Travelling wave solution for the Landau-Ginburg-Higgs model via the inverse scattering transformation method

The Landau-Ginzburg-Higgs (LGH) equation explains the ocean engineering models, superconductivity and drift cyclotron waves in radially inhomogeneous plasma for coherent ion-cyclotron waves. In this paper, with a simple modification of the Ablowitz-Kaup-Newell-Segur (AKNS) formalism, the integrability of LGH equation is proved by deriving the Lax pair. Hence for that, the inverse scattering transformation (IST) is applied, and the travelling wave solutions are obtained and graphically represented in 2d and 3d profiles.

Depending on the arbitrary constants a; b and c, typical forms of Eq. (2) are specified, one of them is the Landau-Ginzburg-Higgs (LGH) equation. When a ¼ À1; b ¼ Àm 2 and c ¼ n 2 , the LGH equation is stated as where uðX; TÞ symbolizes the electrostatic potential of the ion-cyclotron wave, X and T stand for the nonlinearized spatial and temporal coordinates and m and n are real parameters. The LGH Eq. (3) was formulated by Lev Devidovich Landau and Vitaly Lazarevich Ginzburg with broad applications for the internal processes of complex physical phenomena which occur to explain superconductivity and drift cyclotron waves in radially inhomogeneous plasma for coherent ion-cyclotron waves [11]. Another typical form of Eq. (2) can be stated, such as / 4 equation(a ¼ À1; b ¼ 1; c ¼ À1), Klein-Gordon equation (a ¼ À1; b ¼ m 2 ; c ¼ n), Duffing equation (a ¼ 0 with b and c arbitrary), Sine-Gordon equation (a ¼ À1; b ¼ 1; c ¼ À1=6) [8].
It is worth mentioning that there are many attempts in the literature to obtain the exact solutions of Eq. (2), as well as the special case of it, LGH Eq. (3), using different analytical methods. Considering the more general model in Eq. (2), there are different schemes to obtain the exact and explicit solutions, such as the extended mapping method, the hyperbola function method, the Improved tanh-method, the modified extended tanh-function method, a direct and unified algebraic method [8,12,13,16,29], while for the particular case, LGH Eq. (3), the travelling wave solutions have been investigated in different contexts using different approaches such as solitary wave ansatz method in [4], the first integral method in [9], the G 0 =G; 1=G ð Þ -expansion method in [26], the improved Bernoulli sub-equation function (IBSEFM) method in [1], the sine-Gordon expansion (SGE) method in [7], the extended tanh scheme in [6], the generalized Kudryashov technique in [11].
In this paper, under some conditions, the integrability of LGH Eq. (3) is proved by deriving the Lax pair using AKNS scheme. It is worth noting that, there are many different methods for deducing the Lax pair for integrable NLEE's, such as the prolongation method [30], the extended homogeneous balance method [31], the singular manifold method [32][33][34], and the AKNS approach [21,22,35]. Accordingly, and using the inverse scattering transformation (IST) method, we obtain a closed form solution to Eq. (3) of type Kink soliton solution.
The residue of the paper is organized as follows: in Sect. 2, we investigate the Lax pair for Eq. (3) using AKNS approach. The inverse scattering transformation is applied to Eq. (3) in Sect. 3. In Sect. 4, the kink type soliton solution is obtained and graphically represented in 2d and 3d plots and a comparison between our solution and different solutions in the literature is represented in tabularized form.

The derivation of Lax pair
In this section, we derive the Lax pair in matrix form for Eq. (3) by applying a simple modification of the standard AKNS formalism.
Make the following transformation to Eq. (3): Then by chain rule, we have From which we have According to the above, the differential terms can be written in the form Then Eq. (3) become Consider the following linear spectral problems w x ¼ Àiak qðx; tÞ rðx; tÞ iak ! w ð9Þ where w x; t ð Þ ¼ w 1 x; t ð Þ; w 2 x; t ð Þ ð Þ T and k is the spectral parameter with k t ¼ 0.
From Eq. (9) we have and from Eq. (10) we have From Eqs. (11) and (13) we have From Eqs. (12) and (14) we have The compatibility condition w 1xt ¼ w 1tx yields While the compatibility condition w 2xt ¼ w 2tx yields Equating the coefficients of w 1 and w 2 to zero, we obtain the following system of equations Now, expand A; B and C as follows Then Substituting from Eq. (23) into Eq. (21), we obtain the following system of equations Equating the coefficients of k 0 to zero gives While equating the coefficients of k À1 to zero gives Multiply Eq. (30) by 2i, and use Eqs. (26,27), we have Suppose the following quantities for a; a x; t ð Þ; b x; t ð Þ; c x; t ð Þ; qðx; tÞ and rðx; tÞ Under these considerations, Eq. (30) become i.e., which is the LGH equation given in Eq. (3). We also noted that under the considerations given in Eqs. (31)(32)(33)(34), Eqs. (29) and (30) are satisfied. Therefore, The Lax pair for LGH Eq. (3) can be written as Remark (1) Under the assumptions given in Eqs. (31)(32)(33)(34), at x j j ! 1 with the initial condition u ! 0, the limits of A; B and C defined in Eq. (22) are evaluated as.
3 The inverse scattering transformation for LGH Eq. (3) In this section, the inverse scattering transform (IST) procedures will be followed for Eq. (3). Starting from Eq. (9), which may be written in the form Assume that q x; t ð Þ; rðx; tÞ and it's derivative with respect to x are decay sufficiently rapidly as.
x j j ! 1, then we can introduce the following four solutions to Eq. (9), which are defined by their asymptotic behaviors at infinity as These solutions may be written in matrix form as Since solutions given in Eq. (42) and Eq. (43) are linearly dependent, where there Wronskian denoted Then solutions W þ and W À may be connected via the scattering matrix denoted SðkÞ as follows i.e., The solution W þ x; k ð Þ may always be represented by an integral over an appropriate Kernel, while W À can be obtained using the relation (47) i.e., where at x j j ! 1 and Using integration by parts for R 1 x KW 0 dy we have Then Eq. (55) become Equation (58) Then Assume that Then Eq. (47) may be rewritten as Substituting from Eq. (50) into Eq. (66) gives To get an integral equation for K, we multiply Eq. (67) by 1 2p W 0 z; k ð Þ for z [ x, then we have Integrate with respect to k a long appropriate contour in the complex k-plane from À1 to þ1. This contour is indented into the upper half-plane for terms involving e iakz and into the lower half-plane for e Àiakz , we call these contours C þ and C À , respectively. One can arrive to the matrix Marchenko equation where q k ð Þ and q k ð Þ are defined as reflection coefficients, while c n t ð Þ and c m t ð Þ are defined as the normalizing coefficients given by The solution of the matrix Marchenko Eq. (69) gives Kðx; zÞ from which and by using Eqs. (61, 62) we can recover the potentials qðxÞ and rðxÞ.when A $ dðkÞ, B $ 0 and C $ 0 as x j j ! 1 and u ! 0, the time evolution of scattering data can be evaluated as follow (see e.g. [20,21]) Then from Remark (1), one can obtain the time evolution of the scattering data for Eq. (3) as follows 4 Travelling wave solutions for LGH Eq. (3) In this section, we consider the reflectionless potential q k ð Þ ¼ q k ð Þ ¼ 0 and N ¼ N ¼ 1. Substitute by these considerations in Eqs. (70-78) we have Then Eq. (81) become Inserting Eq. (86) in Eq. (82), we have Assume that k 1 À k 1 is pure imaginary, then Eq. (87) become Inserting Eq. (88) into Eq. (85), we obtain Inserting Eq. (89) into Eq. (88), we obtain Then from Eq. (61) In similar way, by solving Eqs. (83, 84) and using Eq. (62) we obtain the potential rðx; tÞ as Case (1) Let k 1 ¼ ib; k 1 ¼ Àib, then according to the symmetry q ¼ Àr we have from Eq. (91) and Eq. (92) that c 1 0 ð Þ ¼ Àc 1 0 ð Þ, let c 1 0 ð Þ ¼ 2ib, then at these symmetries, and since a ¼ m 2 , the potentials qðx; tÞ in Eq. (91) and rðx; tÞ in Eq. (92) can be written as And Since qðx; tÞ ¼ À ffiffi Use the transformation (4) we obtain the solution u 1 X; T ð Þ for the LGH Eq. (3) as where m and n are real parameters. Case (2) Let k 1 and k 1 as defined in case (1) and assume that Since qðx; tÞ ¼ À ffiffi As we mentioned earlier that there exist some attempts for obtaining travelling wave solutions for LGH Eq. (3) in the literature.

Conclusion
In this article, we have investigated the Landau-Ginzburg-Higgs (LGH) equation which explain the superconductivity and drift cyclotron waves in radially inhomogeneous plasma for coherent ion-cyclotron waves by the aid of the inverse scattering transformation (IST) method. The integrability has been proved by deriving the Lax pair under a simple modification of Ablowitz-Kaup-Newell-Segur (AKNS) formalism. Different types of travelling wave solutions have been established and graphically represented in 2d and 3d profiles.