1. Introduction
Partial differential equations (PDEs) particularly their nonlinear regimes are applied to model a wide variety of phenomena in the extensive areas of science and engineering. As useful tools for simulating many nonlinear phenomena, such models play a pivotal role in progressing the real world. During the past few decades, one of the major goals has been the construction of novel approaches to extract solitons for PDEs. In the last years, several well-organized methods have been proposed to derive solitons of PDEs, the hyperbolic function method [1–4], the modified Jacobi methods [5–8], the Kudryashov method [9–14], and the exponential method [15–20], are samples to point out.
It is noteworthy that all conservation laws of PDEs do not include physical meanings, however, such laws are essential to explore the integrability and reduction of PDEs [21, 22]. Conservation laws of PDEs can be found by using a wide range of methods such as the multiplier approach, the Noether approach, the new conservation theorem, and so on. Of these, the new conservation theorem was first established by Ibragimov and is associated with the formal Lagrangian, Lie symmetries, and adjoint equations of PDEs. Conservation laws can be derived for every symmetry of PDEs and the obtained conservation laws are referred to as trivial or non-trivial [23–29].
In the present study, the authors deal with the following Sharma–Tasso–Olver–Burgers equation [30]1 ), such a model consists of the Sharma–Tasso–Olver and Burgers equations. These equations have been the main concern of a lot of research works. Or-Roshid and Rashidi [31] employed an exponential method to derive the solitons of these equations. In another investigation, multiple solitons of these equations were obtained by Wazwaz in [32] through the simplified Hirota’s method. Very recently, Hu et al [33] constructed soliton, lump, and interaction solutions of a 2D-STOBE using a series of systematic ansatzes.
$\begin{eqnarray}\begin{array}{l}{u}_{t}+{c}_{1}\left(3{u}_{x}^{2}+3{u}^{2}{u}_{x}+3u{u}_{xx}+{u}_{xxx}\right)+{c}_{2}\left(2u{u}_{x}+{u}_{xx}\right)\,=\,0,\end{array}\end{eqnarray}$
and derive its conservation laws and kink solitons. As is clear from the name of equation (Kudryashov and exponential methods, as privileged approaches, have been applied by many scholars to retrieve solitons of PDEs. Particularly, the effectiveness of these methods has been demonstrated by Hosseini et al in several papers. Hosseini et al [34] derived solitons of the cubic-quartic nonlinear Schrödinger equation using the Kudryashov method. The exponential method was utilized in [35] by Hosseini et al to acquire solitons of the unstable nonlinear Schrödinger equation.
The rest of the current study is as follows: In section 2 , the conservation theorem and the foundation of Kudryashov and exponential methods are given. In section 3 , the formal Lagrangian, Lie symmetries, and adjoint equations of the STOBE are established to derive its conservation laws. In section 4 , Kudryashov and exponential methods are adopted to seek solitons of the STOBE. Section 4 further gives diverse graphs in 2 and 3D postures to demonstrate the dynamical features of kink solitons. The achievements of the present paper are provided in section 5 .
2. The conservation theorem and methods
In the current section, the conservation theorem and the foundation of Kudryashov and exponential methods are formally given.
2.1. The conservation theorem
To start, suppose that a PDE can be expressed as2 ) is given by3 ) is retrieved by
$\begin{eqnarray}P(u,{u}_{x},{u}_{t},\mathrm{...})=0,\end{eqnarray}$
where $P$ is a polynomial. The Lie point symmetry generator of equation ( $\begin{eqnarray}\begin{array}{l}X={\xi }^{x}\left(x,t,u\right)\displaystyle \frac{\partial }{\partial x}+{\xi }^{t}\left(x,t,u\right)\displaystyle \frac{\partial }{\partial t}\\ \,\,+\,\eta \left(x,t,u\right)\displaystyle \frac{\partial }{\partial u},\end{array}\end{eqnarray}$
where ${\xi }^{x}\left(x,t,u\right),$ ${\xi }^{t}\left(x,t,u\right),$ and $\eta \left(x,t,u\right)$ are known as the infinitesimals. The kth prolongation of equation ( $\begin{eqnarray*}{X}^{\left(k\right)}=X+{\eta }_{i}^{\left(1\right)}\displaystyle \frac{\partial }{\partial {u}_{i}^{\alpha }}+\mathrm{...}+{\eta }_{{i}_{1}{i}_{2}\mathrm{...}{i}_{k}}^{\left(k\right)}\displaystyle \frac{\partial }{\partial {u}_{{i}_{1}{i}_{2}\mathrm{...}{i}_{k}}},k\geqslant 1,\end{eqnarray*}$
where $\begin{eqnarray*}{\eta }_{i}^{\left(1\right)}={D}_{i}\eta -\left({D}_{i}{\xi }^{j}\right){u}_{j},\end{eqnarray*}$
$\begin{eqnarray*}{\eta }_{{i}_{1}{i}_{2}\mathrm{...}{i}_{k}}^{\left(k\right)}={D}_{{i}_{k}}{\eta }_{{i}_{1}{i}_{2}\mathrm{...}{i}_{k-1}}^{\left(k-1\right)}-\left({D}_{{i}_{k}}{\xi }^{j}\right){u}_{{i}_{1}{i}_{2}\mathrm{...}{i}_{k-1}j}.\end{eqnarray*}$
Above $i,j=1,2$ and ${i}_{l}=1$ for $l=1,2,\mathrm{...},k.$ The total derivative operator is indicated by ${D}_{i}.$The formal Lagrangian is given by multiplying equation (2 ) by $w$ as follows
$\begin{eqnarray*}L=wP,\end{eqnarray*}$
where $w$ is the adjoint variable. It is noteworthy that the adjoint equation is retrieved by $\begin{eqnarray}P* \,=\,\displaystyle \frac{\delta L}{\delta u},\end{eqnarray}$
where $\tfrac{\delta }{\delta u}$ is the variational derivative.If the solution of equation (4 ) is found, then, a finite number of conservation laws for equation (2 ) are derived.
Every Lie point, Lie-Bäcklund, and nonlocal symmetry of equation (
$\begin{eqnarray}{T}^{i}={\xi }^{i}L+W\,\frac{\delta L}{\delta {u}_{i}}+\displaystyle \sum _{s\geqslant 1}{D}_{{i}_{1}}\ldots {D}_{{i}_{s}}(W)\frac{\delta L}{\delta {u}_{i{i}_{1}{i}_{2}\mathrm{...}{i}_{s}}},\end{eqnarray}$
where $W=\eta -{\xi }^{j}{u}_{j},$ and ${\xi }^{i}\,$ and $\eta $ are the infinitesimal functions. The conserved vectors generated by equation ( Derived conserved vectors using (
$\begin{eqnarray*}{D}_{i}\left({T}^{i}\right)=0.\end{eqnarray*}$
Here, ${D}_{i}$ is referred to as the total derivative [37].2.2. Foundation of methods
The Kudryashov method applies the following series [9, 10]
$\begin{eqnarray}\begin{array}{l}U\left({\epsilon }\right)={a}_{0}+{a}_{1}K\left({\epsilon }\right)+{a}_{2}{K}^{2}\left({\epsilon }\right)+\ldots \\ \,\,\,+\,{a}_{N}{K}^{N}\left({\epsilon }\right),\,{a}_{N}\ne 0,\end{array}\end{eqnarray}$
as the nontrivial solution of $\begin{eqnarray}O\left(U\left({\epsilon }\right),U^{\prime} \left({\epsilon }\right),U^{\prime\prime} \left({\epsilon }\right),\ldots \right)=0.\end{eqnarray}$
Above, ${a}_{i},i=0,1,\mathrm{...},N$ are unknowns, $N$ is retrieved by the balance approach, and6 ) and (7 ) together result in a nonlinear system which by solving it, solitons of equation (7 ) are derived.
$\begin{eqnarray*}K\left({\epsilon }\right)=\displaystyle \frac{1}{1+d{a}^{{\epsilon }}}\end{eqnarray*}$
is the solution of $\begin{eqnarray*}K^{\prime} \left({\epsilon }\right)=K\left({\epsilon }\right)\left(K\left({\epsilon }\right)-1\right)\mathrm{ln}\left(a\right).\end{eqnarray*}$
Equations (The exponential method investigates a solution for equation (7 ) as [15, 16]
$\begin{eqnarray}\begin{array}{l}U\left({\epsilon }\right)=\displaystyle \frac{{a}_{0}+{a}_{1}{a}^{{\epsilon }}+{a}_{2}{a}^{2{\epsilon }}+\ldots +{a}_{N}{a}^{N{\epsilon }}}{{b}_{0}+{b}_{1}{a}^{{\epsilon }}+{b}_{2}{a}^{2{\epsilon }}+\ldots +{b}_{N}{a}^{N{\epsilon }}},\\ {a}_{N}\ne 0,\,{b}_{N}\ne 0,\end{array}\end{eqnarray}$
where the unknowns are acquired later and $N\in {{\mathbb{Z}}}^{+}.$Again, equations (7 ) and (8 ) together yield a nonlinear system which by solving it, solitons of equation (7 ) are obtained.
3. The STOBE and its conservation laws
In the present section, the conservation theorem is applied to the STOBE to derive its conservation laws. First, the formal Lagrangian is derived in the following form
$\begin{eqnarray}\begin{array}{l}L=w\left({u}_{t}+{c}_{1}\left(3{u}_{x}^{2}+3{u}^{2}{u}_{x}+3u{u}_{xx}+{u}_{xxx}\right)\right.\\ \,\,+\,\left.{c}_{2}\left(2u{u}_{x}+{u}_{xx}\right)\right),\end{array}\end{eqnarray}$
where $w$ is the adjoint variable.The adjoint equation is acquired with the aid of the variational derivative as10 ), then, equation (1 ) is not obtained. Thus, equation (1 ) is not self-adjoint. In such a case, one can say that $w=1$ is a solution of equation (10 ).
$\begin{eqnarray}\begin{array}{l}F* \,=\,-{w}_{t}+{c}_{1}\left(-3{u}^{2}{w}_{x}+3u{w}_{xx}-{w}_{xxx}\right)\\ \,\,+\,{c}_{2}\left({w}_{xx}-2u{w}_{x}\right).\end{array}\end{eqnarray}$
If $u$ is replaced by $w$ in equation (A one-parameter Lie group for equation (1 ) is given by1 ) admits following Lie point symmetry generator1 ), an equivalent condition is obtained as1 ) are given by11 )) has been given in table 1.
$\begin{eqnarray*}x\to x+\varepsilon {\xi }^{x}\left(x,t,u\right),\end{eqnarray*}$
$\begin{eqnarray*}t\to t+\varepsilon {\xi }^{t}\left(x,t,u\right),\end{eqnarray*}$
$\begin{eqnarray*}u\to u+\varepsilon \eta \left(x,t,u\right),\end{eqnarray*}$
where $\varepsilon $ is the group parameter, and consequently equation ( $\begin{eqnarray*}X={\xi }^{x}\displaystyle \frac{\partial }{\partial x}+{\xi }^{t}\displaystyle \frac{\partial }{\partial t}+\eta \displaystyle \frac{\partial }{\partial u}.\end{eqnarray*}$
Obviously, $X$ must be satisfied the Lie symmetry condition as follow $\begin{eqnarray*}\begin{array}{l}{X}^{\left(3\right)}\left({u}_{t}+{c}_{1}\left(3{u}_{x}^{2}+3{u}^{2}{u}_{x}+3u{u}_{xx}+{u}_{xxx}\right)\right.\\ \,+\left.{c}_{2}\left(2u{u}_{x}+{u}_{xx}\right)\right)=0.\end{array}\end{eqnarray*}$
The third prolongation of the Lie point symmetry generator is given by $\begin{eqnarray*}\begin{array}{l}{X}^{\left(3\right)}={\xi }^{x}\displaystyle \frac{\partial }{\partial x}+{\xi }^{t}\displaystyle \frac{\partial }{\partial t}+\eta \displaystyle \frac{\partial }{\partial u}+{\eta }^{x}\displaystyle \frac{\partial }{\partial {u}_{x}}\\ \,\,+\,{\eta }^{t}\displaystyle \frac{\partial }{\partial {u}_{t}}+{\eta }^{xx}\displaystyle \frac{\partial }{\partial {u}_{xx}}+{\eta }^{xxx}\displaystyle \frac{\partial }{\partial {u}_{xxx}},\end{array}\end{eqnarray*}$
where $\begin{eqnarray*}{\eta }^{x}={D}_{x}\left(\eta \right)-{u}_{x}{D}_{x}\left({\xi }^{x}\right)-{u}_{t}{D}_{x}\left({\xi }^{t}\right),\end{eqnarray*}$
$\begin{eqnarray*}{\eta }^{t}={D}_{t}\left(\eta \right)-{u}_{x}{D}_{t}\left({\xi }^{x}\right)-{u}_{t}{D}_{t}\left({\xi }^{t}\right),\end{eqnarray*}$
$\begin{eqnarray*}{\eta }^{xx}={D}_{x}\left({\eta }^{x}\right)-{u}_{xx}{D}_{x}\left({\xi }^{x}\right)-{u}_{xt}{D}_{x}\left({\xi }^{t}\right),\end{eqnarray*}$
$\begin{eqnarray*}{\eta }^{xxx}={D}_{x}\left({\eta }^{xx}\right)-{u}_{xxx}{D}_{x}\left({\xi }^{x}\right)-{u}_{xxt}{D}_{x}\left({\xi }^{t}\right).\end{eqnarray*}$
If we apply ${X}^{\left(3\right)}$ to equation ( $\begin{eqnarray*}\begin{array}{l}{\eta }^{t}+{c}_{1}\left(6{u}_{x}{\eta }^{x}+6u\eta {u}_{x}+3{u}^{2}{\eta }^{x}+3\eta {u}_{xx}+3u{\eta }^{xx}+{\eta }^{xxx}\right)\\ \,+{c}_{2}\left(2\eta {u}_{x}+2u{\eta }^{x}+{\eta }^{xx}\right)=0.\end{array}\end{eqnarray*}$
Now, after some operations, we find the infinitesimal functions as follows $\begin{eqnarray*}\begin{array}{l}{\xi }^{t}={k}_{1}+{k}_{3}t,\,{\xi }^{x}={k}_{2}+{k}_{3}\left(\displaystyle \frac{3{c}_{1}x-2{c}_{2}^{2}t}{9{c}_{1}}\right),\\ \eta ={k}_{3}\left(\displaystyle \frac{-3{c}_{1}u-{c}_{2}}{9{c}_{1}}\right),\end{array}\end{eqnarray*}$
where ${k}_{1},$ ${k}_{2},$ and ${k}_{3}$ are arbitrary constants. Finally, the Lie point symmetry generators of equation ( $\begin{eqnarray}\begin{array}{l}{X}_{1}=\displaystyle \frac{\partial }{\partial t},\,{X}_{2}=\displaystyle \frac{\partial }{\partial x},\,{X}_{3}=\left(\displaystyle \frac{3{c}_{1}x-2{c}_{2}^{2}t}{9{c}_{1}}\right)\displaystyle \frac{\partial }{\partial x}\\ \,\,+\,t\displaystyle \frac{\partial }{\partial t}+\left(\displaystyle \frac{-3{c}_{1}u-{c}_{2}}{9{c}_{1}}\right)\displaystyle \frac{\partial }{\partial u}.\end{array}\end{eqnarray}$
The commutator table for the above symmetries (see equation (Table 1. The commutator table for the above symmetries. |
| $[{X}_{i},{X}_{j}]$ | ${X}_{1}$ | ${X}_{2}$ | ${X}_{3}$ |
|---|---|---|---|
| ${X}_{1}$ | $0$ | $0$ | ${X}_{1}-\tfrac{2{c}_{2}^{2}}{9{c}_{1}}{X}_{2}$ |
| ${X}_{2}$ | $0$ | $0$ | $\tfrac{{X}_{2}}{3}$ |
| ${X}_{3}$ | $-{X}_{1}+\tfrac{2{c}_{2}^{2}}{9{c}_{1}}{X}_{2}$ | $-\tfrac{{X}_{2}}{3}$ | $0$ |
Now, conservation laws of the STOBE for all founded Lie point symmetry generators are derived. Conservation laws formulae for equation (1 ) are as follows
$\begin{eqnarray}\begin{array}{l}{T}^{x}={\xi }^{x}L+W\left[\displaystyle \frac{\partial L}{\partial {u}_{x}}-{D}_{x}\left(\displaystyle \frac{\partial L}{\partial {u}_{xx}}\right)+{D}_{x}^{2}\left(\displaystyle \frac{\partial L}{\partial {u}_{xxx}}\right)\right]\\ \,+{D}_{x}\left(W\right)\left[\displaystyle \frac{\partial L}{\partial {u}_{xx}}-{D}_{x}\left(\displaystyle \frac{\partial L}{\partial {u}_{xxx}}\right)\right]+{D}_{x}^{2}\left(W\right)\left[\displaystyle \frac{\partial L}{\partial {u}_{xxx}}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}{T}^{t}={\xi }^{t}L+W\left(\displaystyle \frac{\partial L}{\partial {u}_{t}}\right).\end{eqnarray}$
If we employ (
$\begin{eqnarray*}\begin{array}{l}{T}_{1}^{x}=-3{c}_{1}\left({u}_{t}w{u}_{x}+{u}_{t}w{u}^{2}+{u}_{xt}uw\right)-2{c}_{2}{u}_{t}wu\\ \,-{c}_{2}{u}_{xt}w+{c}_{1}\left(3{u}_{t}u{w}_{x}-{u}_{t}{w}_{xx}+{u}_{xt}{w}_{x}-{u}_{xxt}w\right)\\ \,+{c}_{2}{u}_{t}{w}_{x},\\ {T}_{1}^{t}=3{c}_{1}\left({u}^{2}{u}_{x}w+wu{u}_{xx}+w{u}_{x}^{2}\right)+{c}_{2}w\left(2u{u}_{x}+{u}_{xx}\right)\\ \,\,+\,{c}_{1}w{u}_{xxx}.\end{array}\end{eqnarray*}$
Due to the satisfaction of the divergence condition, these conservation laws are called local conservation laws. Such conservation laws are infinite trivial conservation laws. In this case, we have14 ) and (15 ) satisfy the divergence condition, so they are trivial conservation laws.
$\begin{eqnarray*}{D}_{x}\left({T}_{1}^{x}\right)+{D}_{t}\left({T}_{1}^{t}\right)={u}_{t}{w}_{t}-{u}_{t}{w}_{t}=0.\end{eqnarray*}$
It is noted that for $w=1,$ from ${T}_{1}^{x}$ and ${T}_{1}^{t},$ one can find $\begin{eqnarray}\begin{array}{l}\tilde{{T}_{1}^{x}}=-3{c}_{1}{u}^{2}{u}_{t}-3{c}_{1}u{u}_{xt}-3{c}_{1}{u}_{x}{u}_{t}-2{c}_{2}u{u}_{t}\\ \,\,-\,{c}_{2}{u}_{xt}-{c}_{1}{u}_{xxt},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\tilde{{T}_{1}^{t}}=3{c}_{1}{u}^{2}{u}_{x}+3{c}_{1}u{u}_{xx}+3{c}_{1}{u}_{x}^{2}+2{c}_{2}u{u}_{x}+{c}_{2}{u}_{xx}\\ \,\,+\,{c}_{1}{u}_{xxx}.\end{array}\end{eqnarray}$
Finite conserved vectors ( If we apply (
$\begin{eqnarray*}\begin{array}{l}{T}_{2}^{x}={c}_{1}{u}_{xx}{w}_{x}-{c}_{1}{u}_{x}{w}_{xx}+{u}_{x}{w}_{x}\left(3{c}_{1}u+{c}_{2}\right)\,+\,w{u}_{t},\\ {T}_{2}^{t}=-w{u}_{x}.\end{array}\end{eqnarray*}$
Owing to the satisfaction of the divergence condition, these conservation laws are called local conservation laws. Such conservation laws are infinite trivial conservation laws. In this case, we have $\begin{eqnarray*}\begin{array}{l}{D}_{x}\left({T}_{2}^{x}\right)+{D}_{t}\left({T}_{2}^{t}\right)=-{w}_{x}\left(3{c}_{1}{u}^{2}{u}_{x}+2{c}_{2}u{u}_{x}\right)\\ \,+{u}_{x}\left(3{c}_{1}{u}^{2}{w}_{x}+2{c}_{2}u{w}_{x}\right)=0.\end{array}\end{eqnarray*}$
It is notable that for $w=1,$ one can get $\begin{eqnarray*}\tilde{{T}_{2}^{x}}={u}_{t},\end{eqnarray*}$
$\begin{eqnarray*}\tilde{{T}_{2}^{t}}=-{u}_{x}.\end{eqnarray*}$
The above finite conserved vectors satisfy the divergence condition, consequently, they are trivial conservation laws. If we employ (
$\begin{eqnarray*}\begin{array}{l}{T}_{3}^{x}=3t{c}_{1}u{u}_{t}{w}_{x}-\displaystyle \frac{2{c}_{2}^{2}tu{u}_{x}{w}_{x}}{3}-\displaystyle \frac{2{c}_{2}^{2}tw{u}_{t}}{9\alpha }\\ \,-3{c}_{1}tw{u}^{2}{u}_{t}-2{c}_{2}twu{u}_{t}-3{c}_{1}tw{u}_{x}{u}_{t}\\ \,+{c}_{1}xu{u}_{x}{w}_{x}-\displaystyle \frac{2{c}_{2}^{3}t{u}_{x}{w}_{x}}{9{c}_{1}}+{c}_{1}{u}^{2}{w}_{x}+\displaystyle \frac{2{c}_{2}u{w}_{x}}{3}\\ \,-{c}_{1}w{u}_{xx}-\displaystyle \frac{{c}_{1}u{w}_{xx}}{3}-\displaystyle \frac{{c}_{2}{w}_{xx}}{9}\\ \,-3{c}_{1}wu{u}_{x}-\displaystyle \frac{2{c}_{2}^{2}uw}{9{c}_{1}}+\displaystyle \frac{{c}_{2}x{u}_{x}{w}_{x}}{3}+{c}_{2}t{u}_{t}{w}_{x}+\displaystyle \frac{2{c}_{1}{u}_{x}{w}_{x}}{3}\\ \,+\displaystyle \frac{xw{u}_{t}}{3}-{c}_{1}w{u}^{3}-{c}_{2}w{u}^{2}\\ \,-{c}_{2}w{u}_{x}+\displaystyle \frac{{c}_{2}^{2}{w}_{x}}{9{c}_{1}}-3{c}_{1}tuw{u}_{xt}+\displaystyle \frac{2{c}_{2}^{2}t{u}_{x}{w}_{xx}}{9}-\displaystyle \frac{2{c}_{2}^{2}t{w}_{x}{u}_{xx}}{9}\\ \,-\displaystyle \frac{{c}_{1}x{u}_{x}{w}_{xx}}{3}-{c}_{1}t{u}_{t}{w}_{xx}\\ \,-{c}_{2}tw{u}_{xt}+\displaystyle \frac{{c}_{1}x{w}_{x}{u}_{xx}}{3}+{c}_{1}t{w}_{x}{u}_{xt}-{c}_{1}tw{u}_{xxt},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{T}_{3}^{t}=3{c}_{1}tw{u}_{x}^{2}+3{c}_{1}tw{u}^{2}{u}_{x}+3{c}_{1}tuw{u}_{xx}+{c}_{1}tw{u}_{xxx}\\ \,\,+\,2{c}_{2}tuw{u}_{x}+{c}_{2}tw{u}_{xx}\\ \,\,-\,\displaystyle \frac{uw}{3}-\displaystyle \frac{{c}_{2}w}{9{c}_{1}}-\displaystyle \frac{xw{u}_{x}}{3}+\displaystyle \frac{2{c}_{2}^{2}tw{u}_{x}}{9{c}_{1}}.\end{array}\end{eqnarray*}$
Due to the satisfaction of the divergence condition, these conservation laws are called local conservation laws. Such conservation laws are infinite trivial conservation laws.It is noted that for $w=1,$ one can find
$\begin{eqnarray*}\begin{array}{l}\tilde{{T}_{3}^{x}}=-{c}_{1}t{u}_{xxt}-3{c}_{1}u{u}_{x}-\displaystyle \frac{2{c}_{2}^{2}u}{9{c}_{1}}-{c}_{2}{u}_{x}-{c}_{2}t{u}_{xt}\\ \,-{c}_{1}{u}_{xx}-{c}_{1}{u}^{3}-{c}_{2}{u}^{2}-\displaystyle \frac{2{c}_{2}^{2}t{u}_{t}}{9{c}_{1}}\\ \,-3{c}_{1}t{u}^{2}{u}_{t}-2{c}_{2}tu{u}_{t}-3{c}_{1}t{u}_{x}{u}_{t}-3{c}_{1}tu{u}_{xt}+\displaystyle \frac{u{u}_{t}}{3},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}\tilde{{T}_{3}^{t}}=3{c}_{1}t{u}^{2}{u}_{x}+2{c}_{2}tu{u}_{x}+3{c}_{1}tu{u}_{xx}+3{c}_{1}t{u}_{x}^{2}\\ \,+{c}_{1}t{u}_{xxx}+{c}_{2}t{u}_{xx}-\displaystyle \frac{u}{3}-\displaystyle \frac{{c}_{2}}{9{c}_{1}}-\displaystyle \frac{x{u}_{x}}{3}+\displaystyle \frac{2{c}_{2}^{2}t{u}_{x}}{9{c}_{1}}.\end{array}\end{eqnarray*}$
Such conserved vectors satisfy the divergence condition, so, they are trivial conservation laws.4. The STOBE and its solitons
In the present section, Kudryashov and exponential methods are adopted to seek solitons of the STOBE. The present section further gives diverse graphs in 2 and 3D postures to demonstrate the dynamical features of kink solitons. To start, we establish a transformation as follows16 ) and (1 ) together result in17 ), one can get [10]
$\begin{eqnarray}u\left(x,t\right)=U\left({\epsilon }\right),\,{\epsilon }=x-wt,\end{eqnarray}$
which $w$ is the soliton velocity. Equations ( $\begin{eqnarray}\begin{array}{l}{c}_{1}\displaystyle \frac{{{\rm{d}}}^{3}U\left({\epsilon }\right)}{{\rm{d}}{{\epsilon }}^{3}}+{c}_{2}\displaystyle \frac{{{\rm{d}}}^{2}U\left({\epsilon }\right)}{{\rm{d}}{{\epsilon }}^{2}}-w\displaystyle \frac{{\rm{d}}U\left({\epsilon }\right)}{{\rm{d}}{\epsilon }}\\ \,+3{c}_{1}U\left({\epsilon }\right)\displaystyle \frac{{{\rm{d}}}^{2}U\left({\epsilon }\right)}{{\rm{d}}{{\epsilon }}^{2}}+3{c}_{1}{\left(\displaystyle \frac{{\rm{d}}U\left({\epsilon }\right)}{{\rm{d}}{\epsilon }}\right)}^{2}\\ \,+3{c}_{1}{U}^{2}\left({\epsilon }\right)\displaystyle \frac{{\rm{d}}U\left({\epsilon }\right)}{{\rm{d}}{\epsilon }}+2{c}_{2}U\left({\epsilon }\right)\displaystyle \frac{{\rm{d}}U\left({\epsilon }\right)}{{\rm{d}}{\epsilon }}=0.\end{array}\end{eqnarray}$
From $\tfrac{{{\rm{d}}}^{3}U\left({\epsilon }\right)}{{\rm{d}}{{\epsilon }}^{3}}$ and ${U}^{2}\left({\epsilon }\right)\tfrac{{\rm{d}}U\left({\epsilon }\right)}{{\rm{d}}{\epsilon }}$ in equation ($N+3=3N+1\Rightarrow N=1.$
4.1. Kudryashov method
Equation (6 ) and $N=1$ offer taking the following solution17 ). From equations (17 ) and (18 ), we will attain a nonlinear system as
$\begin{eqnarray}U\left({\epsilon }\right)={a}_{0}+{a}_{1}K\left({\epsilon }\right),\,{a}_{1}\ne 0,\end{eqnarray}$
for equation ( $\begin{eqnarray*}\begin{array}{l}6{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}+9\,\mathrm{ln}\left(a\right){a}_{1}{c}_{1}+3{a}_{1}^{2}{c}_{1}=0,\\ \,-12{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}+6\,\mathrm{ln}\left(a\right){a}_{0}{c}_{1}-15\,\mathrm{ln}\left(a\right){a}_{1}{c}_{1}\\ \,+6{a}_{0}{a}_{1}{c}_{1}-3{a}_{1}^{2}{c}_{1}+2\,\mathrm{ln}\left(a\right){c}_{2}\\ \,+2{a}_{1}{c}_{2}=0,\\ 7{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}-9\,\mathrm{ln}\left(a\right){a}_{0}{c}_{1}+6\,\mathrm{ln}\left(a\right){a}_{1}{c}_{1}+3{a}_{0}^{2}{c}_{1}\\ \,-6{a}_{0}{a}_{1}{c}_{1}-3\,\mathrm{ln}\left(a\right){c}_{2}\\ \,+2{a}_{0}{c}_{2}-2{a}_{1}{c}_{2}-w=0,\\ \,-{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}+3\,\mathrm{ln}\left(a\right){a}_{0}{c}_{1}-3{a}_{0}^{2}{c}_{1}+\,\mathrm{ln}\left(a\right){c}_{2}\\ \,-2{a}_{0}{c}_{2}+w=0,\end{array}\end{eqnarray*}$
where its solution yieldsCase 1:
$\begin{eqnarray*}\begin{array}{l}{a}_{1}=-\,\mathrm{ln}\left(a\right),\,w={\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}-3\,\mathrm{ln}\left(a\right){a}_{0}{c}_{1}+3{a}_{0}^{2}{c}_{1}\\ \,\,-\,\mathrm{ln}\left(a\right){c}_{2}+2{a}_{0}{c}_{2}.\end{array}\end{eqnarray*}$
Thus, the following soliton to the STOBE is acquired $\begin{eqnarray*}\begin{array}{l}{u}_{1}\left(x,t\right)\,={a}_{0}-\displaystyle \frac{\mathrm{ln}\left(a\right)}{1+d{a}^{x-\left({\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}-3\,\mathrm{ln}\left(a\right){a}_{0}{c}_{1}+3{a}_{0}^{2}{c}_{1}-\,\mathrm{ln}\left(a\right){c}_{2}+2{a}_{0}{c}_{2}\right)t}}.\end{array}\end{eqnarray*}$
Case 2:
$\begin{eqnarray*}\begin{array}{l}{a}_{0}=\displaystyle \frac{3{c}_{1}\,\mathrm{ln}\left(a\right)-{c}_{2}}{3{c}_{1}},\,{a}_{1}=-2\,\mathrm{ln}\left(a\right),\\ w=\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}.\end{array}\end{eqnarray*}$
Thus, the following soliton to the STOBE is derived $\begin{eqnarray*}{u}_{2}\left(x,t\right)=\displaystyle \frac{3{c}_{1}\,\mathrm{ln}\left(a\right)-{c}_{2}}{3{c}_{1}}-\displaystyle \frac{2\,\mathrm{ln}\left(a\right)}{1+d{a}^{x-\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}t}}.\end{eqnarray*}$
The dynamical features of the kink soliton ${u}_{2}\left(x,t\right)$ are given in figure 1 for ${c}_{1}=0.15,$ ${c}_{2}=0.15,$ $d=5,$ and $a=2.7.$
Figure 1. The kink soliton ${u}_{2}\left(x,t\right)$ for ${c}_{1}=0.15,$ ${c}_{2}=0.15,$ $d=5,$ and $a=2.7.$ |
4.2. Exponential method
Taking $N=1$ in equation (8 ) leads to
$\begin{eqnarray}U\left({\epsilon }\right)=\displaystyle \frac{{a}_{0}+{a}_{1}{a}^{{\epsilon }}}{{b}_{0}+{b}_{1}{a}^{{\epsilon }}},\,{a}_{1}\ne 0,\,{b}_{1}\ne 0.\end{eqnarray}$
From equations (17 ) and (19 ), we will achieve a nonlinear system as
$\begin{eqnarray*}\begin{array}{l}-{\left(\mathrm{ln}\left(a\right)\right)}^{2}{a}_{0}{b}_{0}^{2}{b}_{1}{c}_{1}+{\left(\mathrm{ln}\left(a\right)\right)}^{2}{a}_{1}{b}_{0}^{3}{c}_{1}-3\,\mathrm{ln}\left(a\right){a}_{0}^{2}{b}_{0}{b}_{1}{c}_{1}\\ \,+3\,\mathrm{ln}\left(a\right){a}_{0}{a}_{1}{b}_{0}^{2}{c}_{1}-\,\mathrm{ln}\left(a\right){a}_{0}{b}_{0}^{2}{b}_{1}{c}_{2}+\,\mathrm{ln}\left(a\right){a}_{1}{b}_{0}^{3}{c}_{2}\\ \,+w{a}_{0}{b}_{0}^{2}{b}_{1}-w{a}_{1}{b}_{0}^{3}-3{a}_{0}^{3}{b}_{1}{c}_{1}\\ \,+3{a}_{0}^{2}{a}_{1}{b}_{0}{c}_{1}-2{a}_{0}^{2}{b}_{0}{b}_{1}{c}_{2}+2{a}_{0}{a}_{1}{b}_{0}^{2}{c}_{2}=0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}4{\left(\mathrm{ln}\left(a\right)\right)}^{2}{a}_{0}{b}_{0}{b}_{1}^{2}{c}_{1}-4{\left(\mathrm{ln}\left(a\right)\right)}^{2}{a}_{1}{b}_{0}^{2}{b}_{1}{c}_{1}\\ \,+6\,\mathrm{ln}\left(a\right){a}_{0}^{2}{b}_{1}^{2}{c}_{1}-12\,\mathrm{ln}\left(a\right){a}_{0}{a}_{1}{b}_{0}{b}_{1}{c}_{1}\\ \,+6\,\mathrm{ln}\left(a\right){a}_{1}^{2}{b}_{0}^{2}{c}_{1}+2w{a}_{0}{b}_{0}{b}_{1}^{2}-2w{a}_{1}{b}_{0}^{2}{b}_{1}\\ \,-6{a}_{0}^{2}{a}_{1}{b}_{1}{c}_{1}-2{a}_{0}^{2}{b}_{1}^{2}{c}_{2}\\ \,+6{a}_{0}{a}_{1}^{2}{b}_{0}{c}_{1}+2{a}_{1}^{2}{b}_{0}^{2}{c}_{2}=0,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}-{\left(\mathrm{ln}\left(a\right)\right)}^{2}{a}_{0}{b}_{1}^{3}{c}_{1}+{\left(\mathrm{ln}\left(a\right)\right)}^{2}{a}_{1}{b}_{0}{b}_{1}^{2}{c}_{1}\\ \,+3\,\mathrm{ln}\left(a\right){a}_{0}{a}_{1}{b}_{1}^{2}{c}_{1}+\,\mathrm{ln}\left(a\right){a}_{0}{b}_{1}^{3}{c}_{2}-3\,\mathrm{ln}\left(a\right){a}_{1}^{2}{b}_{0}{b}_{1}{c}_{1}\\ \,-\,\mathrm{ln}\left(a\right){a}_{1}{b}_{0}{b}_{1}^{2}{c}_{2}+w{a}_{0}{b}_{1}^{3}-w{a}_{1}{b}_{0}{b}_{1}^{2}-3{a}_{0}{a}_{1}^{2}{b}_{1}{c}_{1}\\ \,-2{a}_{0}{a}_{1}{b}_{1}^{2}{c}_{2}+3{a}_{1}^{3}{b}_{0}{c}_{1}+2{a}_{1}^{2}{b}_{0}{b}_{1}{c}_{2}=0,\end{array}\end{eqnarray*}$
where its solution results in $\begin{eqnarray*}\begin{array}{l}{a}_{0}=-\displaystyle \frac{{b}_{0}\left(3\,\mathrm{ln}\left(a\right){c}_{1}+{c}_{2}\right)}{3{c}_{1}},\,{a}_{1}=\displaystyle \frac{{b}_{1}\left(3\,\mathrm{ln}\left(a\right){c}_{1}-{c}_{2}\right)}{3{c}_{1}},\\ w=\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}.\end{array}\end{eqnarray*}$
Thus, the following soliton to the STOBE is acquired $\begin{eqnarray*}{u}_{1}\left(x,t\right)=\displaystyle \frac{{a}_{0}+{a}_{1}{a}^{x-\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}t}}{{b}_{0}+{b}_{1}{a}^{x-\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}t}},\end{eqnarray*}$
where $\begin{eqnarray*}{a}_{0}=-\displaystyle \frac{{b}_{0}\left(3\,\mathrm{ln}\left(a\right){c}_{1}+{c}_{2}\right)}{3{c}_{1}},\,{a}_{1}=\displaystyle \frac{{b}_{1}\left(3\,\mathrm{ln}\left(a\right){c}_{1}-{c}_{2}\right)}{3{c}_{1}}.\end{eqnarray*}$
Taking $N=2$ in equation (8 ) yields17 ) and (20 ) together result in a nonlinear system whose solution gives
$\begin{eqnarray}U\left({\epsilon }\right)=\displaystyle \frac{{a}_{0}+{a}_{1}{a}^{{\epsilon }}+{a}_{2}{a}^{2{\epsilon }}}{{b}_{0}+{b}_{1}{a}^{{\epsilon }}+{b}_{2}{a}^{2{\epsilon }}},\,{a}_{2}\ne 0,\,{b}_{2}\ne 0.\end{eqnarray}$
Equations (Case 1:
$\begin{eqnarray*}\begin{array}{l}{a}_{0}=-\displaystyle \frac{{b}_{0}\left(3\,\mathrm{ln}\left(a\right){c}_{1}+{c}_{2}\right)}{3{c}_{1}},\,{a}_{2}=\displaystyle \frac{{b}_{2}\left(3\,\mathrm{ln}\left(a\right){c}_{1}-{c}_{2}\right)}{3{c}_{1}},\\ {b}_{1}=-\displaystyle \frac{3{a}_{1}{c}_{1}}{{c}_{2}},\,w=\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}.\end{array}\end{eqnarray*}$
Thus, the following soliton to the STOBE is derived $\begin{eqnarray*}\begin{array}{l}{u}_{2}\left(x,t\right) =\displaystyle \frac{{a}_{0}+{a}_{1}{a}^{x-\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}t}+{a}_{2}{a}^{2\left(x-\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}t\right)}}{{b}_{0}+{b}_{1}{a}^{x-\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}t}+{b}_{2}{a}^{2\left(x-\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}t\right)}},\end{array}\end{eqnarray*}$
where $\begin{eqnarray*}\begin{array}{l}{a}_{0}=-\displaystyle \frac{{b}_{0}\left(3\,\mathrm{ln}\left(a\right){c}_{1}+{c}_{2}\right)}{3{c}_{1}},\,{a}_{2}=\displaystyle \frac{{b}_{2}\left(3\,\mathrm{ln}\left(a\right){c}_{1}-{c}_{2}\right)}{3{c}_{1}},\,\\ {b}_{1}=-\displaystyle \frac{3{a}_{1}{c}_{1}}{{c}_{2}}.\end{array}\end{eqnarray*}$
Case 2:
$\begin{eqnarray*}\begin{array}{l}{a}_{0}=-\displaystyle \frac{{b}_{0}\left(3\,\mathrm{ln}\left(a\right){c}_{1}+{c}_{2}\right)}{3{c}_{1}},\,{a}_{1}=-\displaystyle \frac{{b}_{1}{c}_{2}}{3{c}_{1}},\\ {a}_{2}=-\displaystyle \frac{{b}_{1}^{2}\left(3\,\mathrm{ln}\left(a\right){c}_{1}-{c}_{2}\right)}{6{b}_{0}{c}_{1}},\\ {b}_{2}=-\displaystyle \frac{{b}_{1}^{2}}{2{b}_{0}},\,w=\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}.\end{array}\end{eqnarray*}$
Consequently, the following exact solution to the STOBE is obtained $\begin{eqnarray*}\begin{array}{l}{u}_{3}\left(x,t\right) \,=\displaystyle \frac{{a}_{0}+{a}_{1}{a}^{x-\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}t}+{a}_{2}{a}^{2\left(x-\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}t\right)}}{{b}_{0}+{b}_{1}{a}^{x-\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}t}+{b}_{2}{a}^{2\left(x-\displaystyle \frac{3{\left(\mathrm{ln}\left(a\right)\right)}^{2}{c}_{1}^{2}-{c}_{2}^{2}}{3{c}_{1}}t\right)}},\end{array}\end{eqnarray*}$
where $\begin{eqnarray*}\begin{array}{l}{a}_{0}=-\displaystyle \frac{{b}_{0}\left(3\,\mathrm{ln}\left(a\right){c}_{1}+{c}_{2}\right)}{3{c}_{1}},\,{a}_{1}=-\displaystyle \frac{{b}_{1}{c}_{2}}{3{c}_{1}},\\ {a}_{2}=-\displaystyle \frac{{b}_{1}^{2}\left(3\,\mathrm{ln}\left(a\right){c}_{1}-{c}_{2}\right)}{6{b}_{0}{c}_{1}},\,{b}_{2}=-\displaystyle \frac{{b}_{1}^{2}}{2{b}_{0}}.\end{array}\end{eqnarray*}$
Figure 2 gives the dynamical features of the kink soliton ${u}_{2}\left(x,t\right)$ for ${a}_{1}=-1,$ ${b}_{0}=1,$ ${b}_{2}=1,$ ${c}_{1}=0.15,$ ${c}_{2}=0.15,$ and $a=2.7.$ Furthermore, the physical behaviors of ${u}_{2}^{{\rm{K}}{\rm{M}}}$ and ${u}_{2}^{{\rm{E}}{\rm{M}}}$ for above parameters have been given in figure 3 when $t=0.$According to the authors’ knowledge, the outcomes of the current investigation are new and have been listed for the first time.
The authors successfully used a symbolic computation system to check the correctness of the outcomes of the current paper.
Figure 2. The kink soliton ${u}_{2}\left(x,t\right)$ for ${a}_{1}=-1,$ ${b}_{0}=1,$ ${b}_{2}=1,$ ${c}_{1}=0.15,$ ${c}_{2}=0.15,$ and $a=2.7.$ |
Figure 3. ${u}_{2}^{{\rm{K}}{\rm{M}}}$ and ${u}_{2}^{{\rm{E}}{\rm{M}}}$ for above parameters when $t=0.$ |
5. Conclusion
The principal aim of the current paper was to explore a newly well-established model known as the Sharma–Tasso–Olver–Burgers equation and derive its conservation laws and kink solitons. The study proceeded systematically by constructing the formal Lagrangian, Lie symmetries, and adjoint equations of STOBE to acquire its conservation laws. Besides, kink solitons of the STOBE were formally established using Kudryashov and exponential methods. Various plots in 2 and 3D postures were graphically represented to observe the dynamical characteristics of kink solitons. Based on information from the authors, the outcomes of the current investigation are new and have been listed for the first time. The authors’ suggestion for future works is employing newly well-organized methods [38–44] to acquire other wave structures of STOBE.
Declaration of competing interest
The authors declare no conflict of interest.