Multitime sine-gordon solitons via geometric characteristics

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1 Multitime sine-gordon solitons via geometric characteristics Laura Gabriela Matei, Constantin Udrişte Abstract. Our paper introduces and studies the idea of multitime evolution in the context of solitons. Section 1 presents some historical data about solitons. Section defines the multitime sine-gordon PDE, using a fundamental tensor and a linear connection. Section 3 describes the multitime sine-gordon scalar solitons as special solutions of the multitime sine-gordon PDE. Section 4 proves the existence of the multitime sine-gordon biscalar solitons. Section 5 analyzes the geometric characteristics fundamental tensor, linear connection of the sine-gordon PDE, showing the existence of an infinity of Riemannian or semi-riemannian structures such that the new PDE is a prolongation of the sine-gordon PDE. The two-time sine-gordon geometric dynamics, which is presented here for the first time, shows that the sine-gordon soliton is generated. M.S.C. 010: 35Q51, 37K40, 53C1. Key words: sine-gordon PDE, multitime soliton, methods of Riemannian geometry. 1 Single-time solitons The term soliton was introduced in the 1960 s, but the idea to study the solitons had arisen in the 19th century, when Russell observed a large solitary wave in a canal near Edinburgh. Nowadays, many model PDEs of nonlinear phenomena are known to possess soliton solutions. Physically, solitons are very stable independent solitary waves which have finite energy, and which behave like particles. When they are located mutually far apart, each of them is approximately a traveling wave with constant shape and velocity. When two such solitary waves get closer, they interact, they gradually deform and finally merge into a single wave packet, a -soliton. However, this wave packet soon splits into two solitary waves with the same shape and velocity before collision. When a -soliton meet a 1-soliton, they merge into a 3-soliton and so on. This happens with the solutions of some nonlinear partial differential equations, which are named soliton equations. These PDEs include the sine-gordon equations, Balkan Journal of Geometry and Its Applications, Vol.16, No., 011, pp c Balkan Society of Geometers, Geometry Balkan Press 011.

2 8 Laura Gabriela Matei, Constantin Udrişte with some versions and related equations, and the Korteweg-de Vries equations, with related equations too, etc. Reformulating, the soliton is a solution of a non-linear evolution equation, which at every moment of time is localized in a bounded domain of space, such that the size of the domain remains bounded in time while the movement of the center of the domain can be interpreted as the movement of a particle. Issued from the geometry of surfaces of constant negative curvature and found later in the study of some physical phenomena, the single-time sine-gordon PDE is 1.1 u x 1 u c = sin u. t Multitime sine-gordon PDE Partial differential equations PDEs arise in many areas of science and technology. Usually these equations have several variables. Some of them are called spatial variables and are denoted with x i, i = 1,..., n or by a vector x = x i and others are called time variables, denoted with t α, α = 1,..., m or by a vector t = t α. There are PDEs that contain only spatial variables and there are PDEs that contain both spatial variables and time variables. A PDE is called a multitime partial differential equation if the time evolution parameter t is multidimensional. In the last decade the study of multitime PDEs has received considerable attention see [3]. This paper introduces and studies a multitime version for techniques, see also [1] - [10] of the single-time sine-gordon PDE see [1], [5], []. We search for some particular solutions of the new equation, or some information about the form of these solutions. We study also some geometric characteristics of the single-time sine-gordon PDE and of multitime sine-gordon PDE. A multitime t = t 1,..., t m is a point in the manifold R m. We endow the manifold R m with a symmetric linear connection Γ γ αβ and with a fundamental symmetric contravariant tensor field g = g αβ of constant signature r, z, s, r + z + s = m. Using a C function u : R R m R, we build the Hessian operator Hess Γ u αβ = t α t β u t γ, u and its trace, called ultra-parabolic-hyperbolic operator, Γ,g u = g αβ u t α t β u t γ, α, β, γ {1,..., m}. Define the multitime sine-gordon PDE as.1 u x x, t gαβ u t α t β x, t u x, t = sin ux, t. tγ 3 Multitime sine-gordon scalar solitons For the multitime sine-gordon PDE.1, we seek for solutions in the scalar form 3.1 ux, t = φx v α t α = φξ,

3 Multitime sine-gordon solitons via geometric characteristics 83 where v α, α {1,..., m}, is a constant vector and ξ = x v α t α. These solutions do not depend on x, t coordinates in explicit form. The partial derivatives of the unknown function ux, t are u x = φ ξ, u t α = φ ξ v α, u t α t β = φ ξv α v β. We substitute these derivatives in the PDE.1 and we obtain φ ξ[1 g αβ v α v β ] φ ξg αβ Γ γ αβ v γ = sin φξ. Let v α be a constant vector. We choose a constant fundamental tensor g αβ such that g αβ v α v β = V 1, and the constant connection Γ γ αβ = 0. With this choice, the PDE.1 becomes φ ξ1 V = sin φξ. We multiply both sides by φ ξ and write the equation in the form φ φ 1 1 V φ sin φ = 0. Taking the primitive, we find φ + cos φ 1 V = B, B R, that is φ = ± which is equivalent to dφ dξ = ± 1 V A cos φ, where A = B1 V. We integrate again, selecting A = 1. We get the next sequence of equivalences: dφ dφ ± = dξ ± 1 cos φ 1 V sin φ = ± ln tgφ 4 = ξ + a, a R. 1 V 1 V B1 V cos φ 1 V, Finally, we find two families of solutions, as follows: a ln tg φ 4 = ξ 1 V + a tg φ 4 = c e ξ 1 V, c R, which leads to ξ x v αt α 1 g φξ = 4 arctan c e 1 V, ux, t = 4 arctan c e αβ vαv β, c R. b ln tg φ 4 = ξ 1 V + a ctan φ 4 = c e φξ = 4arcctan c e ξ 1 V ξ 1 V,, ux, t = 4arcctan c R, which leads to c e dξ x vαtα 1 g αβ vαv β, c R. Thus, in the particular case g αβ = constant, Γ γ αβ = 0, we found two families of solutions of the equation.1, which corresponds to the multitime solitons and respectively to the multitime anti-solitons. Theorem 3.1. The scalar solutions of the PDE.1 split as family of multitime solitons and family of multitime anti-solitons.

4 84 Laura Gabriela Matei, Constantin Udrişte 4 Multitime sine-gordon biscalar solitons Let us consider again the multitime sine-gordon PDE.1. We search for solutions in the biscalar form 4.1 ux, t = 4 arctan φx. The partial derivatives are u t α = u x = 4φ x + φ x, 4φx + φ x t α, u xx = 4 φ x + φ x φ x φx + φ x, 4. u t α t β = 4φx + φ x t α tβ tα t β + φ x. After replacing the derivatives 4. and ranging the terms, the equation.1 becomes [ 4 + φ x φ x + g αβ ] φx t α t β t γ + φ x 8φx + φ x φ x + g αβ tα t β = 4.3 = sin 4 arctan φx. On the other hand, tg arctan α sin4 arctan α = 1 + tg arctan α = 1 + α 1 α α 1 α = 4α1 α 1 + α. In view of this, the right side of 4.3 takes the form sin 4 arctan φx 4 φx 1 φ x = = 4φx φ x 1 + φ x + φ x. The equation 4.3 becomes, after some simplifications [ φ x + g αβ ] φx t α t β t γ + φ x

5 Multitime sine-gordon solitons via geometric characteristics 85 φx φ x + g αβ tα t β = φx φ x. Dividing by φx, we obtain [ φ x φx + gαβ 1 ] t α t β t γ + φ x It follows that φ x g αβ tα t β = φ x. φ x φx + gαβ φ x t α t β t γ = [ = + g αβ ] tα t β gαβ t α t β t γ φ x + φ x φxφ x. Differentiating 4.4 with respect to x, the equation becomes φ x φx φ xφ x φ x + gαβ φxφ x t α t β t γ = = 0 + Ax. To differentiate with respect to t δ, δ = 1,..., m, we consider the particular case when g αβ = constant and Γ γ αβ = constant. We obtain the equation or, equivalently, φ x t δ φ xφ x φx g αβ φxφ x +g αβ φxφ x [ 3 t α t β t δ t δ φ x t δ t α t β t γ + 3 t α t β t δ t γ t δ = 0, t γ t δ t δ ] t α t β t γ g αβ φxφ x + φ xφx φ xφ x φ x = 0, tδ δ {1,..., m}.

6 86 Laura Gabriela Matei, Constantin Udrişte We divide the both sides by φxφ x t δ : φ x 1 φx φxφ x + t δ t α t β t γ g αβ 1 = 0. t δ It follows that 4.5 φ x φx φxφ x = gαβ t δ t α t β t γ t δ = 4k, k R, δ {1,..., m}. A function in x and a function in t can be equal if they are equal in fact with the same constant. Let this constant be 4k, for a suitable form of the relations. We also note that in 4.5 the function in x is the same for every δ {1,..., m} and from here it follows that the value of k is the same for every choice of δ. Thus, we obtain a system of equations g αβ t δ d dx φ x φx = 4kφxφ x t α t β t γ = 4k, δ {1,..., m}, tδ an ODE in the unknown φ = φx and m PDEs in the unknown =. Let us transform the ODE 4.6. After integrating the two sides, the equation becomes φ x φx = 4k φ x + a, a R, that is φ x = kφ 3 x+aφx. We multiply by φ x, and apply a new integration of the two sides of the equation. We find φ xφ x = kφ 3 xφ x + aφxφ x 4.8 φ x = kφ 4 x + aφ x + b, k, a, b R. Since always we can relate the constants k, a, b R via the condition a +4kb = 0, the equation 4.8 is a Riccati equation. Taking the curvilinear primitive, the PDE system 4.7 is transformed into the PDE g αβ t α t β t γ = 4k + c, c R,

7 Multitime sine-gordon solitons via geometric characteristics 87 that is 4.9 g αβ t α t β t γ = k 3 + c, c R. Theorem 4.1. The biscalar solutions of the multitime sine-gordon PDE are determined by the solutions of ODE 4.8 and PDEs Geometric characteristics of the sine-gordon PDE The geometric characteristics of some PDEs give relevant link between Differential Geometry and Applied Sciences, for geometrical methods in Statistics, for mathematical modeling in Ecology, for optimization methods on Riemannian or semi-riemannian manifolds etc see also [1] - [10]. 5.1 Case of single-time sine-gordon PDE Let g ij be a fundamental symmetric contravariant tensor field of constant signature 1, 1 and Γ i jk be a symmetric linear connection on the manifold R R. Denoting x 1 = x, x = t; i, j, k = 1,, and using a C function u : R R R, we introduce the hyperbolic PDE g ij u x i x j u Γk ij x k = sin u. Identifying this PDE to the single-time sine-gordon PDE 1.1, we obtain g 11 = 1, g 1 = g 1 = 0, g = 1 c, Γk 11 1 c Γk = 0. The Lorentzian connection Γ i jk = 0, associated to the previous Lorentzian metric g ij, is a particular case. 5. Two-time sine-gordon geometric dynamics Now let us show that the Lorentzian metric {g 11 = 1, g 1 = g 1 = 0, g = 1 c } connects the theory of geometric dynamics to the theory of solitons see [1]. For that we introduce the source as a Lorentzian manifold R R, g 11 = 1, g 1 = = g 1 = 0, g = 1 c and the target as the Riemannian manifold R, g = 1. Let u : R R R, x, t ux, t be a C function. On the Riemannian manifold R, g = 1, we introduce two vector fields X 1 u = sin u, X u = sin u, which determine the -flow 5.1 u x = sin u, u t = sin u.

8 88 Laura Gabriela Matei, Constantin Udrişte The complete integrability condition is the sine-gordon PDE u xt = sin u. The flow is characterized by ln tg u 4 = x + t + ln k, which yields the general solution 5. ux, t = 4 arctank e x+t. We build the Lagrangian L = u x sin u 1 c u t sin u. The Euler-Lagrange PDE generated by L is the sine-gordon PDE u x 1 u c t = 1 1 sin u. c This sine-gordon PDE is a prolongation of the -flow 5.1 since the function 5. is a solution. The theory in this subsection shows that the sine-gordon soliton PDE is generated by a -flow and an appropriate geometric structure. 5.3 Case of multitime sine-gordon PDE Let Γ γ αβ be a symmetric linear connection and g = gαβ be a fundamental symmetric contravariant tensor field of constant signature r, z, s, r +z +s = m on the manifold R m. If the tensor field g = g αβ is nondegenerate a Riemannian or semi-riemannian metric; z = 0 on R m and g αβ is it inverse, then the induced linear connection is Γ γ αβ = 1 gλγ gαλ t β + g βλ t α g αβ t λ Now we look for pairs g, Γ, that is fundamental tensor, connection, such that the solutions of multitime sine-gordon PDE.1 are just the solutions of the single-time PDE 1.1. Theorem 5.1. There exists an infinity of Riemannian or semi-riemannian structures g αβ on R m such that a solution of the PDE 1.1 is also a solution of the PDE.1. Proof. Suppose t 1 = t and u = ux, t 1 a solution of PDE 1.1. The function u = ux, t 1 is a solution of the PDE.1 if the family of Riemannian structures g αβ is fixed by g 11 = 1 c ; Γ γ 11 = 0, γ = 1,..., m. A family of solutions is g 11 = c ; g 1α = 0, for α = 1,..., m; g αβ = arbitrary for α, β. In the sense of the previous Theorem, we can say that the PDE.1 is a prolongation of the PDE Conclusion This paper gives solutions to a generalized sine-gordon PDE, different from the supersymmetric extension of the sine-gordon model. The general expression of this equation is determined by a geometric structure given by a fundamental tensor field g αβ and a linear connection Γ γ αβ. The geometry induced by some of these geometrical objects as well as vector solitons will be studied in a further coming paper. In fact, the integrable models and their multitime formulation find a natural and universal setting in terms of Differential Geometry..

9 Multitime sine-gordon solitons via geometric characteristics 89 Acknowledgements. Partially supported by University Politehnica of Bucharest, and by Academy of Romanian Scientists, Romania. Some ideas were presented at The International Conference of Differential Geometry and Dynamical Systems DGDS- 010, 5-8 August 010, University Politehnica of Bucharest, Bucharest, Romania. References [1] W. B. Cardosoa, A. T. Avelara, D. Bazeiab, M. S. Hussein, Solitons of two-component Bose-Einstein condensates modulated in space and time, arxiv.org/abs/ v [quant-ph], 19 Jan 010. [] A. Degasperis, S. V. Manakov, P. M. Santini, Initial-boundary value problems for linear and soliton PDEs, arxiv: nlin/005030v1, 14 May 00. [3] L. C. Evans, Entropy and Partial Differential Equations, math.berkeley.edu/ evans/entropy.and.pde.pdf, 010. [4] M. Pîrvan, C. Udrişte, Optimal control of electromagnetic energy, Balkan J. Geom. Appl. 15, 1 010, [5] W. Satsanit, A. Kananthai, Ultra-hyperbolic wave operator related to nonlinear wave equation, Int. J. Contemp. Math. Sciences, 5, 3 010, [6] C. Udrişte, Equivalence of multitime optimal control problems, Balkan J. Geom. Appl. 15, 1 010, [7] C. Udrişte, Multitime controllability, observability and bang-bang principle, Journal of Optimization Theory and Applications, 139, 1 008, [8] C. Udrişte, Nonholonomic approach of multitime maximum principle, Balkan J. Geom. Appl. 14, 009, [9] C. Udrişte, Simplified multitime maximum principle, Balkan J. Geom. Appl. 14, 1 009, [10] C. Udrişte, V. Arsinte, C. Cipu, Von Neumann analysis of linearized discrete Tzitzeica PDE, Balkan J. Geom. Appl. 15, 010, [11] C. Udrişte, O. Dogaru, I. Ţevy, D. Bala, Elementary work, Newton law and Euler-Lagrange equations, Balkan J. Geom. Appl. 15, 010, [1] C. Udrişte, M. Ferrara, D. Opriş, Economic Geometric Dynamics, Geometry Balkan Press, Bucharest, 004. [13] C. Udrişte, L. Matei, I. Duca, Multitime Hamilton-Jacobi Theory, Proc. of 8-th WSEAS Int. Conf. on Applied Comp. Sci. ACACOS-09, , Hangzhou, China, May 0-, 009; Proc. of the 9-th WSEAS Int. Conf. on Systems Theory and Sci. Comp. ISTASC-09, Moscow, Russia, August 0-, 009, [14] C. Udrişte, I. Ţevy, Multitime dynamic programming for curvilinear integral actions, Journal of Optimization Theory and Applications, , [15] C. Udrişte, I. Ţevy, Multitime linear-quadratic regulator problem based on curvilinear integral, Balkan J. Geom. Appl. 14, 009, Authors address: Laura Gabriela Matei, Constantin Udrişte University POLITEHNICA of Bucharest, Faculty of Applied Sciences, Department Mathematics-Informatics I, Splaiul Independentei 313, RO Bucharest, Romania. mateiglaura@yahoo.com ; udriste@mathem.pub.ro, anet.udri@yahoo.com