1. Introduction
Solitons have been widely studied in theory and experiment in recent years. Nowadays, the investigation of the soliton solutions of a number of complex nonlinear equations plays a considerable role due to the expectant effectuation in the real world, especially in different aspects of mathematical and physical phenomena [1–9]. Most complex phenomena arising in applied science, such as nuclear physics, chemical reactions, signal processing, optical fibers, fluid mechanics, plasma, nonlinear optics and ecology, can be sometimes modeled and described by these equations. Hereby, a massive number of mathematicians and physicists have attempted to invent various approaches by which one can obtain the soltion solutions of such equations. Among several present methods, we mention the Riccati-Bernoulli sub-ODE method [10, 11], exp-function method [12, 13], sine-cosine method [14, 15], tanh-sech method [16, 17], extended tanh-method [18, 19], F-expansion method [20–22], homogeneous balance method [23, 24], Jacobi elliptic function method [25, 26], the unified method and its generalized form [27–33], and so on. This work is established to utilize the extended Fan Sub-equation technique [34, 35] in determining the soliton and elliptic solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain (HFSC) equation [36–40].
The HFSC equation [36–40] is given by:
$\begin{eqnarray}{\rm{i}}{\psi }_{t}+{\varrho }_{1}{\psi }_{{xx}}+{\varrho }_{2}{\psi }_{{yy}}+{\varrho }_{3}{\psi }_{{xy}}-{\varrho }_{4}| \psi {| }^{2}\psi =0.\end{eqnarray}$
Here, ψ = ψ(x, y, t) is a complex valued function, x, y and t denote the scaled spatial and time coordinates, respectively and the coefficients ϱj for j = 1, 2, 3, 4; are real constants given by [7, 39] $\begin{eqnarray*}\begin{array}{rcl}{\varrho }_{1} & = & {\kappa }^{4}({\rm{\Lambda }}+{{\rm{\Lambda }}}_{2}),{\varrho }_{2}={\kappa }^{4}({{\rm{\Lambda }}}_{1}+{{\rm{\Lambda }}}_{2}),\\ {\varrho }_{3} & = & 2{\kappa }^{4}{{\rm{\Lambda }}}_{2},{\varrho }_{4}=2{\kappa }^{4}{\rm{\Omega }},\end{array}\end{eqnarray*}$
where the parameters Λ, Λ1 represent the coefficients of bilinear exchange interactions in the xy-plane, Λ2 denotes the neighboring interaction along the diagonal, Ω is the uniaxial crystal field anisotropy parameter, and κ is a lattice parameter.Heisenberg ferromagnetic spin chain equation with different magnetic interactions in the classical and semi-classical continuum limit have been identified as interesting nonlinear model systems exhibiting integrability properties including soliton spin excitations. This equation can be used to depict the propagation of long waves, which has many applications in the percolation of water.
The rest of this continuing article is methodized as follows: In section 2 , we propound the formation of the extended Fan Sub-equation method and we implement this technique to find new soliton and elliptic solutions of the HFSC equation. The physical behavior of the solutions together with their graphical illustration is within section 3 . Finally, section 4 is comprised of conclusions in a suitable manner.
2. Mathematical analysis
To solve equation (1 ), we first need to apply the traveling wave transformation
$\begin{eqnarray}\psi ={ \mathcal V }(\xi ){{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}},\quad \xi ={ax}+{by}-\mu t,\quad {\rm{\Phi }}={px}+{qy}-{rt},\end{eqnarray}$
where a, b, μ, p, q, and r are constants to be determined.Utilizing the wave transformation (2 ) in equation (1 ), we attain the following imaginary and real parts, respectively:4 ), we have n = 1. Suppose equation (4 ) has the solution of the form
$\begin{eqnarray}\mu =2a{\varrho }_{1}p+2b{\varrho }_{2}q+{\varrho }_{3}\left({bq}+{aq}\right),\end{eqnarray}$
$\begin{eqnarray}{\delta }_{1}{ \mathcal V }^{\prime\prime} +{\delta }_{2}{{ \mathcal V }}^{3}+{\delta }_{3}{ \mathcal V }=0,\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}{\delta }_{1} & = & {\varrho }_{4}{a}^{2}+{\varrho }_{2}{b}^{2}+{\varrho }_{3}{ab},\\ {\delta }_{2} & = & -{\varrho }_{4},\\ {\delta }_{3} & = & r-{\varrho }_{1}{p}^{2}-q\left({\varrho }_{2}q+{\varrho }_{3}p\right).\end{array}\end{eqnarray*}$
By applying the homogeneous balance to equation ( $\begin{eqnarray}{ \mathcal V }={a}_{0}+{a}_{1}\phi (\xi ),\end{eqnarray}$
where φ satisfies the following general elliptic equation, $\begin{eqnarray}{\left(\displaystyle \frac{{\rm{d}}\phi (\xi )}{{\rm{d}}\xi }\right)}^{2}={\zeta }_{0}+{\zeta }_{1}\phi (\xi )+{\zeta }_{2}{\phi }^{2}(\xi )+{\zeta }_{3}{\phi }^{3}(\xi )+{\zeta }_{4}{\phi }^{4}(\xi ),\end{eqnarray}$
ζ i (i = 0, 1, 2, 3, 4) are real constants.Substituting (5 ) along (6 ) in (4 ) and collecting the coefficients of ${\phi }^{j}{\phi }^{(k)}$,
$\begin{eqnarray*}\begin{array}{rcl}{a}_{0}\left({a}_{0}^{2}{\delta }_{2}+{\delta }_{3}\right)+\displaystyle \frac{1}{2}{a}_{1}{\delta }_{1}{\zeta }_{1} & = & 0,\\ {a}_{1}\left(3{a}_{0}^{2}{\delta }_{2}+{\delta }_{3}\right)+{a}_{1}{\delta }_{1}{\zeta }_{2} & = & 0,\\ 3{a}_{0}{a}_{1}^{2}{\delta }_{2}+\displaystyle \frac{3}{2}{a}_{1}{\delta }_{1}{\zeta }_{3} & = & 0,\\ {a}_{1}^{3}{\delta }_{2}+2{a}_{1}{\delta }_{1}{\zeta }_{4} & = & 0,\end{array}\end{eqnarray*}$
we select variables suitably, to have the most of ζi, (i = 0, 1, 2, 3, 4), $\begin{eqnarray*}\begin{array}{rcl}{\zeta }_{1} & = & -\displaystyle \frac{2{a}_{0}\left({a}_{0}^{2}{\delta }_{2}+{\delta }_{3}\right)}{{a}_{1}{\delta }_{1}},\\ {\zeta }_{2} & = & -\displaystyle \frac{3{a}_{0}^{2}{\delta }_{2}+{\delta }_{3}}{{\delta }_{1}},\\ {\zeta }_{3} & = & -\displaystyle \frac{2{a}_{0}{a}_{1}{\delta }_{2}}{{\delta }_{1}},\\ {\zeta }_{4} & = & -\displaystyle \frac{{a}_{1}^{2}{\delta }_{2}}{2{\delta }_{1}},\end{array}\end{eqnarray*}$
which give $\begin{eqnarray*}\begin{array}{rcl}{a}_{0} & = & \displaystyle \frac{\sqrt{{\delta }_{1}\left(-{\zeta }_{2}\right)-{\delta }_{3}}}{\sqrt{3}\sqrt{{\delta }_{2}}},\\ {a}_{1} & = & \displaystyle \frac{\sqrt{2}\sqrt{-{\delta }_{1}}\sqrt{{\zeta }_{4}}}{\sqrt{{\delta }_{2}}},\end{array}\end{eqnarray*}$
therefore, $\begin{eqnarray}\psi =({a}_{0}+{a}_{1}\phi (\xi )){{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{eqnarray}$
We have following solutions, for more details see also [34, 35].Case I.
If ${\zeta }_{0}={\vartheta }_{3}^{2},{\zeta }_{1}=2{\vartheta }_{1}{\vartheta }_{3}$, ${\zeta }_{2}=2{\vartheta }_{2}{\vartheta }_{3}+{\vartheta }_{1}^{2},{\zeta }_{3}=2{\vartheta }_{1}{\vartheta }_{2},{\zeta }_{4}={\vartheta }_{2}^{2}$, where ${\vartheta }_{1},{\vartheta }_{2},$ and ${\vartheta }_{3}$ are arbitrary constants. The solutions of (1 ) are ${\psi }_{\eta }^{I},(\eta =1,2,\,\ldots ,\,24).$ Some of important solitons are listed below.
Type I: when ${\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}\gt 0$, ${\vartheta }_{1}{\vartheta }_{2}\ne 0$, ${\vartheta }_{2}{\vartheta }_{3}\ne 0$. The following family of dark solitons is obtained as
$\begin{eqnarray}\begin{array}{l}{\psi }_{1}^{I}(\xi )=\left[{a}_{0}\right.\\ \left.+\,{a}_{1}\left(-\displaystyle \frac{\sqrt{{\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}}\tanh \left(\tfrac{1}{2}\xi \sqrt{{\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}}\right)+{\vartheta }_{1}}{2{\vartheta }_{2}}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{array}\end{eqnarray}$
The family of combined bright-dark soliton is obtained as, $\begin{eqnarray}\begin{array}{rcl}{\psi }_{3}^{I}(\xi ) & = & \left[{a}_{0}-\displaystyle \frac{{a}_{1}}{2{\vartheta }_{2}}\left(\sqrt{{\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}}\right.\right.\\ & & \times \,\left(i\,{\rm{sech}} \left(\xi \sqrt{{\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}}\right)\right.\\ & & \left.\left.\left.+\tanh \left(\xi \sqrt{{\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}}\right)\right)+{\vartheta }_{1}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{array}\end{eqnarray}$
The family of combined dark-singular solitons is obtained as $\begin{eqnarray}\begin{array}{rcl}{\psi }_{5}^{I}(\xi ) & = & \left[{a}_{0}-\displaystyle \frac{{a}_{1}}{2{\vartheta }_{2}}\left(\sqrt{{\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}}\right.\right.\\ & & \times \,\left(\tanh \left(\displaystyle \frac{1}{4}\xi \sqrt{{\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}}\right)\right.\\ & & \left.\left.\left.+\coth \left(\displaystyle \frac{1}{4}\xi \sqrt{{\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}}\right)\right)+{\vartheta }_{1}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{array}\end{eqnarray}$
The family of solitons is obtained as $\begin{eqnarray}\begin{array}{rcl}{\psi }_{10}^{I}(\xi ) & = & \left[{a}_{0}+{a}_{1}\left(2\cosh \left(\xi \sqrt{{\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}}\right)\right.\right.\\ & & \times \left(\sqrt{{\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}}\sinh \left(\xi \sqrt{{\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}}\right)\right.\\ & & -\left({\vartheta }_{1}\cosh \left(\xi \sqrt{{\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}}\right)\right.\\ & & \left.\left.{\left.\left.\pm i\sqrt{{\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}}\right)\right)}^{-1}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{array}\end{eqnarray}$
Type II: when ${\vartheta }_{1}^{2}-4{\vartheta }_{2}{\vartheta }_{3}\lt 0$, ${\vartheta }_{1}{\vartheta }_{2}\ne 0$, ${\vartheta }_{2}{\vartheta }_{3}\ne 0$. The following families of periodic solitons are obtained $\begin{eqnarray}\begin{array}{l}{\psi }_{13}^{I}(\xi )=\left[{a}_{0}\right.\\ \left.+\,{a}_{1}\left(-\displaystyle \frac{\sqrt{4{\vartheta }_{2}{\vartheta }_{3}-{\vartheta }_{1}^{2}}\tan \left(\tfrac{1}{2}\xi \sqrt{4{\vartheta }_{2}{\vartheta }_{3}-{\vartheta }_{1}^{2}}\right)-{\vartheta }_{1}}{2{\vartheta }_{2}}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\psi }_{20}^{I}(\xi )\,=\,\left[{a}_{0}+{a}_{1}\left(-\displaystyle \frac{2{\vartheta }_{3}\cos \left(\tfrac{1}{2}\xi \sqrt{4{\vartheta }_{2}{\vartheta }_{3}-{\vartheta }_{1}^{2}}\right)}{\sqrt{4{\vartheta }_{2}{\vartheta }_{3}-{\vartheta }_{1}^{2}}\sin \left(\tfrac{1}{2}\xi \sqrt{4{\vartheta }_{2}{\vartheta }_{3}-{\vartheta }_{1}^{2}}\right)+{\vartheta }_{1}\cos \left(\tfrac{1}{2}\xi \sqrt{4{\vartheta }_{2}{\vartheta }_{3}-{\vartheta }_{1}^{2}}\right)}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\psi }_{24}^{I}(\xi ) & = & \left[{a}_{0}+{a}_{1}\left(\left(4r\sin \left(\displaystyle \frac{1}{4}\xi \sqrt{4{\vartheta }_{2}{\vartheta }_{3}-{\vartheta }_{1}^{2}}\right)\right.\right.\right.\\ & & \times \left.\cos \left(\displaystyle \frac{1}{4}\xi \sqrt{4{\vartheta }_{2}{\vartheta }_{3}-{\vartheta }_{1}^{2}}\right)\right)\\ & & \times \,\left(2\sqrt{4{\vartheta }_{2}{\vartheta }_{3}-{\vartheta }_{1}^{2}}{\cos }^{2}\left(\displaystyle \frac{1}{4}\xi \sqrt{4{\vartheta }_{2}{\vartheta }_{3}-{\vartheta }_{1}^{2}}\right)\right.\\ & & -2{\vartheta }_{1}\sin \left(\displaystyle \frac{1}{4}\xi \sqrt{4{\vartheta }_{2}{\vartheta }_{3}-{\vartheta }_{1}^{2}}\right)\\ & & \times \,\cos \left(\displaystyle \frac{1}{4}\xi \sqrt{4{\vartheta }_{2}{\vartheta }_{3}-{\vartheta }_{1}^{2}}\right)\\ & & \left.\left.{\left.-\sqrt{4{\vartheta }_{2}{\vartheta }_{3}-{\vartheta }_{1}^{2}}\right)}^{-1}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{array}\end{eqnarray}$
Case II.
If ${\zeta }_{0}={\vartheta }_{3}^{2},{\zeta }_{1}=2{\vartheta }_{1}{\vartheta }_{3},{\zeta }_{2}=0$, ${\zeta }_{3}=2{\vartheta }_{1}{\vartheta }_{2},{\zeta }_{4}={\vartheta }_{2}^{2}$, the solutions of (1 ) are ${\psi }_{\eta }^{{II}},(\eta =1,2,\ldots ,12).$ A family of dark soliton is obtained
$\begin{eqnarray}\begin{array}{l}{\psi }_{1}^{{II}}(\xi )=\left[{a}_{0}\right.\\ \left.+\,{a}_{1}\left(-\displaystyle \frac{\sqrt{-6{\vartheta }_{2}{\vartheta }_{3}}\tanh \left(\tfrac{1}{2}\xi \sqrt{-6{\vartheta }_{2}{\vartheta }_{3}}\right)+\sqrt{-2{\vartheta }_{2}{\vartheta }_{3}}}{2{\vartheta }_{2}}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{array}\end{eqnarray}$
Another form of dark-singular soliton is obtained $\begin{eqnarray}\begin{array}{l}{\psi }_{5}^{{II}}(\xi )=\left[{a}_{0}\right.\left.\,+\,{a}_{1}\left(-\displaystyle \frac{\sqrt{-6{qr}}\left(\tanh \left(\tfrac{1}{4}\xi \sqrt{-6{qr}}\right)+\coth \left(\tfrac{1}{4}\xi \sqrt{-6{qr}}\right)\right)+2\sqrt{-2{qr}}}{4q}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}},\end{array}\end{eqnarray}$
Case III.
If ζ0 = ζ1 = 0, we have the following solution of (1 ) in the form ${\psi }_{\eta }^{{III}},(\eta =1,2,\ldots ,10)$ .
Type I: ${\zeta }_{2}=1,{\zeta }_{3}=\tfrac{-2{\lambda }_{3}}{{\lambda }_{1}},{\zeta }_{4}=\tfrac{{\lambda }_{3}^{2}-{\lambda }_{2}^{2}}{{\lambda }_{1}^{2}}$, where ${\lambda }_{1},{\lambda }_{2},{\lambda }_{3}$ are arbitrary constants.
$\begin{eqnarray}{\psi }_{1}^{{III}}(\xi )=\left[{a}_{0}+{a}_{1}\left(\displaystyle \frac{{\lambda }_{1}{\rm{sech}} (\xi )}{{\lambda }_{2}{\rm{sech}} (\xi )+{\lambda }_{3}}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{eqnarray}$
Type II: ${\zeta }_{2}=1,{\zeta }_{3}=\tfrac{-2{\lambda }_{3}}{{\lambda }_{1}},{\zeta }_{4}=\tfrac{{\lambda }_{3}^{2}+{\lambda }_{2}^{2}}{{\lambda }_{1}^{2}}$, where λ1, λ2, λ3 are arbitrary constants.17 )–(18 ). We obtain the families of bright and singular solitons as follows
$\begin{eqnarray}{\psi }_{2}^{{III}}(\xi )=\left[{a}_{0}+{a}_{1}\left(\displaystyle \frac{{\lambda }_{1}\mathrm{csch}(\xi )}{{\lambda }_{2}\mathrm{csch}(\xi )+{\lambda }_{3}}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{eqnarray}$
In particular, if we take ${\lambda }_{2}=0$ in above equations ( $\begin{eqnarray}{\psi }_{1}^{{III}}(\xi )=\left[{a}_{0}+{a}_{1}\left(\displaystyle \frac{{\lambda }_{1}{\rm{sech}} (\xi )}{{\lambda }_{3}}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{eqnarray}$
$\begin{eqnarray}{\psi }_{2}^{{III}}(\xi )=\left[{a}_{0}+{a}_{1}\left(\displaystyle \frac{{\lambda }_{1}\mathrm{csch}(\xi )}{{\lambda }_{3}}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{eqnarray}$
Type III: ${\zeta }_{2}=4,{\zeta }_{3}=-\tfrac{4\left(2{\lambda }_{2}+{\lambda }_{4}\right)}{{\lambda }_{1}}$, ${\zeta }_{4}=\tfrac{4{\lambda }_{2}^{2}+4{\lambda }_{4}{\lambda }_{2}+{\lambda }_{3}^{2}}{{\lambda }_{1}^{2}}$, where λ1, λ2, λ3, λ4 are arbitrary constants. $\begin{eqnarray}{\psi }_{3}^{{III}}(\xi )=\left[{a}_{0}+{a}_{1}\left(\displaystyle \frac{{\lambda }_{1}{{\rm{sech}} }^{2}(\xi )}{{\lambda }_{2}\tanh (\xi )+{\lambda }_{3}+{\lambda }_{4}{{\rm{sech}} }^{2}(\xi )}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{eqnarray}$
Type IV: ${\zeta }_{2}=4,{\zeta }_{3}=\tfrac{4\left({\lambda }_{4}-2{\lambda }_{2}\right)}{{\lambda }_{1}}$, ${\zeta }_{4}=\tfrac{4{\lambda }_{2}^{2}-4{\lambda }_{4}{\lambda }_{2}+{\lambda }_{3}^{2}}{{\lambda }_{1}^{2}}$, where λ1, λ2, λ3, λ4 are arbitrary constants. $\begin{eqnarray}{\psi }_{4}^{{III}}(\xi )=\left[{a}_{0}+{a}_{1}\left(\displaystyle \frac{{\lambda }_{1}{\mathrm{csch}}^{2}(\xi )}{{\lambda }_{2}\coth (\xi )+{\lambda }_{3}+{\lambda }_{4}{\mathrm{csch}}^{2}(\xi )}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{eqnarray}$
In particular, if we consider λ2 = λ4; another family of dark and singular solitons are obtained as follows $\begin{eqnarray}{\psi }_{4}^{{III}}(\xi )=\left[{a}_{0}+{a}_{1}\left(\displaystyle \frac{{\lambda }_{1}{\mathrm{csch}}^{2}(\xi )}{{\lambda }_{2}\coth (\xi )+{\lambda }_{3}+{\lambda }_{2}{\mathrm{csch}}^{2}(\xi )}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{eqnarray}$
Type V: ${\zeta }_{2}=-1,{\zeta }_{3}=\tfrac{2{\lambda }_{3}}{{\lambda }_{1}},{\zeta }_{4}=\tfrac{{\lambda }_{3}^{2}-{\lambda }_{2}^{2}}{{\lambda }_{1}^{2}}$, where λ1, λ2, λ3 are arbitrary constants. $\begin{eqnarray}\begin{array}{l}{\psi }_{6}^{{III}}(\xi )\,=\,\left[{a}_{0}+{a}_{1}\left(-\displaystyle \frac{{\lambda }_{1}(\sinh ({\lambda }_{1}\xi )+\cosh ({\lambda }_{1}\xi ))(\sinh ({\lambda }_{1}\xi )+\cosh ({\lambda }_{1}\xi )+{\lambda }_{2})}{{\lambda }_{3}}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{array}\end{eqnarray}$
Type VI: ${\zeta }_{2}=4,{\zeta }_{3}=\tfrac{-2{\lambda }_{3}}{{\lambda }_{1}},{\zeta }_{4}=\tfrac{{\lambda }_{3}^{2}-{\lambda }_{2}^{2}}{{\lambda }_{1}^{2}}$, where ${\lambda }_{1},{\lambda }_{2},{\lambda }_{3}$ are arbitrary constants. $\begin{eqnarray}{\psi }_{8}^{{III}}(\xi )=\left[{a}_{0}+{a}_{1}\left(\displaystyle \frac{{\lambda }_{1}\csc (\xi )}{{\lambda }_{2}\csc (\xi )+{\lambda }_{3}}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{eqnarray}$
Type VII: ${\zeta }_{2}=-4,{\zeta }_{3}=\tfrac{4\left(2{\lambda }_{2}+{\lambda }_{4}\right)}{{\lambda }_{1}}$, ${\zeta }_{4}=-\tfrac{4{\lambda }_{2}^{2}+4{\lambda }_{4}{\lambda }_{2}-{\lambda }_{3}^{2}}{{\lambda }_{1}^{2}}$, where λ1, λ2, λ3, λ4 are arbitrary constants. $\begin{eqnarray}{\psi }_{9}^{{III}}(\xi )=\left[{a}_{0}+{a}_{1}\left(\displaystyle \frac{{\lambda }_{1}{\sec }^{2}(\xi )}{{\lambda }_{2}\tan (\xi )+{\lambda }_{3}+{\lambda }_{4}{\sec }^{2}(\xi )}\right)\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{eqnarray}$
Case IV.
For ${\zeta }_{0}=\tfrac{1}{4},{\zeta }_{2}=\tfrac{1-2{m}^{2}}{2},{\zeta }_{4}=\tfrac{1}{4}$, the solution of (1 ) is of the form1 ) is of the form
$\begin{eqnarray}{\psi }_{3}^{{IV}}(\xi )=\left[{a}_{0}+{a}_{1}(\mathrm{cn}\xi )\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}},\end{eqnarray}$
gives the bright soliton for $m\to 1$, $\begin{eqnarray}{\psi }_{3}^{{IV}}(\xi )=\left[{a}_{0}+{a}_{1}{\rm{sech}} (\xi )\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}},\end{eqnarray}$
and the periodic singular solution for $m\to 0$, $\begin{eqnarray}{\psi }_{3}^{{IV}}(\xi )=\left[{a}_{0}+{a}_{1}\cos (\xi )\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}},\end{eqnarray}$
for ${\zeta }_{0}=\tfrac{1}{4},{\zeta }_{2}=\tfrac{1-2{m}^{2}}{2},{\zeta }_{4}=\tfrac{1}{4}$, the solution of ( $\begin{eqnarray}{\psi }_{13}^{{IV}}(\xi )=\left[{a}_{0}+{a}_{1}(\mathrm{ns}\xi \pm \mathrm{cs}\xi )\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}},\end{eqnarray}$
gives the combined dark-singular wave solution for $m\to 1$, $\begin{eqnarray}{\psi }_{13}^{{IV}}(\xi )=\left[{a}_{0}+{a}_{1}(\coth (\xi )+\mathrm{csch}(\xi ))\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}},\,\end{eqnarray}$
and the periodic singular solution for $m\to 0$, $\begin{eqnarray}{\psi }_{13}^{{IV}}(\xi )=\left[{a}_{0}+{a}_{1}(\cot (\xi )+\csc (\xi ))\right]{{\rm{e}}}^{{\rm{i}}{\rm{\Phi }}}.\end{eqnarray}$
3. Physical description
The graphical representation of solitons has been illustrated in the following figures, for various values of the parameters. Mathematica 11 is used to carry out simulations and to visualize the behavior of nonlinear waves observed by the equation (1 ).
Figures 1(a), (b), and (c) illustrate the 3D chart of the absolute value of ${\psi }_{1}^{I}(x,y,t)$ established in Case I (Type I) when t = −0.5, t = 0, and t = 0.5 respectively. Figure 1 represents complex solitary wave solution with the parameters ϑ1 = 1, ϑ2 = −1, ϑ3 = 1, ϱ1 = 1, ϱ2 = 3, ϱ3 = 4, ϱ4 = −1, a = 1, b = −1, p = −2, q = 1, and r = −3.
Figure 1. $| {\psi }_{1}^{I}(x,y,t)| :$ The complex solitary wave solution when (a) t = −0.5 (b) t = 0 (c) t = 0.5. |
Figures 2(a), (b), and (c) show the 3D chart of the absolute value of ${\psi }_{1}^{{III}}(x,y,t)$ established in Case III (Type I) when t = −0.5, t = 0, and t = 0.5 respectively. Figure 2 represents complex bright soliton wave solution with the parameters λ1 = −1, λ2 = −1, λ3 = −2, ϱ1 = 1, ϱ2 = 3, ϱ3 = 4, ϱ4 = −1, a = 1, b = −1, p = −2, q = 1, and r = −3.
Figure 2. $| {\psi }_{1}^{{III}}(x,y,t)| :$ The complex bright soliton wave solution when (a) t = −0.5 (b) t = 0 (c) t = 0.5. |
Figures 3(a), (b), and (c) show the 3D chart of the absolute value of ${\psi }_{3}^{{III}}(x,y,t)$ established in Case III (Type III) when t = −0.5, t = 0, and t = 0.5 respectively. Figure 3 represents complex dark soliton wave (a ’W ’ shape wave) solution with the parameters λ1 = 1, λ2 = −1, λ3=−2, λ4 = 1, ϱ1 = 1, ϱ2 = 3, ϱ3 = 4, ϱ4 = −1, a = 1, b = −1, p = −2, q = 1, and r = −3.
Figure 3. $| {\psi }_{3}^{{III}}(x,y,t)| :$ The complex dark soliton wave solution when (a) t = −0.5 (b) t = 0 (c) t = 0.5. |
Figures 4(a), 4(b), and 4(c) show the 3D chart of the absolute value of ${\psi }_{3}^{{IV}}(x,y,t)$ established in Case IV when t = −0.5, t = 0, and t = 0.5 respectively. Figure 4 represents complex elliptic wave solution with the parameters λ1 = −1, ${\lambda }_{2}$=-1, λ3 = −2, ϱ1 = 1, ϱ2 = 3, ϱ3 = 4, ϱ4 = −1, ${\zeta }_{0}=\tfrac{1}{4},{\zeta }_{2}=\tfrac{1-2{m}^{2}}{2},{\zeta }_{4}=\tfrac{1}{4},m=\tfrac{1}{3},a=1$, b = −1, p =−2, q = 1, and r = −3.
Figure 4. $| {\psi }_{3}^{{IV}}(x,y,t)| :$ The complex elliptic wave solution when (a) t = −0.5 (b) t = 0 (c) t = 0.5. |
4. Conclusions
In this study, new soliton and elliptic wave solutions with different wave structures for the Heisenberg ferromagnetic spin chain equation have been constructed via the extended FAN sub-equation method. A set of new exact solutions is found corresponding to various parameters. The graphical representations of the solutions are also demonstrated by figures 1–4, to investigate the behavior of the nonlinear model. Moreover, it is observed that the proposed approach can also be applied to other types of more complex models of contemporary science.