Cotton-Type and Joint Invariants for Linear Elliptic Systems

Abstract

Cotton-type invariants for a subclass of a system of two linear elliptic equations, obtainable from a complex base linear elliptic equation, are derived both by spliting of the corresponding complex Cotton invariants of the base complex equation and from the Laplace-type invariants of the system of linear hyperbolic equations equivalent to the system of linear elliptic equations via linear complex transformations of the independent variables. It is shown that Cotton-type invariants derived from these two approaches are identical. Furthermore, Cotton-type and joint invariants for a general system of two linear elliptic equations are also obtained from the Laplace-type and joint invariants for a system of two linear hyperbolic equations equivalent to the system of linear elliptic equations by complex changes of the independent variables. Examples are presented to illustrate the results.

1. Introduction

The general scalar linear second order partial differential equation (PDE) in two independent variables (𝑥,𝑦) is of the form 𝑎𝑤_𝑥𝑥+2𝑏𝑤_𝑥𝑦+𝑐𝑤_𝑦𝑦+𝑑𝑤_𝑥+𝑒𝑤_𝑦+𝑓𝑤=𝑔, (1) where 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, and 𝑔 are given 𝐶² functions of 𝑥 and 𝑦. All linear PDEs of the form (1) are hyperbolic, elliptic, or parabolic depending on whether 𝑏²−𝑎𝑐 is positive, negative, or zero, respectively, and thus can be simplified to one of the three canonical forms by introducing new coordinates [1]. The general linear second-order PDE (1) was first classified by Lie [2] in terms of its symmetry properties. He obtained seven canonical forms according to their point symmetries and types of equations. Of these, four belonged to the hyperbolic class and three to the parabolic class.

The scalar linear second order elliptic equation in two independent variables in canonical form is 𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝑎𝑢_𝑥+𝑏𝑢_𝑦+𝑐𝑢=0. (2) It is well known that by means of the linear complex transformations [1, 3] 1𝑥=2(𝑡+𝑧),𝑦=−𝑖2(𝑡−𝑧), (3) the elliptic equation (2) can be mapped to the linear hyperbolic equation 𝑢_𝑡𝑧+𝐴𝑢_𝑡+𝐵𝑢_𝑧+𝐶𝑢=0, (4) where 1𝐴=4()1𝑎+𝑖𝑏,𝐵=4()1𝑎−𝑖𝑏,𝐶=4𝑐. (5) In 1773, Laplace [4] in his fundamental memoir deduced two semi-invariants ℎ=𝐴_𝑡+𝐴𝐵−𝐶,𝑘=𝐵_𝑍+𝐴𝐵−𝐶, (6) for (4), known as the Laplace invariants.

These Laplace invariants (6) can be transformed, by use of the inverse of the transformations (3) as well as after the substitution of (5) into (6) and then splitting the real and imaginary parts, to arrive at the Cotton invariants 𝜇=𝑎_𝑦−𝑏_𝑥,𝐻=𝑎_𝑥+𝑏_𝑦+12𝑎²+𝑏²−2𝑐. (7) These invariants (7) were first derived by Cotton [5]. Laplace and Cotton invariants remain unaltered under linear transformations of the dependent variable which, respectively, map the linear hyperbolic and elliptic equations into themselves. The corresponding invariant quantities for the linear parabolic equations can be found in [6–8]. Ovsiannikov [9] used the Laplace invariants in the group classification of the hyperbolic equation (4) by writing the determining equations for the symmetries of (4) in terms of these invariants. The solution of the equivalence problem for scalar linear (1+1) hyperbolic equations and some new invariants are given in [10, 11]. Laplace-type and joint invariants for a system of two linear hyperbolic equations are derived in [12] and Laplace-type invariants for a subclass of a system of two linear hyperbolic equations obtained from a complex linear hyperbolic equation are presented in [13]. The approach of complex symmetry analysis (CSA), was utilized in [14]. This method provides a connection between a complex scalar ordinary differential equation (ODE)/PDE and a system of real ODEs/PDEs by a complex split of the base complex equation into real and imaginary parts. In this work, we derive Cotton-type invariants for a subclass of a system of two linear hyperbolic PDEs. Cotton-type and joint invariants for a general linear system of elliptic equations are also determined. Examples are provided as illustration.

The outline of this note is as follows. In Section 2, Cotton-type invariants are derived for a subsystem of two linear elliptic equations by split of the complex Cotton invariants for the corresponding scalar complex linear elliptic equation. Cotton-type invariants for the same class are also obtained by transforming the Laplace-type invariants for the corresponding system of two linear hyperbolic equations which are equivalent to the system of two linear elliptic equations by means of complex linear transformations of the independent variables. These are shown to be the same. Moreover, Cotton-type and joint invariants for a general linear system of two elliptic equations are derived in Section 3. Then in Section 4 some examples are given to illustrate the results. Finally, in Section 5, a brief conclusion is given.

2. Cotton Invariants for a Subclass

In this section, Cotton-type invariants for a subsystem of two linear elliptic equations are first obtained from a complex scalar linear elliptic equation by splitting the complex Cotton invariants of the base complex equation into real and imaginary parts. Then for such a system, we determine invariants from the Laplace-type invariants for the equivalent system of two linear hyperbolic equations. This is achieved by performing complex splits of the Laplace-type invariants. It is concluded, as a proposition, that the Cotton-type invariants are the same for both the approaches. The subsystem of two elliptic equations 𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝛼₁𝑢_𝑥−𝛼₂𝑣_𝑥+𝛽₁𝑢_𝑦−𝛽₂𝑣_𝑦+𝛾₁𝑢−𝛾₂𝑣𝑣=0,_𝑥𝑥+𝑣_𝑦𝑦+𝛼₂𝑢_𝑥+𝛼₁𝑣_𝑥+𝛽₂𝑢_𝑦+𝛽₁𝑣_𝑦+𝛾₂𝑢+𝛾₁𝑣=0 (8) is obtained by splitting of the complex linear elliptic equation 𝑤_𝑥𝑥+𝑤_𝑦𝑦+𝑎𝑤_𝑥+𝑏𝑤_𝑦+𝑐𝑤=0, (9) where 𝑎=𝛼₁+𝑖𝛼₂,𝑏=𝛽₁+𝑖𝛽₂,𝑐=𝛾₁+𝑖𝛾₂,𝑤=𝑢+𝑖𝑣. (10) The Cotton invariants, corresponding to the complex elliptic equation (9), are (7) which split into the four invariants 𝜇₁=𝛼_1𝑦−𝛽_1𝑥,𝜇₂=𝛼_2𝑦−𝛽_2𝑥,𝐻₁=𝛼_1𝑥+𝛽_1𝑦+12𝛼²₁+𝛽²₁−12𝛼²₂+𝛽²₂−2𝛾₁,𝐻₂=𝛼_2𝑥+𝛽_2𝑦+𝛼₁𝛼₂+𝛽₁𝛽₂−2𝛾₂. (11) These are precisely the Cotton-type invariants for the linear elliptic system (8). The simplest case is when the semiinvariants (11) are zero. In this case the elliptic PDE system (8) reduces to the Laplace system by linear transformation of the dependent variables. This is similar to the scalar linear elliptic PDE case.

Now for the system of elliptic equations (8), we derive the Cotton-type invariants by transforming the system of equations to the corresponding linear hyperbolic equations and then using the inverse transformations of the independent variables to convert the Laplace-type invariants to the Cotton-type invariants. By means of the transformations (3), the system of elliptic equations (8) can be mapped to the system of two linear hyperbolic type equations as follows: 𝑢_𝑡𝑧+𝐴₁𝑢_𝑡−𝐴₂𝑣_𝑡+𝐵₁𝑢_𝑧−𝐵₂𝑣_𝑧+𝐶₁𝑢−𝐶₂𝑣𝑣=0,_𝑡𝑧+𝐴₂𝑢_𝑡+𝐴₁𝑣_𝑡+𝐵₂𝑢_𝑧+𝐵₁𝑣_𝑧+𝐶₂𝑢+𝐶₁𝑣=0, (12) where 𝐴₁=14𝛼₁+𝑖𝛽₁,𝐵₁=14𝛼₁−𝑖𝛽₁,𝐶₁=14𝛾₁,𝐴₂=14𝛼₂+𝑖𝛽₂,𝐵₂=14𝛼₂−𝑖𝛽₂,𝐶₂=14𝛾₂. (13) The system of hyperbolic equations (12) has four Laplace-type invariants [13]: ℎ₁=𝐴_1𝑡+𝐴₁𝐵₁−𝐴₂𝐴₂−𝐶₁,ℎ₂=𝐴_2𝑡+𝐴₁𝐵₂+𝐴₂𝐵₁−𝐶₂,𝑘₁=𝐵_1𝑧+𝐴₁𝐵₁−𝐴₂𝐵₂−𝐶₁,𝑘₂=𝐵_2𝑧+𝐴₁𝐵₂+𝐴₂𝐵₁−𝐶₂. (14) Now by the application of the transformations (3) and complex splits, the Laplace-type invariants (14) become the Cotton-type invariants (11). We therefore conclude the following result.

Proposition 1. For a class of a system of two linear elliptic equations (8) obtained from a complex base linear elliptic equation (9) or equivalent to a subsystem of two linear hyperbolic equations (12) by complex linear transformations of the independent variables (3), Cotton-type invariants either constructed by splitting of the complex Cotton invariants (7) of the complex base elliptic equation into real and imaginary parts or those computed by split of the Laplace-type invariants (14) of the system of linear hyperbolic equations are identical to (11).

3. Cotton-Type and Joint Invariants in General

In this section, Cotton-type and joint invariants for a general system of two linear elliptic equations are obtained. A general system of two linear elliptic equations is 𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝑎₁𝑢_𝑥+𝑎₂𝑣_𝑥+𝑏₁𝑢_𝑦+𝑏₂𝑣_𝑦+𝑐₁𝑢+𝑐₂𝑣𝑣=0,_𝑥𝑥+𝑣_𝑦𝑦+𝑎₃𝑢_𝑥+𝑎₄𝑣_𝑥+𝑏₃𝑢_𝑦+𝑏₄𝑣_𝑦+𝑐₃𝑢+𝑐₄𝑣=0. (15) By means of the complex transformations of the independent variables (3), this system (15) is transformed into the system of two linear hyperbolic equations as follows: 𝑢_𝑡𝑧+𝐴₁𝑢_𝑡+𝐴₂𝑣_𝑡+𝐵₁𝑢_𝑧+𝐵₂𝑣_𝑧+𝐶₁𝑢+𝐶₂𝑣𝑣=0,_𝑡𝑧+𝐴₃𝑢_𝑡+𝐴₄𝑣_𝑡+𝐵₃𝑢_𝑧+𝐵₄𝑣_𝑧+𝐶₃𝑢+𝐶₄𝑣=0, (16) where 𝐴₁=14𝑎₁+𝑖𝑏₁,𝐵₁=14𝑎₁−𝑖𝑏₁,𝐶₁=14𝑐₁,𝐴₂=14𝑎₂+𝑖𝑏₂,𝐵₂=14𝑎₂−𝑖𝑏₂,𝐶₂=14𝑐₂,𝐴₃=14𝑎₃+𝑖𝑎₃,𝐵₃=14𝑎₃−𝑖𝑏₃,𝐶₃=14𝑐₃,𝐴₄=14𝑎₄+𝑖𝑏₄,𝐵₄=14𝑎₄−𝑖𝑏₄,𝐶₄=14𝑐₄. (17) This system of linear hyperbolic equations (16) has five semi-invariants [12] under the linear change of dependent variables. They are [12] 𝐼₁=𝑘₁+𝑘₄,𝐼₂=𝑘₅+𝑘₈,𝐼₃=𝑘₁𝑘₄−𝑘₂𝑘₃,𝐼₄=𝑘₅𝑘₈−𝑘₆𝑘₇,𝐼₅=𝑘₁𝑘₅+𝑘₂𝑘₇+𝑘₃𝑘₆+𝑘₄𝑘₈, (18) where 𝑘₁=𝐴₁𝐵₁+𝐴₃𝐵₂+𝐴_1𝑡−𝐶₁,𝑘₂=𝐴₁𝐵₃+𝐴₃𝐵₄+𝐴_3𝑡−𝐶₃,𝑘₃=𝐴₂𝐵₁+𝐴₄𝐵₂+𝐴_2𝑡−𝐶₂,𝑘₄=𝐴₂𝐵₃+𝐴₄𝐵₄+𝐴_4𝑡−𝐶₄,𝑘₅=𝐴₁𝐵₁+𝐴₂𝐵₃+𝐵_1𝑧−𝐶₁,𝑘₆=𝐴₃𝐵₁+𝐴₄𝐵₃+𝐵_3𝑧−𝐶₃,𝑘₇=𝐴₁𝐵₂+𝐴₂𝐵₄+𝐵_2𝑧−𝐶₂,𝑘₈=𝐴₃𝐵₂+𝐴₄𝐵₄+𝐵_4𝑧−𝐶₄. (19) The system of linear hyperbolic equations (16) also has the four joint invariants [12] 𝐽₁=𝐼₂𝐼₁,𝐽₂=𝐼₃𝐼²₁,𝐽₃=𝐼₄𝐼²₁,𝐽₄=𝐼₅𝐼²₁. (20) We utilize the same approach as in the previous section. Indeed via the transformations (3), the Laplace-type invariants (18) transform to the five Cotton-type invariants 𝐻₁=Im𝐾₁+𝐾₄=Im𝐾₅+𝐾₈,𝐻₂=Re𝐾₁+𝐾₄=Re𝐾₅+𝐾₈,𝐻₃=Im𝐾₁𝐾₄−𝐾₂𝐾₃=Im𝐾₅𝐾₈−𝐾₆𝐾₇,𝐻₄=Re𝐾₁𝐾₄−𝐾₂𝐾₃=Re𝐾₅𝐾₈−𝐾₆K₇,𝐻₅=Re𝐾₁𝐾₅+𝐾₂𝐾₇+𝐾₃𝐾₆+𝐾₄𝐾₈. (21) And the invariant equation is Im𝐾₁𝐾₅+𝐾₂𝐾₇+𝐾₃𝐾₆+𝐾₄𝐾₈=0, (22) where 𝐾₁=14²𝑎²₁+𝑏²₁+𝑎₂𝑎₃+𝑏₂𝑏₃+2𝑎_1𝑥+2𝑏_1𝑦−4𝑐₁+𝑖4²𝑎₂𝑏₃−𝑎₃𝑏₂−2𝑎_1𝑦+2𝑏_1𝑥,𝐾₂=14²𝑎₁𝑎₃+𝑏₁𝑏₃+𝑎₃𝑎₄+𝑏₃𝑏₄+2𝑎_3𝑥+2𝑏_3𝑦−4𝑐₃+𝑖4²𝑎₃𝑏₁−𝑎₁𝑏₃+𝑎₄𝑏₃−𝑎₃𝑏₄−2𝑎_3𝑦+2𝑏_3𝑥,𝐾₃=14²𝑎₁𝑎₂+𝑏₁𝑏₂+𝑎₂𝑎₄+𝑏₂𝑏₄+2𝑎_2𝑥+2𝑏_2𝑦−4𝑐₂+𝑖4²𝑎₁𝑏₂−𝑎₂𝑏₁+𝑎₂𝑏₄−𝑎₄𝑏₂−2𝑎_2𝑦+2𝑏_2𝑥,𝐾₄=14²𝑎₂𝑎₃+𝑏₂𝑏₃+𝑎²₄+𝑏²₄+2𝑎_4𝑥+2𝑏_4𝑦−4𝑐₄+𝑖4²𝑎₃𝑏₂−𝑎₂𝑏₃−2𝑎_4𝑦+2𝑏_4𝑥,𝐾₅=14²𝑎²₁+𝑏²₁+𝑎₂𝑎₃+𝑏₂𝑏₃+2𝑎_1𝑥+2𝑏_1𝑦−4𝑐₁+𝑖4²𝑎₃𝑏₂−𝑎₂𝑏₃+2𝑎_1𝑦−2𝑏_1𝑥,𝐾₆=14²𝑎₁𝑎₃+𝑏₁𝑏₃+𝑎₃𝑎₄+𝑏₃𝑏₄+2𝑎_3𝑥+2𝑏_3𝑦−4𝑐₃+𝑖4²𝑎₁𝑏₃−𝑎₃𝑏₁+𝑎₃𝑏₄−𝑎₄𝑏₃+2𝑎_3𝑦−2𝑏_3𝑥,𝐾₇=14²𝑎₁𝑎₂+𝑏₁𝑏₂+𝑎₂𝑎₄+𝑏₂𝑏₄+2𝑎_2𝑥+2𝑏_2𝑦−4𝑐₂+𝑖4²𝑎₂𝑏₁−𝑎₁𝑏₂+𝑎₄𝑏₂−𝑎₂𝑏₄+2𝑎_2𝑦−2𝑏_2𝑥,𝐾₈=14²𝑎₂𝑎₃+𝑏₂𝑏₃+𝑎²₄+𝑏²₄+2𝑎_4𝑥+2𝑏_4𝑦−4𝑐₄+𝑖4²𝑎₂𝑏₃−𝑎₃𝑏₂+2𝑎_4𝑦−2𝑏_4𝑥. (23) Note that we have an invariant equation here. This differs from the invariants of the split elliptic system of Section 2. We therefore have the following result.

Proposition 2. A general system of two linear elliptic equations (15) has the five Cotton-type invariants (21) and its coefficients satisfy the invariant condition (22).

Now the four joint invariants (20) reduce to the four invariants of the elliptic equations (15) and they are 𝜇₁=𝐻²₁−𝐻²₂𝐻₃+2𝐻₁𝐻₂𝐻₄𝐻²₁−𝐻²₂²+4𝐻²₁𝐻²₂,𝜇₂=𝐻²₁−𝐻²₂𝐻₄−2𝐻₁𝐻₂𝐻₃𝐻²₁−𝐻²₂²+4𝐻²₁𝐻²₂,𝜇₃=𝐻²₁−𝐻²₂𝐻₅𝐻²₁−𝐻²₂²+4𝐻²₁𝐻²₂,𝜇₄=−2𝐻₁𝐻₂𝐻₅𝐻²₁−𝐻²₂²+4𝐻²₁𝐻²₂, (24) where the semi-invariants 𝐻²₁ and 𝐻²₂ are both not zero. The situation when both are zero occur for the Laplace system discussed earlier. We have thus obtained the Cotton-type and joint invariants for a general linear elliptic system of two equations (15) by using the Laplace-type and joint invariants of the general system of linear hyperbolic equations (16) by utilizing the known semi- and joint invariants of [12]. We thus state the following proposition.

Proposition 3. A general system of two linear elliptic equations (15) has the four joint invariants (24).

4. Applications

Here we present some examples for illustration. We consider 𝑢, 𝑣, 𝑢, and 𝑣 as dependent variables and 𝑥, 𝑦, 𝑠, and 𝑡 as independent variables.

Example 4. Consider the system of two linear elliptic equations 𝑢_𝑥𝑥+𝑢_𝑦𝑦+2𝑥𝑢_𝑥+4𝑦𝑢_𝑦+2𝑦²𝑣𝑢=0,_𝑥𝑥+𝑣_𝑦𝑦−4𝑥𝑣_𝑥−2𝑦𝑣_𝑦1+2𝑦²+3𝑥²𝑣=0. (25) This system transforms to the simplest elliptic equations 𝑢_𝑥𝑥+𝑢_𝑦𝑦=0,𝑣_𝑥𝑥+𝑣_𝑦𝑦=0, (26) under the transformation 𝑢=𝑥𝑦²𝑢,𝑣𝑣=𝑥²𝑦. (27) The systems of elliptic equations (25) and (26) are transformable into each other as these systems have the same Cotton-type semi-invariants 𝐻₁=𝐻₂=𝐻₃=𝐻₄=𝐻₅=0. (28)

Example 5. The system of elliptic equations 𝑢_𝑥𝑥+𝑢_𝑦𝑦+1−2𝑦𝑢_𝑥+(1−2𝑥)𝑢_𝑦+𝑥²+𝑦²𝑣−𝑥−𝑦𝑢=0,_𝑥𝑥+𝑣_𝑦𝑦+1−2𝑦𝑣_𝑥+(1−2𝑥)𝑣_𝑦+𝑥²+𝑦²−𝑥−𝑦𝑣=0, (29) with the Cotton-type invariants 𝐻₁=0,𝐻₂=14,𝐻₃=0,𝐻₄=164,𝐻₅=1,32 (30) reduces to the simple system of elliptic equations 𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝑢_𝑥+𝑢_𝑦=0,𝑣_𝑥𝑥+𝑣_𝑦𝑦+𝑣_𝑥+𝑣_𝑦=0, (31) by the application of the transformation 𝑢=exp−𝑥𝑦𝑢,𝑣=exp−𝑥𝑦𝑣. (32) The system (31) also has the Cotton-type invariants (30).

Example 6. The uncoupled system two of elliptic equations 𝑢_𝑥𝑥+𝑢_𝑦𝑦+2𝑥+1𝑢_𝑥+41−𝑦𝑢_𝑦+1𝑥+6𝑦²−2𝑦𝑣𝑢=0,_𝑥𝑥+𝑣_𝑦𝑦+21−𝑥𝑣_𝑥+41+𝑦𝑣_𝑦+2𝑦²+2𝑦−1𝑥+2𝑥²𝑣=0, (33) has the Cotton-type invariants 𝐻₁=0,𝐻₂=14,𝐻₃=0,𝐻₄=164,𝐻₅=1.32 (34) Therefore it is reducible to the simple system 𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝑢_𝑥+𝑢_𝑦=0,𝑣_𝑥𝑥+𝑣_𝑦𝑦+𝑣_𝑥+𝑣_𝑦=0, (35) by means of the transformation 𝑥𝑢=𝑦²𝑢,𝑦𝑣=²𝑥𝑣. (36)

Example 7. Consider now the linear system of elliptic equations 𝑢_𝑥𝑥+𝑢_𝑦𝑦+2𝑥+12𝑢_𝑥+12−2𝑦𝑢_𝑦+124𝑦²−1𝑦+1𝑥𝑣𝑢=0,_𝑥𝑥+𝑣_𝑦𝑦+2𝑥+12𝑣_𝑥+12−2𝑦𝑣_𝑦+124𝑦²−1𝑦+1𝑥𝑣=0, (37) which has the joint invariants 𝜇₁=0,𝜇₂1=−4,𝜇₃1=−2,𝜇₄=0. (38) By using the transformation 𝑥𝑠=2𝑦,𝑡=2,𝑥𝑢=𝑦𝑢,𝑥𝑣=𝑦𝑣, (39) the above system reduces to the simple system 𝑢_𝑠𝑠+𝑢_𝑡𝑡+𝑢_𝑠+𝑢_𝑡=0,𝑣_𝑠𝑠+𝑣_𝑡𝑡+𝑣_𝑠+𝑣_𝑡=0, (40) because this system has the joint invariant identical to the system (37).

Example 8. Finally, the coupled system of elliptic equations 𝑢_𝑥𝑥+𝑢_𝑦𝑦+12+𝑥𝑢_𝑥+2𝑦³𝑥^−3/2𝑣_𝑥−2𝑦𝑢_𝑦+1𝑥−14𝑥²+2𝑦²𝑢−2𝑦³𝑥^−5/2𝑣𝑣=0,_𝑥𝑥+𝑣_𝑦𝑦+2𝑥^3/2𝑦³𝑢_𝑥1+21−𝑥𝑣_𝑥+4𝑦𝑣_𝑦+𝑥^1/2𝑦³1𝑢+2𝑦²−1𝑥+1𝑥²𝑣=0, (41) with the joint invariants 𝜇₁=0,𝜇₂=0,𝜇₃=−1,𝜇₄=0, (42) simplifies to the system 𝑢_𝑠𝑠+𝑢_𝑡𝑡+𝑢_𝑠+𝑢_𝑡+𝑣_𝑠+𝑣_𝑡=0,𝑣_𝑠𝑠+𝑣_𝑡𝑡+𝑢_𝑠+𝑢_𝑡+𝑣_𝑠+𝑣_𝑡=0, (43) which has the same joint invariants as the system (41). The transformation that does this reduction is 𝑠=𝑥+𝑦,𝑡=𝑥−𝑦,√𝑢=𝑥𝑦𝑢,𝑦𝑣=²𝑥𝑣. (44)

5. Conclusion

In this paper, we have derived the Cotton-type invariants for a special class of a system of two linear elliptic equations in two independent variables which arises from the complex split of a base complex linear elliptic equation. Moreover, the Cotton-type and joint invariants for a general system of two linear elliptic equations are also obtained. Laplace 1773, in his fundamental memoir, discussed two semi-invariants under the change of dependent variables of the scalar linear hyperbolic equation, known as Laplace invariants. Later, Cotton 1900 derived semi-invariants for the scalar linear elliptic equation, known as Cotton invariants. Linear hyperbolic and elliptic equations can be transformed into each other by the application of linear complex transformation of the independent variables. So do Laplace and Cotton invariants.

By a complex split, a complex scalar linear elliptic equation has been transformed into a system of two linear elliptic equations, which is a subclass of the general system of two linear elliptic equations. Cotton-type semi-invariants for this system of elliptic equations are obtained by two approaches. One is by splitting of the complex Cotton invariants that correspond to the complex base scalar linear elliptic equation into real and imaginary parts and the second by transformation of the subsystem of the linear elliptic equations into linear hyperbolic equations and application of the linear inverse transformations on the Laplace-type semi-invariants of the hyperbolic equations to deduce the Cotton-type invariants for the required subsystem of linear elliptic equations. It is found that the Cotton-type invariants by both approaches are the same. For a general system of linear elliptic equations, the Cotton-type and joint invariants have been constructed by transformation of the system of two linear elliptic equations into a system of two linear hyperbolic equations and thereafter applying the linear inverse transformations on the Laplace-type and joint invariants of [12] to deduce the Cotton-type and joint invariants for the linear system of elliptic equations.

Conflict of Interests

It is declared by the authors that there is no conflict of interests in the publication of this paper.

Acknowledgments

A. Aslam is grateful to DECMA for hospitality and financial support for his stay at DECMA during which this work was completed as well as NUST for travel support. FMM is thankful to the NRF of South Africa for a research grant support.