New Complex and Hyperbolic Forms for Ablowitz–Kaup–Newell–Segur Wave Equation with Fourth Order

The study of nonlinear differential equation (NLDE) solutions attracts the attention of scientists. NLDE is used in many areas such as physics and chemistry. We need to explore the solutions of NLDE that have an important place in applied mathematics. Some scientists have explored these solutions.

In recent years, several effective methods, including extended tanh method [1, 2], first integral method [3, 4], He’s semi-inverse method [5, 6], sine–cosine method [7, 8], dynamical system method [9], modified simple equation method [10, 11], Bell-polynomial method [12], simplified Hirota’s method [13], Cole–Hopf transformation method [14], sine–cosine method [15], tanh method [16], generalized tanh function method [17], improved F-expansion method with Riccati equation [18, 19], modified exp(–Ω (ξ))-expansion function method [20, 21], and improved Bernoulli subequation function method [22], have been successfully considered to find the exact solutions of a wide variety of NLDEs and many others [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88].

In this study, we obtain new complex hyperbolic function solutions to the nonlinear Ablowitz-Kaup–Newell–Segur wave equation (AKNSWE) with fourth order [23], which is defined as

4uxt+uxxxt+8uxuxy+4uxxuy−γuxx=0, $$\begin{array}{} \displaystyle 4{u_{{\rm{xt}}}} + {u_{{\rm{xxxt}}}} + 8{u_x}{u_{{\rm{xy}}}} + 4{u_{{\rm{xx}}}}{u_y} - {\rm{\gamma }}{{\rm{u}}_{{\rm{xx}}}} = 0, \end{array}$$(1)

where γ is a real constant with a non-zero value, by using the sine-Gordon expansion method (SGEM).

Let us consider the following sine-Gordon equation [24, 25, 26]:

uxx−utt=m2sinu, $$\begin{array}{} \displaystyle {u_{xx}} - {u_{tt}} = {m^2}\sin \left( u \right), \end{array}$$(2)

where u = u(x, t) and m is a real constant. When we apply the wave transform ξ = μ (x – ct) to Eq. (2), we obtain the nonlinear ordinary differential equation (NODE) as follows:

U″=m2μ21−c2sinU, $$\begin{array}{} \displaystyle U'' = \frac{{{m^2}}}{{{\mu ^2}\left( {1 - {c^2}} \right)}}\sin \left( U \right), \end{array}$$(3)

where U = U(ξ), ξ is the amplitude of the travelling wave, and c is the velocity of the travelling wave. If we reconsider Eq. (3), we can write it in the full simplified version as follows:

U2′2=m2μ21−c2sin2U2+K, $$\begin{array}{} \displaystyle {\left[ {{{\left( {\frac{U}{2}} \right)}^\prime }} \right]^2} = \frac{{{m^2}}}{{{\mu ^2}\left( {1 - {c^2}} \right)}}{\sin ^2}\left( {\frac{U}{2}} \right) + K, \end{array}$$(4)

where K is the integration constant. When we resubmit as K=0,wξ=U2,anda2=m2μ21−c2 $\begin{array}{} \displaystyle K = 0,{\rm{ }}w\left( \xi \right) = \frac{U}{2}, ~\text{and}~ {a^2} = \frac{{{m^2}}}{{{\mu ^2}\left( {1 - {c^2}} \right)}} \end{array}$ in Eq. (4), we can obtain the following equation:

w′=asinw. $$\begin{array}{} \displaystyle {w^\prime } = a\sin \left( w \right). \end{array}$$(5)

If we put a = 1 in Eq. (5), we can obtain the following equation:

w′=sinw. $$\begin{array}{} \displaystyle {w^\prime } = \sin \left( w \right). \end{array}$$(6)

If we solve Eq. (6) by using separation of variables, we find the following two significant equations:

sinw=sinwξ=2peξp2e2ξ+1p=1=sec⁡hξ, $$\begin{array}{} \displaystyle \sin \left( w \right) = \sin \left( {w\left( \xi \right)} \right) = {\left. {\frac{{2p{e^\xi }}}{{{p^2}{e^{2\xi }} + 1}}} \right|_{p = 1}} = \sec h\left( \xi \right), \end{array}$$(7)

cosw=coswξ=p2e2ξ−1p2e2ξ+1p=1=tanhξ, $$\begin{array}{} \displaystyle \cos \left( w \right) = \cos \left( {w\left( \xi \right)} \right) = {\left. {\frac{{{p^2}{e^{2\xi }} - 1}}{{{p^2}{e^{2\xi }} + 1}}} \right|_{p = 1}} = \tanh \left( \xi \right), \end{array}$$(8)

where p is the integral constant having a non-zero value. For the solution of following nonlinear partial differential equation

Pu,ux,ut,⋯=0, $$\begin{array}{} \displaystyle P\left( {u,{u_x},{u_t}, \cdots } \right) = 0, \end{array}$$(9)

let us consider

Uξ=∑i=1δtanhi−1ξBisec⁡hξ+Aitanhξ+A0. $$\begin{array}{} \displaystyle U\left( \xi \right) = \sum\limits_{i = 1}^\delta {{{\tanh }^{i - 1}}} \left( \xi \right)\left[ {{B_i}\sec h\left( \xi \right) + {A_i}\tanh \left( \xi \right)} \right] + {A_0}. \end{array}$$(10)

We can rewrite Eq. (10) according to Eqs. (7 and 8) as follows:

Uw=∑i=1δcosi−1wBisinw+Aicosw+A0. $$\begin{array}{} \displaystyle U\left( w \right) = \sum\limits_{i = 1}^\delta {{{\cos }^{i - 1}}} \left( w \right)\left[ {{B_i}\sin \left( w \right) + {A_i}\cos \left( w \right)} \right] + {A_0}. \end{array}$$(11)

Under the terms of homogenous balance technique, we can determine the values of n under the terms of NODE. Let the coefficients of sinⁱ(w) cos^j(w) all be zero; it yields a system of equations. Solving this system by using Wolfram Mathematica 9 gives the values of .A_i, B_i, μ and c Finally, substituting the values of A_i, B_i, μ and c in Eq.(10), we can find new analytical solutions to Eq. (9).

SGEM has been successfully used to obtain analytical solutions to the AKNSWE. Using u(x, y, t) = U(ξ), ξ = x + y + ωt in Eq. (1), we get

4ω−γU′+ωU‴+6U′2=0. $$\begin{array}{} \displaystyle \left( {4\omega - \gamma } \right)U' + \omega U''' + 6{\left( {U'} \right)^2} = 0. \end{array}$$

If we consider V = U′, we can obtain the following nonlinear differential equation:

4ω−γV+6V2+ωV″=0. $$\begin{array}{} \displaystyle \left( {4\omega - \gamma } \right)V + 6{V^2} + \omega V'' = 0. \end{array}$$(12)

After balancing, we find δ = 2. For this value, Eq. (11) can be written as

Vw=B1sinw+A1cosw+B2coswsinw+A2cos2w+A0. $$\begin{array}{} \displaystyle V\left( w \right) = {B_1}{\rm{sin}}\left( w \right) + {A_1}{\rm{cos}}\left( w \right) + {B_2}{\rm{cos}}\left( w \right){\rm{sin}}\left( w \right) + {A_2}{\rm{co}}{{\rm{s}}^2}\left( w \right) + {A_0}. \end{array}$$(13)

If we put Eq. (13) with second derivation into Eq. (12), we can find a trigonometric equation. Solving this, we can choose the following coefficients:

Case 1

A0=ω3;A1=0;A2=−ω2;B1=0;B2=−iω2;γ=3ω. $$\begin{array}{} \displaystyle {A_0} = \frac{\omega }{3};{A_1} = 0;{A_2} = - \frac{\omega }{2};{B_1} = 0;{B_2} = - \frac{{{\rm{i}}\omega }}{2};\gamma = 3\omega . \end{array}$$

Inserting these values into Eq. (10) with V = U′ yields

U1x,y,t=−16ωx+y+tω−3isechx+y+tω−3tanhx+y+tω. $$\begin{array}{} \displaystyle {U_1}\left( {x,y,t} \right) = - \frac{1}{6}\omega \left( {x + y + t\omega - 3{\rm{i}}s{\rm{ech}}\left[ {x + y + t\omega } \right] - 3{\rm{tanh}}\left[ {x + y + t\omega } \right]} \right). \end{array}$$(14)

Case 7

Finally, if we take into account the following coefficient into Eq. (10) with V = U′,

A0=γ10;A1=0;A2=−γ10;B1=0;B2=iγ10;ω=γ5, $$\begin{array}{} \displaystyle {A_0} = \frac{\gamma }{{10}};{A_1} = 0;{A_2} = - \frac{\gamma }{{10}};{B_1} = 0;{B_2} = \frac{{{\rm{i}}\gamma }}{{10}};\omega = \frac{\gamma }{5}, \end{array}$$

we obtain another complex hyperbolic function solutions as follows:

U7x,y,t=γ−5i+5coth12γt5+x+y. $$\begin{array}{} \displaystyle {U_7}\left( {x,y,t} \right) = \frac{\gamma }{{ - 5i + 5{\rm{coth}}\left( {\frac{1}{2}\left( {\frac{{\gamma t}}{5} + x + y} \right)} \right)}}. \end{array}$$(20)

U7x,y,t=γ−5i+5coth12γt5+x+y. $$\begin{array}{} \displaystyle {U_7}\left( {x,y,t} \right) = \frac{\gamma }{{ - 5{\rm{i}} + 5{\rm{coth}}\left[ {\frac{1}{2}\left( {\frac{{\gamma t}}{5} + x + y} \right)} \right]}}. \end{array}$$(21)

Fig. 7

Fig. 8

Fig. 9

In summary, we have successfully applied the SGEM to Eq. (1) to find new complex hyperbolic function solutions. We have plotted 2D and 3D surfaces of the solutions along with contour surfaces under the suitable values of parameters by using computational program along with contour surfaces of them. It has been observed that the travelling wave solutions obtained in this paper are entirely new complex hyperbolic function solutions compared with [23]. To the best of our knowledge, the application of SGEM to the nonlinear Ablowitz–Kaup–Newell–Segur wave equation with fourth order has not been submitted to literature before.