On a three-dimensional Riccati differential equation and its symmetries
aa r X i v : . [ m a t h - ph ] J un On a three-dimensional Riccatidifferential equation and its symmetries
Charles Papillon and S´ebastien TremblayD´epartement de math´ematiques et d’informatiqueUniversit´e du Qu´ebec, Trois-Rivi`eres, Qu´ebec, G9A 5H7, CanadaNovember 13, 2018
Abstract
A three-dimensional Riccati differential equation of complex quaternion-valued functions is studied. Many properties similar to those of the ordi-nary differential Riccati equation such that linearization and Picard theo-rem are obtained. Lie point symmetries of the quaternionic Riccati equa-tion are calculated as well as the form of the associated three-dimensionalpotential of the Schr¨odinger equation. Using symmetry reductions and re-lations between the three-dimensional Riccati and the Schr¨odinger equa-tion, examples are given to obtain solutions of both equations.
Keywords : Spatial Riccati equation, Complex quaternions (biquater-nions), Lie point symmetries, Schr¨odinger operatorPACS 2010: 02.30.Jr, 02.10.Hh, 02.20.Sv
The Riccati nonlinear ordinary differential equation [8, 12] is given by y ′ ( x ) = p ( x ) + p ( x ) y ( x ) + p ( x ) y ( x ) , p ( x ) = 0 , (1)where the prime represents the derivative, plays a very important role in math-ematical physics. This equation is the simplest nonlinear differential equationwhich being linearizable can be completely solved. Indeed, a classical resultfor this equation comes from the fact that this equation can be reduced to asecond-order linear ordinary differential equation [12] u ′′ − (cid:18) p + p ′ p (cid:19) u ′ + p p u = 0 , (2)where the solution of this equation will lead to a solution y = − u ′ / ( p u ) ofthe Riccati equation (1). Conversely by using the Cole-Hopf transformation y = u ′ /u , for which a generalization is used in this article, the equation u ′′ + P ( x ) u ′ + Q ( x ) u = 01s transformed into the Riccati equation y ′ = − Q ( x ) − P ( x ) y − y . The theoryof the Riccati equation is therefore equivalent to the theory of the homoge-neous linear equation of the second-order. An interesting particular case is thehomogeneous one-dimensional Schr¨odinger equation − u ′′ + q ( x ) u = 0 , (3)related to the Riccati equation y ′ + y = q ( x ) (4)using the Cole-Hopf transformation.Moreover, the Schr¨odinger operator in (3) can be factorized in the form − d dx + q ( x ) = − (cid:18) ddx + y ( x ) (cid:19) (cid:18) ddx − y ( x ) (cid:19) (5)if and only if (4) holds. This observation is the key for the original Darbouxtransformation [7, 18] for the one-dimensional Schr¨odinger equation. The Ric-cati equation (4) is the equation that we consider for generalization using com-plex quaternions in this paper.The general solution of the Riccati equation can be obtained in differentways, depending on the number of particular solutions which are known. Fromthe first Euler theorem’s we know that given one particular solution y of theRiccati equation (1) the general solution y may be obtained using the trans-formation u = 1 / ( y − y ). This yields equation (1) to the linear first-ordernonhomogeneous equation u ′ + (2 y p + p ) u + p = 0 (6)and a set of solutions to the Riccati equation is then given by y = y + 1 /u .Thus given a particular solution of the Riccati equation, it can be linearized andthe general solution can be found in two integrations. From the second Eulertheorem’s we know that giving two particular solutions y , y of the Riccatiequation (1) the general solution can be found in one integration, i.e. y = cy exp (cid:2) R p ( y − y ) dx (cid:3) − y c exp (cid:2) R p ( y − y ) dx (cid:3) − , (7)where c is a constant. Finally, as was shown by S. Lie [17], the Riccati equationis the only ordinary nonlinear differential equation of first-order which possessesa (nonlinear) superposition formula. For any three solutions y , y , y of the Ric-cati equation (1), the general solution y can be obtained without any integrationas y = y ( y − y ) + ky ( y − y ) y − y + k ( y − y ) , (8)where k is a constant. Now from Picard’s theorem [22], given a fourth particularsolution y , we have ( y − y )( y − y )( y − y )( y − y ) = k. (9)2ifferentiating (9) and dividing the result by ( y − y )( y − y )( y − y )( y − y ),see [13], the Picard’s theorem is equivalent to( y − y ) ′ y − y + ( y − y ) ′ y − y − ( y − y ) ′ y − y − ( y − y ) ′ y − y = 0 . (10)Some generalizations of the Riccati equation have been considered in theliterature. In [13] authors have studied a complex Riccati equation related to thestationnary Schr¨odinger equation. This generalized Riccati equation possessesmany interesting properties similar to its one-dimensional prototype and appearsfor instance in the two-dimensional SUSY quantum mechanics [5]. Moreover,up to change of signs, the spatial Riccati equation that we consider in thispaper was studied in [3, 14, 15] where many interesting results was obtained.In this paper additional results similar to the properties of the (classical) one-dimensional Riccati equation are found: among others generalizations of thefirst Euler theorem’s as well as the Picard’s theorem are obtained. Moreover,the Lie point symmetries of the generalized equation are found and used toobtain solutions of the Riccati and the Schr¨odinger equation in the space.For the reasons mentioned above, Riccati equation (1) appears naturally inquantum physics, but this equation also appears in statistical thermodynamicsas well as in cosmology [11, 20, 21]. In the fields of applied mathematics, thisequation appears in many instances when we can find exact solutions of nonlin-ear partial differential equations. Indeed, it plays a very important role in thesolution of integrable nonlinear partial differential equations. These equationsare characterized by being the compatibility conditions between two linear par-tial differential equations (the Lax pair) for an auxiliary function, the so calledwave function [1, 6]. It is also the only first-order nonlinear ordinary differentialequation which possesses the Painlev´e property [12], i.e. which has no movablesingularity. The algebra of complex quaternions, also called biquaternions, denoted by H ( C )consists of elements of the form x = X ℓ =0 x ℓ e ℓ , (11)for x ℓ ∈ C and the basis elements e ℓ satisfying the following commutativity andmultiplication rules e e ℓ = e ℓ e , e p e q = − δ pq + ε pqr e r , ie ℓ = e ℓ i, ℓ = 0 , , , , p, q, r = 1 , , δ pq is the Kronecker delta, ε pqr is the Levi-Civita symbol (i.e. ε pqr is 1if ( pqr ) is an even permutation or (123), − i is the complex imaginary element. It is useful to3epresent an element x ∈ H ( C ) as being the sum of a scalar and a vector part,i.e. x = x + x , where x := Sc x and x := Vec x = X j =1 x j e j . (13)The conjugate of an element x = x + x ∈ H ( C ) is defined by x := x − x andthe biquaternionic modulus is given by | x | := xx = P ℓ =0 x ℓ ∈ C .The multiplication of two elements x, y ∈ H ( C ) can be expressed in the form xy = ( x + x )( y + y ) = x y + x y + y x − h x , y i + x × y , (14)where h x , y i and x × y represent the standard inner and cross products in C ,respectively.By M x we denote the operator of multiplication by a complex quaternion x from the right-hand side, i.e. M x y = yx, for all y ∈ H ( C ) . (15)In what follows and for the rest of the paper we suppose that Ω is a subdo-main of R . The Dirac operator D , also called the Moisil-Theodoresco operator,is defined as [9] Dϕ := X k =1 e k ∂ k ϕ, (16)where ∂ k := ∂∂x k and ϕ is a complex quaternionic valued-function ϕ ∈ C (Ω , H ( C )).By a straightforward calculation, we find Dϕ = − div ϕ + grad ϕ + rot ϕ . (17)A right Dirac operator D r can also be defined as D r ϕ := P k =1 ∂ k ϕ e k andis represented by D r ϕ = − div ϕ + grad ϕ − rot ϕ . Defining the quaternionicconjugate operator C H as C H x = x for all x ∈ H ( C ), the left and the right Diracoperators are related by C H D = − D r C H . Moreover, by direct calculations wefind that D ϕ = − ∆ ϕ (18)and obtain the following quaternionic Leibniz rule [10] D [ ϕ ψ ] = D [ ϕ ] ψ + ϕD [ ψ ] − X k =1 ϕ k ∂ k ψ (19)for ϕ, ψ ∈ C (Ω , H ( C )). In particular, we observe that if Vec ϕ = , i.e. ϕ = ϕ ,then D [ ϕ ψ ] = D [ ϕ ] ψ + ϕ D [ ψ ].Given a known function ψ = P k =1 ψ k e k ∈ C (Ω , Vec H ( C )), we considerthe linear partial differential equationgrad ϕ = ψ , (20)4here rot ψ = (the compatibility condition) and the unknown function ϕ is in C (Ω , C ). The complex-valued scalar function ϕ is then said to be the potentialof the vectorial function ψ . It is well known that we can reconstruct ϕ , up toan arbitrary complex constant C , in the following way A [ ψ ]( x, y, z ) = Z xx ψ ( ξ, y , z ) dξ + Z yy ψ ( x, η, z ) dη + Z zz ψ ( x, y, ζ ) dζ + C, (21)where ( x , y , z ) is an arbitrary point in the domain of interest Ω [16]. Let us consider the homogeneous Schr¨odinger equation (cid:0) − ∆ + q ( x, y, z ) (cid:1) ψ = 0 , (22)where ∆ is the three-dimensional laplacian, q, ψ are complex valued-functionssuch that q ∈ C (Ω , C ) and ψ ∈ C (Ω , C ). Let us also consider the followingnonlinear partial differential equation D Q + | Q | = q ( x, y, z ) , (23)where Q is a vectorial complex quaternionic functions in C (Ω , Vec H ( C )) and | Q | = − Q = P j =1 Q j . We call equation (23) the three-dimensional Riccatiequation or the (bi)quaternionic Riccati equation for Q in the (bi)quaternionsand q a (complex)real-valued function. Decomposing this equation into scalarand vector parts, we find − div Q + | Q | = q, rot Q = . (24)In [15] authors have shown a natural counterpart of the relation mentionedin the introduction, between the one-dimensional Schr¨odinger equation (3) andthe Riccati equation (4): Theorem 1 [15] The complex-valued function ψ is a solution of the Schr¨odingerequation (22) if and only if Q = − Dψψ is a solution of the Riccati equation (23).
Theorem 2
The complex quaternionic function Q is a solution of the Riccatiequation (23) if and only if ψ = exp ( −A [ Q ]) is a solution of the Schr¨odingerequation (22), where A [ Q ] is the scalar function defined by (21). Proof.
For a given solution Q of (23), we have D e −A [ Q ] = − D (cid:16) D ( A [ Q ]) e −A [ Q ] (cid:17) = − D (cid:16) Q e −A [ Q ] (cid:17) = e −A [ Q ] (cid:16) − D Q + Q (cid:17) = − e −A [ Q ] (cid:16) D Q + | Q | (cid:17) . − ∆ + q ) e −A [ Q ] = ( D + q ) e −A [ Q ] and q = D Q + | Q | we find thedesired result.The last two theorems represent direct generalization of the relation betweenthe one-dimensional Schr¨odinger equation (3) and Riccati equation (4) using theCole-Hopf transformation.A factorization of the Schr¨odinger operator can be written in the followingform [2, 4] (cid:0) − ∆ + q ( x, y, z ) (cid:1) ψ = (cid:0) D + M Q (cid:1)(cid:0) D − Q C H (cid:1) ψ = (cid:0) D r + Q (cid:1)(cid:0) D r − M Q C H (cid:1) ψ (25)if and only if Q is a solution of the Riccati equation (23). Here the operators C H , which do not act on the (scalar) complex function ψ , are inserted forconvenience in what follows. The second factorization obtained in (25) consistsof the application of the operator C H on the first factorization.This last result generalizing factorization (5), combining with theorems 1and 2, suggest that (23) can be considered a good generalisation of (4).Considering now the first factorization in (25) given in terms of two linearfirst-order partial differential operators, we obtain the following two proposi-tions. Proposition 3 [16]
Let ϕ be a non-vanishing solution of the Schr¨odinger equa-tion (22) . If W = W + W ∈ C (Ω , H ( C )) is a solution of the complex quater-nionic equation (cid:16) D − Dϕϕ C H (cid:17) W = 0 , (26) then W is a solution of the same Schr¨odinger equation (22) . Moreover, thecomponents of W satisfy the equations div (cid:20) ϕ grad (cid:18) W ϕ (cid:19)(cid:21) = 0 (27) and rot (cid:2) ϕ − rot ( ϕ W ) (cid:3) = . (28)Using theorem 1, an immediate consequence of this last proposition is that Q = − DW W is solution of the Riccati equation (23) for a nonvanishing complex-valued function W in Ω.For a given purely vectorial function F ( x ) defined for all x = ( x , x , x ) ∈ Ω,we define B [ F ]( x ) by B [ F ]( x ) := 14 π Z Ω F ( y ) | x − y | d Ω . roposition 4 Let ϕ be a non-vanishing solution of the Schr¨odinger equation (22) . If w ∈ C (Ω , Vec H ( C )) is a purely vectorial solution of the complexquaternionic equation (cid:16) D + M Dϕϕ (cid:17) w = 0 , (29) then a solution of equation (26) is given by W = 12 (cid:18) ϕ A (cid:20) w ϕ (cid:21) − ϕ − rot (cid:0) B [ ϕ w ] (cid:1) + grad hϕ (cid:19) , (30) where h is an arbitrary harmonic function in Ω . Moreover, a solution of theSchr¨odinger equation (22) is given by W = 12 ϕ A (cid:20) w ϕ (cid:21) and a solution of theRiccati equation (23) is given by Q = − ϕ − (cid:16) grad ϕ + w A h w ϕ i (cid:17) . (31) Proof.
The solution W given by equation (30) was obtained in [16] (see theorem9) and, using theorem 1, the solution W of the Schr¨dinger equation (22) is thescalar part of W . Finally for a non-vanishing scalar function A h w ϕ i , usingtheorem 2, we find Q = − D (cid:16) ϕ A h w ϕ i(cid:17) ϕ A h w ϕ i = − A h w ϕ i grad ϕ + w ϕ A h w ϕ i which is the desired result (31).The following proposition will be usefull in the demonstration of the nexttheorem. Proposition 5 [16]
Let W be a scalar solution of the Schr¨odinger equation (22) with q = ∆ ϕϕ in Ω . Then the vector function W satisfying (28) , such that W = W + W is a solution of (26) , is constructed according to the formula W = − ϕ − (cid:26) rot (cid:18) B (cid:20) ϕ grad (cid:18) W ϕ (cid:19)(cid:21)(cid:19) + grad h (cid:27) , (32) where h is an arbitrary harmonic function.Given a solution W of (28) , the corresponding solution W of the Schr¨odingerequation (22) with q = ∆ ϕϕ such that W = W + W is a solution of (26) is con-structed as follows W = − ϕ A (cid:2) ϕ − rot( ϕ W ) (cid:3) . (33)We are now able to consider generalizations of the first Euler’s theorem aswell as the Picard’s theorem (or its equivalent form (10)).7 heorem 6 Let Q be a bounded particular solution of the Riccati equation (23) . Then the Riccati equation (23) is reduced to the following first-order equa-tion DW = − Q W (34) in the following sense. Any solution of the Riccati equation (23) has the form Q = − D (Sc W )Sc W (35) and vice-versa; any solution of equation (34) can be expressed via a correspond-ing solution Q of (23) as follows W = e −A [ Q ] − e A [ Q ] n rot (cid:16) B h e − A [ Q ] grad( e −A [ Q − Q ] ) i(cid:17) + grad h o , (36) where h is an arbitrary harmonic function. Proof.
Let Q be a bounded solution of (23). By using theorem 2 we have that Dϕϕ = − Q , where ϕ is a nonvanishing solution of the Schr¨odinger equation (22)with q = ∆ ϕϕ . Now equation (34) can be rewritten as (26). In the same waywe obtain Q = − Dψψ , where ψ is a solution of the Schr¨odinger equation (22).Using proposition 5, the function ψ is the scalar part of a solution W of (26),i.e. ψ = Sc W . The first part of the theorem is therefore shown.Let us suppose now that W = W + W is a solution of (34). Again, we havethat Dϕϕ = − Q . According to proposition 3, W is a solution of the Schr¨odingerequation (22) with q = ∆ ϕϕ and Q = − DW W is a solution of the Riccati equation(23) by theorem 1. Using proposition 5 we obtain W = W − ϕ − (cid:26) rot (cid:18) B (cid:20) ϕ grad (cid:18) W ϕ (cid:19)(cid:21)(cid:19) + grad h (cid:27) . (37)From theorem 2, it is now sufficient to make the substitutions ϕ = e −A [ Q ] and W = e −A [ Q ] to obtain equation (36). Theorem 7
Let Q k , k = 1 , , , , be four solutions of the Riccati equation (23) . Then we have D ( Q − Q ) − Q × Q ) Q − Q + D ( Q − Q ) − Q × Q ) Q − Q − D ( Q − Q ) − Q × Q ) Q − Q − D ( Q − Q ) − Q × Q ) Q − Q = . (38) Proof.
From the trivial equation C H h ( Q + Q ) + ( Q + Q ) − ( Q + Q ) − ( Q + Q ) i = , (39)8et us consider the first term: C H ( Q + Q ) = − ( Q + Q )( Q − Q ) Q − Q = | Q | + Q Q − Q Q − | Q | Q − Q = ( q − | Q | ) − ( q − | Q | ) + Q Q − Q Q Q − Q = D Q − D Q + Q Q − Q Q Q − Q = − D ( Q − Q ) − Q × Q ) Q − Q . Doing similar calculus for the other terms, we obtain the desired result.In this last proof, we see that the cross product terms in (38) are a conse-quence of the noncommutativity of (complex) quaternions. The analogue partin the classical Picard’s theorem are obviously null.
The purpose to find Lie point symmetries of a system of PDE’s is to obtain alocal group of transformations that maps every solution of the same system. Inother words, it map the solution set of the system to itself.Let us consider a general system of differential equations E i ( x, u, u (1) , . . . , u ( n ) ) = 0 ,x ∈ R p , u ∈ R q , i = 1 , . . . , m, p, q, m, n ∈ Z > , (40)where x and u are independent and dependent variables, respectively, and u ( k ) denotes all partial derivatives of order k of all components of u . We are nowlooking to obtain the local Lie group G of local Lie point transformations e x = Λ λ ( x, u ) , e u = Ω λ ( x, u ) , (41)taking solutions of (40) into solutions, where λ are the group parameters. Thefunctions Λ λ , Ω λ are well defined for λ close to the identity, λ = e , and for x and u close to some origin in the space M ⊂ X × U ( X ∼ R p , U ∼ R q ) ofindependent and dependent variables. In other words, the group G acts on amanifold M such a manner that whenever u = f ( x ) is a solution of (40), thenso is e u ( e x ) = λ · f ( x ).Following S. Lie, as presented e.g. in [19], we shall look for infinitesimalgroup transformations, i.e. construct the Lie algebra L of the Lie group G . Forthe purpose of the paper, we shall present the algorithm in a purely utilitarian9anner. For the proofs we refer to [19]. The Lie algebra L will be realized interms of the vector fields on M , i.e. differential operators b v = p X i =1 ξ i ( x, u ) ∂ x i + q X α =1 φ α ( x, u ) ∂ u α , (42)where ∂ x i := ∂∂x i and ∂ u α := ∂∂u α . The functions ξ i and φ α are to be determined.The algorithm for determining the Lie algebra L of the symmetry group G ofthe system (40) is given bypr ( n ) b v · E i (cid:12)(cid:12)(cid:12) E j =0 = 0 , i, j = 1 , . . . , m, (43)where the n -th prolongation of the vector fields (42) is given bypr ( n ) b v = b v + q X α =1 X J φ Jα ∂∂u αJ J ≡ J ( k ) = ( j , . . . , j k ) , ≤ j k ≤ p, k = j + · · · + j k . (44)The prolongation is defined on the corresponding jet space M ( n ) ⊂ X × U ( n ) .Here the coefficients φ Jα depend on x, u and the derivatives of u up to order k for J = J ( k ). These coefficients are obtained by the following formula [19] φ Jα = D J φ α + p X i =1 ξ i u αi ! + p X i =1 ξ i u αJ,i , (45)where D J is the total derivative operator, u αi := ∂u α ∂x i and u αJ,i := ∂u αJ ∂x i .Condition (43) provides a system of linear ordinary differential equationsof order n for the coefficients ξ i and φ α . The determining equations for thesymmetry operator b v are obtained by setting equal to zero the coefficients ofeach linearly independent expressions in the derivative of the u α . Each vectorfield (42) provides a one parameter Lie group G ( λ ) obtained by integrating thesystem d e x i dλ = ξ i ( e x, e u ) , e x i (cid:12)(cid:12)(cid:12) λ =0 = x i , d e u α dλ = φ α ( e x, e u ) , e u α (cid:12)(cid:12)(cid:12) λ =0 = u α . (46) We now apply the Lie symmetry tool, presented in the last subsection, for thecase of the quaternionic Riccati equation (23). Decomposing explicitly equation(23) in terms of its components, where for convenience we set Q := ue + ve + we for u, v, w, q ∈ C (Ω , R ), the system (24) is then equivalent to E ≡ − ( ∂ u + ∂ v + ∂ w ) + ( u + v + w ) − q = 0 , (47a) E ≡ ∂ w − ∂ v = 0 , (47b) E ≡ ∂ u − ∂ w = 0 , (47c) E ≡ ∂ v − ∂ u = 0 . (47d)10or this system of equations the vector fields (42) can be expressed as b v = ξ∂ x + η∂ y + τ ∂ z + φ∂ u + ψ∂ v + ζ∂ w , (48)where the real-valued coefficient functions ξ, . . . , ζ depend on the three indepen-dent variables x, y, z and the three dependent variables u, v, w . Considering nowthe algorithm given by (43), i.e. applying the vector fields (48) on the systemof equations (47a),...,(47d) on the solution set of the Riccati equation (23), wefind a system of 37 determining equations (the calculation was checked using PDEtools in Maple 16). We find that the generator b v must actually have theform b v = h a (cid:0) x − ( y + z ) (cid:1) + 2 a xy + 2 a xz − a z + a x − a y + a i ∂ x + h a xy + a (cid:0) y − ( x + z ) (cid:1) + 2 a yz + a y + a x − a z + a i ∂ y + h a xz + 2 a yz + a (cid:0) z − ( x + y ) (cid:1) + a x + a z + a y + a i ∂ z + h a (cid:0) − xu + yv + zw ) (cid:1) + 2 a ( xv − yu ) + 2 a ( xw − zu ) − a w − a u − a v i ∂ u + h a ( yu − xv ) + a (cid:0) − xu + yv + zw ) (cid:1) − a ( zv − yw ) − a v + a u − a w i ∂ v + h a ( zu − xw )+2 a ( zv − yw ) + a (cid:0) − xu + yv + zw ) (cid:1) + a u − a w + a v i ∂ w , (49)for some real parameters a , . . . , a . The potential q being an arbitrary real-valued function is subject to one further determining equation that involve q explicitly: h a ( y + z − x ) − a xy − a xz + a z − a x + a y − a i q x + h − a xy + a ( x + z − y ) − a yz − a y − a x + a z − a i q y + h − a xz − a yz + a ( x + y − z ) − a x − a z − a y − a i q z − h a + 2( a x + a y + a z ) i q = 0 . (50)For each non-zero parameters a k , k = 1 , . . . ,
10, a Lie point symmetry of theRiccati equation (23) exists if and only if a solution of (50) is obtained. Hence,solving equation (50) for each parameter a k , i.e. obtaining the form of thepotential function q , the associate generator b v is given from (49). For con-venience in what follows we define x := ( x, y, z ), Q ( x ) := ( u ( x ) , v ( x ) , w ( x )), r := p x + y + z and the function c ( x , y ) := 1 − h x , y i . We present theresults in the following table: 11 ector fields Potentials q ( x, y, z ) b v = ∂ x q = F ( y, z ) b v = ∂ y q = F ( x, z ) b v = ∂ z q = F ( x, y ) b v = y∂ z − z∂ y + v∂ w − w∂ v q = F (cid:0) x, p y + z (cid:1)b v = z∂ x − x∂ z + w∂ u − u∂ w q = F (cid:0) y, √ x + z (cid:1)b v = x∂ y − y∂ x + u∂ v − v∂ u q = F (cid:0) z, p x + y (cid:1)b v = x∂ x + y∂ y + z∂ z − u∂ u − v∂ v − w∂ w q = x − F (cid:16) yx , zx (cid:17)b v = [ x − ( y + z )] ∂ x + 2 xy∂ y + 2 xz∂ z q = r − F (cid:16) yr , zr (cid:17) + c (cid:0) x , Q ( x ) (cid:1) ∂ u + 2( yu − xv ) ∂ v + 2( zu − xw ) ∂ w b v = 2 xy∂ x + [ y − ( x + z )] ∂ y + 2 yz∂ z q = r − F (cid:16) xr , zr (cid:17) +2( xv − yu ) ∂ u + c (cid:0) x , Q ( x ) (cid:1) ∂ v + 2( zv − yw ) ∂ w b v = 2 xz∂ x + 2 yz∂ y + [ z − ( x + y )] ∂ z q = r − F (cid:16) xr , yr (cid:17) +2( xw − zu ) ∂ u + 2( yw − zv ) ∂ v + c (cid:0) x , Q ( x ) (cid:1) ∂ w Table 1
Generators b v , . . . , b v represent translations in the space of independent vari-ables. The generators b v , . . . , b v represent simultaneous rotations in the spaceof independent and dependent variables and b v represents dilations. Finally, thegenerators b v , . . . , b v represent conical symmetries.In what follows, we define the function α ( x, r, λ ) := r λ − xλ + 1 and x ⊤ represents the column vector of x . In particular, we note that α ( x, r, λ )is a quadratic polynomial in λ without real roots except on the x -axis. Theone-parameter groups G k generated by the b v k are obtained using (46).12 : e x = x + λ e , e Q = Q , e := (1 , , , G : e x = x + λ e , e Q = Q , e := (0 , , , G : e x = x + λ e , e Q = Q , e := (0 , , , G : e x ⊤ = R ( λ ) x ⊤ , e Q ⊤ = R ( λ ) Q ⊤ , R ( λ ) := λ sin λ − sin λ cos λ , G : e x ⊤ = R ( λ ) x ⊤ , e Q ⊤ = R ( λ ) Q ⊤ , R ( λ ) := cos λ − sin λ λ λ , G : e x ⊤ = R ( λ ) x ⊤ , e Q ⊤ = R ( λ ) Q ⊤ , R ( λ ) := cos λ sin λ − sin λ cos λ
00 0 1 , G : e x = e λ x , e Q = e − λ Q , G : e x = x − λr e α ( x, r, λ ) , e Q ⊤ = u + c ( x , Q ) λ − (cid:0) r u + c ( x , Q ) x (cid:1) λ α ( x, r, λ ) v + y (cid:0) uλ + c ( x , Q ) λ (cid:1) α ( x, r, λ ) w + z (cid:0) uλ + c ( x , Q ) λ (cid:1) , y + z = 0 , e x = (cid:0) x − xλ , , (cid:1) , e Q ⊤ = (1 − xλ ) (cid:2) u + λ (1 − xu )] v (1 − xλ ) w (1 − xλ ) , y + z = 0 , G : e x = x − λr e α ( y, r, λ ) , e Q ⊤ = α ( y, r, λ ) u + x (cid:0) vλ + c ( x , Q ) λ (cid:1) v + c ( x , Q ) λ − (cid:0) r v + c ( x , Q ) y (cid:1) λ α ( y, r, λ ) w + z (cid:0) vλ + c ( x , Q ) λ (cid:1) , x + z = 0 , e x = (cid:0) , y − yλ , (cid:1) , e Q ⊤ = u (1 − yλ ) (1 − yλ ) (cid:2) v + λ (1 − yv )] w (1 − yλ ) , x + z = 0 , G : e x = x − λr e α ( z, r, λ ) , e Q ⊤ = α ( z, r, λ ) u + x (cid:0) wλ + c ( x , Q ) λ (cid:1) α ( z, r, λ ) v + y (cid:0) wλ + c ( x , Q ) λ (cid:1) w + c ( x , Q ) λ − (cid:0) r w + c ( x , Q ) z (cid:1) λ , x + y = 0 , e x = (cid:0) , , z − zλ (cid:1) , e Q ⊤ = u (1 − zλ ) v (1 − zλ ) (1 − zλ ) (cid:2) w + λ (1 − zw )] , x + y = 0 . Since each group G k is a symmetry group associated with a potential q written in term of F k , if Q ( x ) = (cid:16) f ( x ) , g ( x ) , h ( x ) (cid:17) is a solution of (23) so arethe functions 13 (1) = (cid:16) f ( x − λ e ) , g ( x − λ e ) , h ( x − λ e ) (cid:17) , Q (2) = (cid:16) f ( x − λ e ) , g ( x − λ e ) , h ( x − λ e ) (cid:17) , Q (3) = (cid:16) f ( x − λ e ) , g ( x − λ e ) , h ( x − λ e ) (cid:17) , Q (4) = R ( λ ) (cid:16) f ( R ⊤ x ) , g ( R ⊤ x ) , h ( R ⊤ x ) (cid:17) ⊤ , Q (5) = R ( λ ) (cid:16) f ( R ⊤ x ) , g ( R ⊤ x ) , h ( R ⊤ x ) (cid:17) ⊤ , Q (6) = R ( λ ) (cid:16) f ( R ⊤ x ) , g ( R ⊤ x ) , h ( R ⊤ x ) (cid:17) ⊤ , Q (7) = e − λ (cid:16) f ( e − λ x ) , g ( e − λ x ) , h ( e − λ x ) (cid:17) , Q (8 a ) = (cid:16) f ( x + r λ e α ( − x,r,λ ) ) + c (cid:0) x , Q ( x + r λ e α ( − x,r,λ ) ) (cid:1) ( λ − xλ ) − r λ f (cid:0) x + r λ e α ( − x,r,λ ) (cid:1) ,α ( x, r, λ ) g ( x + r λ e α ( − x,r,λ ) ) + y h f ( x + r λ e α ( − z,r,λ ) ) λ + c (cid:0) x , Q ( x + r λ e α ( − x,r,λ ) ) (cid:1) λ i ,α ( x, r, λ ) h ( x + r λ e α ( − x,r,λ ) ) + z h f ( x + r λ e α ( − x,r,λ ) ) λ + c (cid:0) x , Q ( x + r λ e α ( − x,r,λ ) ) (cid:1) λ i(cid:17) , for y + z = 0 , Q (8 b ) = (cid:16) (1 − xλ ) f (0 , , x xλ ) + (1 − xλ ) λ, (1 − xλ ) g (0 , , x xλ ) , (1 − xλ ) h (0 , , x xλ ) (cid:17) , for y + z = 0 Q (9 a ) = (cid:16) α ( y, r, λ ) f ( x + r λ e α ( − y,r,λ ) ) + x h g ( x + r λ e α ( − y,r,λ ) ) λ + c (cid:0) x , Q ( x + r λ e α ( − y,r,λ ) ) (cid:1) λ i ,g ( x + r λ e α ( − y,r,λ ) ) + c (cid:0) x , Q ( x + r λ e α ( − y,r,λ ) ) (cid:1) ( λ − yλ ) − r λ g (cid:0) x + r λ e α ( − y,r,λ ) (cid:1) ,α ( y, r, λ ) h ( x + r λ e α ( − y,r,λ ) ) + z h g ( x + r λ e α ( − y,r,λ ) ) λ + c (cid:0) x , Q ( x + r λ e α ( − y,r,λ ) ) (cid:1) λ i(cid:17) , for x + z = 0 , Q (9 b ) = (cid:16) (1 − yλ ) f (0 , , y yλ ) , (1 − yλ ) g (0 , , y yλ ) + (1 − yλ ) λ, (1 − yλ ) h (0 , , y yλ ) (cid:17) , for x + z = 0 Q (10 a ) = (cid:16) α ( z, r, λ ) f ( x + r λ e α ( − z,r,λ ) ) + x h h ( x + r λ e α ( − z,r,λ ) ) λ + c (cid:0) x , Q ( x + r λ e α ( − z,r,λ ) ) (cid:1) λ i ,α ( z, r, λ ) g ( x + r λ e α ( − z,r,λ ) ) + y h h ( x + r λ e α ( − z,r,λ ) ) λ + c (cid:0) x , Q ( x + r λ e α ( − z,r,λ ) ) (cid:1) λ i ,h ( x + r λ e α ( − z,r,λ ) ) + c (cid:0) x , Q ( x + r λ e α ( − z,r,λ ) ) (cid:1) ( λ − zλ ) − r λ h (cid:0) x + r λ e α ( − z,r,λ ) (cid:1)(cid:17) , for x + y = 0 , Q (10 b ) = (cid:16) (1 − zλ ) f (0 , , z zλ ) , (1 − zλ ) g (0 , , z zλ ) , (1 − zλ ) h (0 , , z zλ ) + (1 − zλ ) λ (cid:17) , for x + y = 0 , where each new solution Q ( k ) is associated with a potential function q associated14ith the generator b v k in Table 1. Another application of symmetry methods is to reduce systems of differentialequations, finding equivalent system of simpler form. This application is called reduction . In this section we perform some symmetry reductions of the quater-nionic Riccati equation (23) using symmetry groups obtained in the last section.We consider solutions of these symmetry reductions and combine them with theresults found in section 2. We refere to [19] chapter 3 for symmetry reductionusing Lie group symmetries.
For the group of rotations about de z -axis and the w -axis generated by b v = x∂ y − y∂ x + u∂ v − v∂ u , invariants are provided by ρ = p x + y , z , ˆ u = u cos θ + v sin θ , ˆ v = − u sin θ + v cos θ and ˆ w = w . The reduced equations(47a) , . . . , (47d) are then E ≡ − (cid:18) ˆ u ρ + ˆ uρ + ˆ w z (cid:19) + ˆ u + ˆ v + ˆ w − F ( z, ρ ) = 0 , (51a) E ≡ ˆ v z + (ˆ u z − ˆ w ρ ) tan θ = 0 , (51b) E ≡ ˆ u z − ˆ w ρ − ˆ v z tan θ = 0 , (51c) E ≡ ˆ v ρ + ˆ vρ = 0 , (51d)where F is the potential for the generator b v in Table 1.The system of equation (51b) and (51c) can be simplified as E ′ ≡ ˆ u z − ˆ w ρ = 0 (52)and E ′ ≡ ˆ v z = 0 . (53)From (53) and (51d) we find ˆ v = c ρ , where c is a real constant. If we furtherassume translational invariance under b v = ∂ z , equation (52) implies that ˆ w = c , where c is a real constant. We have now to solve equation (51a) which canbe rewritten as ˆ u ρ = ˆ u − ˆ uρ − e F ( ρ ) , (54)where e F ( ρ ) := F ( ρ ) − (cid:16) c ρ (cid:17) − c . Since F is an arbitrary function of ρ , theconstants c , c can be absorbed such that c = c = 0. Equation (54) is a one-dimensional (classical) Riccati equation of the form (1). Therefore, following152) equation (54) is equivalent to g ρρ + g ρ ρ − e F g = 0 , (55)where solution g ( ρ ) of (55) leads to solution ˆ u = − g ρ /g of (54).Assuming now a specific form of the potential, we consider e F ( ρ ) = k ρ , (56)where k is a real constant. Solution of equation (54) is given byˆ u ( ρ ) = − kρ (cid:0) ρ k + e ck (cid:1) ( ρ k − e ck ) (57)where c is a real constant. Since we have found ˆ u ( ρ ) and that ˆ v ( ρ ) = 0 , ˆ w ( ρ ) = 0,we can now obtain the inverse transformations u = ˆ ux p x + y − ˆ vy p x + y ,v = ˆ uy p x + y + ˆ vx p x + y ,w = ˆ w, (58)such that u = − kx h(cid:0) x + y (cid:1) k + e ck i ( x + y ) h ( x + y ) k − e ck i ,v = − ky h(cid:0) x + y (cid:1) k + e ck i ( x + y ) h ( x + y ) k − e ck i ,w = 0 . (59)Thus for a potential of the form q = k x + y , a solution of the quaternionic Riccatiequation (23) is given by Q = ue + ve + we .Now using theorem 2 a solution ψ of the Schr¨odinger equation (22) can befound using this solution Q of the three-dimensional Riccati equation: ψ ( x, y, z ) = exp ( −A [ Q ])= exp (cid:18) − (cid:18)Z x u ( ξ, , dξ + Z y v ( x, η, dη + Z z w ( x, y, ζ ) dζ − ln C (cid:19)(cid:19) = C (cid:16)(cid:0) x + y (cid:1) k e − ck − (cid:17) ( x + y ) k ( e − ck − . (60)16herefore, a solution of the Schr¨odinger equation (22) is given by ψ ( x, y, z ) = C (cid:16)(cid:0) x + y (cid:1) k − e ck (cid:17) ( x + y ) k (1 − e ck ) (61)for a potential q = k x + y . Let us now look at the one-parameter group G of conical symmetry generatedby b v = 2 xz∂ x + 2 yz∂ y + [ z − ( x + y )] ∂ z + 2( xw − zu ) ∂ u + 2( yw − zv ) ∂ v + c (cid:0) x , Q ( x ) (cid:1) ∂ w (62)for the potential q = r − F (cid:16) xr , yr (cid:17) and x + y = 0.Independent invariants can be easily calculated, we obtain s = xr and t = yr . (63)We have now to solve the following system dx = xzduxw − zu = xzdvyw − zv = xzdw − xu − yv − zw , (64)where an immediate solution of v is given by v = ts u . The system (64) thenbecomes dr = du sw √ − r ( s + t ) − ur = dw r √ − r ( s + t ) − s + t s ) u √ − r ( s + t ) − wr . (65)We find the following solutions u = − − r ( s + t ) r U + √ − r ( s + t ) r iV + sw = s + t ) √ − r ( s + t ) sr U + − r ( s + t )2 r s iV + √ − r ( s + t ) r . (66)The dependent invariants are therefore given by U = − r (cid:0) − r ( s + t ) (cid:1) u + 2 r s p − r ( s + t ) w − r s = ( x + y − z ) u + (2 xz ) w − x,V = i h − r ( s + t ) p − r ( s + t ) u − r s (cid:0) − r ( s + t ) (cid:1) w + 2 rs p − r ( s + t ) i = i h − z ( x + y ) r u + xr (cid:0) x + y − z (cid:1) w + xzr i . (67)17alculating now the terms u x , . . . , w z and substituing into the system ofequations (47a) , . . . , (47d), we find E ≡ − s V + (cid:18) s + t s (cid:19) U + U + tU t + sU s = sF ( s, t ) ,E ≡ − (cid:2) r t (cid:3) iV + h rt p − r ( s + t ) i U + (cid:2) − r s (cid:3) iV t + (cid:2) r st (cid:3) iV s + h rs p − r ( s + t ) i U t − h rst p − r ( s + t ) i U s = 0 ,E ≡ (cid:2) − r t (cid:3) iV + h rt p − r ( s + t ) i U − (cid:2) r s t (cid:3) iV t + (cid:2) s (2 r t − (cid:3) iV s + h rs t p − r ( s + t ) i U t − h rst p − r ( s + t ) i U s = 0 ,E ≡ − h rt p − r ( s + t ) i iV + (cid:2) t (cid:0) − r ( s + t ) (cid:1)(cid:3) U − h rs p − r ( s + t ) i iV t + h rst p − r ( s + t ) i iV s + (cid:2) s (cid:0) − r ( s + t ) (cid:1)(cid:3) U t − (cid:2) st (cid:0) − r ( s + t ) (cid:1)(cid:3) U s = 0 . Considering the new equations E ′ = tE − E , E ′ = tE + E , as well as E ′′ = p − r ( s + t ) E ′ − rtE , E ′ = p − r ( s + t ) E ′ + 4 rtE , we obtain V = sV s + tV t . We find V t = 0 and the simplified system of equations (47a) , . . . , (47d)becomes s + t s U + 1 s U + ts U t + U s = F − (cid:18) C (cid:19) , ts U + sU t − tU s = 0 . (68)with V ( s ) = iC s, C a real constant. (69)To go further in this symmetry reduction let us assume that F = ( C ) ,i.e. the potential is of the form q = (cid:18) C r (cid:19) . Then we obtain U ( s, t ) = 2 s ( s + t ) ln (cid:0) C ( s + t ) (cid:1) , (70)18here C is a real constant, such that u = − C xzr + 2 x ( x + y − z ) r ( x + y ) ln (cid:18) C (cid:18) x + y r (cid:19)(cid:19) + xr ,v = − C yzr + 2 y ( x + y − z ) r ( x + y ) ln (cid:18) C (cid:18) x + y r (cid:19)(cid:19) + yr ,w = C ( x + y − z )2 r + 4 zr ln (cid:18) C (cid:18) x + y r (cid:19)(cid:19) + zr . (71)Therefore, Q = ue + ve + we is a quaternionic solution of the three-dimensionalRiccati equation with potential q ( x, y, z ) = (cid:18) C r (cid:19) .Again here the solution Q can be used to obtain a solution ψ of the Schr¨odingerequation (22) using theorem 2. We find ψ ( x, y, z ) = exp ( −A [ Q ])= exp (cid:18) − (cid:18)Z x u ( ξ, , dξ + Z y v ( x, η, dη + Z z w ( x, y, ζ ) dζ − ln C (cid:19)(cid:19) = C ln (cid:16) C x + y ( x + y + z ) (cid:17) ln C p x + y + z exp (cid:16) C z x + y + z ) (cid:17) (72)i.e. ψ ∈ ker (cid:0) − ∆ + q ( x, y, z ) (cid:1) for q ( x, y, z ) = (cid:18) C x + y + z ) (cid:19) . Acknowledgement
CP and ST would like to thank Benoit Huard from Northumbria University forhis help in the use of softwares that were used to calculate Lie symmetries. CPacknowledges a scholarship from the Institut des Sciences Math´ematiques.19 eferences [1] Ablowitz M Clarkson P 1991
Solitons, nonlinear evolution equations andinverse scattering
Cambridge University Press Cambridge[2] Bernstein S 1996 Proceedings of the symposium Analytical and numericalmethods in quaternionic and Clifford analysis Seiffen[3] Bernstein S 2006
Compl. var. ell. eq. Compl. var. J. Phys. A: Math. Theor. Spectral transform and solitons
North-Holland Publ. Comp. Amsterdam[7] Darboux G 1882
Comptes Rendus Introduction to nonlinear differential and integral equations
New York: Dover Publications[9] G¨urlebeck K Habetha K Spr¨osig W 2007
Holomorphic functions in theplane and n -dimensional space Birkh¨auser[10] G¨urlebeck K Spr¨osig W 1989
Quaternionic analysis and elliptic boundaryvalue problems
Berlin: Akademie-Verlag[11] Henkel M Unterberger J 2006
Nucl. Phys. B
Ordinary differential equations
New York: Dover Publica-tions[13] Khmelnytskaya K V Kravchenko V V 2008
J. Phys. A: Math. Theor. J. Phys. A: Math. Theor. Cliff. Analy. Appl. edBrackxs F et al (Dordrecht: Kluwer) 143–154[16] Kravchenko V Tremblay S 2011 Math. Meth. Appl. Sc. Vorlesungen ¨uber continuierliche Gruppen mil ge-ometrischen und anderen Anwend , Teubner B Leipzig (reprinted by ChelseaPubl. Comp.) New York[18] Matveev V and Salle M 1991
Darboux transformations and solitons
NewYork: Springer[19] Olver P J 1993
Applications of Lie groups to differential equations
SpringerNew York 2020] Schuch D 2014
J. Phys. A: Conf. Ser.