Exact solutions of (2 + 1)-Ablowitz-Kaup-Newell-Segur equation

Different studies can be seen on NPDEs in various disciplines such as physical events, applied sciences, natural events and engineering. Scientists have used many methods to find analytical solutions of NPDEs. Some of these methods are new sub equation method [1, 2], (1/G′)-expansion method [3, 4], Homotopy analysis and Homotopy-Pade methods [5], (G′/G)-expansion method [6, 7], Variational Iteration Algorithm-I [8], decomposition method [9,10,11], sumudu transform method [12], sub equation method [13, 14], collocation method [15], the auto-Bäcklund transformation method [16], the Clarkson-Kruskal (CK) direct method [17], first integral method [18], homogeneous balance method [19], SGEEM [20], residual power series method [21], Modified Kudryashov method [22], sine-Gordon expansion method (SGEM) [23, 24] and so on [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46].

Many analytical methods have been added to the literature to find the exact solution of the NPDEs. We know that each method produces different types of solutions. For example, sub equation method produces dark solitons, Hirota bilinear form lump solitons. In this study, we aim to produce hyperbolic type wave solutions using the (1/G′)-expansion method. For this, we have dealt with the AKNS equation with strong nonlinearity. So let's take the AKNS equation [47], (1) 4uxt+uxxxz+8uxzux+4uzuxx=0. 4{u_{xt}} + {u_{xxxz}} + 8{u_{xz}}{u_x} + 4{u_z}{u_{xx}} = 0. Scientists related to the equation of AKNS equation have produced several studies in the literature. Some of these are obtained traveling wave reductions of AKNS equation applying classical Lie symmetries [48], new soliton like solutions of AKNS equation obtained using the extended auxiliary equation method [49], the truncated Painlevè analysis of AKNS equation obtained with their symmetries [50], complex combined dark-bright soliton solutions of AKNS equation attained using SGEM [51], general solutions of the AKNS system have been obtained via standard truncated Painlevè expansions [52].

Consider form of NPDEs, (2) †(u,∂u∂t,∂u∂x,∂u∂z,∂2u∂x2,...)=0. \dagger \left( {u,{{\partial u} \over {\partial t}},{{\partial u} \over {\partial x}},{{\partial u} \over {\partial z}},{{{\partial ^2}u} \over {\partial {x^2}}},...} \right) = 0. Let ξ = x + z − ct, u(x,z,t) = u(ξ) = u, where c is a constant and the speed of the wave. We can convert it into the following nODE (3) θ(u,−cu′,u″,u‴,...)=0. \theta \left( {u, - cu',u'',u''',...} \right) = 0. The solution that makes the Eq. (3) correct is assumed to have the form (4) u(ξ)=a0+∑i=1mai(1G′)i, u\left( \xi \right) = {a_0} + \sum\limits_{i = 1}^m {a_i}{\left( {{1 \over {G'}}} \right)^i}, where a_i, (i = 0,1,...,m) are constants, G = G(ξ) provides the following second order IODE (5) G″+λG′+μ=0, G'' + \lambda G' + \mu = 0, where λ and μ are constants to be determined after, (6) 1G′(ξ)=1−μλ+Bcosh(ξλ)−Bsinh(ξλ), {1 \over {G'\left( \xi \right)}} = {1 \over { - {\mu \over \lambda } + B\cosh \left( {\xi \lambda } \right) - B\sinh \left( {\xi \lambda } \right)}}, where B is integral constant. If the desired derivatives of the Eq. (4) are calculated and substituting in the Eq. (3), a polynomial with the argument (1/G′) is attained. An algebraic equation system is created by equalizing the coefficients of this polynomial to zero. The equation are solved using package program and put into place in the default Eq. (3) solution function. Lastly, the solutions of Eq. (2) are found.

Let's consider the Eq. (1). By applying the transmutation ξ = x + z − ct, allows us to convert Eq. (1) into an nonlinear ODE (7) −4cu′+u‴+6(u′)2=0. - 4cu' + u''' + 6{\left( {u'} \right)^2} = 0. In Eq. (7), m = 1 is found from the definition of balancing term and written according to Eq. (4) (8) u(ξ)=a0+a1(1G′), a1≠0. u\left( \xi \right) = {a_0} + {a_1}\left( {{1 \over {G'}}} \right),\quad {a_1} \ne 0. If Eq. (8) is replacing in Eq. (7) and the coefficients are equal to zero, the following system is attained. (9) (1G′[ξ])1:−4cλa1+λ3a1=0,(1G′[ξ])2:−4cμa1+7λ2μa1+6λ2a12=0,(1G′[ξ])3:12λμ2a1+12λμa12=0,(1G′[ξ])4:6μ3a1+6μ2a12=0. \matrix{ {{{\left( {{1 \over {G'\left[ \xi \right]}}} \right)}^1}: - 4c\lambda {a_1} + {\lambda ^3}{a_1} = 0,} \hfill \cr {{{\left( {{1 \over {G'\left[ \xi \right]}}} \right)}^2}: - 4c\mu {a_1} + 7{\lambda ^2}\mu {a_1} + 6{\lambda ^2}a_1^2 = 0,} \hfill \cr {{{\left( {{1 \over {G'\left[ \xi \right]}}} \right)}^3}:12\lambda {\mu ^2}{a_1} + 12\lambda \mu a_1^2 = 0,} \hfill \cr {{{\left( {{1 \over {G'\left[ \xi \right]}}} \right)}^4}:6{\mu ^3}{a_1} + 6{\mu ^2}a_1^2 = 0.} \hfill \cr } When we solve Eq. (9) system with the help of computer ready package program, we can present the following situations.

Case1. (10) a1=−μ, c=λ24, {a_1} = - \mu ,\quad c = {{{\lambda ^2}} \over 4}, if the values in Eq. (10) are written in place of Eq. (8), the hyperbolic type solution below are obtain (11) u1(x,z,t)=−μ−μλ+B(cosh[λ(x+z−tλ24)]−sinh[λ(x+z−tλ24)])+a0. {u_1}\left( {x,z,t} \right) = - {\mu \over { - {\mu \over \lambda } + B\left( {\cosh \left[ {\lambda \left( {x + z - {{t{\lambda ^2}} \over 4}} \right)} \right] - \sinh \left[ {\lambda \left( {x + z - {{t{\lambda ^2}} \over 4}} \right)} \right]} \right)}} + {a_0}. As seen in figure 1, we can observe asymptotic behavior in the 0 < x < 5 range. Different interpretations and studies can be made for this situation.

Fig. 1

Case2. (12) a1=−μ, λ=2c, {a_1} = - \mu ,\quad \lambda = 2\sqrt c , if the values in Eq. (12) are written in place of Eq. (8), the hyperbolic type solution below are obtain (13) u2(x,z,t)=−μ−μ2c+B(cosh[(−ct+x+z)2c]−sinh[(−ct+x+z)2c])+a0. {u_2}\left( {x,z,t} \right) = - {\mu \over { - {\mu \over {2\sqrt c }} + B\left( {\cosh \left[ {\left( { - ct + x + z} \right)2\sqrt c } \right] - \sinh \left[ {\left( { - ct + x + z} \right)2\sqrt c } \right]} \right)}} + {a_0}.

Fig. 2

In this study, analytical solutions of the AKNS equation were obtained via the (1/G′)-expansion method. Although the nonlinear terms of this equation are strong, traveling wave solutions are obtained. The solutions obtained are of the hyperbolic type. By giving special values to the parameters in these solutions, 3D, 2D and contour graphics are presented. We consider these wave solutions are especially important for scientists studying shock wave structure and asymptotic behavior. In addition, it will be much more valuable when the physical meaning is loaded on the constants in these solutions. Algebraic equation obtained by classical expansion methods is simpler. However, the disadvantage of the expansion method we use is that it only provides hyperbolic type solution. The advantage of this method is that it is different from the solutions obtained in other expansion methods. While obtaining solutions in many NPDEs, we encountered less complex operations while solving this equation. However, this study also offers solutions with two cases. It is seen that the method applied in the study is easy, reliable, useful and can be used in future studies. Ready-made package programs are used to facilitate the complexity of the procedure in this package.