The sine-Gordon (sG) equation, expressed in cone coordinates as
$\begin{eqnarray}{u}_{xt}=\gamma \sin u,\end{eqnarray}$
where γ is an arbitrary constant, originates from the field of differential geometry. Its applicability, however, has greatly expanded across numerous disciplines of physics, underscoring its profound utility as a model for the dynamics of slowly varying wave amplitudes. For instance, the sG equation effectively describes the propagation of dislocations within crystalline structures, characterized by a periodic potential represented by $\sin u$ [1]. Furthermore, the sG equation serves as a theoretical framework for modeling elementary particles, demonstrating equivalence to the Thirring model [2, 3]. It also plays a crucial role in elucidating the propagation of magnetic flux in long Josephson-junction transmission lines [4, 5]. Furthermore, the sG equation is instrumental in describing essential properties of the two-dimensional Coulomb gas [6]. It governs the modulation of weakly unstable baroclinic wave packets in two-layer fluid systems, offering insights into analogous wave dynamics in moving media [7, 8]. The sG equation has emerged as a foundational model in both theoretical and applied physics, owing to its capacity to describe a diverse array of physical phenomena..Since its establishment, the sG equation has attracted considerable attention and has been the focus of extensive scholarly research. The equation is characterized by the presence of Lax pairs, which facilitate its resolution using the inverse scattering transform. A variety of analytical methods have been successfully applied to the sG equation, including Darboux transformations, Bäcklund transformations, and the linear superposition principle. Furthermore, the sG equation is recognized as CRE integrable, as evidenced in the work by Lou [9]. Various solutions of the sG equation have been derived, including n-soliton solutions, breather solutions, soliton molecule solutions [10], negaton and complexiton solutions [11], and rogue periodic wave solutions [12]. Additionally, the sG equation has been generalized to encompass nonlocal cases [13–15], discrete settings [16, 17], fractional equation [18], and a supersymmetric framework [19]. Furthermore, the duality problems associated with the potential modified Korteweg–de Vries (mKdV) hierarchy and the sG hierarchy have been explored via relativistically invariant fields [20].
Investigating the symmetries associated with the sG equation is of particular interest and significance. The symmetry σ of the sG equation (1 ) can be described by 1 ) is invariant under the transformation u → u + εσ, where ε is an infinitesimal parameter. The recursion operator, also referred to as the strong symmetry Φ, of the sG equation (1 ) is defined by [20–22]2 ). By applying the recursion operator Φ to these symmetries, one can derive an infinite sequence of symmetries as follows:
$\begin{eqnarray}{\sigma }_{xt}-\gamma \sigma \cos u=0,\end{eqnarray}$
indicating the sG equation ( $\begin{eqnarray}{\rm{\Phi }}={D}^{2}+{u}_{x}^{2}-{u}_{x}{D}^{-1}{u}_{xx},\end{eqnarray}$
where D = ∂x and ${D}^{-1}={\partial }_{x}^{-1}$ satisfy the relations DD−1 = D−1D = 1. The existence of this recursion operator suggests that the sG equation possesses an infinite number of symmetries, thereby categorizing it as a symmetry integrable system. Notably, various symmetries can be demonstrated, including those representing spatial translational invariance ux, temporal translational invariance ut, and scale transformation invariance tut − xux, all of which satisfy the linearized equation ( $\begin{eqnarray}\begin{array}{rcl}{K}_{m+1} & = & {{\rm{\Phi }}}^{m}{u}_{x},\qquad {G}_{m+1}={{\rm{\Phi }}}^{m}{u}_{t},\\ {\tau }_{m+1} & = & {{\rm{\Phi }}}^{m}(t{u}_{t}-x{u}_{x}),\,m=0,1,\,\ldots ,\,\infty .\end{array}\end{eqnarray}$
Here, the symmetries Km+1 are assigned as K-symmetries, while the symmetries τm+1 are referred to as τ-symmetries. The first few K-symmetries are articulated as follows $\begin{eqnarray}{K}_{1}={u}_{x},\end{eqnarray}$
$\begin{eqnarray}{K}_{2}={u}_{xxx}+\frac{1}{2}{u}_{x}^{3},\end{eqnarray}$
$\begin{eqnarray}{K}_{3}={u}_{xxxxx}+\frac{5}{2}{u}_{x}^{2}{u}_{xxx}+\frac{5}{2}{u}_{x}{u}_{xx}^{2}+\frac{3}{8}{u}_{x}^{5},\end{eqnarray}$
and the initial three τ-symmetries are delineated as $\begin{eqnarray}{\tau }_{1}=t{u}_{t}-x{u}_{x},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\tau }_{2} & = & -x\left({u}_{xxx}+\frac{1}{2}{u}_{x}^{3}\right)+t\left({u}_{x}^{2}{u}_{t}-{u}_{x}{D}^{-1}{u}_{xx}{u}_{t}+{u}_{xxt}\right)\\ & & -2{u}_{xx}-\frac{1}{2}{u}_{x}{D}^{-1}{u}_{x}^{2},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\tau }_{3} & = & -x\left({u}_{xxxxx}+\frac{5}{2}{u}_{x}^{2}{u}_{xxx}+\frac{5}{2}{u}_{x}{u}_{xx}^{2}+\frac{3}{8}{u}_{x}^{5}\right)\\ & & +t\left[{u}_{x}{D}^{-1}{u}_{xxx}{u}_{xx}+\left(\frac{1}{2}{u}_{x}^{2}+{u}_{xxx}\right){D}^{-1}{u}_{xt}{u}_{x}\right.\\ & & +\frac{1}{2}{u}_{x}{D}^{-1}{u}_{x}^{3}{u}_{xt}+{u}_{xxxxt}\\ & & \left.+2{u}_{xt}{u}_{x}{u}_{xx}+2{u}_{x}^{2}{u}_{xt}\right]+\frac{3}{2}{u}_{x}{D}^{-1}{u}_{xx}^{2}\\ & & -\left(\frac{1}{4}{u}_{x}^{3}+\frac{1}{2}{u}_{xxx}\right){D}^{-1}{u}_{x}^{2}-\frac{3}{8}{u}_{x}{D}^{-1}{u}_{x}^{4}\\ & & -4{u}_{xxxx}-7{u}_{x}^{2}{u}_{xx}.\end{array}\end{eqnarray}$
The Gm+1 symmetry described in (4 ) is independent of the K-symmetries and τ-symmetries. This independent can be demonstrated through an examination of the initial symmetries, which are enumerated as follows:
$\begin{eqnarray}\begin{array}{rcl}{G}_{1} & = & {u}_{t},\\ {G}_{2} & = & {u}_{ttt}+\frac{1}{2}{u}_{t}^{3},\\ {G}_{3} & = & {u}_{ttttt}+\frac{5}{2}{u}_{t}^{2}{u}_{ttt}+\frac{5}{2}{u}_{t}{u}_{tt}^{2}+\frac{3}{8}{u}_{t}^{5}.\end{array}\end{eqnarray}$
Despite significant advancements in the study of symmetries of the sG equation, numerous open questions remain that warrant further investigation. For example, it has been known that K1 symmetry is regarded as spatial translational invariance and the τ1 is responsible for scaling transformation invariance. While K1 and τ1 yield clear physical interpretations, the implications of the remaining infinite symmetries have yet to be systematically analyzed. Moreover, the completeness of the identified symmetries has not been thoroughly explored. Additionally, the potential for generating multi-soliton solutions through the utilization of generalized symmetries remains insufficiently examined.
Recent inquiries addressing the infinite symmetries highlighted in [23] and [24] have made progress toward resolving these unresolved issues. It has been established that the known K-symmetries and τ-symmetries of both the Korteweg–de Vries (KdV) equation and the Burgers equation can be represented as linear combinations of parameter translation symmetries, including center translation and wave number translation symmetries. For a fixed n-wave solution, wherein n is a finite quantity, only a restricted subset of the K-symmetries and τ-symmetries remains independent; the remaining symmetries can be expressed as linear combinations derived from this finite collection, with the exception of some special symmetry.
It is then natural to explore the infinite number of symmetries of the sG equation using the method proposed in [23] and [24]. Let us start from the symmetries of the sG equation related to the n-soliton solutions. The n-soliton solution of the sG equation (1 ) possesses the form
$\begin{eqnarray}{u}_{ns}=2{\rm{i}}\,{\mathrm{ln}}\,\left(\frac{{f}^{* }}{f}\right),\end{eqnarray}$
where ${\rm{i}}=\sqrt{-1}$, f* is the conjugate of f, given by $\begin{eqnarray}\begin{array}{l}f=\displaystyle \sum _{\mu =0,1}\exp \left(\displaystyle \sum _{j=1}^{n}{\mu }_{j}\left({\xi }_{i}+{\rm{i}}\frac{\pi }{2}\right)+\displaystyle \sum _{1\leqslant j\lt l}^{n}{\mu }_{j}{\mu }_{l}{A}_{jl}\right),\\ {\xi }_{j}={k}_{j}x+\frac{\gamma }{{k}_{j}}t+{c}_{j},\,\\ {{\rm{e}}}^{{A}_{jl}}={\left(\frac{{k}_{j}-{k}_{l}}{{k}_{j}+{k}_{l}}\right)}^{2},\,j=1,2,\,\ldots ,\,n,\end{array}\end{eqnarray}$
where the sum over μ = 0, 1 refers to each of the μj, j = 1, 2, …, n, and kj and cj are arbitrary constants. In wave theory, the parameters ki and ci have distinct physical interpretations. The parameter ki represents the wave number, which characterizes the spatial frequency of the wave, determining its wavelength and propagation properties. In contrast, ci denotes the central position of the wave in the spatial domain, specifying its localization.For example, the two-soliton solution is written as 13 ).
$\begin{eqnarray}{u}_{2s}=2{\rm{i}}\,{\mathrm{ln}}\,\left(\frac{{F}_{2}}{{G}_{2}}\right),\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{F}_{2} & = & -{({k}_{1}+{k}_{2})}^{2}[\exp ({\xi }_{1})+\exp ({\xi }_{2})-1]\\ & & +{({k}_{1}-{k}_{2})}^{2}\exp ({\xi }_{1}+{\xi }_{2}),\\ {G}_{2} & = & {({k}_{1}+{k}_{2})}^{2}[\exp ({\xi }_{1})+\exp ({\xi }_{2})+1]\\ & & +{({k}_{1}-{k}_{2})}^{2}\exp ({\xi }_{1}+{\xi }_{2}),\end{array}\end{eqnarray}$
with ξ1 and ξ2 given by (For the two-soliton solution (14 )–(15 ), we can find the sG equation (1 ) possesses four special symmetries 16 ) and (17 ), are solutions to the symmetry equation (2 ) with u = u2s given by (14 )-(15 ). The physical significance of these four symmetries is readily discernible. The parameters k1 and k2 denote the wave numbers of the solitons, while c1 and c2 represent their respective center positions. Consequently, the symmetries presented in (16 ) correspond to translations of the wave numbers, whereas the symmetries specified in (17 ) relate to translations of the soliton centers.
$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{{k}_{1}} & = & {\partial }_{{k}_{1}}{u}_{2s}=\frac{4{\rm{i}}({k}_{1}+{k}_{2})\exp ({\xi }_{1})}{{k}_{1}^{2}{F}_{2}{G}_{2}}\\ & & \times [4{k}_{1}^{2}{k}_{2}({k}_{1}-{k}_{2})\exp ({\xi }_{1}+{\xi }_{2})\\ & & +({k}_{1}-{k}_{2})({k}_{1}^{4}x-{k}_{1}^{2}{k}_{2}^{2}x-\gamma {k}_{1}^{2}t+\gamma {k}_{2}^{2}t+4{k}_{1}^{2}{k}_{2})\\ & & \times \exp (2{\xi }_{2})-{({k}_{1}+{k}_{2})}^{3}({k}_{1}^{2}x-\gamma t))],\\ {\sigma }_{{k}_{2}} & = & {\partial }_{{k}_{2}}{u}_{2s}=\frac{4{\rm{i}}({k}_{1}+{k}_{2})\exp ({\xi }_{2})}{{k}_{2}^{2}{F}_{2}{G}_{2}}\\ & & \times [4{k}_{1}{k}_{2}^{2}({k}_{2}-{k}_{1})\exp ({\xi }_{1}+{\xi }_{2})\\ & & +({k}_{2}-{k}_{1})({k}_{2}^{4}x-{k}_{1}^{2}{k}_{2}^{2}x+\gamma {k}_{1}^{2}t-\gamma {k}_{2}^{2}t+4{k}_{1}{k}_{2}^{2})\\ & & \times \exp (2{\xi }_{1})-{({k}_{1}+{k}_{2})}^{3}({k}_{2}^{2}x-\gamma t)],\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{\sigma }_{{c}_{1}} & = & {\partial }_{{c}_{1}}{u}_{2s}=\frac{4{\rm{i}}{({k}_{1}+{k}_{2})}^{2}\exp ({\xi }_{1})}{{F}_{2}{G}_{2}}\\ & & \times [{({k}_{1}-{k}_{2})}^{2}\exp (2{\xi }_{2})-{({k}_{1}+{k}_{2})}^{2}],\\ {\sigma }_{{c}_{2}} & = & {\partial }_{{c}_{2}}{u}_{2s}=\frac{4{\rm{i}}{({k}_{1}+{k}_{2})}^{2}\exp ({\xi }_{2})}{{F}_{2}{G}_{2}}\\ & & \times [{({k}_{1}+{k}_{2})}^{2}-{({k}_{1}-{k}_{2})}^{2}\exp (2{\xi }_{1})].\end{array}\end{eqnarray}$
These symmetries, expressed by (The symmetry equation (2 ) is characterized by linearity, indicating that linear combinations of these four special symmetries also constitute symmetries of the sG equation associated with two-soliton solutions. Consequently, it is relatively straightforward to derive that the infinitely many K-symmetries delineated in (4 ) can be expressed as 17 ). The relations outlined in (18 ) demonstrate the infinite K-symmetries corresponding to the two-soliton solution are linear combinations of the two center translation symmetries, ${\sigma }_{{c}_{1}}$ and ${\sigma }_{{c}_{2}}$. Additionally, it is important to note that the K-symmetries are not incomplete. Among the total K-symmetries, only K1 and K2 are independent and not derivable from other symmetries, while the remaining K-symmetries can be constructed through linear combinations as indicated by
$\begin{eqnarray}\begin{array}{rcl}{K}_{1}{| }_{u={u}_{2s}} & = & {u}_{x}{| }_{u={u}_{2s}}={k}_{1}{\sigma }_{{c}_{1}}+{k}_{2}{\sigma }_{{c}_{2}},\\ {K}_{2}{| }_{u={u}_{2s}} & = & {\rm{\Phi }}{u}_{x}{| }_{u={u}_{2s}}={k}_{1}^{3}{\sigma }_{{c}_{1}}+{k}_{2}^{3}{\sigma }_{{c}_{2}},\\ {K}_{3}{| }_{u={u}_{2s}} & = & {{\rm{\Phi }}}^{2}{u}_{x}{| }_{u={u}_{2s}}={k}_{1}^{5}{\sigma }_{{c}_{1}}+{k}_{2}^{5}{\sigma }_{{c}_{2}},\\ & \vdots & \\ {K}_{p}{| }_{u={u}_{2s}} & = & {{\rm{\Phi }}}^{p-1}{u}_{x}{| }_{u={u}_{2s}}=\displaystyle \sum _{j=1}^{2}{k}_{j}^{2p-1}{\sigma }_{{c}_{j}}\\ & = & {k}_{1}^{2p-1}{\sigma }_{{c}_{1}}+{k}_{2}^{2p-1}{\sigma }_{{c}_{2}},\\ p & = & 4,5,\ldots ,\,\infty ,\end{array}\end{eqnarray}$
where ${\sigma }_{{c}_{1}}$ and ${\sigma }_{{c}_{2}}$ are given by ( $\begin{eqnarray}\begin{array}{l}{K}_{m\geqslant 3}\,=\,{k}_{1}^{2}{k}_{2}^{2}({A}_{1m}{K}_{1}+{A}_{2m}{K}_{2}),\\ {A}_{1m}\,=\,\frac{{k}_{2}^{2m-4}-{k}_{1}^{2m-4}}{{k}_{1}^{2}-{k}_{2}^{2}},\,{A}_{2m}=\frac{{k}_{2}^{2m-2}-{k}_{1}^{2m-2}}{{k}_{1}^{2}{k}_{2}^{2}({k}_{1}^{2}-{k}_{2}^{2})}.\end{array}\end{eqnarray}$
For the infinitely many τ-symmetries given by (4 ) correspond to the two-soliton solution (14 )-(15 ) of the sG equation, we can find the following relations 16 ) and the two center translation symmetries (17 ). Furthermore, the τ-symmetries described in (20 ) are not entirely independent, as the symmetries K1, K2, τ1 and τ2 can be regarded as the independent components. The linear combinations of these independent symmetries collectively compose the full τ-symmetries as
$\begin{eqnarray}\begin{array}{rcl}{\tau }_{1}{| }_{u={u}_{2s}} & = & -{k}_{1}{\sigma }_{{k}_{1}}-{k}_{2}{\sigma }_{{k}_{2}},\\ {\tau }_{2}{| }_{u={u}_{2s}} & = & (2{k}_{1}^{2}+4{k}_{1}{k}_{2}){\sigma }_{{c}_{1}}\\ & & +(4{k}_{1}{k}_{2}+2{k}_{2}^{2}){\sigma }_{{c}_{2}}-{k}_{1}^{3}{\sigma }_{{k}_{1}}-{k}_{2}^{3}{\sigma }_{{k}_{2}},\\ {\tau }_{3}{| }_{u={u}_{2s}} & = & (4{k}_{1}^{4}+4{k}_{1}^{3}{k}_{2}+4{k}_{1}{k}_{2}^{3}){\sigma }_{{c}_{1}}\\ & & +(4{k}_{2}^{4}+4{k}_{1}^{3}{k}_{2}+4{k}_{1}{k}_{2}^{3}){\sigma }_{{c}_{2}}-{k}_{1}^{5}{\sigma }_{{k}_{1}}-{k}_{2}^{5}{\sigma }_{{k}_{2}},\\ & \vdots & \\ {\tau }_{m}{| }_{u={u}_{2s}} & = & 2(m-1)\displaystyle \sum _{j=1}^{2}{k}_{j}^{2(m-1)}{\sigma }_{{c}_{j}}\\ & & +4\left(\displaystyle \sum _{q=1}^{m-1}{k}_{1}^{2q-1}{k}_{2}^{2m-2q-1}\right)\displaystyle \sum _{j=1}^{2}{\sigma }_{{c}_{j}}-\displaystyle \sum _{j=1}^{2}{k}_{j}^{2m-1}{\sigma }_{{k}_{j}},\\ m & = & 4,5,\,\ldots .\end{array}\end{eqnarray}$
These relationships indicate that the τ-symmetries can be represented as linear combinations of the two wave number translation symmetries ( $\begin{eqnarray}\begin{array}{rcl}{\tau }_{m\geqslant 3} & = & {k}_{1}^{2}{k}_{2}^{2}{A}_{1m}{\tau }_{1}+{k}_{1}^{2}{k}_{2}^{2}{A}_{2m}{\tau }_{2}\\ & & +\frac{2{B}_{1m}}{{k}_{1}+{k}_{2}}{K}_{1}+\frac{2{B}_{2m}}{{k}_{1}+{k}_{2}}{K}_{2},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{B}_{1m} & = & -(m-1){k}_{1}{k}_{2}{({k}_{1}+{k}_{2})}^{2}({k}_{1}^{2}-{k}_{1}{k}_{2}+{k}_{2}^{2}){A}_{2m}\\ & & -m{k}_{1}^{3}{k}_{2}^{3}{A}_{2m}+(m-1)({k}_{1}+{k}_{2})({k}_{1}^{2}+{k}_{1}{k}_{2}+{k}_{2}^{2}){A}_{3m},\\ {B}_{2m} & = & (m-1){k}_{1}{k}_{2}({k}_{1}^{2}+{k}_{2}^{2}){A}_{2m}+(m-2){k}_{1}^{2}{k}_{2}^{2}{A}_{2m}\\ & & -(m-1){A}_{3m},\,{A}_{3m}=\frac{{k}_{1}^{2m}-{k}_{2}^{2m}}{{k}_{1}{k}_{2}({k}_{1}^{2}-{k}_{2}^{2})}.\end{array}\end{eqnarray}$
The results obtained in this study are quite different from those reported in [23] and [24]. Certain symmetries that lack physical significance have been systematically excluded from the complete set of symmetries associated with the KdV equation, the Burgers equation, and the STO equation. In contrast, the infinitely many K- and τ-symmetries of the sG equation encompass all established symmetries, as the foundational symmetries ux and tut − xux exhibit well-defined physical significance.
For the n-soliton solution described by (12 )-(13 ), the K-symmetries of the sG equation can be articulated as presented in 23 ) are not autonomous; only n of these, specifically Ki, i = 1, 2, …, n, are independent. The remaining symmetries can be constructed linearly from these independent symmetries, where
$\begin{eqnarray}\begin{array}{rcl}{K}_{m}{| }_{u={u}_{ns}} & = & \displaystyle \sum _{j=1}^{n}{k}_{j}^{2m-1}{\sigma }_{{c}_{j}},\qquad {\sigma }_{{c}_{j}}={\partial }_{{c}_{j}}{u}_{ns},\\ m & = & 1,2,\,\ldots ,\infty .\end{array}\end{eqnarray}$
Notably, the infinitely many K-symmetries identified in ( $\begin{eqnarray}\begin{array}{rcl}{K}_{m}{| }_{u={u}_{ns}} & = & {{\rm{\Phi }}}^{m-1}{u}_{x}{| }_{u={u}_{ns}}=\displaystyle \sum _{p=1}^{n}{k}_{p}^{2m-1}\frac{{{\rm{\Gamma }}}_{np}}{{{\rm{\Gamma }}}_{n}}{\left|\right.}_{u={u}_{ns}},\\ m & = & n+1,n+2,\,\ldots ,\,\infty .\end{array}\end{eqnarray}$
Here, Γnp represents the determinant of the n × n matrix Mp, expressed as ${{\rm{\Gamma }}}_{np}\equiv {\rm{\det }}({M}_{p})$, with $\begin{eqnarray}\begin{array}{rcl} & & {M}_{p}\\ & = & \left(\begin{array}{ccccccc}{k}_{1} & {k}_{2} & \cdots \, & {K}_{1} & \cdots \, & {k}_{n-1} & {k}_{n}\\ {k}_{1}^{3} & {k}_{2}^{3} & \cdots \, & {K}_{2} & \cdots \, & {k}_{n-1}^{3} & {k}_{n}^{3}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {k}_{1}^{2n-1} & {k}_{2}^{2n-1} & \cdots \, & {K}_{2n-1} & \cdots \, & {k}_{n-1}^{2n-1} & {k}_{n}^{2n-1}\end{array}\right),\end{array}\end{eqnarray}$
while Γn denotes the determinant of another n × n matrix M, defined as ${{\rm{\Gamma }}}_{n}\equiv {\rm{\det }}(M)$ with $\begin{eqnarray}\begin{array}{rcl} & & M\\ & = & \left(\begin{array}{ccccccc}{k}_{1} & {k}_{2} & \cdots \, & {k}_{p} & \cdots \, & {k}_{n-1} & {k}_{n}\\ {k}_{1}^{3} & {k}_{2}^{3} & \cdots \, & {k}_{p}^{3} & \cdots \, & {k}_{n-1}^{3} & {k}_{n}^{3}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {k}_{1}^{2n-1} & {k}_{2}^{2n-1} & \cdots \, & {k}_{p}^{2n-1} & \cdots \, & {k}_{n-1}^{2n-1} & {k}_{n}^{2n-1}\end{array}\right).\end{array}\end{eqnarray}$
In addition to the n-soliton solution, the sG equation (1 ) admits a diverse range of solution types, including complexitons, positons, and negatons. Of particular interest is a unique one-complex solution, which is given by
$\begin{eqnarray}\begin{array}{rcl}{u}_{1c} & = & 2{\rm{i}}\,{\mathrm{ln}}\,\left(\frac{\sinh (\eta )-\theta }{\sinh (\eta )+\theta }\right),\qquad \eta =\lambda x+\frac{\gamma t}{\lambda }+{\delta }_{1},\\ \theta & = & \lambda x-\frac{\gamma t}{\lambda }+{\delta }_{2},\end{array}\end{eqnarray}$
where λ, δ1 and δ2 are arbitrary constants. Similar to the soliton solution, λ represents the wave number of the complexitons, while δi, i = 1, 2 signify the central position of the complexitons.It can be readily demonstrated that there are three special symmetries associated with the one-complex solution (27 ), reading as 28 )-(30 ) constitute solutions to the symmetry equation (2 ) with u = u1c. Specifically, the symmetries (28 )-(29 ) correspond to the real and imaginary components of the wave center translation, while the symmetry (30 ) is associated with the wave number translation.
$\begin{eqnarray}\begin{array}{rcll}{\sigma }_{{\delta }_{1}} & = & {\partial }_{{\delta }_{1}}{u}_{1c}= & \frac{8{\rm{i}}\lambda ({\lambda }^{2}x-\gamma t+{\delta }_{2}\lambda )\cosh (\eta )}{{\lambda }^{2}\cosh (\eta )-2{\lambda }^{4}{x}^{2}-2{\gamma }^{2}{t}^{2}-4{\lambda }^{3}{\delta }_{2}x+4({\lambda }^{2}\gamma x+{\delta }_{2}\lambda \gamma )t-{\lambda }^{2}(2{\delta }_{2}^{2}+1)},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcll}{\sigma }_{{\delta }_{2}} & = & {\partial }_{{\delta }_{2}}{u}_{1c}= & \frac{-8{\rm{i}}{\lambda }^{2}\sinh (\eta )}{{\lambda }^{2}\cosh (\eta )-2{\lambda }^{4}{x}^{2}-2{m}^{2}{t}^{2}-4{\lambda }^{3}{\delta }_{2}x+4({\lambda }^{2}\gamma x+{\delta }_{2}\lambda \gamma )t-{\lambda }^{2}(2{\delta }_{2}^{2}+1)},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rlll}{\sigma }_{\lambda } & = & {\partial }_{\lambda }{u}_{1c}= & \frac{8{\rm{i}}[({\lambda }^{2}x-\gamma t)({\lambda }^{2}x-\gamma t+{\delta }_{2}\lambda )\cosh (\eta )-\lambda ({\lambda }^{2}x+\gamma t)\sinh (\eta )]}{\lambda [{\lambda }^{2}\cosh (\eta )-2{\lambda }^{4}{x}^{2}-2{\gamma }^{2}{t}^{2}-4{\lambda }^{3}{\delta }_{2}x+4({\lambda }^{2}\gamma x+{\delta }_{2}\lambda \gamma )t-{\lambda }^{2}(2{\delta }_{2}^{2}+1)]}.\end{array}\end{eqnarray}$
In other words, the symmetries described in (The K-symmetries associated with the one-complex solution (27 ), can be expressed as 31 ) and τ-symmetries (32 ) associated with the one-complex solution (27 ) can be expressed as linear combinations of the center translation symmetries ${\sigma }_{{\delta }_{1}}$, ${\sigma }_{{\delta }_{2}}$, and the wave number translation symmetry σλ. Among these, only K1, K2 and τ1 are independent, with the complete set of symmetries described by
$\begin{eqnarray}\begin{array}{rcl}{K}_{1} & = & \lambda {\sigma }_{{\delta }_{1}}+\lambda {\sigma }_{{\delta }_{2}},\\ {K}_{2} & = & {k}_{1}^{3}{\sigma }_{{\delta }_{1}}+3{\lambda }^{3}{\sigma }_{{\delta }_{2}},\\ {K}_{3} & = & {\lambda }^{5}{\sigma }_{{\delta }_{1}}+5{\lambda }^{5}{\sigma }_{{\delta }_{2}},\\ & \vdots & \\ {K}_{m} & = & {\lambda }^{2m-1}{\sigma }_{{\delta }_{1}}+(2m-1){\lambda }^{2m-1}{\sigma }_{{\delta }_{2}},\,m=4,5,\,\ldots .\end{array}\end{eqnarray}$
whereas the τ-symmetries are represented by $\begin{eqnarray}\begin{array}{rcl}{\tau }_{1} & = & -\lambda {\sigma }_{\lambda },\\ {\tau }_{2} & = & -{\lambda }^{3}{\sigma }_{\lambda }+4{\lambda }^{2}{\sigma }_{{\delta }_{1}}+(2{\delta }_{2}{\lambda }^{2}+4{\lambda }^{2}){\sigma }_{{\delta }_{2}},\\ {\tau }_{3} & = & -{\lambda }^{5}{\sigma }_{\lambda }+8{\lambda }^{4}{\sigma }_{{\delta }_{1}}+(4{\delta }_{2}{\lambda }^{4}+16{\lambda }^{4}){\sigma }_{{\delta }_{2}},\\ & \vdots & \\ {\tau }_{m} & = & -{\lambda }^{2m-1}{\sigma }_{\lambda }+4(m-1){\lambda }^{2(m-1)}{\sigma }_{{\delta }_{1}}\\ & & +2(m-1){\lambda }^{2(m-1)}[{\delta }_{2}+2(m-1)]{\sigma }_{{\delta }_{2}},\\ m & = & 4,5,\,\ldots .\end{array}\end{eqnarray}$
Consequently, the K-symmetries ( $\begin{eqnarray}{K}_{m\geqslant 3}=-(m-2){\lambda }^{2m-2}{K}_{1}+(m-1){\lambda }^{2m-4}{K}_{2},\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{\tau }_{m\geqslant 3} & = & {\lambda }^{2(m-1)}{\tau }_{1}-(m-1)(2m+{\delta }_{2}-8){\lambda }^{2m-3}{K}_{1}\\ & & +(m-1)(2m+{\delta }_{2}-4){\lambda }^{2m-5}{K}_{2}.\end{array}\end{eqnarray}$
The construction of multi-soliton solutions can be systematically achieved through the application of generalized symmetries, such as the K-symmetry. For instants, the general n-wave solution of the sG equation can be expressed as 35 ) into the sG equation (1 ) yields the constraint equation 35 ) are given by 3 ). By substituting the n-wave solution (35 ) into the K-symmetries (37 ), we obtain a sequence of expressions that characterize the symmetries of the sG equation, denoted as
$\begin{eqnarray}\begin{array}{l}u=U({\xi }_{1},\,{\xi }_{2},\,\ldots ,{\xi }_{n})\equiv U,\,\\ {\xi }_{j}={k}_{j}x+{\omega }_{j}t+{c}_{j},\,j=1,2,\,\ldots ,\,n,\end{array}\end{eqnarray}$
where kj, ωj, j = 1, 2, …, n are constants to be further determined. Substituting the general n-wave solution ( $\begin{eqnarray}\displaystyle \sum _{p,\,q=1}^{n}{k}_{p}{\omega }_{q}{U}_{{\xi }_{p}{\xi }_{q}}=\sin U.\end{eqnarray}$
Here, for simplicity, the parameter γ in sG equation is set to 1. The infinitely many K-symmetries related to the n-wave solution ( $\begin{eqnarray}{K}_{m}={{\rm{\Phi }}}^{m-1}{u}_{x},\,m=1,2,\,\ldots ,\,\infty ,\end{eqnarray}$
where the recursion operator Φ is defined by ( $\begin{eqnarray}{K}_{1}:{K}_{1}=\displaystyle \sum _{p=1}^{n}{k}_{p}{U}_{{\xi }_{p}},\end{eqnarray}$
$\begin{eqnarray}{K}_{2}:\displaystyle \sum _{j=1}^{n}{a}_{2j}{U}_{{\xi }_{j}}=\frac{1}{2}\displaystyle \sum _{p=1}^{n}{k}_{p}^{3}{U}_{{\xi }_{p}}^{3}+\displaystyle \sum _{p,q,r}^{n}{k}_{p}{k}_{q}{k}_{r}{U}_{{\xi }_{p}{\xi }_{q}{\xi }_{r}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{K}_{3}:\displaystyle \sum _{j=1}^{n}{a}_{3j}{U}_{{\xi }_{j}} & = & \frac{3}{8}\displaystyle \sum _{p,q,r,s,l}^{n}{k}_{p}{k}_{q}{k}_{r}{k}_{s}{k}_{l}{U}_{{\xi }_{p}}{U}_{{\xi }_{q}}{U}_{{\xi }_{r}}{U}_{{\xi }_{s}}{U}_{{\xi }_{l}}\\ & & +\displaystyle \sum _{p,q,r,s,l}^{n}{k}_{p}{k}_{q}{k}_{r}{k}_{s}{k}_{l}{U}_{{\xi }_{i}{\xi }_{q}{\xi }_{r}{\xi }_{s}{\xi }_{l}}\\ & & +\frac{5}{2}\displaystyle \sum _{p,q,r,s,r}^{n}{k}_{p}{k}_{q}{k}_{r}{k}_{s}{k}_{l}{U}_{{\xi }_{p}{\xi }_{q}{\xi }_{r}}{U}_{{\xi }_{s}}{U}_{{\xi }_{l}}\\ & & +\frac{2}{5}\displaystyle \sum _{p,q,r,s,l}^{n}{k}_{p}{k}_{q}{k}_{r}{k}_{s}{k}_{l}{U}_{{\xi }_{p}{\xi }_{q}}{U}_{{\xi }_{r}{\xi }_{s}}{U}_{{\xi }_{l}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{K}_{m}:\displaystyle \sum _{m=1}^{n}{a}_{mj}{U}_{{\xi }_{j}} & = & {K}_{m}{| }_{u=U,{u}_{x}=\displaystyle \sum _{p}{k}_{p}{U}_{{\xi }_{p}},{U}_{xx}=\displaystyle \sum _{p,q}{k}_{p}{k}_{q}{U}_{{\xi }_{p}{\xi }_{q}},\,\ldots ,}\\ & & m\gt 3,\end{array}\end{eqnarray}$
where the parameter amj, m ≥ 2, j = 1, 2, …, n are constants that must be determined through the dispersion relations.The enforcement of the infinite constraints (38 )–(41 ) within the n-wave assumption (35 ) enables the precise determination of the n-wave solutions of the sG equation (1 ). In particular, by imposing dispersion relations of the form 36 ), (39 ) and (42 ) with n = 1, the one-soliton (kink) solution for sG equation (1 ) with γ = 1 takes the form
$\begin{eqnarray}{\omega }_{j}=\frac{1}{{k}_{j}},\,{a}_{mj}={k}_{j}^{2m-1},\,m=2,3,\,\ldots ,\end{eqnarray}$
one can explicitly derive the n-soliton solutions. A particularly interesting case arises when n = 1. By solving ( $\begin{eqnarray}\begin{array}{rcl}u & = & 2{\rm{i}}\,{\mathrm{ln}}\,\left[\tanh \left({\xi }_{1}+{c}_{0}\right)+{\rm{i}}\sqrt{1-{\tanh }^{2}\left({\xi }_{1}+{c}_{0}\right)}\right],\\ {\xi }_{1} & = & {k}_{1}x+\frac{t}{{k}_{1}}+{c}_{1},\end{array}\end{eqnarray}$
where k1, c1 and c0 are arbitrary constants. To the best of our knowledge, this particular form of the kink solution has not been previously documented in the literature. Nonetheless, owing to the inherent challenges associated with solving the equation and its complex structure, we have been unsuccessful in deriving multi-soliton solutions for n ≥ 2.Regarding the τ-symmetries, we hypothesize that employing a similar methodology could enable the derivation of multi-soliton solutions. However, due to the inherent complexities associated with solving these equations, we have not yet succeeded in obtaining these solutions.
In summary, this paper examines the physical significance of the infinitely many K- and τ-symmetries associated with both soliton and complex solutions of the sG equation. The K-symmetries are expressed as linear combinations of wave center translation symmetries, while the τ-symmetries are linear combinations of both wave center translation and wave number translation symmetries. Notably, only a subset of the K- and τ-symmetries are independent, implying that these symmetries are not fully independent but rather constrained. The n-soliton solutions of the sG equation are derived through the application of generalized symmetries.
We have yet to ascertain the corresponding physical significance of the specific Gm+1 symmetry as outlined in equation (4 ). Further research is warranted to elucidate this aspect.
It is important to emphasize that the potential modified Korteweg–de Vries (pmKdV) hierarchy and the sG hierarchy exhibit a duality, sharing a common recursion operator. We postulate that analogous results can be derived for the pmKdV equation. Further comprehensive investigation is required to solidify these findings.
An interesting observation is that solutions to integrable systems may involve additional free parameters. Future research will be necessary to explore the full range of symmetries and their physical interpretations.