1. Introduction
The study of symmetry is a crucial and impactful approach in various fields of the natural sciences. In particular, for physical systems characterized by differential equations, identifying symmetries can lead to significant reductions in both the order of ordinary differential equations (ODEs) and the dimensionality of partial differential equations (PDEs) associated with nonlinear systems. This reduction process ultimately facilitates the simplification of complex problems [1]. A system that displays an infinite number of symmetries is classified as an integrable system [2].
In the field of integrable systems, the existence of these infinite symmetries highlights the critical importance of symmetry analysis. A variety of methods have been developed to identify and explore these symmetries, including the Lie point symmetry method [1, 3], the master-symmetry method [4], the formal series symmetry method [5], and the recursion operator method [6–8]. Notably, research by Olver [9] demonstrated that recursion operators can be effectively utilized to derive the infinitely many symmetries of several integral PDEs, including the well-known Korteweg–de Vries (KdV) equation, modified Korteweg–de Vries (mKdV) equation, and sine-Gordon (sG) equations.
Moreover, the method of symmetry analysis is intricately connected to other important techniques for solving integrable systems, such as the inverse scattering transform (IST) [10], the Bäcklund transformation [11], the Darboux transformation [12], Hirota's bilinear method [13], as well as classical and non-classical Lie group methods [1, 14]. Additional methods, including the mapping deformation method [15], standard and non-standard truncated Painlevé expansion methods [16, 17], and function expansion methods [18, 19], further illustrate the integral role that symmetry plays in the investigation and resolution of integrable systems.
The exploration of symmetry principles is crucial for advancing our understanding of integrable systems and plays a significant role in the development and application of mathematical physics. Despite notable progress in symmetry studies, several key challenges remain unresolved. One major issue is the physical interpretation of the infinitely many symmetries associated with these systems. While some results elucidating these interpretations are available, comprehensive insights into the implications of these symmetries on physical phenomena are still lacking.
Recent research [20–22] has yielded promising advancements in this area, particularly regarding the classification of infinite symmetries in integrable systems that admit multi-wave solutions. A conjecture has been promoted suggesting that these infinite symmetries can be expressed as linear combinations of translational symmetries linked to the wave parameters. Furthermore, this study introduces a novel method for constructing n-wave solutions, which may provide deeper insights into the underlying structure of integrable systems and their symmetries. Nevertheless, significant gaps remain in our understanding, highlighting the ongoing need for further investigation into the physical implications of these symmetries and their broader applications in the field.
The potential mKdV equation
$\begin{eqnarray}\begin{array}{r}{u}_{t}={u}_{xxx}-2{u}_{x}^{3},\end{array}\end{eqnarray}$
is intricately linked to the mKdV equation by a special transformation which can be applied in various fields of physics, including nonlinear optics [23], relativistic degenerate magnetoplasma [24], dusty plasma [25], optical fibers [26], fluid dynamics [27], and so on. Furthermore, the pmKdV equation shares a common recursion operator with the sine-Gordon (sG) equation, which is extensively utilized in both quantum and classical field theories [28]. Notably, the pmKdV equation is also recognized for governing the fundamental field that emerges on the two-dimensional boundary of the full dynamics of AdS3 in gravitational theory [29].Due to its connections with both the mKdV and sG equations, significant research has been conducted on the pmKdV equation (1 ), such as the derivation of Lie point symmetries, conservation laws [3], multi-soliton solutions [30], and complex solutions. However, the substantial gaps remain in the understanding of the physical interpretations of its infinitely many symmetries and the corresponding conservation laws.
Thus, we undertake a comprehensive investigation of the pmKdV equation, building upon ideas proposed in [20]. The structure of this manuscript is organized as follows: in section 2 , we provide some results related to the symmetries of the pmKdV equation, such as its recursive operator along with the corresponding K-symmetries and τ-symmetries, the physical significance of some known symmetries. The n-soliton solutions are also shown. Section 3 starts from the K-symmetries and τ-symmetries associated with two-soliton solutions of the pmKdV equation, analyzes their physical implications, and demonstrates the incompleteness of the infinitely many symmetries. The physical meaning of the K-symmetries and τ-symmetries associated with n-soliton solution is further studied. Section 4 explore the K-symmetries and τ-symmetries associated with two-wave complexiton solution. In section 5 , we apply the new method outlined in [20] to solve the multi-wave solution of the pmKdV equation. The final section is dedicated to the discussion and summary.
2. Recursion operator and n-soliton solutions
The symmetry σ of the pmKdV equation (1 ) is characterized as the solution to its linearized equation, expressed as 1 ) retains its form under the transformation u → u + εσ, with ε representing an infinitesimal parameter.
$\begin{eqnarray}\begin{array}{r}{\sigma }_{t}={G}^{{\prime} }\sigma =({D}^{3}-6{u}_{x}^{2}D)\sigma ,\end{array}\end{eqnarray}$
where D = ∂x. This relation indicates that the pmKdV equation (A powerful method for identifying symmetries of a given system involves employing a strong symmetry, represented by a recursion operator, to act on known symmetries. A strong symmetry is defined by the condition 3 ), is also a symmetry of the same equation. For the pmKdV equation (1 ), one such recursion operator [28, 31, 32] is given by
$\begin{eqnarray}\begin{array}{r}\frac{{\rm{d}}{\rm{\Phi }}}{{\rm{d}}t}=[{G}^{{\prime} },{\rm{\Phi }}]={G}^{{\prime} }{\rm{\Phi }}-{\rm{\Phi }}{G}^{{\prime} }.\end{array}\end{eqnarray}$
In other words, if σ is a symmetry of an evolution equation, then Φσ, where Φ is determined by equation ( $\begin{eqnarray}\begin{array}{r}{\rm{\Phi }}={D}^{2}-4{u}_{x}{D}^{-1}{u}_{x}D,\end{array}\end{eqnarray}$
where the relations DD−1 = D−1D = 1 hold.The seed symmetries of the pmKdV equation (1 ) can be readily identified, such as 1 ), can be systematically generated through the iterative application of recursive operators to the known symmetries K0 and τ0. These symmetries can be expressed as follows:
$\begin{eqnarray}\begin{array}{r}{K}_{0}={u}_{x},\quad {\tau }_{0}=3t{u}_{x}+\frac{1}{2}{D}^{-1}\left(\exp (2u){D}^{-1}\exp (-2u)\right).\end{array}\end{eqnarray}$
Consequently, the K-symmetries and τ-symmetries of pmKdV equation ( $\begin{eqnarray}\begin{array}{rcl}{K}_{m} & = & {{\rm{\Phi }}}^{m}{u}_{x},\,n=0,\,1,\,\ldots ,\,\infty ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\tau }_{m} & = & {{\rm{\Phi }}}^{m}\left[3t{u}_{x}+\frac{1}{2}{D}^{-1}\left(\exp (2u){D}^{-1}\exp (-2u)\right)\right],\\ m & = & 0,\,1,\,\ldots ,\,\infty .\end{array}\end{eqnarray}$
For example, the first four K-symmetries can be denoted as
$\begin{eqnarray}\begin{array}{rcl}{K}_{0} & = & {u}_{x},\\ {K}_{1} & = & {u}_{xxx}-2{u}_{x}^{3}={u}_{t},\\ {K}_{2} & = & {u}_{xxxxx}-10{u}_{x}{u}_{xx}^{2}-10{u}_{x}^{2}{u}_{xxx}+6{u}_{x}^{5},\\ {K}_{3} & = & {u}_{xxxxxxx}-70{u}_{xx}^{2}{u}_{xxx}-42{u}_{x}{u}_{xxx}^{2}\\ & & -56{u}_{x}{u}_{xx}{u}_{xxxx}-14{u}_{x}^{2}{u}_{xxxxx}+140{u}_{x}^{3}{u}_{xx}^{2}\\ & & +70{u}_{x}^{4}{u}_{xxx}-20{u}_{x}^{7},\end{array}\end{eqnarray}$
while the first four τ-symmetries are represented as $\begin{eqnarray}\begin{array}{rcl}{\tau }_{0} & = & 3t{u}_{x}+\frac{1}{2}{D}^{-1}\left(\exp (2u){D}^{-1}\exp (-2u)\right)\\ & = & 3t{K}_{0}+\frac{1}{2}{D}^{-1}\left(\exp (2u){D}^{-1}\exp (-2u)\right),\\ {\tau }_{1} & = & 3t{u}_{xxx}-6{u}_{x}^{3}t+{u}_{x}x=3t{K}_{1}+x{K}_{0},\\ {\tau }_{2} & = & 18{u}_{x}^{5}t-30{u}_{x}t{u}_{xx}^{2}-2{u}_{x}^{3}x-30{u}_{x}^{2}t{u}_{xxx}\\ & & -2{u}_{x}{D}^{-1}{u}_{x}^{2}+3t{u}_{xxxxx}+{u}_{xxx}x+2{u}_{xx}\\ & = & 3t{K}_{2}+x{K}_{1}-2{u}_{x}{D}^{-1}{u}_{x}^{2}+2{u}_{xx},\\ {\tau }_{3} & = & -60{u}_{x}^{7}t+210{u}_{x}^{4}t{u}_{xxx}+6{u}_{x}^{5}x+420{u}_{x}^{3}t{u}_{xx}^{2}\\ & & -126{u}_{x}t{u}_{xxx}^{2}-10{u}_{x}^{2}x{u}_{xxx}-210{u}_{xx}^{2}t{u}_{xxx}\\ & & +4{D}^{-1}{u}_{x}^{2}{u}_{x}^{3}-42{u}_{x}^{2}t{u}_{xxxxx}-10{u}_{x}x{u}_{xx}^{2}\\ & & -168{u}_{x}t{u}_{xx}{u}_{xxxx}-28{u}_{x}^{2}{u}_{xx}-2{u}_{xxx}{D}^{-1}{u}_{x}^{2}\\ & & +6{u}_{x}{D}^{-1}{u}_{xx}^{2}+6{u}_{x}{D}^{-1}{u}_{x}^{4}\\ & & +3t{u}_{xxxxxxx}+{u}_{xxxxx}x+4{u}_{xxxx}\\ & = & 3t{K}_{3}+x{K}_{2}+4{u}_{x}^{3}{D}^{-1}{u}_{x}^{2}-28{u}_{x}^{2}{u}_{xx}\\ & & -2{u}_{xxx}{D}^{-1}{u}_{x}^{2}+6{u}_{x}{D}^{-1}{u}_{xx}^{2}\\ & & +6{u}_{x}{D}^{-1}{u}_{x}^{4}+4{u}_{xxxx}.\end{array}\end{eqnarray}$
Among these symmetries, only K0, K1, and τ1 exhibit physical significance. Specifically, K0 is associated with spatial translational invariance, K1 correlates with temporal translational invariance, and τ1 governs scaling transformation invariance. The physical implications of the remaining symmetries, however, remain unclear.The n-soliton solution of the pmKdV equation (1 ) possesses the form [33, 34]
$\begin{eqnarray}{u}_{ns}=\mathrm{ln}\left(\frac{{f}_{ns}}{{f}_{ns}^{* }}\right),{f}_{ns}=\displaystyle \sum _{\mu =0,1}\exp \left(\displaystyle \sum _{j=1}^{N}{\mu }_{j}\left({\xi }_{j}+{\rm{i}}\frac{\pi }{2}\right)+\displaystyle \sum _{1\leqslant j\lt l}^{n}{\mu }_{j}{\mu }_{l}{A}_{jl}\right),\end{eqnarray}$
where $\begin{eqnarray}{\xi }_{j}={k}_{j}x+{k}_{j}^{3}t+{c}_{j},{{\rm{e}}}^{{A}_{jl}}={\left(\frac{{k}_{j}-{k}_{l}}{{k}_{j}+{k}_{l}}\right)}^{2}.\end{eqnarray}$
Here ${f}_{ns}^{* }$ denotes the complex conjugate of fns, and the summation over μ = 0, 1 accounts for each μj where j = 0, 1, 2, …, N. The real arbitrary constants kj and cj serve distinct roles in wave theory: kj represents the wave number, while cj indicates the central position of the waves.The one-, two-, and three-soliton solutions can be precisely expressed as
$\begin{eqnarray}\begin{array}{r}{u}_{1s}=\mathrm{ln}\left(\frac{1+{\rm{i}}{{\rm{e}}}^{{k}_{1}x+{k}_{1}^{3}t+{c}_{1}}}{1-{\rm{i}}{{\rm{e}}}^{{k}_{1}x+{k}_{1}^{3}t+{c}_{1}}}\right)=2{\rm{i\; arctan}}({{\rm{e}}}^{{k}_{1}x+{k}_{1}^{3}t+{c}_{1}}),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{u}_{2s}=2{\rm{i\; arctan}}\left[\frac{{({k}_{1}+{k}_{2})}^{2}({{\rm{e}}}^{{\xi }_{1}}+{{\rm{e}}}^{{\xi }_{2}})}{{({k}_{1}-{k}_{2})}^{2}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}}-{({k}_{1}+{k}_{2})}^{2}}\right],\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{u}_{3s} & & =2{\rm{i\; arctan}}\left(\frac{{f}_{30}}{{f}_{31}}\right),\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}{f}_{30} & = & {({k}_{1}+{k}_{2})}^{2}{({k}_{2}+{k}_{3})}^{2}{({k}_{1}+{k}_{3})}^{2}({{\rm{e}}}^{{\xi }_{1}}+{{\rm{e}}}^{{\xi }_{2}}+{{\rm{e}}}^{{\xi }_{3}})\\ & & -{({k}_{1}-{k}_{2})}^{2}{({k}_{2}-{k}_{3})}^{2}{({k}_{1}-{k}_{3})}^{2}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}+{\xi }_{3}},\\ {f}_{31} & = & {({k}_{1}-{k}_{2})}^{2}{({k}_{2}+{k}_{3})}^{2}{({k}_{1}+{k}_{3})}^{2}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}}\\ & & +{({k}_{2}-{k}_{3})}^{2}{({k}_{1}+{k}_{2})}^{2}{({k}_{1}+{k}_{3})}^{2}{{\rm{e}}}^{{\xi }_{2}+{\xi }_{3}}\\ & & +{({k}_{1}-{k}_{3})}^{2}{({k}_{1}+{k}_{2})}^{2}{({k}_{2}+{k}_{3})}^{2}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{3}}\\ & & -{({k}_{1}+{k}_{2})}^{2}{({k}_{2}+{k}_{3})}^{2}{({k}_{1}+{k}_{3})}^{2}.\end{array}\end{eqnarray}$
3. K-symmetries and τ-symmetries of the pmKdV equation related to n-soliton solution
To investigate the physical implications of K-symmetry and τ-symmetry of the pmKdV equation, we consider the two-soliton solution given by (13 ). From this two-soliton solution, we can derive four specific symmetries, denoted as follows 16 )–(19 ) into the symmetry definition equation (2 ), one can readily verify that the equation is satisfied. This confirms that expressions (16 )–(19 ) represent four notable symmetries of the pmKdV equation.
$\begin{eqnarray}{\sigma }_{{c}_{1}}={\partial }_{{c}_{1}}{u}_{2s}=\frac{-2{\rm{i}}{({k}_{1}+{k}_{2})}^{2}{{\rm{e}}}^{{\xi }_{1}}\left[{({k}_{1}-{k}_{2})}^{2}{{\rm{e}}}^{2{\xi }_{2}}+{({k}_{2}+{k}_{1})}^{2}\right]}{{({k}_{1}+{k}_{2})}^{4}({{\rm{e}}}^{2{\xi }_{1}}+{{\rm{e}}}^{2\xi 2})+8{k}_{1}{k}_{2}{({k}_{1}+{k}_{2})}^{2}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}}+{({k}_{1}-{k}_{2})}^{4}{{\rm{e}}}^{2{\xi }_{1}+2{\xi }_{2}}+{({k}_{1}+{k}_{2})}^{4}},\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{{c}_{2}}={\partial }_{{c}_{2}}{u}_{2s}=\frac{-2{\rm{i}}{({k}_{1}+{k}_{2})}^{2}{{\rm{e}}}^{{\xi }_{2}}\left[{({k}_{1}-{k}_{2})}^{2}{{\rm{e}}}^{2{\xi }_{1}}+{({k}_{2}+{k}_{1})}^{2}\right]}{{({k}_{1}+{k}_{2})}^{4}({{\rm{e}}}^{2{\xi }_{1}}+{{\rm{e}}}^{2\xi 2})+8{k}_{1}{k}_{2}{({k}_{1}+{k}_{2})}^{2}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}}+{({k}_{1}-{k}_{2})}^{4}{{\rm{e}}}^{2{\xi }_{1}+2{\xi }_{2}}+{({k}_{1}+{k}_{2})}^{4}},\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{\sigma }_{{k}_{1}} & = & {\partial }_{{k}_{1}}{u}_{2s}=-2{\rm{i}}({k}_{1}+{k}_{2}){{\rm{e}}}^{{\xi }_{1}}\{4{k}_{2}({k}_{1}-{k}_{2}){{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}}\\ & & +({k}_{1}-{k}_{2})[({k}_{1}^{2}-{k}_{2}^{2})x+3({k}_{1}^{4}-{k}_{1}^{2}{k}_{2}^{2})t+4{k}_{2}]{{\rm{e}}}^{2{\xi }_{2}}\\ & & +{({k}_{1}+{k}_{2})}^{3}(x+3{k}_{1}^{2}t)\}\times \left[{({k}_{1}+{k}_{2})}^{4}{{\rm{e}}}^{2{\xi }_{1}}\right.\\ & & +{({k}_{1}+{k}_{2})}^{4}{{\rm{e}}}^{2{\xi }_{2}}+8{k}_{1}{k}_{2}{({k}_{1}+{k}_{2})}^{2}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}}\\ & & {\left.+{({k}_{1}-{k}_{2})}^{4}{{\rm{e}}}^{2{\xi }_{1}+2{\xi }_{2}}+{({k}_{1}+{k}_{2})}^{4}\right]}^{-1},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{{k}_{2}} & & ={\partial }_{{k}_{2}}{u}_{2s}=-2{\rm{i}}({k}_{1}+{k}_{2}){{\rm{e}}}^{{\xi }_{1}}\{4{k}_{1}({k}_{2}-{k}_{1}){{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}}\\ & & +({k}_{1}-{k}_{2})[({k}_{1}^{2}-{k}_{2}^{2})x+3({k}_{1}^{2}{k}_{2}^{2}-{k}_{1}^{4})t-4{k}_{1}]{{\rm{e}}}^{2{\xi }_{2}}\\ & & +{({k}_{1}+{k}_{2})}^{3}(x+3{k}_{2}^{2}t)\}\times \left[{({k}_{1}+{k}_{2})}^{4}{{\rm{e}}}^{2{\xi }_{1}}\right.\\ & & +{({k}_{1}+{k}_{2})}^{4}{{\rm{e}}}^{2{\xi }_{2}}+8{k}_{1}{k}_{2}{({k}_{1}+{k}_{2})}^{2}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}}\\ & & {\left.+{({k}_{1}-{k}_{2})}^{4}{{\rm{e}}}^{2{\xi }_{1}+2{\xi }_{2}}+{({k}_{1}+{k}_{2})}^{4}\right]}^{-1}.\end{array}\end{eqnarray}$
By substituting these four symmetries (Specifically, the parameters c1 and c2 correspond to the center position of the solitons, while k1 and k2 denote the wave numbers. Consequently, the physical interpretations of the symmetries ${\sigma }_{{c}_{1}}$ and ${\sigma }_{{c}_{2}}$ are related to the invariance under translations in the center of the soliton, while ${\sigma }_{{k}_{1}}$ and ${\sigma }_{{k}_{2}}$ are associated with the invariance under translations in the wave number.
Using the K-symmetries defined by (6 ), we obtain the following special K-symmetries with u = u2s, say 17 ). The results in (20 ) indicate that the infinite number of K-symmetries for the two-soliton solution consists of linear combinations of the center translation symmetries ${\sigma }_{{c}_{1}}$ and ${\sigma }_{{c}_{2}}$. This observation implies that the set of K-symmetries may not be incomplete. In fact, among the infinitely many K-symmetries associated with the special two-soliton solution given by equation (20 ), only K0 and K1 are independent; all other symmetries can be expressed through the linear relationship
$\begin{eqnarray}\begin{array}{rcl}{K}_{0}{| }_{u={u}_{2s}} & = & {u}_{x}{| }_{u={u}_{2s}}={k}_{1}{\sigma }_{{c}_{1}}+{k}_{2}{\sigma }_{{c}_{2}},\\ {K}_{1}{| }_{u={u}_{2s}} & = & {\rm{\Phi }}{u}_{x}{| }_{u={u}_{2s}}={k}_{1}^{3}{\sigma }_{{c}_{1}}+{k}_{2}^{3}{\sigma }_{{c}_{2}},\\ {K}_{2}{| }_{u={u}_{2s}} & = & {{\rm{\Phi }}}^{2}{u}_{x}{| }_{u={u}_{2s}}={k}_{1}^{5}{\sigma }_{{c}_{1}}+{k}_{2}^{5}{\sigma }_{{c}_{2}},\\ & \vdots & \\ {K}_{m}{| }_{u={u}_{2s}} & = & {{\rm{\Phi }}}^{m}{u}_{x}{| }_{u={u}_{2s}}=\displaystyle \sum _{j=1}^{2}{k}_{j}^{2m+1}{\sigma }_{{c}_{j}}\\ & = & {k}_{1}^{2m+1}{\sigma }_{{c}_{1}}+{k}_{2}^{2m+1}{\sigma }_{{c}_{2}},\\ m & = & 3,\,4,\,\ldots ,\,\infty ,\end{array}\end{eqnarray}$
where ${\sigma }_{{c}_{1}}$ and ${\sigma }_{{c}_{2}}$ are the symmetries given by ( $\begin{eqnarray}\begin{array}{r}{K}_{p\geqslant 2}{| }_{u={u}_{2s}}=\left(\frac{{k}_{1}^{2}{k}_{2}^{2p}-{k}_{2}^{2}{k}_{1}^{2p}}{{k}_{1}^{2}-{k}_{2}^{2}}\right){K}_{0}+\left(\frac{{k}_{1}^{2p}-{k}_{2}^{2p}}{{k}_{1}^{2}-{k}_{2}^{2}}\right){K}_{1}.\end{array}\end{eqnarray}$
Similarly, starting from (7 ), the τ-symmetries associated with the two-soliton solution (13 ) can be expressed in the form 16 )–(19 ). Analysis of the result (22 ) reveals that the infinitely many τ-symmetries of the two-soliton solution are linear combinations of the center translation symmetries described in (16 )–(17 ) and the wave number translation symmetries delineated in (18 )–(19 ).
$\begin{eqnarray}\begin{array}{l}{\tau }_{1}{| }_{u={u}_{2s}}={\rm{\Phi }}{\tau }_{0}{| }_{u={u}_{2s}}={k}_{1}{\sigma }_{{k}_{1}}+{k}_{2}{\sigma }_{{k}_{2}},\\ {\tau }_{2}{| }_{u={u}_{2s}}={{\rm{\Phi }}}^{2}{\tau }_{0}{| }_{u={u}_{2s}}=(-2{k}_{1}^{2}-4{k}_{1}{k}_{2}){\sigma }_{{c}_{1}}\\ \quad +\,(-4{k}_{1}{k}_{2}-2{k}_{2}^{2}){\sigma }_{{c}_{2}}+{k}_{1}^{3}{\sigma }_{{k}_{1}}+{k}_{2}^{3}{\sigma }_{{k}_{2}},\\ {\tau }_{3}{| }_{u={u}_{2s}}={{\rm{\Phi }}}^{3}{\tau }_{0}{| }_{u={u}_{2s}}=(-4{k}_{1}^{4}-4{k}_{1}^{3}{k}_{2}-4{k}_{1}{k}_{2}^{3}){\sigma }_{{c}_{1}}\\ \quad +\,(-4{k}_{1}^{3}{k}_{2}-4{k}_{1}{k}_{2}^{3}-4{k}_{2}^{4}){\sigma }_{{c}_{2}}+{k}_{1}^{5}{\sigma }_{{k}_{1}}+{k}_{2}^{5}{\sigma }_{{k}_{2}},\\ \vdots \\ {\tau }_{m}{| }_{u={u}_{2s}}={{\rm{\Phi }}}^{m}{\tau }_{0}{| }_{u={u}_{2s}}\\ =-2(m-1)\displaystyle \sum _{j=1}^{2}{k}_{j}^{2m-2}{\sigma }_{{c}_{j}}-4\left(\displaystyle \sum _{p=1}^{m-1}{k}_{1}^{2p-1}{k}_{2}^{2(m-p)-1}\right)\displaystyle \sum _{j=1}^{2}{\sigma }_{{c}_{j}}\\ \quad +\displaystyle \sum _{j=1}^{2}{k}_{j}^{2m-1}{\sigma }_{{k}_{j}},\quad m=4,\,5,\,\ldots ,\,\infty ,\end{array}\end{eqnarray}$
where the special symmetries ${\sigma }_{{c}_{j}}$ and ${\sigma }_{{k}_{j}}$, j = 1, 2 are given by (Furthermore, a thorough investigation into the incompleteness of the τ-symmetries outlined in (22 ) confirms that this set is indeed incomplete, as any given τ-symmetries can be articulated as a linear combination of τ1, τ2, K0, and K1, in the form
$\begin{eqnarray}\begin{array}{rcl}{\tau }_{q\geqslant 3}{| }_{u={u}_{2s}} & = & -\frac{{k}_{2}^{2}{A}_{q3}}{{k}_{1}^{2}}{\tau }_{1}+\frac{{A}_{q2}}{{k}_{1}^{2}}{\tau }_{2}\\ & & +\frac{2{B}_{q1}}{{k}_{1}{k}_{2}({k}_{1}^{2}-{k}_{2}^{2})}{K}_{0}+\frac{2{B}_{q2}}{{k}_{1}{k}_{2}({k}_{1}+{k}_{2})}{K}_{1},\end{array}\end{eqnarray}$
with the corresponding relationship given by $\begin{eqnarray}\begin{array}{rcl}{A}_{qr} & = & \displaystyle \sum _{j=0}^{q-r}{k}_{1}^{2(q-j-1)}{k}_{2}^{2j},\,r=2,\,3,\\ {B}_{q1} & = & -q\left[({k}_{1}^{5}-{k}_{2}^{5}){C}_{q1}+{k}_{1}^{2}{k}_{2}^{2}({k}_{1}^{3}-{k}_{2}^{3}){C}_{q2}\right]\\ & & +({k}_{1}^{3}+{k}_{2}^{3})({k}_{1}^{2}-{k}_{2}^{2}){C}_{q1}+{k}_{1}^{2}{k}_{2}^{2}({k}_{1}^{3}-{k}_{2}^{3}){C}_{q2},\\ {B}_{q2} & = & q\left[({k}_{1}^{2}+{k}_{1}{k}_{2}+{k}_{2}^{2}){C}_{q1}+{k}_{1}^{2}{k}_{2}^{2}{C}_{q2}\right]\\ & & -{({k}_{1}+{k}_{2})}^{2}{C}_{q1}-{k}_{1}^{2}{k}_{2}^{2}{C}_{q2},\end{array}\end{eqnarray}$
where Cq1 and Cq2 are provided by $\begin{eqnarray}\begin{array}{r}{C}_{q1}=\frac{{k}_{2}^{2q-2}-{k}_{1}^{2q-2}}{{k}_{1}^{2}-{k}_{2}^{2}},\qquad {C}_{q2}=\frac{{k}_{1}^{2q}-{k}_{2}^{2q}}{{k}_{1}^{2}{k}_{2}^{2}({k}_{1}^{2}-{k}_{2}^{2})}.\end{array}\end{eqnarray}$
Based on these findings, we extend our analysis to encompass the general n-soliton solutions u = uns. We successfully derive expressions for infinite sets of K- and τ-symmetries and clarify their physical interpretations.
First, for the n-soliton solution, we identify 2n special symmetries, which are represented as follows: 10 ). The expressions for ${({F}_{ns})}_{{c}_{m}}$, ${({F}_{ns}^{* })}_{{c}_{m}}$, ${({F}_{ns})}_{{k}_{m}}$, and ${({F}_{ns}^{* })}_{{k}_{m}}$ are given by
$\begin{eqnarray}{\sigma }_{{c}_{m}}={\partial }_{{c}_{m}}{u}_{ns}=\frac{{({F}_{ns})}_{{c}_{m}}}{{F}_{ns}}-\frac{{({F}_{ns}^{* })}_{{c}_{m}}}{{F}_{ns}^{* }},m=1,\,2,\,\ldots ,\,n,\end{eqnarray}$
and $\begin{eqnarray}{\sigma }_{{k}_{m}}={\partial }_{{k}_{m}}{u}_{ns}=\frac{{({F}_{ns})}_{{k}_{m}}}{{F}_{ns}}-\frac{{({F}_{ns}^{* })}_{{k}_{m}}}{{F}_{ns}^{* }},m=1,\,2,\,\ldots ,\,n,\end{eqnarray}$
where Fns is defined in ( $\begin{eqnarray}\begin{array}{rcl}{({F}_{ns})}_{{c}_{m}} & = & \displaystyle \sum _{\mu =0,1}{\mu }_{m}\exp \left(\displaystyle \sum _{j=1}^{n}{\mu }_{j}\left({\xi }_{j}+{\rm{i}}\frac{\pi }{2}\right)+\displaystyle \sum _{1\leqslant j\lt l}^{n}{\mu }_{j}{\mu }_{l}{A}_{jl}\right),\\ {({F}_{ns}^{* })}_{{c}_{m}} & = & \displaystyle \sum _{\mu =0,1}{\mu }_{m}\exp \left(\displaystyle \sum _{j=1}^{n}{\mu }_{j}\left({\xi }_{j}-{\rm{i}}\frac{\pi }{2}\right)+\displaystyle \sum _{1\leqslant j\lt l}^{n}{\mu }_{j}{\mu }_{l}{A}_{jl}\right),\\ {({F}_{ns})}_{{k}_{m}} & = & \displaystyle \sum _{\mu =0,1}\left({\mu }_{m}{\xi }_{m,{k}_{m}}+\displaystyle \sum _{1\leqslant i\lt p}^{n}{\mu }_{i}{\mu }_{p}{A}_{ip,{k}_{m}}\right)\\ & & \times \exp \left(\displaystyle \sum _{j=1}^{n}{\mu }_{j}\left({\xi }_{j}+{\rm{i}}\frac{\pi }{2}\right)+\displaystyle \sum _{1\leqslant j\lt l}^{n}{\mu }_{j}{\mu }_{l}{A}_{jl}\right),\\ {({F}_{ns}^{* })}_{{k}_{m}} & = & \displaystyle \sum _{\mu =0,1}\left({\mu }_{m}{\xi }_{m,{k}_{m}}+\displaystyle \sum _{1\leqslant i\lt p}^{n}{\mu }_{i}{\mu }_{p}{A}_{ip,{k}_{m}}\right)\\ & & \times \exp \left(\displaystyle \sum _{j=1}^{n}{\mu }_{j}\left({\xi }_{j}-{\rm{i}}\frac{\pi }{2}\right)+\displaystyle \sum _{1\leqslant j\lt l}^{n}{\mu }_{j}{\mu }_{l}{A}_{jl}\right),\end{array}\end{eqnarray}$
for m = 1, 2, …, n. These symmetries have direct physical interpretations: ${\sigma }_{{c}_{m}}$ embodies the invariance under translation of the m-th soliton's center cm, and ${\sigma }_{{k}_{m}}$ corresponds to the invariance under translation of its wave number km.Because 6 ) associated with u = uns (10 ), it is natural to obtain 10 ), the relations (32 ) shows that the K-symmetries (6 ) are linear combinations of center translation invariance of the m-th soliton's center cm. Moreover, it can be determined that the infinite symmetries of the n-soliton solution are not autonomous, only 2n of them are independent with the remaining symmetries being linearly constructed by them 33 ) denote the determinant of matrix R and Rj with order n, respectively, which are defined as ${{\rm{\Theta }}}_{n}=\det ({\boldsymbol{R}})$ and ${{\rm{\Theta }}}_{nj}=\det ({{\boldsymbol{R}}}_{j})$, and the structures of the matrices are as follows:
$\begin{eqnarray}\begin{array}{rcl}{({F}_{ns})}_{x} & = & {\partial }_{x}{F}_{ns}=\displaystyle \sum _{m=1}^{n}{k}_{m}{({F}_{ns})}_{{c}_{m}},\\ {({F}_{ns}^{* })}_{x} & = & {\partial }_{x}{F}_{ns}^{* }=\displaystyle \sum _{m=1}^{n}{k}_{m}{({F}_{ns}^{* })}_{{c}_{m}},\\ {({F}_{ns})}_{t} & = & {\partial }_{t}{F}_{ns}=\displaystyle \sum _{m=1}^{n}{k}_{m}^{3}{({F}_{ns})}_{{c}_{m}},\\ {({F}_{ns}^{* })}_{t} & = & {\partial }_{t}{F}_{ns}^{* }=\displaystyle \sum _{m=1}^{n}{k}_{m}^{3}{({F}_{ns}^{* })}_{{c}_{m}},\\ x{({F}_{ns})}_{x}+3t{({F}_{ns})}_{t} & = & x{\partial }_{x}{F}_{ns}+3t{\partial }_{t}{F}_{ns}=\displaystyle \sum _{m=1}^{n}{k}_{m}{({F}_{ns})}_{{k}_{m}},\\ x{({F}_{ns}^{* })}_{x}+3t{({F}_{ns}^{* })}_{t} & = & x{\partial }_{x}{F}_{ns}^{* }+3t{\partial }_{t}{F}_{ns}^{* }=\displaystyle \sum _{m=1}^{n}{k}_{m}{({F}_{ns}^{* })}_{{k}_{m}},\end{array}\end{eqnarray}$
we can find the following relations hold $\begin{eqnarray}\begin{array}{rcl}{({u}_{ns})}_{x} & = & {\partial }_{x}{u}_{ns}=\frac{{({F}_{ns})}_{x}}{{F}_{ns}}-\frac{{({F}_{ns}^{* })}_{x}}{{F}_{ns}^{* }}\\ & = & \displaystyle \sum _{m=1}^{n}{k}_{m}\left(\frac{{({F}_{ns})}_{{c}_{m}}}{{F}_{ns}}-\frac{{({F}_{ns}^{* })}_{{c}_{m}}}{{F}_{ns}^{* }}\right)=\displaystyle \sum _{m=1}^{n}{k}_{m}{\sigma }_{{c}_{m}},\\ & & {({u}_{ns})}_{t}={\partial }_{t}{u}_{ns}=\frac{{({F}_{ns})}_{t}}{{F}_{ns}}-\frac{{({F}_{ns}^{* })}_{t}}{{F}_{ns}^{* }}\\ & = & \displaystyle \sum _{m=1}^{n}{k}_{m}^{3}\left(\frac{{({F}_{ns})}_{{c}_{m}}}{{F}_{ns}}-\frac{{({F}_{ns}^{* })}_{{c}_{m}}}{{F}_{ns}^{* }}\right)=\displaystyle \sum _{m=1}^{n}{k}_{m}^{3}{\sigma }_{{c}_{m}},\\ & & 3t{({u}_{ns})}_{t}+x{({u}_{ns})}_{x}=\displaystyle \sum _{m=1}^{n}{k}_{m}{\sigma }_{{k}_{m}},\\ & & m=1,\,2,\,\ldots ,\,n.\end{array}\end{eqnarray}$
Thus for the K-symmetries ( $\begin{eqnarray}\begin{array}{rcl} & & {K}_{0}={{\rm{\Phi }}}^{0}{u}_{x}{| }_{u={u}_{ns}}=\displaystyle \sum _{m=1}^{n}{k}_{m}{\sigma }_{{c}_{m}},\\ & & {K}_{1}={\rm{\Phi }}{u}_{x}{| }_{u={u}_{ns}}=\displaystyle \sum _{m=1}^{n}{k}_{m}^{3}{\sigma }_{{c}_{m}},\\ & & {K}_{2}={{\rm{\Phi }}}^{2}{u}_{x}{| }_{u={u}_{ns}}=\displaystyle \sum _{m=1}^{n}{k}_{m}^{5}{\sigma }_{{c}_{m}}.\end{array}\end{eqnarray}$
By recursively using Kn+1 = ΦKn, the expressions of the symmetries Ki can be directly written down as $\begin{eqnarray}\begin{array}{r}{K}_{i}={{\rm{\Phi }}}^{i}{u}_{x}{| }_{u={u}_{ns}}=\displaystyle \sum _{m=1}^{n}{k}_{m}^{2i+1}{\sigma }_{{c}_{m}},\qquad i=3,\,4,\,\ldots ,\,\infty .\end{array}\end{eqnarray}$
For the n-soliton solution u = uns ( $\begin{eqnarray}\begin{array}{r}{K}_{i\gt n-1}={{\rm{\Phi }}}^{i}{u}_{x}{| }_{u={u}_{ns}}=\displaystyle \sum _{j=1}^{n}{k}_{j}^{2i+1}\frac{{{\rm{\Theta }}}_{nj}}{{{\rm{\Theta }}}_{n}}{\left|\right.}_{u={u}_{ns}},\\ i=n,\,n+1,\,\ldots ,\,\infty ,\end{array}\end{eqnarray}$
where Θn, and Θnj in ( $\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{R}} & = & \left(\begin{array}{ccccccc}{k}_{1} & {k}_{2} & \cdots \, & {k}_{m} & \cdots \, & {k}_{n-1} & {k}_{n}\\ {k}_{1}^{3} & {k}_{2}^{3} & \cdots \, & {k}_{m}^{3} & \cdots \, & {k}_{n-1}^{3} & {k}_{n}^{3}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {k}_{1}^{2n+1} & {k}_{2}^{2n+1} & \cdots \, & {k}_{m}^{2n+1} & \cdots \, & {k}_{n-1}^{2n+1} & {k}_{n}^{2n+1}\end{array}\right),\\ {{\boldsymbol{R}}}_{j} & = & \left(\begin{array}{ccccccc}{k}_{1} & {k}_{2} & \cdots \, & {K}_{0} & \cdots \, & {k}_{n-1} & {k}_{n}\\ {k}_{1}^{3} & {k}_{2}^{3} & \cdots \, & {K}_{1} & \cdots \, & {k}_{n-1}^{3} & {k}_{n}^{3}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {k}_{1}^{2n+1} & {k}_{2}^{2n+1} & \cdots \, & {K}_{n-1} & \cdots \, & {k}_{n-1}^{2n+1} & {k}_{n}^{2n+1}\end{array}\right).\end{array}\end{eqnarray}$
For the τ-symmetries related to the n-soliton solution, it is easy to find τ1 can be directly written down as 12 ) is 32 ), we know the K-symmetries of the one-soliton solution is 38 ) and ${\tau }_{1}{| }_{u={u}_{1s}}$ (36 ) as
$\begin{eqnarray}\begin{array}{r}{\tau }_{1}={\rm{\Phi }}{\tau }_{0}{| }_{u={u}_{ns}}=3t{K}_{1}+x{K}_{0}{| }_{u={u}_{ns}}=\displaystyle \sum _{m=1}^{n}{k}_{m}{\sigma }_{{k}_{m}}.\end{array}\end{eqnarray}$
For the other high-order τ-symmetries, it is necessary to show the results for different n. For instance, the τ-symmetries for the one-soliton solution u = u1s ( $\begin{eqnarray}\begin{array}{r}{\tau }_{1}{| }_{u={u}_{1s}}={\rm{\Phi }}{\tau }_{0}{| }_{u={u}_{1s}}=3t{K}_{1}+x{K}_{0}{| }_{u={u}_{1s}}={k}_{1}{\sigma }_{{k}_{1}},\end{array}\end{eqnarray}$
while the other τ-symmetries can be obtained $\begin{eqnarray}\begin{array}{l}{\tau }_{2}{| }_{u={u}_{1s}}={{\rm{\Phi }}}^{2}{\tau }_{0}{| }_{u={u}_{1s}}\\ \quad =(3t{K}_{2}+x{K}_{1}-2{u}_{x}{D}^{-1}{u}_{x}^{2}+2{u}_{xx}){| }_{u={u}_{1s}}={k}_{1}^{3}{\sigma }_{{k}_{1}}-2{k}_{1}^{2}{\sigma }_{{c}_{1}},\\ {\tau }_{3}{| }_{u={u}_{1s}}={{\rm{\Phi }}}^{3}{\tau }_{0}{| }_{u={u}_{1s}}={k}_{1}^{5}{\sigma }_{{k}_{1}}-4{k}_{1}^{4}{\sigma }_{{c}_{1}},\\ \quad \vdots \\ \quad {\tau }_{i}{| }_{u={u}_{1s}}={{\rm{\Phi }}}^{i}{\tau }_{0}{| }_{u={u}_{1s}}={k}_{1}^{2i-1}{\sigma }_{{k}_{1}}-2(i-1){k}_{1}^{2i-2}{\sigma }_{{c}_{1}}.\end{array}\end{eqnarray}$
From ( $\begin{eqnarray}\begin{array}{rcl}{K}_{0}{| }_{u={u}_{1s}} & = & {k}_{1}{\sigma }_{{c}_{1}},\\ {K}_{1}{| }_{u={u}_{1s}} & = & {k}_{1}^{3}{\sigma }_{{c}_{1}},\\ & & \vdots \\ {K}_{i\geqslant 2}{| }_{u={u}_{1s}} & = & {k}_{1}^{2i+1}{\sigma }_{{c}_{1}},\end{array}\end{eqnarray}$
and the whole K-symmetries set can be expressed by the independent symmetry K0, say $\begin{eqnarray}\begin{array}{r}{K}_{i\geqslant 1}{| }_{u={u}_{1s}}={k}_{1}^{2}{K}_{0}{| }_{u={u}_{1s}}.\end{array}\end{eqnarray}$
Thus, the τ-symmetries can be given by the two independent symmetries ${K}_{0}{| }_{u={u}_{1s}}$ ( $\begin{eqnarray}\begin{array}{rcl}{\tau }_{2}{| }_{u={u}_{1s}} & = & {k}_{1}^{2}{\tau }_{1}{| }_{u={u}_{1s}}-2{k}_{1}{K}_{0}{| }_{u={u}_{1s}},\\ & & \vdots \\ {\tau }_{i\geqslant 3}{| }_{u={u}_{1s}} & = & {k}_{1}^{2i-2}{\tau }_{1}{| }_{u={u}_{1s}}-2(i-1){k}_{1}^{2i-3}{K}_{0}{| }_{u={u}_{1s}},\end{array}\end{eqnarray}$
by recursively using τi = Φiτi−1.The result (40 ) shows the τ-symmetries of the one-soliton solution are related to K0 and τ1. In other words, the τ-symmetries of the one-soliton solution are related to the wave number translational symmetries ${\sigma }_{{k}_{1}}$ and the center translational symmetries ${\sigma }_{{c}_{1}}$, and can be considered as the linear combinations of ${\sigma }_{{k}_{1}}$ and ${\sigma }_{{c}_{1}}$.
For the two-soliton solution, the results of the τ-symmetries (7 ) associated with u = u2s have been shown in (22 )–(23 ). It can be determined that the infinite symmetries of the two-soliton solution are not autonomous, only four of them are independent, as K0, K1, τ1, τ2 and other symmetries can be expressed by these independent symmetries.
In general, the infinitely many K- and τ-symmetries can be generated by recursively using Ki = ΦiKi−1 and τi = Φiτi−1.
The findings presented in this section demonstrate that the infinitely many K-symmetries and τ-symmetries associated with the n-soliton solution can be expressed as linear combinations of the wave number translational symmetries ${\sigma }_{{k}_{m}}$ and the center translational symmetries ${\sigma }_{{c}_{m}}$. Furthermore, it is shown that these infinitely many K-symmetries and τ-symmetries do not form a complete set, as only a subset of them are independent.
It should be mentioned that the τ0 symmetry does not correlate with the symmetries ${\sigma }_{{k}_{m}}$ and ${\sigma }_{{c}_{m}}$. The physical meaning of τ0 may need more related study.
4. K-symmetries and τ-symmetries related to two-wave complexiton solution
It is known that the potential modified Korteweg–de Vries (pmKdV) equation (1 ) admits a rich variety of solution structures, including soliton, breather, and complexiton solutions. For the n-soliton solution, the associated K-symmetries and τ-symmetries have been shown to be linear combinations of wave number translational symmetries and the center translational symmetries. It is natural to expect that similar symmetry patterns hold for other types of solutions. In this section, we focus on the physical interpretation of the K-symmetries and τ-symmetries associated with complexiton solutions.
The two-wave complexiton solution of the pmKdV equation can be written as
$\begin{eqnarray}\begin{array}{r}{u}_{2com}=2{\rm{arctanh}}\left[\frac{{k}_{2}({k}_{1}x-3{k}_{1}{k}_{2}^{2}t+{c}_{1})}{{k}_{1}\cos ({k}_{2}x-{k}_{2}^{3}t+{c}_{2})}\right],\end{array}\end{eqnarray}$
where k1, k2 are wave numbers and c1, and c2 are phase constants representing the initial positions of the waves.For this solution one can verify the existence of four elementary symmetries
$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{{c}_{1}} & = & {\partial }_{{c}_{1}}{u}_{2com}=\frac{2{k}_{1}{k}_{2}\cos ({\eta }_{2})}{{k}_{1}^{2}{\cos }^{2}({\eta }_{1})-{k}_{2}^{2}{\eta }_{1}^{2}},\\ {\sigma }_{{c}_{2}} & = & {\partial }_{{c}_{2}}{u}_{2com}=\frac{2{k}_{1}{k}_{2}{\eta }_{1}\sin ({\eta }_{2})}{{k}_{1}^{2}{\cos }^{2}({\eta }_{1})-{k}_{2}^{2}{\eta }_{1}^{2}},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{\sigma }_{{k}_{1}} & = & {\partial }_{{k}_{1}}{u}_{2com}=-\frac{2{c}_{1}{k}_{2}\cos ({\eta }_{2})}{{k}_{1}^{2}{\cos }^{2}({\eta }_{1})-{k}_{2}^{2}{\eta }_{1}^{2}},\\ {\sigma }_{{k}_{2}} & = & {\partial }_{{k}_{2}}{u}_{2com}\\ & = & \frac{[({k}_{1}{k}_{2}(8{\eta }_{1}-6{c}_{1})-6{k}_{1}^{2}({\eta }_{2}-{c}_{2})]\cos ({\eta }_{2})+2{k}_{2}^{2}{\eta }_{1}({\eta }_{1}-{c}_{1})\sin ({\eta }_{2})}{{k}_{2}[{k}_{1}^{2}{\cos }^{2}({\eta }_{1})-{k}_{2}^{2}{\eta }_{1}^{2}]},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{r}{\eta }_{1}={k}_{1}x-3{k}_{1}{k}_{2}^{2}t+{c}_{1},\qquad {\eta }_{2}={k}_{2}x-{k}_{2}^{3}t+{c}_{2}.\end{array}\end{eqnarray}$
Clearly, ${\sigma }_{{c}_{1}}$ and ${\sigma }_{{c}_{2}}$ correspond to center translations, while ${\sigma }_{{k}_{1}}$ and ${\sigma }_{{k}_{2}}$ reflect invariance under shifts of the wave numbers.Thus, the first three K-symmetries of the pmKdV equation generated from the two-wave complexiton solution are 46 ) shows that every K-symmetry associated with the two-wave complexiton solution is a linear combination of the center translational symmetries ${\sigma }_{{c}_{1}}$ and ${\sigma }_{{c}_{2}}$. Moreover, the family of K-symmetries is non-autonomous, only K0 and K1 are independent, and all higher members can be written in terms of them as 45 ).
$\begin{eqnarray}\begin{array}{rcl}{K}_{0} & = & {{\rm{\Phi }}}^{0}{u}_{x}{| }_{u={u}_{2com}}=\frac{2{k}_{1}{k}_{2}[{k}_{1}\cos ({\eta }_{2})+{k}_{1}{\eta }_{1}\sin ({\eta }_{2})]}{{k}_{1}^{2}{\cos }^{2}({\eta }_{1})-{k}_{2}^{2}{\eta }_{1}^{2}}\\ & = & {k}_{1}{\sigma }_{{c}_{1}}+{k}_{2}{\sigma }_{{c}_{2}},\\ {K}_{1} & = & {{\rm{\Phi }}}^{1}{u}_{x}{| }_{u={u}_{2com}}=-\frac{6{k}_{1}^{2}{k}_{2}^{3}\cos ({\eta }_{2})+2{k}_{1}{k}_{2}^{4}{\eta }_{1}\sin ({\eta }_{2})}{{k}_{1}^{2}{\cos }^{2}({\eta }_{1})-{k}_{2}^{2}{\eta }_{1}^{2}}\\ & = & -3{k}_{1}{k}_{2}^{2}{\sigma }_{{c}_{1}}-{k}_{2}^{3}{\sigma }_{{c}_{2}},\\ {K}_{2} & = & {{\rm{\Phi }}}^{2}{u}_{x}{| }_{u={u}_{2com}}=\frac{2{k}_{1}{k}_{2}^{5}[5{k}_{1}\cos ({\eta }_{2})+{k}_{2}\eta \sin ({\eta }_{2})]}{{k}_{1}^{2}{\cos }^{2}({\eta }_{1})-{k}_{2}^{2}{\eta }_{1}^{2}}\\ & = & 5{k}_{1}{k}_{2}^{4}{\sigma }_{{c}_{1}}+{k}_{2}^{5}{\sigma }_{{c}_{2}}.\end{array}\end{eqnarray}$
Using the recursion operator Φ via Ki+1 = ΦKi, all higher K-symmetries of the pmKdV equation can be expressed as $\begin{eqnarray}\begin{array}{r}{K}_{m\geqslant 3}={(-1)}^{m}(2m+1){k}_{1}{k}_{2}^{2m}{\sigma }_{{c}_{1}}+{(-1)}^{m}{k}_{2}^{2m+1}{\sigma }_{{c}_{2}}.\end{array}\end{eqnarray}$
( $\begin{eqnarray}\begin{array}{rcl} & & {K}_{2}=-{k}_{2}^{4}{K}_{0}-2{k}_{2}^{2}{K}_{1},\\ & & {K}_{3}=2{k}_{2}^{6}{K}_{0}+3{k}_{2}^{4}{K}_{1},\\ & & \vdots \\ & & {K}_{m\geqslant 4}={(-1)}^{m+1}{k}_{2}^{2m-2}[(m-1){k}_{2}^{2}{K}_{0}+m{K}_{1}],\end{array}\end{eqnarray}$
with K0 and K1 given in (In a similar way, the first three τ-symmetries of the pmKdV equation derived from the two-wave complexiton solution are
$\begin{eqnarray}\begin{array}{rcl}{\tau }_{1} & = & {\rm{\Phi }}{\tau }_{0}{| }_{u={u}_{2com}}={k}_{1}{\sigma }_{{k}_{1}}+{k}_{2}{\sigma }_{{k}_{2}},\\ {\tau }_{2} & = & {\rm{\Phi }}{\tau }_{1}{| }_{u={u}_{2com}}=-4{\rm{i}}{k}_{1}{k}_{2}{\sigma }_{{c}_{1}}\\ & & -4{\rm{i}}{k}_{2}^{2}{\sigma }_{{c}_{2}}-3{k}_{1}{k}_{2}^{2}{\sigma }_{{k}_{1}}-{k}_{2}^{3}{\sigma }_{{k}_{2}},\\ & & {\tau }_{3}={\rm{\Phi }}{\tau }_{2}{| }_{u={u}_{2com}}=16{\rm{i}}{k}_{1}{k}_{2}^{3}{\sigma }_{{c}_{1}}\\ & & +8{\rm{i}}{k}_{2}^{4}{\sigma }_{{c}_{2}}+5{k}_{1}{k}_{2}^{4}{\sigma }_{{k}_{1}}+{k}_{2}^{5}{\sigma }_{{k}_{2}},\end{array}\end{eqnarray}$
each of which is a linear combination of the four elementary symmetries ${\sigma }_{{c}_{1}}$, ${\sigma }_{{c}_{2}}$, ${\sigma }_{{k}_{1}}$ and ${\sigma }_{{k}_{2}}$, with i2 = −1. Applying the recursion operator repeatedly, τm+1 = Φτm, yields the infinite hierarchy $\begin{eqnarray}\begin{array}{rcl}{\tau }_{m\geqslant 4} & = & {(-4)}^{m-1}{\rm{i}}{k}_{1}{k}_{2}^{2m-3}{\sigma }_{{c}_{1}}-{\rm{i}}{(-2)}^{m}{k}_{2}^{2m-2}{\sigma }_{{c}_{2}}\\ & & +{(-1)}^{m-1}(2m-1){k}_{1}{k}_{2}^{2m-2}{\sigma }_{{k}_{1}}\\ & & +{(-1)}^{m-1}{k}_{2}^{2m-1}{\sigma }_{{k}_{2}}.\end{array}\end{eqnarray}$
Thus, the whole set of τ-symmetries consists of linear combinations of the center translational symmetries and the wave-number symmetries.A closer inspection reveals that only four symmetries are independent: K0, K1, τ1, and τ2. All higher τ-symmetries can be expressed as linear combinations of these four, i.e.
$\begin{eqnarray}\begin{array}{l}{\tau }_{3}=-4{\rm{i}}{k}_{2}^{3}{K}_{0}-4{\rm{i}}{k}_{2}{K}_{1}-{k}_{2}^{4}{\tau }_{1}-2{k}_{2}^{2}{\tau }_{2},\\ {\tau }_{4}=20{\rm{i}}{k}_{2}^{5}{K}_{0}+24{\rm{i}}{k}_{2}^{3}{K}_{1}+2{k}_{2}^{6}{\tau }_{1}+3{k}_{2}^{4}{\tau }_{2},\\ \vdots \\ {\tau }_{m\geqslant 5}={a}_{m}{K}_{0}+{b}_{m}{K}_{1}+{c}_{m}{\tau }_{1}+{d}_{m}{\tau }_{2},\end{array}\end{eqnarray}$
where the coefficients are given by $\begin{eqnarray}\begin{array}{rcl} & & {a}_{m}={\rm{i}}{(-1)}^{m}[{2}^{2m-3}-3\cdot {2}^{m-1}+4(m-1)]{k}_{2}^{2m-3},\\ & & {b}_{m}={\rm{i}}{(-1)}^{m}({2}^{2m-3}-{2}^{m-1}){k}_{2}^{2m-5},\\ & & {c}_{m}={(-1)}^{m-1}(2-m){k}_{2}^{2m-2},\\ & & {d}_{m}={(-1)}^{m}(m-1){k}_{2}^{2m-4}.\end{array}\end{eqnarray}$
For the two-wave complexiton solution of the pmKdV equation, both the K-symmetries and τ-symmetries are linear combinations of the elementary center translational symmetries ${\sigma }_{{c}_{1}}$, ${\sigma }_{{c}_{2}}$, and wave number symmetries ${\sigma }_{{k}_{1}}$ and ${\sigma }_{{k}_{2}}$. Only four of these symmetries {K0, K1, τ1, τ2} are independent; the entire infinite hierarchy can be reconstructed from them through the recursion operator Φ. This result extends the pattern previously observed for n-soliton solutions and confirms the universal role of parameter-translation symmetries in organizing the symmetry structure of integrable equations.
5. n-wave solutions of the pmKdV equation
Lie point symmetry serves as a foundational tool in the study of integrable systems, providing a powerful framework for deriving similarity solutions and uncovering underlying integrable structures through invariance analysis under continuous transformations. This approach not only facilitates the systematic construction of exact solutions but also enables symmetry reduction to lower the dimensionality of the governing equations, thereby simplifying their analysis. Building upon these principles and inspired by the novel methodology proposed in [20] for generating n-wave solutions via generalized K-symmetries and symmetry conjectures, this section employs the K-symmetries of the pmKdV equation to systematically investigate its n-wave solutions.
The general form of an n-wave solution to the pmKdV equation (1 ) can be expressed as
$\begin{eqnarray}u=U({\xi }_{1},\,{\xi }_{2},\,\ldots ,\,{\xi }_{n})\equiv U,{\xi }_{i}={k}_{i}x+{\omega }_{i}t+{c}_{i},\end{eqnarray}$
where ki, ωi and ci, i = 1, 2, …, n are arbitrary constants. Physically, ki denotes the wave number, governing spatial periodicity; ωi represents the frequency, controlling temporal evolution; and ci phase shift, which determines the initial wave profile. Substituting this ansatz into the pmKdV equation yields the governing relation: $\begin{eqnarray}\displaystyle \sum _{i=1}^{n}{\omega }_{i}{U}_{{\xi }_{i}}=\displaystyle \sum _{i,\,j,\,m}{k}_{i}{k}_{j}{k}_{m}{U}_{{\xi }_{i}{\xi }_{j}{\xi }_{m}}-2{\left(\displaystyle \sum _{i}{k}_{i}{U}_{{\xi }_{i}}\right)}^{3}.\end{eqnarray}$
Furthermore, substituting the same ansatz into the infinite set of K-symmetries, Km = Φmux, which are expressible as linear combinations of center translational symmetries for the n-wave solution, leads to
$\begin{eqnarray}{K}_{m}^{{\prime} }=\displaystyle \sum _{j=1}^{n}{a}_{mj}{U}_{{\xi }_{j}},m=0,\,1,\,2,\,\ldots ,\,\infty .\end{eqnarray}$
This process thus yields a series of constraint equations that the n-wave solution must satisfy: $\begin{eqnarray}\begin{array}{r}{K}_{0}:{K}_{0}=\displaystyle \sum _{i=1}^{n}{k}_{i}{U}_{{\xi }_{i}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{K}_{1}:{K}_{1}=\displaystyle \sum _{i=1}^{n}{\omega }_{i}{U}_{{\xi }_{i}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{K}_{2}:\displaystyle \sum _{j=1}^{n}{a}_{2j}{U}_{{\xi }_{j}} & = & \displaystyle \sum _{i,j,m,s,r}^{n}{k}_{i}{k}_{j}{k}_{m}{k}_{s}{k}_{r}{U}_{{\xi }_{i}{\xi }_{j}{\xi }_{m}{\xi }_{s}{\xi }_{r}}\\ & & -10\displaystyle \sum _{i,j,m,s,r}^{n}{k}_{i}{k}_{j}{k}_{m}{k}_{s}{k}_{r}{U}_{{\xi }_{i}}{U}_{{\xi }_{j}{\xi }_{m}}{U}_{{\xi }_{s}{\xi }_{r}}\\ & & -10\displaystyle \sum _{i,j,m,s,r}^{n}{k}_{i}{k}_{j}{k}_{m}{k}_{s}{k}_{r}{U}_{{\xi }_{i}}{U}_{{\xi }_{j}}{U}_{{\xi }_{m}{\xi }_{s}{\xi }_{r}}\\ & & +6\displaystyle \sum _{i,j,m,s,r}^{n}{k}_{i}{k}_{j}{k}_{m}{k}_{s}{k}_{r}{U}_{{\xi }_{i}}{U}_{{\xi }_{j}}{U}_{{\xi }_{m}}{U}_{{\xi }_{s}}{U}_{{\xi }_{r}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{K}_{3}:\displaystyle \sum _{j=1}^{n}{a}_{3j}{U}_{{\xi }_{j}}=\displaystyle \sum _{i,j,m,s,r,p,q}^{n}{k}_{i}{k}_{j}{k}_{m}{k}_{s}{k}_{r}{k}_{p}{k}_{q}{U}_{{\xi }_{i}{\xi }_{j}{\xi }_{m}{\xi }_{s}{\xi }_{r}{\xi }_{p}{\xi }_{q}}\\ \quad -70\displaystyle \sum _{i,j,m,s,r,p,q}^{n}{k}_{i}{k}_{j}{k}_{m}{k}_{s}{k}_{r}{k}_{p}{k}_{q}{U}_{{\xi }_{i}{\xi }_{j}}{U}_{{\xi }_{m}{\xi }_{s}}{U}_{{\xi }_{r}{\xi }_{p}{\xi }_{q}}\\ \quad -42\displaystyle \sum _{i,j,m,s,r,p,q}^{n}{k}_{i}{k}_{j}{k}_{m}{k}_{s}{k}_{r}{k}_{p}{k}_{q}{U}_{{\xi }_{i}}{U}_{{\xi }_{j}{\xi }_{m}{\xi }_{s}}{U}_{{\xi }_{r}{\xi }_{p}{\xi }_{q}}\\ \quad -56\displaystyle \sum _{i,j,m,s,r,p,q}^{n}{k}_{i}{k}_{j}{k}_{m}{k}_{s}{k}_{r}{k}_{p}{k}_{q}{U}_{{\xi }_{i}}{U}_{{\xi }_{j}{\xi }_{m}}{U}_{{\xi }_{s}{\xi }_{r}{\xi }_{p}{\xi }_{q}}\\ \quad -14\displaystyle \sum _{i,j,m,s,r,p,q}^{n}{k}_{i}{k}_{j}{k}_{m}{k}_{s}{k}_{r}{k}_{p}{k}_{q}{U}_{{\xi }_{i}}{U}_{{\xi }_{j}}{U}_{{\xi }_{m}{\xi }_{s}{\xi }_{r}{\xi }_{p}{\xi }_{q}}\\ \quad +140\displaystyle \sum _{i,j,m,s,r,p,q}^{n}{k}_{i}{k}_{j}{k}_{m}{k}_{s}{k}_{r}{k}_{p}{k}_{q}{U}_{{\xi }_{i}}{U}_{{\xi }_{j}}{U}_{{\xi }_{m}}{U}_{{\xi }_{s}{\xi }_{r}}{U}_{{\xi }_{p}{\xi }_{q}}\\ \quad +70\displaystyle \sum _{i,j,m,s,r,p,q}^{n}{k}_{i}{k}_{j}{k}_{m}{k}_{s}{k}_{r}{k}_{p}{k}_{q}{U}_{{\xi }_{i}}{U}_{{\xi }_{j}}{U}_{{\xi }_{m}}{U}_{{\xi }_{s}}{U}_{{\xi }_{r}{\xi }_{p}{\xi }_{q}}\\ \quad -20\displaystyle \sum _{i,j,m,s,r,p,q}^{n}{k}_{i}{k}_{j}{k}_{m}{k}_{s}{k}_{r}{k}_{p}{k}_{q}{U}_{{\xi }_{i}}{U}_{{\xi }_{j}}{U}_{{\xi }_{m}}{U}_{{\xi }_{s}}{U}_{{\xi }_{r}}{U}_{{\xi }_{p}}{U}_{{\xi }_{q}},\end{array}\end{eqnarray}$
$\begin{eqnarray}{K}_{i}:\displaystyle \sum _{j=1}^{n}{a}_{ij}{U}_{{\xi }_{j}}={K}_{i}{| }_{u=U,{u}_{x}=\displaystyle \sum _{i}{k}_{i}{U}_{{\xi }_{i}},{u}_{xx}=\displaystyle \sum _{i,\,j}{k}_{i}{k}_{j}{U}_{{\xi }_{i}{\xi }_{j}},\,\ldots ,}\,,\,i\gt 3.\end{eqnarray}$
The coefficients aij (i≥2, j = 1, 2, …, n) are to be determined from the symmetry structure. These multiple constraints, inherent to the n-wave hypothesis, can be systematically linked to the underlying symmetries of the system to facilitate the construction of explicit n-wave solutions.By selecting specific dispersion relations and associated parameters, various types of solutions to the pmKdV equation can be derived. For instance, starting from the standard dispersion relation for solitons 53 ) and (57 ) as
$\begin{eqnarray}{\omega }_{j}={k}_{j}^{3},\,{a}_{2j}={k}_{j}^{5},{a}_{3j}={k}_{j}^{7},\,{a}_{ij}={k}_{j}^{2i+1},=2,\,3,\,\ldots ,\end{eqnarray}$
the n-soliton solution can be directly constructed. As an illustrative example, the one-soliton solution is governed by the reduced system of ( $\begin{eqnarray}\begin{array}{r}{U}_{1{\xi }_{1}}-{U}_{1{\xi }_{1}{\xi }_{1}{\xi }_{1}}+2{U}_{1{\xi }_{1}}^{3}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}-{U}_{1{\xi }_{1}}^{5}-{U}_{1{\xi }_{1}}^{3}+{U}_{1{\xi }_{1}}{U}_{1{\xi }_{1}{\xi }_{1}}^{2}=0.\end{array}\end{eqnarray}$
Solving these equations yields the one-soliton solution $\begin{eqnarray}\begin{array}{r}u=\mathrm{ln}\left(\frac{{{\rm{e}}}^{{\xi }_{1}+{b}_{1}}-1}{{{\rm{e}}}^{{\xi }_{1}+{b}_{1}}+1}\right)+{b}_{2},\end{array}\end{eqnarray}$
where ${\xi }_{1}={k}_{1}x+{k}_{1}^{3}t+{c}_{1}$, and k1, c1, b1, b2 are arbitrary constants.For two-wave solutions, the constraint (54 ) takes the form 57 )–(59 ). Solving these constraints yields several distinct types of two-wave solutions, classified according to their dispersion relations:
$\begin{eqnarray}\begin{array}{r}{K}_{m}={a}_{m1}{U}_{{\xi }_{1}}+{a}_{m2}{U}_{{\xi }_{2}},\,m=0,\,1,\,2,\,\ldots ,\,\infty ,\end{array}\end{eqnarray}$
and a family of constraint equations is obtained by setting n = 2 in (1. Two-wave complexitons (Type I):
Choosing
$\begin{eqnarray}\begin{array}{r}{\omega }_{1}={k}_{1}^{3}-3{k}_{1}{k}_{2}^{2},\,{\omega }_{2}=-{k}_{2}^{3}+3{k}_{1}^{2}{k}_{2},\end{array}\end{eqnarray}$
with the coefficients am1 and am2 being given by $\begin{eqnarray}\begin{array}{rcl}{a}_{m1} & = & \frac{1}{2}\left[({k}_{1}+{\rm{i}}{k}_{2}){\left(\frac{{k}_{1}^{2}+{k}_{2}^{2}}{{k}_{1}-{\rm{i}}{k}_{2}}\right)}^{2m}\right.\\ & & \left.+({k}_{1}-{\rm{i}}{k}_{2}){\left(\frac{{k}_{1}^{2}+{k}_{2}^{2}}{{k}_{1}+{\rm{i}}{k}_{2}}\right)}^{2m}\right],\\ {a}_{m2} & = & \frac{1}{2}\left[({k}_{2}-{\rm{i}}{k}_{1}){\left(\frac{{k}_{1}^{2}+{k}_{2}^{2}}{{k}_{1}-{\rm{i}}{k}_{2}}\right)}^{2m}\right.\\ & & \left.+({k}_{2}+{\rm{i}}{k}_{1}){\left(\frac{{k}_{1}^{2}+{k}_{2}^{2}}{{k}_{1}+{\rm{i}}{k}_{2}}\right)}^{2m}\right],\end{array}\end{eqnarray}$
we obtain the first kind of two-wave complexiton solution: $\begin{eqnarray}\begin{array}{r}u=\pm 2\,{\rm{arctanh}}\left(\frac{{k}_{2}\sinh ({\xi }_{1})}{{k}_{1}\sin ({\xi }_{2})}\right).\end{array}\end{eqnarray}$
2. Two-wave complexitons (Type II):Selecting
$\begin{eqnarray}\begin{array}{r}{\omega }_{1}=-{k}_{1}^{3}-3{k}_{1}{k}_{2}^{2},\quad {\omega }_{2}=-{k}_{2}^{3}-3{k}_{1}^{2}{k}_{2}.\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}{a}_{m1} & = & \frac{{(-1)}^{m}}{2}\left[{({k}_{1}-{k}_{2})}^{1+2m}+{({k}_{1}+{k}_{2})}^{1+2m}\right],\\ {a}_{m2} & = & \frac{{(-1)}^{m}}{2}\left[{({k}_{1}+{k}_{2})}^{1+2m}-{({k}_{1}-{k}_{2})}^{1+2m}\right],\end{array}\end{eqnarray}$
yields the second kind of two-wave complexiton $\begin{eqnarray}\begin{array}{r}u=\pm 2\,{\rm{arctanh}}\left(\frac{{k}_{2}\sin ({\xi }_{1})}{{k}_{1}\cos ({\xi }_{2})}\right).\end{array}\end{eqnarray}$
3. Two-wave complexitons (Type III):A third kind of solution arises from the dispersion relation
$\begin{eqnarray}\begin{array}{r}{\omega }_{1}=-3{k}_{1}{k}_{2}^{2},\,{\omega }_{2}=-{k}_{2}^{3},\end{array}\end{eqnarray}$
with the corresponding coefficients being $\begin{eqnarray}\begin{array}{r}{a}_{m1}={(-1)}^{m}(1+2m){k}_{1}{k}_{2}^{2m},\,{a}_{m2}={(-1)}^{m}{k}_{2}^{1+2m},\end{array}\end{eqnarray}$
leading to $\begin{eqnarray}\begin{array}{r}u=\pm 2\,{\rm{arctanh}}\left(\frac{{k}_{2}{\xi }_{1}}{{k}_{1}\cos ({\xi }_{2})}\right).\end{array}\end{eqnarray}$
4. Breather-type solutions:With
$\begin{eqnarray}\begin{array}{r}{\omega }_{1}=-{k}_{1}^{3}+3{k}_{1}{k}_{2}^{2},\quad {\omega }_{2}={k}_{2}^{3}-3{k}_{1}^{2}{k}_{2},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{a}_{m1} & = & \frac{{(-1)}^{m+1}}{2}\left[-({k}_{1}+{\rm{i}}{k}_{2}){\left(\frac{{k}_{1}^{2}+{k}_{2}^{2}}{{k}_{1}-{\rm{i}}{k}_{2}}\right)}^{2m}\right.\\ & & \left.+({\rm{i}}{k}_{2}-{k}_{1}){\left(\frac{{k}_{1}^{2}+{k}_{2}^{2}}{{k}_{1}+{\rm{i}}{k}_{2}}\right)}^{2m}\right],\\ {a}_{m2} & = & \frac{{(-1)}^{m+1}}{2}\left[({\rm{i}}{k}_{1}-{k}_{2}){\left(\frac{{k}_{1}^{2}+{k}_{2}^{2}}{{k}_{1}-{\rm{i}}{k}_{2}}\right)}^{2m}\right.\\ & & \left.-({k}_{2}+{\rm{i}}{k}_{1}){\left(\frac{{k}_{1}^{2}+{k}_{2}^{2}}{{k}_{1}+{\rm{i}}{k}_{2}}\right)}^{2m}\right],\end{array}\end{eqnarray}$
a breather solution is obtained $\begin{eqnarray}\begin{array}{r}u=\pm 2{\rm{i}}\,{\rm{\arctan }}\left(\frac{{k}_{2}\sin ({\xi }_{1})}{{k}_{1}\cosh ({\xi }_{2})}\right).\end{array}\end{eqnarray}$
5. Two-soliton solution:This type solution is given by
$\begin{eqnarray}\begin{array}{r}u=\pm 2{\rm{i}}\arctan \left(\frac{{k}_{2}\sinh ({\xi }_{1})}{{k}_{1}\cosh ({\xi }_{2})}\right),\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{r}{\omega }_{1}={k}_{1}^{3}+3{k}_{1}{k}_{2}^{2},\quad {\omega }_{2}={k}_{2}^{3}+3{k}_{1}^{2}{k}_{2},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{a}_{m1} & = & \frac{{({k}_{1}-{k}_{2})}^{1+2m}}{2}+\frac{{({k}_{1}+{k}_{2})}^{1+2m}}{2},\\ {a}_{m2} & = & \frac{{({k}_{1}+{k}_{2})}^{1+2m}}{2}-\frac{{({k}_{1}-{k}_{2})}^{1+2m}}{2}.\end{array}\end{eqnarray}$
6. Double-pole solution:Representing a merged soliton-antisoliton structure [35], this solution is given by
$\begin{eqnarray}\begin{array}{rcl}u & = & \pm 2{\rm{i}}\,{\rm{\arctan }}\left(\frac{{k}_{2}{\xi }_{1}}{{k}_{1}\cosh ({\xi }_{2})}\right),\\ {\omega }_{1} & = & 3{k}_{1}{k}_{2}^{2},\qquad {\omega }_{2}={k}_{2}^{3},\end{array}\end{eqnarray}$
with coefficients $\begin{eqnarray}\begin{array}{r}{a}_{m1}=(1+2m){k}_{1}{k}_{2}^{2m},\,{a}_{m2}={k}_{2}^{1+2m}.\end{array}\end{eqnarray}$
Based on the framework of Lie point symmetries and K-symmetries, this section systematically investigates the n-wave solutions of the pmKdV equation. By substituting the general form of the n-wave solution into both the equation and the infinite-dimensional K-symmetries, the governing relations and constraint equations are derived. Through the selection of different dispersion relations and associated parameters, one-soliton solutions and various types of two-wave solutions-including complexitons, breathers, two-soliton interactions, and double-pole solutions, are explicitly constructed, revealing the rich wave structures admitted by the pmKdV equation. This approach demonstrates the effectiveness and universality of symmetry analysis in solving integrable systems and classifying their exact solutions.
6. Discussion and summary
This manuscript has presented a comprehensive analysis of the infinitely many K-symmetries and τ-symmetries and τ-symmetries of the pmKdV equation, clarifying their physical interpretations. Our results reveal an intrinsic connection between these symmetries and the translational invariance properties of multi-soliton solutions. Specifically, the K-symmetries are linked to center translation invariance, while the τ-symmetries can be expressed as a combination of center and wave number translation invariances. Furthermore, we have demonstrated that the set of symmetries of the pmKdV equation is incomplete. For n-soliton solutions, only 2n symmetries are independent, with all others being expressible in terms of these. For example, in the case of one-soliton solution, only K0 and τ1 are independent; for the two-soliton solutions, only K0, K1, τ1, and τ2 are independent; the remaining symmetries can be represented as combinations of these four symmetries. Similar symmetries structures for the two-wave complexiton solution can be found. Here, the symmetry τ0 should be excluded from our conclusions. The physical interpretation of τ0 requires further investigation.
It is worth noting that the infinitely many K-symmetries and τ-symmetries arising in integrable systems associated with n-wave solutions may share similar physical interpretations. Thus far, it has been observed that the K-symmetries and τ-symmetries of several classical integrable equations, including the KdV, Burgers, STO, mKdV, and sG equations exhibit such analogous physical meaning.
Due to the duality between the pmKdV and the sG equations, the results and solution structures we derive for the pmKdV equation in this work can be directly translated and applied to the sG equation. However, it is important to clarify that this duality does not imply a one-to-one correspondence in all practical aspects. While the underlying mathematical frameworks are related, the sG equation possesses greater intrinsic complexity, which significantly limits the tractability of deriving explicit exact solutions for arbitrary dispersion relations. Consequently, the scope and variety of exact solutions we have obtained for the pmKdV equation may not directly replicable for the sG equation.
The exploration of the physical meaning of the infinitely many K-symmetries and τ-symmetries provides a special way to generate n-wave solutions of the integrable systems. Based on the n-wave hypothesis, we first constructed exact n-wave solutions to the pmKdV equation. By performing symmetry reduction via the infinitely many K-symmetries, we rigorously derived the one-soliton solution along with several types of two-wave solutions, including complexitons, breathers, and double-pole solutions. Specific dispersion relations and coefficient parameters were identified to systematically generate these diverse types of multi-wave solutions.
In summary, this work deepens our understanding of multi-wave solutions and their symmetry structures in integrable systems. Future studies should extend this approach to other types of special solutions, particularly those involving parameters beyond the center and wave number, and further explore the use of symmetry methods for solution generation, as well as the potential physical applications of these solutions.