Kink-soliton, singular-kink-soliton and singular-periodic solutions for a new two-mode version of the Burger–Huxley model: applications in nerve fibers and liquid crystals

New two-mode version of the generalized Burger–Huxley equation is derived using Korsunsky’s operators. The new model arises in the applications of nerve fibers and liquid crystals, and it describes the interaction of two symmetric waves moving simultaneously in the same direction. Solitary wave solutions of types kink-soliton, singular-kink-soliton and singular-periodic are obtained to this model by means of the simplified bilinear method, polynomial-function method and the Kudryashov-expansion method. A comprehensive graphical analysis is conducted to show some physical properties of this new type of nonlinear equations. Finally, all obtained solutions are verified by direct substitutions in the new model.


Introduction
The generalized Burger-Huxley (gBH) equation is classified as a nonlinear partial differential equation (NPDE) of first order in time t, and it admits the propagation of single-wave (single-mode) solution. The gBH takes the following form: where = (x, t) and , , are general real scalars. For the case of = 0 and = 1 , gBH is reduced to the Huxley model which represents the propagation of nerve pulses in nerve fibers and wall motion in liquid crystals (Wang et al. 1990). For = 0 and = 1 , (1.1) is the Burgers' equation which describes the field of wave propagation in nonlinear dissipative models. The Burger-Huxley (1.1), describes the interaction of convection terms against diffusion transmission (Wang et al. 1990). Different schemes are used to find explicit function solutions to gBH; For example, Hyperbolic function solutions, trigonometric function solutions and rational solutions are obtained are obtained by using the (G � ∕G)-expansion method (Manafian and Lakestani 2015) and the auxiliary Riccati equation method (Wang et al. 2013). In Yefimova and Kudryashov (2004), The Cole-Hopf transformation, that transforms the Burgers equation into the heat-conduction equation, is used to obtain exact solutions of the Burgers-Huxley. Finally, kinks and periodic-waves are extracted by means of the tanh-coth method (Wazwaz 2008) and the tanh-expansion method (Wazwaz 2005).
NPDEs of second-order in time t, as in the case of Boussinesq equation, admit the propagation of two-waves (bidirectional-waves) moving, simultaneously, in the same direction. A bidirectional term can be associated with many applications and phenomena such as bidirectional pulses, bidirectional signals, bidirectional turbines and others. Motivated by the terminology of bidirectional-waves, Korsunsky and Wazwaz (Korsunsky 1994;) suggested a two-mode (TM) nonlinear equations in the following form N =nonlinear terms, L =linear terms, s =phase velocity ( s > 0 ), a =nonlinearity parameter ( |a| ≤ 1 ) and b =dispersion parameter ( |b| ≤ 1).
In this work, we present, for the first time, the two-mode extension of gBH (TMgBH) by using Korsunsky-Wazwaz sense. To do so, we consider the following mappings Therefore, the TMgBH takes the following form Kuramoto-Sivashinsky , two-mode perturbed Burgers and Ostrovsky models , dual-mode Schrodinger with nonlinearity Kerr laws ; Alquran and Jaradat 2019) and two-mode complex Hirota model .
Studying the propagation of nonlinear waves is of great interest in the field of nonlinear optics, fluid dynamics and others. An essential tool that support the theoretical study of nonlinear models is to find their solutions. Having closed form solutions give light to investigate the effect of both nonlinear and dispersive terms such models have. For this purpose, researchers have made much efforts in finding explicit (Ali and Hadhoud 2019;Ali and Ma 2019a, b), analytical (Ma et al. 2020;Akgul et al. 2021;Atangana and Akgul 2020;Owolabi et al. 2020;Sulaiman et al. 2020;Alquran et al. 2021[28]) and numerical solitary wave solutions Ali 2019;Yusuf 2020;Baleanu et al. 2020;Qureshi et al. 2020;Qureshi and Yusuf 2019;Mustapha et al. 2020) to a number of nonlinear models.
The contribution of this work is two-fold: First, we seek possible solitary solutions to TMgBH by means by implementing three techniques; the simplified bilinear method, polynomial-function method and the Kudryashov-expansion method. Second, we present some physical properties for the bidirectional waves solutions obtained to the new TMgBH.

Simplified bilinear method
In this section, we are interested in obtaining one-soliton solution to (1.4) by using the simplified bilinear method ). This technique requires the following auxiliary functions: Substitution of (2.6) in the linear terms of (1.4), and solving for the speed c, we get Note that in (2.9), the wave transform = x − ct has two value of c, "two speeds". Then, we insert (2.8) in (1.4) and solve the resulting equation for the unknowns A and to reach at the following outputs: Therefore, the one-soliton solution to the TMBH is � .
(2.10) a =b, It is worth to mention that for > 0 , the solution (2.11) is of type kink. While as, < 0 gives singular-kink. Now, as the speed c has two distinct values (2.9), the dynamic concept of the soliton solutions given in (2.11) can be regarded as the propagations of bidirectionalwaves, "labeled as left-wave and right-wave", moving simultaneously in the same direction and their interaction depends on the increase of the embedded phase-velocity s, see Fig. 1.

Polynomial-function method
To retrieve more solutions to TMgBH, we implement the alternative polynomial-function method (Huang 2006;Deng and Gao 2017). First, we transform (1.4) through the wave transform = (x − ct) into the following reduced differential equation: where = ( ) . The suggested solution to (3.12) is where the variable Y = Y( ) satisfies the auxiliary differential equation For this stage, we need to find the required higher derivatives of ( ) by the aid of (3.14). Implicit differentiation of (3.13) gives Substitution of (3.13) and (3.15) in (3.12), results in a finite series in terms of negative and positive powers of Y. Setting each coefficient of Y i to zero leads to a non-algebraic system in the unknowns b 0 , b 1 , b 2 , and c. Now, we solve the obtained system based on the states of the free constants A and B defined in (3.14). We consider the following two cases: Case I A = 0, B ≠ 0 , leads to the following outputs: (2.11) (x, t) = e x−ct 1 + e x−ct .
(3.12) 2 (c 2 − s 2 ) � + (c + as)( � + (1 − )( − )) + 3 (c + bs) �� = 0, where Ω = √ 2 + 8 . The solution of (3.14) is Y = A k e −A (x−ct) −1 . Accordingly, the solution of TMBH is The solution given in (3.19) is of type kink-soliton when ( k < 0 ) and of type singular-kink for ( k > 0 ). On the other side, it is clear that the solution given in (3.17) is of type singularperiodic and the obtained left-wave and right-wave solutions are presented in Fig. 2. The interaction of these two periodic solutions upon increasing the phase velocity s is shown in Fig. 3. (3.18)

Kudryashov-expansion method
We present a third technique, the Kudryashov-expansion method (Abu et al. 2018; Jaradat and Alquran 2020; Alquran et al. , 2020aAlquran et al. , b, 2021, to seek other new solutions that TMgBH could have. By the wave transform = x − ct , (1.4) is reduced to where = ( ) . The suggested solution to (4.20) is where the variable Y = Y( ) satisfies the auxiliary differential equation Differentiating (4.21) based on (4.22), we get Now, we insert (4.21) and (4.23) in (4.20), to obtain a polynomial of degree 3 in the variable Y whose coefficients are (4.20) (c 2 − s 2 ) � + (c + as)( � + (1 − )( − )) + (c + bs) �� = 0, Since the above coefficients are identical to zeros, the first coefficient is zero if Accordingly, setting the other coefficients to zeros, produce the following results: Therefore, the two-wave solutions of TMBH is We should point here that the sign of the free parameter d in (4.27) determine two different physical shapes for the TMgBH. For example, if d < 0 , the resulting type is kink-soliton which is the same finding obtained in Section 2. While as, for d > 0 , it is of singular-kink type. Figure 4 shows the the so-called left-wave and the right-wave and their interaction. Figure 5, presents the impact of the phase velocity s acting on the overlapping of these two moving left-right waves. (4.24) Y 3 = 1 (as + c) − 2 1 (as + c) + 2 2 (bs + c).
(4.26) The gBH is an example of (5.28) which admits the propagation of single-mode wave, while as, the new TMgBH is an example of (5.29) and admits the propagation of bidirectional (two-mode) waves. To validate the connection between gBH and TMgBH, if we consider s = 0 in (2.9), then c = − 2 which is a single wave speed and leads to a single wavesolution. Same observations can be verified for c given in (3.16), (3.18) and (4.26).
Three functional techniques are used to study the TMgBH and different types of solitary solutions are obtained. Graphical analysis is conducted and reveals some physical properties regarding the extracted bidirectional-waves of the TMgBH; • They are of type bidirectional kinks, bidirectional singular-kinks, bidirectional singular-periodic-waves. • The bidirectional waves are symmetric. • The interaction of the bidirectional waves are dependent on the phase velocity s. By the increase of s, the two waves are approaching to each other.
Finally, the findings of the current work can be visualized in practice as motion of bidirectional-walls moving in liquid crystals, see Figs. 2 and 3. Moreover, in nerve fibers, nerve pulses can propagate through two different paths and maintain its physical shapes.

Conflicts of interest
The authors declares that they have no conflict of interest.