Group classification of Schr¨ odinger equations C´ elestin Kurujyibwami Department of Applied Mathematics, University of Rwanda Department of Mathematics, Link¨ oping University First Network Meeting for Sida- and ISP-funded PhD Students in Mathematics Stockholm 7–8 March 2017 1 / 13
Supervision Ass. Prof Peter Basarab-Horwath Prof Roman O. Popovych Main Supervisor Assistant Supervisor Link¨ oping University Institute of Maths NAS of Ukraine and W.P. Institute Vienna, Austria. 2 / 13
Duration and area of research Started: AUG 2012, End: AUG 2017. Support: ISP through EAUMP. Area of research: Symmetry analysis of differential equations, Lie algebras and group classification problems of differential equations. Thesis title: Group classification of Schr¨ odinger Equations 3 / 13
Outline Introduction Results, Impact and applications Conclusion 4 / 13
Introduction A symmetry of a differential equation (ordinary or partial) is an invertible transformation that maps the set of solutions of the equation to itself: if L ( u ) = 0 is a differential equation, where u stands for a solution of the equation, then we must have L (˜ u ) = 0 whenever L ( u ) = 0 , where ˜ u is the transform of u . A class of of differential equations is L| S := {L θ | θ ∈ S} , where S is the solution set of auxiliary system of differential equations S ( x , u ( p ) , θ ( q ′ ) ( x , u ( p ) )) = 0 and L θ is a system of differential equations L ( x , u ( p ) , θ ( q ) ( x , u ( p ) )) = 0 parameterized by the arbitrary elements θ ( x , u ( p ) ) = ( θ 1 ( x , u ( p ) ) , . . . , θ k ( x , u ( p ) )). 5 / 13
Introduction Cont. Example The class of (1+1)-D LinSchEqs i ψ t + ψ xx + V ( t , x ) ψ = 0 with its conjugate, where θ = ( V , V ∗ ) and S : V ψ = V ψ ∗ = V ψ t = V ψ ∗ t = V ψ x = V ψ ∗ x = 0 V ψ tx = V ψ ∗ tx = V ψ tt = V ψ ∗ tt = V ψ xx = V ψ ∗ xx = 0 , and the same equation for V ∗ . Definition The classification of Lie symmetry groups of systems from the class L| S up to the equivalence relations generated by point transformations between systems from the class is called group classification of the class L| S . 6 / 13
Research Motivation The group classification of the class V of linear Schr¨ odinger equations (SchEqs) with real-valued potentials, i ψ t + ∆ ψ + V ( t , x ) ψ = 0 was carried out. Therefore, this class with arbitrary complex-valued potentials remains an open problem. The study of the class S of nonlinear Schr¨ odinger equations, i ψ t + ∆ ψ + f ( ρ ) ψ + V ( t , x ) ψ = 0 , f ρ � = 0, where ρ = | ψ | was carried out in (1+1)D. Its group classification in multidimensional case remains attractive and open. The existing algebraic method for group classification (G.C) of differential equations was not efficient. So, there is need to construct the new version of this method and test its efficiency for G.C. 7 / 13
Problem statement Consider the class H of SchEqs of the general form i ψ t = H ( t , x , ψ, ψ ∗ , ψ a , ψ ∗ a , ψ aa , ψ ∗ aa ) with | H ψ aa | � = | H ψ ∗ aa | , where H is a complex-valued smooth function of its arguments and ψ is an unknown complex function of the real independent variables t and x . We study the point transformations between equations from the subclasses of the class H and then construct the new version of the algebraic method for G.C called uniform semi-normalization of classes of DE’s. We preliminary study Lie symmetry properties for the classes V and S with arbitrary complex potentials, and solve completely the G.C. of these classes for n = 1 and n = 2. 8 / 13
Results The Developed theory and its applications are summarized in the following papers Paper 1. Algebraic method for group classification of (1+1)D linear Schr¨ odinger equations with complex-valued potentials , 30 pp, Submitted. Paper 2. Equivalence groupoid for (1 + 2) D linear Schr¨ odinger equations with complex potentials , 8 pp, published. Paper 3. Group classification of Multidimensional linear Schr¨ odinger equations with algebraic method , 27 pp, ready for submission. Paper 4. Group classification of multidimensional nonlinear Schr¨ odinger equations , 30 pp, final stage Paper 5. Admissible transformations of (1+1)D Schr¨ odinger equations with variable mass , 13 pp, final stage. 9 / 13
example The G.C for the class of (1+1) LinScHEqs, i ψ t + ψ xx + V ( t , x ) ψ = 0 results in 6 inequivalent families of potentials together with their corresponding Lie symmetry extensions. The famous is the free Schr¨ odinger equation, V = 0 whose Lie symmetry extension is spanned by the vector fields t ∂ t + 1 M , I , ∂ t , ∂ x , 2 x ∂ x , t ∂ x + 1 t 2 ∂ t + tx ∂ x + 1 4 x 2 M − 1 2 xM , 2 tI , where M = i ψ∂ ψ − i ψ ∗ ∂ ψ ∗ and I = ψ∂ ψ + ψ ∗ ∂ ψ ∗ , respectively. 10 / 13
Applications and Impact We developed a new method of the algebraic method ”uniform semi-normalization ” which can be applied to various classes of DE’s not only for Schr¨ odinger equations. The knowledge of point transformations connecting equations from systems of the class can be used to select most relevant models from parameterized families of models. These equations can be used in the field of Physics where Schr¨ odinger equations arise. For Lie symmetries, they can be used to construct invariant reductions for various models: mathematical, biological, . . . . A scientific community will benefit from the new theory based on uniformly semi-normalized of classes of differential equations to solve other problems. The country and the continent in general will benefit from my expertise. I will introduce this new field through teaching, training, seminars, workshops and conferences in the region. It will be the same also for some new features(eg. Latex). 11 / 13
Conclusion We constructed a new version of the algebraic method of group classification which is based on uniformly semi-normalization of classes of differential equations. This is the effective method that can be applied to various classes of differential equations not only Schr¨ odinger equations.. We solved the problem of group classification of linear and nonlinear SchEqs with constant mass. We extended results from real potentials to complex potentials and found missed cases in existing incomplete results. The final lists provide inequivalent families of complex potentials and corresponding Lie symmetry extensions which can be potentially used in many areas of Physics. 12 / 13
Tack s˚ a mycket! Thank you for your attention! 13 / 13
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