On Integrability of Christouâ€™s Sixth Order Solitary Wave Equations

We examine the integrability in terms of PainlevÃ¨ analysis for several models of higher order nonlinear solitary wave equations which were recently derived by Christou. Our results point out that these equations do not possess PainlevÃ¨ property and fail the PainlevÃ¨ test for some special values of the coefficients; and that indicates a non-integrability criteria of the equations by means of the PainlevÃ¨ integrability.


Introduction
A variety of new nonlinear partial differential equations were recently introduced in the work of [1] from applying a different type of techniques; The author managed to exploit fundamental physics laws, Taylor series expansion and Hirota's bilinear operator to derive some higher order solitary wave equations. The first model, the sixth order solitary wave equation using Ohm's law is given by (1) where is a function of and , the subscripts denote to partial derivatives with respect to the independent variables, and , and .
The is the charge on the capacitor, is Faraday's constant, is the length on the capacitor, is a small parameter of the Taylor expansion and controls the triple interactions between sections and must be non-negative. The travelling wave solutions for the equation (1) was obtained in [2] by using the improved generalized tanh-coth method. The second model, the sixth order solitary wave equation using Hirota's bilinear operator is given by (2) The third model, the sixth order Sine-Gordon equation is written as (3)

ISSN: 0067-2904
Allami Iraqi Journal of Science, 2019, Vol. 60, No. 5, pp: 1172-1179 1173 where is a function of and , and are some physical quantities. We shall inspect Painlevè integrability for the equations (1), (2) and (3); and the integrability means here is that the differential equation does have Painlevè property. For a given partial differential equation where is a function of , is said to have the Painlevè property if solutions are single valued about non-characteristic movable singularity manifolds, and these manifolds are determined by the condition of the form , where is an analytic function. In other words, if is a solution for Partial differential equation, then it takes Laurent type expansion ∑ (4) where and are both analytic functions, is an integer number, and the number of arbitrary functions is equal to the order of the differential equation. Wiess, Tabor and Carnevale (WTC) [3] introduced an approach that one can examine singularity structure of partial differential equations directly. In addition, Wiess [4][5][6][7] investigated the Painlevè property for several partial differential equations and he showed how to construct their Bäcklund transformations and Lax pairs. The WTC approach is basically built on three steps. Firstly, the leading order analysis, obtaining the dominant behavior of all possible singularities of the equation. Secondly, finding the resonances where arbitrary constants may occur in the Laurent expansion. Thirdly, verifying the resonance conditions in each Laurent expansion explicitly. The equation survives Painlevè test if all the three steps are satisfied. A concise review of many methods of Painlevè tests can be found in [8], and for recent applications see [9][10][11][12].
The rest of the paper is organized as follows, in section two the Painlevè analysis for the sixth order solitary wave equations using Ohm's law is considered, section three is dedicated to apply the Painlevè test for the nonlinear sixth order equation using Hirota's bilinear operator. We move to section four where the test is performed for solitary wave Sine-Gordon equation. The last section is conclusions.

Painlevè analysis for the sixth order solitary wave equations using Ohm's law
We consider the case when the coefficients of the equation (1)  There exist two families (7) and (8) of Painèlve expansions that needs to be discussed separately. For first family (7), inserting into equation (7) yields when that leads to the two branches √ and √ . In order to find the resonances, where the arbitrary constants may occur in the series, take a linear perturbation of the leading order , (9) where is a small parameter correction to the leading order. Substituting (9) into (7)  A finite number of terms here represent a local solution of the equation (5). Now, we move to deal with the second family (8). Substituting into equation (8) to have The dominant balancing at the singular order leads to the branch . For the sake of finding the resonances, taking a linear perturbation of the leading order , (13) Substituting (13) into (8) to get Using Kruskal's formula [13], that is , and grouping the terms linear in , and also setting which comes from Kruskal's formula, to gain with the benefit of , the resonances polynomial is Solving the algebraic equation for to gain the Fuchs indices √ √ √ . Two of the resonances are non-integer numbers, thus, the equation (5) fails the test and that due to occurring of algebraic or logarithmic branch points.

Painlevè analysis for the Sixth order equation using Hirota's bilinear operator
To perform Painlevè test for equation (2), the equation is given by (14) One can deduce that the most singular terms in the equation (14) are . For more details about negative resonances we refer the reader to [14,15]. We consider only the principle branch, when to verify the compatibility conditions. Substituting ∑ into equation (15) and collecting the coefficients of . Here we write down a few of these equations (17) Allami Iraqi Journal of Science, 2019, Vol. 60, No. 5, pp: 1172-1179 1177 (20) where . Solving equation (17) to obtain, by using Kruskal's simplification [13], , the following branches and From equation (18) we have and and so Also, from equation (19) one can get and since that leads to which is inconsistent. Therefore, the compatibility condition is not satisfied and the equation (14) fails the Painlevè test.

Painlevè analysis for sixth order Sine-Gordon equation
We discuss the case when the coefficients of the equation (3)

Conclusion
After all what we have discussed in the current work, it seems to be that all examined equations do not survive the Painlevè test, and therefore they are not integrable, in the Painlevè sense, for some special values of the coefficients. The Painlevé analysis, in fact, gives us an idea about the nature of solutions of the equations. The bad positions of resonances provide an evidence on occurring of algebraic or logarithmic branch points in their solutions, and the inconsistent of the compatibility conditions making the equations do not pass the test. The test for other special values of the coefficients or even the more general cases of the equations that needs to be considered in the future work.