Articles you may be interested inQuenching phenomena for second-order nonlinear parabolic equation with nonlinear source AIP Conf.Approximation of solutions to an abstract Cauchy problem for a system of parabolic equations AIP Conf.A parabolic approximation method with application to global wave propagation We establish a new type of local asymptotic formula for the Green's function G t ͑x , y͒ of a uniformly parabolic linear operator ‫ץ‬ t − L with nonconstant coefficients using dilations and Taylor expansions at a point z = z͑x , y͒ for a function z with bounded derivatives such that z͑x , x͒ = x R N . Our method is based on dilation at z, Dyson, and Taylor series expansions. We use the Baker-Campbell-Hausdorff commutator formula to explicitly compute the terms in the Dyson series. Our procedure leads to an explicit, elementary, algorithmic construction of approximate solutions to parabolic equations that are accurate to arbitrarity prescribed order in the shorttime limit. We establish mapping properties and precise error estimates in the exponentially weighed, L p -type Sobolev spaces W a s,p ͑R N ͒ that appear in practice.We observe that ␣ uniquely determines k and ᐉ, so that our notation is justified. Let ␣ = ͑␣ j ͒ A k,ᐉ . We remark that if k = n + 1 or some ␣ j = n +1 ͑in which case L ␣ j z stands in fact for L n+1 s,z ͒, then ⌳ ␣,z and ⌳ z ᐉ depend on s, so we shall sometimes denote these terms by ⌳ ␣,s,z and ⌳ s,z ᐉ .Also, in what follows, when no confusion can arise, we will drop the explicit dependence on z. However, in Sec. V, z will be allowed to vary and we will reinstate the full notation. We also observe that each ⌳ z ᐉ or ⌳ s,z ᐉ is well defined as a Riemann integral by Lemma II.9 and by the following lemmas. Let us recall that ͗x͘ w = ͑1+͉x − w͉ 2 ͒ 1/2 . Lemma 3.6: The family ͕͗x͘ z −j L j z ;s ͑0,1͔,z R N , j = 0, ... ,n + 1͖ defines a bounded subset of L.Proof: This is an immediate consequence of Remark 3.3 if j ഛ n and of directly estimating the remainder in the Taylor series for j = n +1.In the following lemma, we shall use an arbitrary center for our weight. Lemma 3.7: For each given ⑀ Ͼ 0, the family ͕e −⑀͗z−w͘ e −⑀͗x͘ w L j z ;s ͑0,1͔,z R N , j = 0, ... ,n + 1͖ is a bounded subset of L.Proof: Let us assume first that w = z. We need to prove that the family ͕e −⑀͗x͘ z L j z ;s ͑0,1͔,z R N , j = 0, ... ,n + 1͖ is bounded in L. Indeed, this follows from Lemma 3.6 and the simple observation that ͗x͘ z j e −⑀͗x͘ z ഛ C, with C independent of z and j.To obtain the statement of the theorem, we then apply the triangle inequality to the vectors ͑0,x͒ , ͑1,z͒ , ͑1,w͒ R 1+N to conclude that ͗x − z͘ − ͗x − w͘ ഛ ͉z − w͉ ഛ ͗z − w͘. This shows that e ⑀͑͗x−z͘−͗x−w͘−͗z−w͒͘ ഛ 1. Hence, the familys ͑0,1͔, z R N , j =0, ... ,n + 1, is bounded in L, as claimed. Lemma 2.9 together with Lemma 3.7 then give the following result.Corollary 3.8: We have ⌳ ␣,z B͑W a s,p , W a−⑀ r,p ͒ for any ␣ A k,ᐉ , z R N , r , s R, 1Ͻ p Ͻϱ, and ⑀ Ͼ 0. Moreover, we have that ʈ⌳ ␣,z ʈ W a,z q,p →W a−⑀,z r,p ഛ C q,r,p,a,⑀ e k⑀͗z−w͘ ...

The construction of a multidifferential complex, i.e. a differential complex with many differential operators, is presented. By a suitable choice of generators, the differentials become anticommuting and can generate a (co)homological resolution. A concrete construction of a n-differential complex is presented and its application to the BRST description of a gauge theory, with a special look at the Yang–Mills model, is considered.

Abstract. We analyze several types of soliton solutions to a family of Tzitzeica equations. To this end we use two methods for deriving the soliton solutions: the dressing method and Hirota method. The dressing method allows us to derive two types of soliton solutions. The first type corresponds to a set of 6 symmetrically situated discrete eigenvalues of the Lax operator L; to each soliton of the second type one relates a set of 12 discrete eigenvalues of L. We also outline how one can construct general N soliton solution containing N 1 solitons of first type and N 2 solitons of second type, N = N 1 + N 2 . The possible singularities of the solitons and the effects of change of variables that relate the different members of Tzitzeica family equations are briefly discussed. All equations allow quasi-regular as well as singular soliton solutions. MSC: 35Q51, 35Q53, 37K40

This paper proposes an algorithm for the Lie symmetries investigation in the case of a 2D Hamiltonian system. General Lie operators are deduced firstly and, in the the next step, the associated Lie invariants are derived. The 2D Yang-Mills mechanical model is chosen as a test model for this method.

The general BRST-anti-BRST construction in the framework of the antifieldantibracket formalism is illustrated in the case of the Freedmann-Townsend model.