β0 + 16ξβ 0 4β 0 ξ v 0 λ 8ξv 0 λ + 9ξ v0 λ + 16, in which v 0 = exp δ 0 ); xξ) = 3 ln ξβ 0 + ) 3 α 0 v 0 λα 0 ξ + ξβ 0 + ) ln α 0 + η, mξ) = 0, ξβ 0 + ) 3 α 0 αξ) = v 0 λα 0 ξ + ξβ 0 + ), δξ) = ln ξβ 0 + ) 4 3 4v 0 λα 0 ξ + ξβ 0 + ) + δ 0, 68) βξ) = ξ +, β 0 in which v 0 = exp δ 0 ). 65)

14 Bies, Górka, and eyes J. Math. Phys. 53, ) We can use these results in order to integrate Eq. 49) for the original variable u of Eq. 3). We obtain the following result. Theorem.3: Let ux, t) be a solution of Eq. 3). Then, the solution ux, t, τ) to the initial value problem u τ = 1 + αx,τ) )e δx,τ), 69) ux, t, 0) = ux, t), 70) in which αx, t, τ) and δx, t, τ) are determined by one of 63) 68), is a one-parameter family of solutions to Eq. 3). Example: We consider Eq. 3) in old variables η and t, and we set u 0 η, t) = 0, the trivial solution to 3). We easily obtain two sets of initial conditions for the variables m, α, δ, and β: and m 0 = 0 α 0 = e η+λ t, δ 0 = η + λ t β 0 = 4 t, 71) λ m 0 = 0 α 0 = 0, δ 0 = η + λ t β 0 = C. 7) First we use 7). Equation 64) imply that x = η and that δx, t,τ) = ln It follows from 69) and 70) that ux, t,τ) = Cτ + ) 4 CCτ + ) C solves Eq. 3) and it satisfies ux, t,0)= u 0 x, t). Now we use 71). Equation 68) yields and so we obtain + x + λ t). ) e x λ t η = x + lnτt + λ) lnλ τ + τt + λ), τt + λ) 3 e x+λ t τt + λ)4 αx, t,τ) =, δx, t,τ) = ln λ τ + τt + λ)τt + λ ) λ τt + λ ) x + t λ. 73) eplacing into 69) and 70) we obtain that the function τ ux, t,τ) = 0 λ st + λ ) st + λ) 4 + is a solution to Eq. 3) and it satisfies ux, t,0)= u 0 x, t). λ st + λ) ) ) ds e x λ t λ s + st + λ) st + λ ) III. LOCAL WELL-POSEDNESS In this section we show the existence of weak solutions to Eq. 1). As stated in Sec. I, this part of our work uses techniques from ef. 13 and, and also ef. 19. First of all, we rewrite Eq. 1) as follows: u u xx ) t = 3u u x + u x ) 3 + 4uu x u xx u x u xx ) + u u xxx u x ) u xxx, 74) u0) = u 0. 75)

15 Bies, Górka, and eyes J. Math. Phys. 53, ) We divide this section in two parts. First, we find aprioriestimates, and then we complete our existence proof. A. Aprioriestimates We regularize problem 74) and 75) by adding ɛu xxxxt to the left hand side of 74) and we also regularize the initial data. We get the initial value problem u ɛ u ɛ xx + ɛu ɛ xxxx) = t 3u ɛ u ɛ x + u ɛ x) 3 + 4u ɛ u ɛ xu ɛ xx u ɛ xu ɛ xx) + u ɛ u ɛ xxx u ɛ x) u ɛ xxx, 76) u ɛ 0) = u ɛ 0. Lemma 3.1: Let q 1/, ) and u ɛ be a solution of problem 76), such that u ɛ W 1, [0, T 1 ); H q+5 ). Then, there exists T 0, T 1 ] and a constant C such that the estimate holds. u ɛ L 0,T ;H q+ ) + u ɛ t L 0,T ;H q+1 ) CT, u 0 H q+) 77) Proof: First of all we recall the very useful operator given by see efs. and 13): = I x ) 1. 78) In terms of we define the Sobolev norm as follows: v H s = s v L. Moreover, it can be shown by using the properties of the Fourier transform that the norms v H s+1 and v H s + v x H s are equivalent. For q > 1 the space Hq is a Banach algebra, that is, there exists a constance C q such that for all u, v H q the estimate uv H q C q u H q v H q holds. Now, applying q u ɛ ) q to Eq. 76) and integrating with respect to x, we get 1 d u ɛ H dt q + uɛ x H + q ɛ uɛ xx ) H = 3 q u ɛ u ɛ x q u ɛ dx q + q u ɛ x) 3 q u ɛ dx + 4 q u ɛ u ɛ xu ɛ xx q u ɛ dx q u ɛ xu ɛ xx) q u ɛ dx + q u ɛ u ɛ xxx q u ɛ dx q u ɛ x) u ɛ xxx q u ɛ dx = 3 q u ɛ u ɛ x q u ɛ dx + q u ɛ x) 3 q u ɛ dx + 4 q u ɛ u ɛ xu ɛ xx q u ɛ dx q u ɛ xu ɛ xx) q u ɛ dx q u ɛ u ɛ xu ɛ xx q u ɛ dx q u ɛ u ɛ xx q u ɛ xdx + q u ɛ xu ɛ xx) q u ɛ dx + q u ɛ x) u ɛ xx q u ɛ xdx = 3 q u ɛ u ɛ x q u ɛ dx + q u ɛ x) 3 q u ɛ dx + q u ɛ u ɛ xu ɛ xx q u ɛ dx + q u ɛ x) u ɛ xx q u ɛ xdx q u ɛ u ɛ xx q u ɛ xdx = 3 q u ɛ u ɛ x q u ɛ dx + q u ɛ u ɛ x) q u ɛ xdx 1 q u ɛ x) 3 q u ɛ xxdx 3 + q u ɛ u ɛ x q u ɛ xxdx = I 1 + I + I 3 + I 4.

16 Bies, Górka, and eyes J. Math. Phys. 53, ) Using the Cauchy-Schwartz inequality, the fact that H q forms a Banach algebra, integration by parts, and the inequality: which is true for q > 0seeef.19), we obtain: fg H q C f L g H q + f H q g L ), I 1 c 1 u ɛ 3 H q uɛ x H q, I u ɛ u ɛ x) H q u ɛ x H q c u ɛ x H q u ɛ x L uɛ H q + u ɛ L u ɛ x L u ɛ ) x H q, I 3 c 3 u ɛ xx H q u ɛ x L uɛ x H q + u ɛ x ) H q uɛ x L, I 4 c 4 u ɛ H q u ɛ L uɛ x H q + u ɛ x L u ɛ ) H. q Now, by properties of Sobolev norms, we get: d u ɛ H dt q + uɛ x H + q ɛ uɛ xx ) H c u 4 q H. 79) q+ Next, we differentiate Eq. 76) with respect to x: u ɛ x u ɛ xxx + ɛu ɛ xxxxx) = t 3u ɛ u ɛ x) x + u ɛ x) 3 ) x + 4u ɛ u ɛ xu ɛ xx) x u ɛ xu ɛ xx) ) x + u ɛ u ɛ xxx) x u ɛ x) u ɛ xxx) x, Applying q u ɛ x q to the above equation and integrating over we obtain 1 d u ɛ x H dt q + uɛ xx H + q ɛ uɛ xxx ) H = 3 q u ɛ u ɛ x) q x q u ɛ xdx + q u ɛ x) 3 x q u ɛ xdx + 4 q u ɛ u ɛ xu ɛ xx) x q u ɛ xdx q u ɛ xu ɛ xx) ) x q u ɛ xdx + q u ɛ u ɛ xxx) x q u ɛ xdx q u ɛ x) u ɛ xxx) x q u ɛ xdx = 3 q u ɛ u ɛ x) x q u ɛ xdx + q u ɛ x) 3 x q u ɛ xdx + q u ɛ u ɛ xu ɛ xx) x q u ɛ xdx q u ɛ u ɛ xx) x q u ɛ xxdx + q u ɛ x) u ɛ xx) x q u ɛ xxdx = J 1 + J + J 3 + J 4 + J 5. We can easily estimate the terms J 1, J as follows: J 1 d 1 u ɛ xx H q u ɛ L uɛ x H q + u ɛ x L u ɛ ) H, q J d u ɛ xx H q u ɛ x L uɛ x H q + u ɛ ) H q uɛ L, Using the Plancherel theorem and basic properties of the Fourier Transform we can show the formula q f x q g x dx = q+1 f q+1 gdx + q f q gdx. 80) J 3 = q+1 u ɛ u ɛ xu ɛ xx q+1 udx + q u ɛ u ɛ xu ɛ xx q u ɛ dx d 3 u ɛ H q+1 u ɛ x L uɛ x H q+1 + u ɛ H q+1 u ɛ x L u ɛ x H q+1 + u ɛ x H u ɛ q+1 L u ɛ x L + u ɛ x H q+1 u ɛ x L uɛ H q+1), J 4 = q+1 u ɛ u ɛ xx q+1 u ɛ dx + q u ɛ u ɛ xx q u ɛ dx = J 41 + J 4. The term J 4 can be estimated in the same way as the terms I i s.

17 Bies, Górka, and eyes J. Math. Phys. 53, ) Now, we recall the Kato-Ponce inequality. Lemma 3. ef. 19): Let us assume that s > 0. Then, there exists c, such that the following inequality holds: [ s, g] L c g x L s 1 L + s g L f L ). Using Lemma 3. we can estimate J 41 in the following manner: J 41 = [ q+1, u ɛ ]u ɛ xx q+1 u ɛ xdx u ɛ ) q+1 u ɛ xx q+1 u ɛ xdx ) d 4 u ɛ x H q+1 u ɛ u ɛ x L u ɛ xx H q + u ɛ H q+1 u ɛ xx L + u ɛ u ɛ x q+1 u ɛ x) dx d 4 u ɛ x H q+1 ) u ɛ u ɛ x L u ɛ xx H q + u ɛ H q+1 u ɛ xx L + u ɛ L u ɛ x L u ɛ x H. q+1 Finally, we turn our attention to the term J 5 and we obtain J 5 = q u ɛ x) u ɛ xx) x q u ɛ xxdx = q+1 u ɛ x) u ɛ xx) q+1 u ɛ xdx + q u ɛ x) u ɛ xx) q u ɛ xdx = J 51 + J 5. The term J 5 can be easy estimated. Using the Kato-Ponce inequality we estimate J 51 as follows: J 51 = q+1 u ɛ x) u ɛ xx) q+1 u ɛ xdx = [ q+1, u ɛ x) ]u ɛ xx q+1 u ɛ xdx u ɛ x) q+1 u ɛ xx q+1 u ɛ xdx = [ q+1, u ɛ x) ]u ɛ xx q+1 u ɛ xdx + u ɛ xu ɛ xx q+1 u ɛ x) dx d 5 u ɛ x H q+1 u ɛ x u ɛ xx L u ɛ xx H q + u ɛ x) H q+1 u ɛ ) xx L + u ɛ L u ɛ x L u ɛ x H. q+1 Thus, we can write d u ɛ x H dt xx H + q ɛ uɛ xxx ) H d u ɛ 4 q H, q+ 81) and if we add inequalities 79) and 81) we obtain: d dt uɛ H q + uɛ x H q ɛ) uɛ xx H + q ɛ uɛ xxx H ) q C uɛ 4 H. q+ 8) Hence, d dt uɛ H + ɛ u ɛ q+ xx H + q ɛ uɛ xxx H ) q C uɛ H + ɛ u ɛ q+ xx H + ɛ u xxx q H q ). 83) Now, integrating the above inequality we obtain u ɛ u 0 + ɛ u H H 0xx H + ɛ u 0xxx q H q q+ 1 Ct u 0 + ɛ u H q+ 0xx H + ɛ u 0xxx q H ) q provided that T < C u 0 H q+ 1 Ct1 + u 0 H q+ ) 1 + u 0 H q+ 1 CT1 + u 0 H q+ ), 84) 1+ u 0 H q+ ) and ɛ is small enough. In order to estimate the norm of u t we apply the operator q u t q to Eq. 76) and we integrate with respect to x u t H q + u xt H q + ɛ u xxt H q = L 1 + L + L 3 + L 4 + L 5 + L 6.

18 Bies, Górka, and eyes J. Math. Phys. 53, ) In a similar way to what we just did, we can show that the following estimates hold: L 1 + L + L 3 + L 4 e 1 u ɛ t H q u ɛ t 3 H, q+ L 5 + L 6 e u ɛ t H q + u ɛ xt H q ) u ɛ x 3 H. q+ Due to the Cauchy inequality, we obtain the estimate u ɛ t H q + uɛ xt H q C u 3 H u ɛ q+ xt H q + uɛ t H q ) 85) C u 6 H + 1 q+ u t H + 1 q u xt H. q Thus, inequality 85) implies u ɛ t H q + uɛ xt H q C u ɛ 6 H. q+ And thanks to 84) we obtain u ɛ t H q+1 CT, u 0 H q+). This finishes the proof of the Lemma. We can also prove the following lemma, using methods similar to the ones appearing in the paper. 14 Lemma 3.3: Let T > 0 be the constant from Lemma 3.1. Then for each ɛ>0 there exists a solution u ɛ of regularized problem 76). B. Existence of solution The technical results obtained in Sec. III A can be now used to prove existence and uniqueness of weak) solutions to Eq. 74). Multiplication of 74) by a smooth function leads us to the following definition of weak solution: Definition 3.1: Let us assume that u 0 H 1. We say that u is a weak solution of problem 74) if there exists T > 0 such that u W 1, 0, T ); H 1 ) L 0, T ); H ), u0) = u 0 and for any v H and for almost all 0 t T the following identity holds: u t v + u tx v x ) dx = 3u u x v + u x uv x + u u x v xx 1 3 u3 x v xx) dx. The main result of this section is the following theorem. Theorem 3.1: If q 1/, ) and u 0 H q+ ), then there exists a unique weak solution of problem 74) such that u W 1, 0, T ; H 1+q ) L 0, T ; H q+ ). Proof: From Lemmas 3.1 and 3.3, we obtain a subsequence {u ɛ n } such that {u ɛ n } is a solution to regularized problem and u ɛ n u ɛ n t u in L 0, T ; H q+ ), u t in L 0, T ; H q+1 ). Now, let us take γ C c ) and φ C1 [0, T]. Let > 0 be such that suppγ ) B0, ). Hence, by the Aubin-Lions Lemma see ef. 9) there exists a subsequence of u ɛ n, which we denote again as u ɛ n, such that u ɛ n u strongly in L 0, T ; H q+1+δ ),

19 Bies, Górka, and eyes J. Math. Phys. 53, ) where δ<1. The regularized solutions u ɛ n of 76) satisfy the following identity: T u ɛ n t φγ + u ɛ n tx φγ x + ɛu ɛ n t φγ xxxx dtdx = T 0 We pass to the limit and obtain: = 0 3u ɛ n u ɛ n x γφ+ u ɛ n x uɛ n γ x φ + u ɛ n u ɛ n x γ xx φ 1 3 uɛ n 3 x γ xxφdtdx. T 0 T 0 u t φγ + u tx φγ x dtdx 3u u x γφ+ u x uγ xφ + u u x γ xx φ 1 3 u3 x γ xxφdtdx, because equation is true for all φ C 1 [0, T] and Cc ) is dense in H, we have that u is weak solution of 74). Moreover, by standard consideration we get that u0) = u 0. Finally, we show the uniqueness of solutions. Let us assume that we have two solutions u 1 and u and denote u = u 1 u. In definition of weak solution we take v = u and we obtain the following equation: 1 d dt u L + u x L ) = 3u u 1 u 1x 3u u u 1x 3u u xu +u x ) u 1x u 1 + u x u x ) u 1 + u x ) uu x 1 3 +uu 1 u 1x u xx + u uu 1x u xx + u u xu xx ux u 1x ) u xx + u x u 1x u x u xx + u x ) u x u xx ) dx. If we integrate by parts equation, we get that there exists C > 0 such that 1 d dt u H C u 1 H. 1 Hence, thanks to the Gronwall lemma we obtain u = 0. ACKNOWLEDGMENTS E. G. eyes is partially supported by FONDECYT Grant No I. M. Anderson, The Vessiot Handbook, Technical eport, Utah State University, Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. ev. Lett. 7111), ). 3 S. S. Chern and K. Tenenblat, Foliations on a surface of constant curvature and the modified Korteweg-de Vries equations, J. Differ. Geom. 16, ). 4 S. S. Chern and K. Tenenblat, Pseudo spherical surfaces and evolution equations, Stud. Appl. Math. 74, ). 5 A. Cheviakov, GeM : A Maple module for symmetry and conservation law computation for PDEs/ODEs, see cheviakov/gem/. 6 A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier Grenoble) 50, ). 7 A. Constantin and J. Escher, Global weak solutions for a shallow water equation, Indiana Univ. Math. J. 47, ). 8 A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181, ). 9 A. S. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math. 39, ). 10 B. Fuchssteiner, The Lie algebra structure of nonlinear evolution equations admitting infinite dimensional abelian symmetry groups, Prog. Theor. Phys. 65, ). 11 B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Phys. D 95, ). 1 B. Fuchssteiner and A. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4, /8). 13 P. Górka and E. G. eyes, The modified Camassa-Holm equation, Int. Math. es. Notices 1, ).

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