1. Introduction
Nonlinear phenomena are general problems in every field of engineering technology, science research, the natural world and human society activities. Nonlinear integrable equations play a crucial role in revealing nonlinear phenomena in various fields due to their fascinating features, such as N-soliton solutions [1–6], Bäcklund transformations [7–9], Lax pairs and the Painlevé test [10–13]. Among these integrable features, the Lax pair is a wonderful representation of integrable systems involving two linear operators, which can be differential operators or matrices [14]. A pair of linear operators L and A related to a given nonlinear partial differential equation may pave a way for solving the equation. It is difficult to find L and A corresponding to a given equation, so assuming that L and A are given and determining which partial differential equation they correspond to is actually simpler. The Painlevé test is widely and successfully used to study the integrability of nonlinear partial differential equations through analyzing the singularity structure of the solution. There are abundant successful examples of the method [14–16].
For integrable equations, in addition to investigating their integrable properties, the study of exact solutions has always been an important foundational topic in nonlinear science. There are many types of effective approaches to solve integrable equations, such as the inverse scattering method [15, 16], the Darboux transformation [17–20], Painlevé series expansion method [21] and Hirota direct method [22]. Among the existing techniques, the Darboux transformation is one of the most important ways for discussing compatibility equations of spectral problems. It is worth mentioning that the Darboux transformation is extremely useful in finding soliton solutions for nonlinear integrable equations from a trivial seed solution. In fact, through iteration, N-soliton solutions represented by special determinants, such as the Wronskian or Grammian, can be generated. Such N-soliton solutions have certain research value and practical significance in various scientific fields. Moreover, a Lax pair is very helpful for constructing Darboux transformations of integrable systems [23–25].
The Korteweg–de Vries (KdV) equation was first derived analytically by Johannes Korteweg together with his student, Gustav de Vries, in 1895 when they developed a theory for nonlinear waves [26]. The standard KdV equation is written as1.1 ) can also be referred to as1.2 ), the (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation is obtained by1.4 ) becomes
$\begin{eqnarray}{u}_{t}+6{{uu}}_{x}+{u}_{{xxx}}=0,\end{eqnarray}$
which describes the disturbance of long, one-dimensional water waves on shallow-water surfaces with small amplitude [27]. By using the recursion operator of the KdV equation [28] $\begin{eqnarray}{\rm{\Phi }}(u)\equiv {\partial }_{x}^{2}+4u+2{u}_{x}{\partial }_{x}^{-1},\end{eqnarray}$
the KdV equation ( $\begin{eqnarray*}{u}_{t}+{\rm{\Phi }}(u){u}_{x}=0,\end{eqnarray*}$
where ${\partial }_{x}^{-1}f=\int f\,{\rm{d}}x.$ Moreover, applying the same form of the KdV recursion operator ( $\begin{eqnarray}{u}_{t}+{\rm{\Phi }}(u){u}_{y}=0,\end{eqnarray}$
which is equivalent to $\begin{eqnarray}{u}_{t}+{u}_{{xxy}}+4{{uu}}_{y}+2{u}_{x}{\partial }_{x}^{-1}{u}_{y}=0.\end{eqnarray}$
Taking the potential u = φx, equation ( $\begin{eqnarray}{\varphi }_{{xt}}+{\varphi }_{{xxxy}}+4{\varphi }_{x}{\varphi }_{{xy}}+2{\varphi }_{{xx}}{\varphi }_{y}=0.\end{eqnarray}$
The CBS equation, also known as the breaking soliton equation, was proposed by Calogero and Degasperis [29], and also constructed by Bogoyavlenskii and Schiff via different techniques [30, 31]. This equation is widely used to describe the (2+1)-dimensional interaction between Riemann waves and long waves in shallow water [30, 31]. The KdV equation and the CBS equation are two well-known integrable models in (1+1) and (2+1)-dimensions, respectively, both possessing N-soliton solutions [22, 32, 33], Painlevé properties and infinitely many conservation laws [14, 34].As a modified form of the standard KdV equation (1.1 ) in the nonlinear term, the modified Korteweg–de Vries (mKdV) equation1.1 ) with the solution v of the mKdV equation (1.6 ) [35]. Similarly, employing the Miura transformation (1.7 ), the modified Calogero–Bogoyavlenskii–Schiff (mCBS) equation1.4 ) [32]. It is easy to see that equation (1.8 ) is reduced to the modified KdV equation (1.6 ) in the case of y = x. The N-soliton solutions for the mCBS equation (1.8 ) can be generated through the Hirota direct method [32, 36].
$\begin{eqnarray}{v}_{t}-6{v}^{2}{v}_{x}+{v}_{{xxx}}=0,\end{eqnarray}$
is also one of the well-known nonlinear integrable equations. Miura transformations exist between the KdV equation and the mKdV equation. The following Miura transformation $\begin{eqnarray}u=-{v}^{2}\pm {v}_{x},\end{eqnarray}$
connects the solution u of the KdV equation ( $\begin{eqnarray}{v}_{t}-4{v}^{2}{v}_{y}-2{v}_{x}{\partial }_{x}^{-1}{\left({v}^{2}\right)}_{y}+{v}_{{xxy}}=0,\end{eqnarray}$
can be derived from the CBS equation (Latterly, a novel (2+1)-dimensional mKdV system1.6 ) and the mCBS equation (1.8 ) by Wang and Wazwaz et al [37–39]. This equation is known as the (2+1)-dimensional modified Korteweg–de Vries–Calogero–Bogoyavlenskii–Schiff (mKdV-CBS) equation, which has attracted the attention of many scholars. The nonlocal symmetry and soliton-cnoidal wave interaction solutions for equation (1.9 ) were established by virtue of the truncated Painlevé expansion and consistent Riccati expansion approach, respectively [37]. A large number of solutions with various physical characteristics, including multiple soliton solutions, kink solutions and singular solutions [38], were given by applying the simplified Hirota’s method and other ways. By means of a direct symbolic computation, three classes of rational solutions for equation (1.9 ) were presented [39]. To the best of our knowledge, the Lax pair with a spectral parameter and Darboux transformation for equation (1.9 ) have not been revealed in previous articles.
$\begin{eqnarray}{u}_{t}-4{u}^{2}{u}_{y}-2{u}_{x}{\partial }_{x}^{-1}{\left({u}^{2}\right)}_{y}+{u}_{{xxy}}-6{u}^{2}{u}_{x}+{u}_{{xxx}}=0,\end{eqnarray}$
was introduced by combining the mKdV equation (In our previous work, a (2+1)-dimensional generalized KdV equation was investigated in the form [40]1.10 ) is a combination of the KdV equation (1.1 ) and the CBS equation (1.4 ), which is also referred to as the (2+1)-dimensional KdV-like model [8]. The bilinear Bäcklund transformation and Lax pair for equation (1.10 ) were obtained [40], implying that a linear combination in a soliton hierarchy is still integrable. The extended form (1.10 ) may potentially describe the propagation of long, two-dimensional solitary waves in the branches of physics, including plasma physics, condensed matter, nonlinear optics and fluid dynamics.
$\begin{eqnarray}\begin{array}{l}a(6\gamma {{uu}}_{x}+{u}_{{xxx}})\\ \quad +\,b({u}_{{xxy}}+2\gamma {u}_{x}{\partial }_{x}^{-1}{u}_{y}+4\gamma {{uu}}_{y})+{{cu}}_{t}=0,\end{array}\end{eqnarray}$
where the constants a, b, c and γ satisfy γc(a2 + b2) ≠ 0. Obviously, taking a = b = c = γ = 1, equation (Motivated by the recent studies mentioned above, we will consider an extension of the mKdV-CBS equation (1.9 ) by connecting equation (1.10 ), written as1.11 ) reduces to the mKdV-CBS equation (1.9 ). Taking a = b = c = 1 and $\gamma =\pm \sqrt{-1}$, equation (1.11 ) yields the following mKdV-CBS equation:1.11 ) contains significant integrable equations as its special cases, such as the mKdV equation and the mCBS equation. The mKdV equation and the mCBS equation possess wide applications in the fields of nonlinear science [41, 42]. In particular, the mKdV equation is used to describe the propagation of solitons in lattices, the motion of nonlinear Alfvén waves in plasma and fluid mechanics [41]. Therefore, we believe that equation (1.11 ) can be effectively applied in practical systems.
$\begin{eqnarray}\begin{array}{l}a({v}_{{xxx}}-6{\gamma }^{2}{v}^{2}{v}_{x})\\ \quad +\,b\left({v}_{{xxy}}-2{\gamma }^{2}{v}_{x}{\partial }_{x}^{-1}{\left({v}^{2}\right)}_{y}-4{\gamma }^{2}{v}^{2}{v}_{y}\right)+{{cv}}_{t}=0,\end{array}\end{eqnarray}$
where the constants a, b, c and γ are arbitrary, which satisfy γc(a2 + b2) ≠ 0. It is obvious that for a = b = c = 1 and γ = ± 1, equation ( $\begin{eqnarray}({v}_{{xxx}}+6{v}^{2}{v}_{x})+\left({v}_{{xxy}}+2{v}_{x}{\partial }_{x}^{-1}{\left({v}^{2}\right)}_{y}+4{v}^{2}{v}_{y}\right)+{v}_{t}=0.\end{eqnarray}$
We can clearly see that the extended form (The paper is structured as follows. In section 2 , based on the Lax pair of equation (1.10 ), the extended mKdV-CBS equation (1.11 ) will be derived, thereby presenting its Lax pair with a spectral parameter. Furthermore, the Darboux transformation will be furnished with the help of the obtained Lax pair. In section 3 , one-soliton, two-soliton solutions and soliton molecules will be explored for equation (1.11 ). Some concluding remarks will be given in the final section.
2. Lax pair and Darboux transformation
In this section, we first present the Lax pair of the generalized KdV equation (1.10 ) obtained through the bilinear Bäcklund transformation [40]. The Lax pair of equation (1.10 ) can be expressed as1.10 ) arises from the compatibility condition [L1, L2] = 0 of the above system. The system (2.1 ) is equivalent to the following representation:1.10 ). We next consider another second-order spectral problem corresponding to (2.2a ) as follows:2.3 ), then (2.3 ) becomes2.5 ) is the spectral problem (2.2a ). Moreover, substituting (2.4 ) and (2.6 ) into (2.2b ), a direct calculation yields2.2a ) and the following constraint2.8 ) is just equation (1.11 ) by taking the derivative with respect to x at both ends of (2.8 ). By using (2.3 ) and (2.7 ), it can directly verify that the compatibility condition ψxxt = ψtxx is nothing but the potential v is a solution of the extended mKdV-CBS equation (1.11 ). Namely, setting1.11 ) is generated from the compatibility condition $[{L}_{1}^{{\prime} },{L}_{2}^{{\prime} }]=0$ of these two operators. This shows the system (2.9 ) is a Lax pair of equation (1.11 ). Further, we also found that formula (2.6 ) is a Miura transformation, which provides a relation between solutions of equation (1.10 ) and equation (1.11 ).
$\begin{eqnarray}{L}_{1}={\partial }_{x}^{2}+\gamma u-\lambda ,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{L}_{2} & = & c{\partial }_{t}+a\left(4{\partial }_{x}^{3}+6\gamma u{\partial }_{x}+3\gamma {u}_{x}\right)\\ & & +\,b\left(2\gamma ({\partial }_{x}^{-1}{u}_{y}){\partial }_{x}+4\lambda {\partial }_{y}-\gamma {u}_{y}\right),\end{array}\end{eqnarray}$
where λ is an arbitrary constant. We see that equation ( $\begin{eqnarray}{\phi }_{{xx}}=\lambda \phi -\gamma u\phi ,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\phi }_{t} & = & -\frac{a}{c}\left(4{\phi }_{{xxx}}+6\gamma u{\phi }_{x}+3\gamma {u}_{x}\phi \right)\\ & & -\,\frac{b}{c}\left(2\gamma ({\partial }_{x}^{-1}{u}_{y}){\phi }_{x}+4\lambda {\phi }_{y}-\gamma {u}_{y}\phi \right),\end{array}\end{eqnarray}$
where φ is an eigenfunction and λ is a spectral parameter. The compatibility condition φxxt = φtxx is nothing but the potential u is a solution of equation ( $\begin{eqnarray}{\psi }_{{xx}}=\lambda \psi -2\gamma v{\psi }_{x},\end{eqnarray}$
where ψ is an eigenfunction and λ is a spectral parameter. Substituting the transformation [22] $\begin{eqnarray}\psi ={{\rm{e}}}^{-\gamma {\partial }^{-1}v}\phi \end{eqnarray}$
between the eigenfunctions φ and ψ into ( $\begin{eqnarray}{\phi }_{{xx}}+(-\gamma {v}_{x}-{\gamma }^{2}{v}^{2})\phi =\lambda \phi .\end{eqnarray}$
Setting $\begin{eqnarray}u=-{v}_{x}-\gamma {v}^{2},\end{eqnarray}$
then we see that ( $\begin{eqnarray}\begin{array}{rcl}{\psi }_{t} & = & \frac{a}{c}\left[(2{\gamma }^{2}{v}^{2}+2\gamma {v}_{x}-4\lambda ){\psi }_{x}-4\lambda \gamma v\psi \right]\\ & & +\,\frac{b}{c}\left[(2{\gamma }^{2}{\partial }_{x}^{-1}{\left({v}^{2}\right)}_{y}+2\gamma {v}_{y}){\psi }_{x}\right.\\ & & \left.-\,4\lambda {\psi }_{y}-4\lambda \gamma ({\partial }_{x}^{-1}{v}_{y})\psi \right],\end{array}\end{eqnarray}$
where ( $\begin{eqnarray}\displaystyle \frac{a}{c}\left[2{\gamma }^{2}{v}^{3}-{v}_{{xx}}\right]+\displaystyle \frac{b}{c}\left[2{\gamma }^{2}v{\partial }_{x}^{-1}{\left({v}^{2}\right)}_{y}-{v}_{{xy}}\right]-{\partial }_{x}^{-1}{v}_{t}=0\end{eqnarray}$
have been applied. It is easy to observe that condition ( $\begin{eqnarray}{L}_{1}^{{\prime} }={\partial }_{x}^{2}+2\gamma v{\partial }_{x}-\lambda ,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{L}_{2}^{{\prime} }=c{\partial }_{t}-a\left[(2{\gamma }^{2}{v}^{2}+2\gamma {v}_{x}-4\lambda ){\partial }_{x}-4\lambda \gamma v\right]\\ \,-b\left[(2{\gamma }^{2}{\partial }_{x}^{-1}{\left({v}^{2}\right)}_{y}+2\gamma {v}_{y}){\partial }_{x}-4\lambda {\partial }_{y}-(4\lambda \gamma {\partial }_{x}^{-1}{v}_{y})\right],\end{array}\end{eqnarray}$
equation (In the following, according to the spectral problem (2.3 ), we derived the Darboux transformation for the extended mKdV-CBS equation (1.11 ) via the same method presented in the existing literature [14, 19, 22]. We consider a linear transformation $\psi \to \bar{\psi }$ 2.3 ) into the following spectral problem with a potential $\bar{v}$ 2.3 ), the above transformation (2.10 ) becomes2.10 ) and its first-order derivative into (2.11 ) gives2.12 ) and (2.13 ), we have2.14 ) into (2.15 ) and integrating the resulting form, we get2.16 ) yields
$\begin{eqnarray}\bar{\psi }=\sigma {\psi }_{x}+\psi ,\end{eqnarray}$
which transforms ( $\begin{eqnarray}{\bar{\psi }}_{{xx}}=\lambda \bar{\psi }-2\gamma \bar{v}{\bar{\psi }}_{x}.\end{eqnarray}$
Applying the spectral problem ( $\begin{eqnarray}\begin{array}{rcl}{\bar{\psi }}_{{xx}} & = & (-2\gamma v-4\gamma v{\sigma }_{x}+4{\gamma }^{2}\sigma {v}^{2}+{\sigma }_{{xx}}-2\gamma \sigma {v}_{x}+\lambda \sigma ){\psi }_{x}\\ & & +\,\lambda (1+2{\sigma }_{x}-2\gamma \sigma v)\psi .\end{array}\end{eqnarray}$
In addition, substituting ( $\begin{eqnarray}\begin{array}{rcl}{\bar{\psi }}_{{xx}} & = & (4{\gamma }^{2}\sigma v\bar{v}-2\gamma {\sigma }_{x}\bar{v}-2\gamma \bar{v}+\lambda \sigma ){\psi }_{x}\\ & & +\,(-2\gamma \lambda \sigma \bar{v}+\lambda )\psi .\end{array}\end{eqnarray}$
Comparing the coefficients of ψx and ψ in ( $\begin{eqnarray}\bar{v}=v-\displaystyle \frac{{\sigma }_{x}}{\gamma \sigma },\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\sigma }_{{xx}}+4\sigma {\gamma }^{2}{v}^{2}-4\gamma {\sigma }_{x}v-2a\gamma {v}_{x}-2\gamma v\\ \,=\,4\sigma {\gamma }^{2}v\bar{v}-2\gamma \bar{v}-2\gamma {\sigma }_{x}\bar{v}.\end{array}\end{eqnarray}$
Substituting ( $\begin{eqnarray}{\sigma }_{x}{\sigma }^{-2}-2\gamma {\sigma }^{-1}v+{\sigma }^{-2}={\lambda }_{1},\end{eqnarray}$
where λ1 is a constant. Furthermore, assuming $\sigma =-{{ff}}_{x}^{-1}$ and substituting it into ( $\begin{eqnarray}{f}_{{xx}}+2\gamma {{vf}}_{x}={\lambda }_{1}f.\end{eqnarray}$
Consequently, the above results can be summarized as the following theorem:Theorem 2.1. Suppose that the linear problems (
$\begin{eqnarray}\bar{\psi }=\psi +\sigma {\psi }_{x},\,\,\sigma =-\displaystyle \frac{f(x,{\lambda }_{1})}{{f}_{x}(x,{\lambda }_{1})}\end{eqnarray}$
converts ( $\begin{eqnarray}\bar{v}=v+\frac{1}{\gamma }{\left[\mathrm{ln}\frac{{f}_{x}(x,{\lambda }_{1})}{f(x,{\lambda }_{1})}\right]}_{x},\end{eqnarray}$
and f is the fixed solution of (As we know, transformation (2.18 ) with (2.19 ), which transforms the linear problem (2.3 ) into the linear problem (2.11 ) with the same form, is defined as the Darboux transformation [14, 22]. It is worth pointing out that the Darboux transformation is very powerful in constructing soliton solutions. Before applying the above theorem to soliton theory, we also need to consider the following proposition:
Proposition 2.1. Suppose that f satisfies (
$\begin{eqnarray}\begin{array}{rcl}{f}_{t} & = & \frac{a}{c}\left[(2{\gamma }^{2}{v}^{2}+2\gamma {v}_{x}-4{\lambda }_{1}){f}_{x}-4{\lambda }_{1}\gamma {vf}\right]\\ & & +\,\frac{b}{c}\left[\left(2{\gamma }^{2}{\partial }_{x}^{-1}{\left({v}^{2}\right)}_{y}+2\gamma {v}_{y}\right){f}_{x}-4{\lambda }_{1}{f}_{y}\right.\\ & & \left.-\,(4{\lambda }_{1}\gamma {\partial }_{x}^{-1}{v}_{y})f\right].\end{array}\end{eqnarray}$
Under the Darboux transformation ( $\begin{eqnarray}\begin{array}{rcl}{\bar{\psi }}_{t} & = & \frac{a}{c}\left[(2{\gamma }^{2}{\bar{v}}^{2}+2\gamma {\bar{v}}_{x}-4\lambda ){\bar{\psi }}_{x}-4\lambda \gamma \bar{v}\bar{\psi }\right]\\ & & +\,\frac{b}{c}\left[(2{\gamma }^{2}{\partial }_{x}^{-1}({\bar{v}}_{y}^{2})+2\gamma {\bar{v}}_{y}){\bar{\psi }}_{x}\right.\\ & & \left.-\,4\lambda {\bar{\psi }}_{y}-(4\lambda \gamma {\partial }_{x}^{-1}{\bar{v}}_{y})\bar{\psi }\right].\end{array}\end{eqnarray}$
Using the same calculation as described in the existing literature [14, 19], the proof of proposition 2.1 will be given in appendix.3. Soliton solutions and soliton molecules
From the presented results in section 2 , it can be seen that if v is a solution to the extended mKdV-CBS equation (1.11 ), then $\bar{v}$ determined by (2.19 ) is also a solution of equation (1.11 ). We now construct soliton solutions for equation (1.11 ) by utilizing the Darboux transformation. Taking v = 0 as the seed solution and choosing ${\lambda }_{1}={k}_{1}^{2}$ in (2.17 ) and (2.20 ), we obtain the following linear partial differential equations: 3.1 ), we have2.19 ), an exact one-soliton solution of equation (1.11 ) can be given as3.4 ) can be expressed as3.3 ) and δ is an arbitrary constant.
$\begin{eqnarray}{f}_{{xx}}={k}_{1}^{2}f,\end{eqnarray}$
$\begin{eqnarray}{f}_{t}=\frac{a}{c}\left(-4{k}_{1}^{2}{f}_{x}\right)+\frac{b}{c}\left(2{\gamma }^{2}{d}_{1}{f}_{x}-4{k}_{1}^{2}{f}_{y}-4\gamma {k}_{1}^{2}{d}_{2}f\right),\end{eqnarray}$
where d1, d2 are arbitrary integral constants, and k1 is a non-zero constant. Solving the above system ( $\begin{eqnarray}f={C}_{1}{e}^{{\xi }_{1}}+{C}_{2}{e}^{-{\xi }_{1}},\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}{\xi }_{1} & = & {k}_{1}x+{l}_{1}y-\left[\frac{4a}{c}{k}_{1}^{3}-\frac{b}{c}\left(2{\gamma }^{2}{d}_{1}{k}_{1}-4{k}_{1}^{2}{l}_{1}\right)\right]t\\ & & +\,{\xi }_{1}^{0},\,{d}_{2}\,=\,0.\end{array}\end{eqnarray}$
Here C1, C2, l1, d1 and ${\xi }_{1}^{0}$ are arbitrary given that every term in the solution makes sense. Through transformation ( $\begin{eqnarray}\bar{v}=\frac{4{C}_{1}{C}_{2}{k}_{1}}{\gamma ({C}_{1}^{2}{e}^{2{\xi }_{1}}-{C}_{2}^{2}{e}^{-2{\xi }_{1}})}.\end{eqnarray}$
In particular, taking ${C}_{2}={\rm{i}}{C}_{1}{e}^{-\delta },{\rm{i}}=\sqrt{-1}$, the above one-soliton solution ( $\begin{eqnarray}\bar{v}=\frac{2{\rm{i}}{k}_{1}}{\gamma }{\rm{sech}} (2{\xi }_{1}+\delta ),\end{eqnarray}$
where ξ1 is defined by (We iterate the above Darboux transformation, a direct computation yields:1.11 ) possesses the form3.7 ) and the parameters involved make the solution meaningful. The N-times iterated or N-fold Darboux transformation leads to3.9 ) and (3.10 ) is an N-soliton solution of equation (1.11 ). Figure 1 shows the evolution of a special two-soliton solution $\bar{\bar{v}}$ determined by (3.8 ) with the parameter selections3.9 ) with (3.10 ) under the parameter selections
$\begin{eqnarray}\begin{array}{rcl}\bar{\bar{v}} & = & \bar{v}+\frac{1}{\gamma }{\left[\mathrm{ln}\frac{{\bar{\psi }}_{x}(x,{\lambda }_{2})}{\bar{\psi }(x,{\lambda }_{2})}\right]}_{x}\\ & = & v+\frac{1}{\gamma }{\left[\mathrm{ln}\frac{{W}_{r}({f}_{x}(x,{\lambda }_{1}),{f}_{x}(x,{\lambda }_{2}))}{{W}_{r}(f(x,{\lambda }_{1}),f(x,{\lambda }_{2}))}\right]}_{x},\end{array}\end{eqnarray}$
where Wr(f(x, λ1), f(x, λ2)) = f(x, λ1)fx(x, λ2) − f(x, λ2)fx(x, λ1) is the standard Wronskian determinant. Let us take $\begin{eqnarray}\begin{array}{rcl}v & = & 0,f(x,{\lambda }_{1})=\cosh {\xi }_{1},\quad f(x,{\lambda }_{2})=\sinh {\xi }_{2},\\ {\xi }_{i} & = & {k}_{i}x+{l}_{i}y-\left[\frac{4a}{c}{k}_{i}^{3}-\frac{b}{c}\left(2{\gamma }^{2}{d}_{1}{k}_{i}-4{k}_{i}^{2}{l}_{i}\right)\right]t\\ & & +\,{\xi }_{i}^{0},i\,=\,1,2.\end{array}\end{eqnarray}$
A two-soliton solution of equation ( $\begin{eqnarray}\begin{array}{l}\bar{\bar{v}}\,=\displaystyle \frac{2({k}_{1}^{2}-{k}_{2}^{2})({k}_{2}\sinh 2{\xi }_{1}+{k}_{1}\sinh 2{\xi }_{2})}{\gamma [{\left({k}_{1}+{k}_{2}\right)}^{2}{\cosh }^{2}({\xi }_{1}-{\xi }_{2})-{\left({k}_{2}-{k}_{1}\right)}^{2}{\cosh }^{2}({\xi }_{1}+{\xi }_{2})]},\end{array}\end{eqnarray}$
with ξi, i = 1, 2 being given by ( $\begin{eqnarray}v[N]=v+\frac{1}{\gamma }{\left[\mathrm{ln}\frac{{W}_{r}({f}_{x}(x,{\lambda }_{1}),{f}_{x}(x,{\lambda }_{2}),\,\cdots \,,{f}_{x}(x,{\lambda }_{N}))}{{W}_{r}(f(x,{\lambda }_{1}),f(x,{\lambda }_{2}),\,\cdots \,,f(x,{\lambda }_{N}))}\right]}_{x},\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{l}{W}_{r}(f(x,{\lambda }_{1}),f(x,{\lambda }_{2}),\,\cdots \,,f(x,{\lambda }_{N}))\\ \quad =\,\det {\left({\partial }_{x}^{i-1}f(x,{\lambda }_{j})\right)}_{1\leqslant i\leqslant N,1\leqslant j\leqslant N}.\end{array}\end{eqnarray*}$
Furthermore, we choose $\begin{eqnarray}\begin{array}{rcl}v & = & 0,f(x,{\lambda }_{2l+1})=\cosh {\xi }_{2l+1},f(x,{\lambda }_{2l})=\sinh {\xi }_{2l},\\ {\xi }_{i} & = & {k}_{i}x+{l}_{i}y-\left[\frac{4a}{c}{k}_{i}^{3}-\frac{b}{c}\left(2{\gamma }^{2}{d}_{1}{k}_{i}-4{k}_{i}^{2}{l}_{i}\right)\right]t\\ & & +\,{\xi }_{i}^{0},1\leqslant i\leqslant N,\end{array}\end{eqnarray}$
then the function v[N] defined by ( $\begin{eqnarray}\begin{array}{rcl}a & = & b=c=\gamma =1,{k}_{1}=1,{l}_{1}=3,{k}_{2}=2,{l}_{2}=-1,\\ {d}_{1} & = & {\xi }_{1}^{0}={\xi }_{2}^{0}=0.\end{array}\end{eqnarray}$
Figure 2 displays the propagation of a special three-soliton solution determined by ( $\begin{eqnarray}\begin{array}{rcl}a & = & b=c=\gamma =1,{k}_{1}=1,{l}_{1}=3,{k}_{2}=2,{l}_{2}=-1,\\ {k}_{3} & = & 3,{l}_{3}=-4,{d}_{1}={\xi }_{1}^{0}={\xi }_{2}^{0}={\xi }_{3}^{0}=0.\end{array}\end{eqnarray}$
It can be observed that two or three solitary waves propagate at a certain speed, collide after a while, and then continue to propagate in their original shapes.
Figure 1. Three-dimensional plots of $\bar{\bar{v}}$ determined by ( |
Figure 2. Three-dimensional plots of v[3] determined by ( |
Soliton molecules are bound states of solitons, which have been observed in systems from optical fibers to mode-locked lasers. At present, soliton molecules have increasingly become an interesting topic in various areas, including nonlinear optics, fluid mechanics and Bose–Einstein condensates, since they provide important insights into the fundamental interactions between solitons and the potential dynamics in complex nonlinear systems. There are different solution methods for finding soliton molecules, for instance, the velocity resonance mechanism proposed by Lou [43, 44]. A new mechanism for discovering soliton molecules, called velocity resonance, was introduced by employing velocity resonance ki/kj = ωi/ωj, where the parameters ki, kj and ωi, ωj are wave numbers and frequencies, respectively. Under the above resonant condition, the ith and jth solitons are bounded and develop a soliton molecule by choosing appropriate solution parameters. This method can not only be extensively applied to (1+1)-dimensional systems [45], but also to higher-dimensional nonlinear systems [44]. In this section, we would like to investigate soliton molecules of equation (1.11 ).
For the (2+1) dimensional equation (1.11 ), the velocity resonant conditions become3.13 ) can be obtained as follows:3.14 ), the two-soliton solution (3.8 ) of equation (1.11 ) yields a two-soliton molecule
$\begin{eqnarray}\frac{{k}_{i}}{{k}_{j}}=\frac{{l}_{i}}{{l}_{j}}=\frac{-\frac{4a}{c}{k}_{i}^{3}+\frac{b}{c}\left(2{\gamma }^{2}{d}_{1}{k}_{i}-4{k}_{i}^{2}{l}_{i}\right)}{-\frac{4a}{c}{k}_{j}^{3}+\frac{b}{c}\left(2{\gamma }^{2}{d}_{1}{k}_{j}-4{k}_{j}^{2}{l}_{j}\right.)},{k}_{i}\ne \pm {k}_{j}.\end{eqnarray}$
The solution of equation ( $\begin{eqnarray}{l}_{i}=-\frac{{{ak}}_{i}}{b},{l}_{j}=-\frac{{{ak}}_{j}}{b}.\end{eqnarray}$
Therefore, via the resonance condition ( $\begin{eqnarray}\begin{array}{rcl}\bar{\bar{v}} & = & \frac{2({k}_{1}^{2}-{k}_{2}^{2})({k}_{2}\sinh 2{\xi }_{1}+{k}_{1}\sinh 2{\xi }_{2})}{\gamma [{\left({k}_{1}+{k}_{2}\right)}^{2}{\cosh }^{2}({\xi }_{1}-{\xi }_{2})-{\left({k}_{2}-{k}_{1}\right)}^{2}{\cosh }^{2}({\xi }_{1}+{\xi }_{2})]},\\ {\xi }_{i} & = & {k}_{i}x-\frac{a}{b}{k}_{i}y+\frac{2b{\gamma }^{2}{d}_{1}}{c}{k}_{i}t+{\xi }_{i}^{0},\,i=1,2,\end{array}\end{eqnarray}$
with arbitrary non-zero constants ki and d1.From solution (3.15 ), a special two-soliton molecule profile of $\bar{\bar{v}}$ with the parameters
$\begin{eqnarray}\begin{array}{rcl}a & = & b=c=\gamma =1,{k}_{1}=1,{k}_{2}=0.5,\\ {d}_{1} & = & 3,{\xi }_{1}^{0}=1,{\xi }_{2}^{0}=2,\end{array}\end{eqnarray}$
is plotted in figure 3. It can be observed that two line soliton waves are parallel to each other in the (x, y)-plane, and they carry different widths and amplitudes due to k1 ≠ k2, l1 ≠ l2. However, the velocities of the two solitons in the molecule are the same. It is also worth explicitly noting that the selection of the parameters ${\xi }_{1}^{0}$ and ${\xi }_{2}^{0}$ will cause a change in the distance between two solitons in the molecule.
Figure 3. Three-dimensional plots of $\bar{\bar{v}}$ determined by ( |
4. Concluding remarks
In summary, an extended mKdV-CBS equation (1.11 ) has been explored by means of the existing results, thereby presenting its Lax pair with a spectral parameter. Meanwhile, a Miura transformation has been found, which provides a relation between solutions of the extended mKdV-CBS equation (1.11 ) and the extended KdV equation (1.10 ). Then, associated with the resulting Lax pair, the Darboux transformation has been derived to the introduced equation in detail. The resultant Darboux transformation has been applied to soliton solutions. Furthermore, the soliton molecules have been given by the velocity resonance mechanism. Our results indicate that equation (1.11 ) is integrable and they provide good supplements to the existing literature. The present study is believed to contribute to a general understanding of the complex dynamical phenomena in areas such as fluids, ocean dynamics and plasmas. In particular, the investigation of soliton solutions would be helpful in describing the behaviors of wave propagations in dispersive wave theories.
We also point out that equation (1.11 ) can be written as a Hirota bilinear form:
$\begin{eqnarray}{D}_{x}^{2}f\cdot f^{\prime} =0,\end{eqnarray}$
$\begin{eqnarray}({{aD}}_{x}^{3}+{{bD}}_{x}^{2}{D}_{y}+{{cD}}_{t})f\cdot f^{\prime} =0,\end{eqnarray}$
under the logarithmic transformations $\begin{eqnarray}v=\frac{1}{\gamma }{\left[\mathrm{ln}(\frac{f}{f^{\prime} })\right]}_{x},\rho =\mathrm{ln}({ff}^{\prime} ),\end{eqnarray}$
where the auxiliary function $\rho$ satisfies $\rho$xx + γ2v2 = 0 and D is the Hirota’s bilinear differential operator [1]. There are any potential extensions or future research directions that could be built upon our work. A large number of interesting solutions generated by the Darboux transformation and the Hirota bilinear form, including lump solutions [46–49], Hirota N-soliton solutions, breathers and Wronskian solutions [4, 7, 50–52], will be discussed in the future.Acknowledgments
The authors express their sincere thanks to the referees and editors for their valuable comments. This work was supported by the National Natural Science Foundation of China (Grant No. 12271488).
Conflict of interest
The authors declare that they have no conflict of interest.
Appendix
We give a proof of proposition 2.1.
Proof. We differentiate the transformation (
$\begin{eqnarray}\begin{array}{rcl}{\bar{\psi }}_{t} & = & \left[\frac{a}{c}\left(\sigma {A}_{1}+{B}_{1}+\sigma {B}_{1,x}-2\gamma \sigma {{vB}}_{1}\right)\right.\\ & & \left.+\,\frac{b}{c}\left(\sigma {A}_{2}+{B}_{2}+\sigma {B}_{2,x}-2\gamma \sigma {{vB}}_{2}\right)+{\sigma }_{t}\right]{\psi }_{x}\\ & & +\left[\frac{a}{c}\left({A}_{1}+\sigma {A}_{1,x}+\lambda \sigma {B}_{1}\right)\right.\\ & & \left.+\,\frac{b}{c}\left({A}_{2}+\sigma {A}_{2,x}+\lambda \sigma {B}_{2}\right)\right]\psi -\frac{b}{c}\left(4\lambda {\psi }_{y}+4\lambda \sigma {\psi }_{{xy}}\right),\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{A}_{1} & = & -4\lambda \gamma v,{B}_{1}=-4\lambda +2\gamma {v}_{x}+2{\gamma }^{2}{v}^{2},\\ {A}_{2} & = & -4\lambda \gamma {\partial }_{x}^{-1}{v}_{y},{B}_{2}=2{\gamma }^{2}{\partial }_{x}^{-1}{\left({v}^{2}\right)}_{y}+2\gamma {v}_{y}.\end{array}\end{eqnarray}$
In addition, substituting ( $\begin{eqnarray}\begin{array}{rcl}{\bar{\psi }}_{t} & = & \left\{\frac{a}{c}\left[\sigma {\bar{A}}_{1}+{\bar{B}}_{1}(1+{\sigma }_{x}-2\gamma \sigma v)\right]\right.\\ & & \left.+\frac{b}{c}\left[\sigma {\bar{A}}_{2}+{\bar{B}}_{2}(1+{\sigma }_{x}-2\gamma \sigma v)-4\lambda {\sigma }_{y}\right]\right\}{\psi }_{x}\\ & & +\left\{\frac{a}{c}({\bar{A}}_{1}+\lambda \sigma {\bar{B}}_{1})+\frac{b}{c}\left({\bar{A}}_{2}+\lambda \sigma {\bar{B}}_{2}\right)\right\}\psi \\ & & -\frac{b}{c}\left(4\lambda {\psi }_{y}+4\lambda \sigma {\psi }_{{xy}}\right),\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{\bar{A}}_{1} & = & -4\lambda \gamma \bar{v},{\bar{B}}_{1}=-4\lambda +2\gamma {\bar{v}}_{x}+2{\gamma }^{2}{\bar{v}}^{2},\\ {\bar{A}}_{2} & = & -4\lambda \gamma {\partial }_{x}^{-1}{\bar{v}}_{y},{\bar{B}}_{2}=2{\gamma }^{2}{\partial }_{x}^{-1}{\left({\bar{v}}^{2}\right)}_{y}+2\gamma {\bar{v}}_{y}.\end{array}\end{eqnarray}$
To prove proposition 2.1, we only need to prove that the above two expressions hold simultaneously.Let us firstly verify that the coefficients of ψ in (A1 ) and (A3 ) are equal. According to (2.18 ) and (2.19 ), we can obtain the following relational expressions:A5 ), we haveA2 ) and (A4 ), respectively. To verifyA2 ) and (A4 ), respectively. By applying (2.14 ), (2.19 ), (A7 ) and (A8 ), a direct computation leads toA8 ) and (A5 ), the left hand side of (A15 ) becomesA5 ) and (A6 ), the right hand side of (A15 ) becomesA11 ) is valid. Hence, from (A10 ) and (A11 ), we can see that the coefficients of ψ in (A1 ) and (A3 ) are equal.
$\begin{eqnarray}{\sigma }_{x}=-1+2\gamma \sigma v+{\lambda }_{1}{\sigma }^{2},\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{{xy}}=2\gamma {\sigma }_{y}v+2\gamma \sigma {v}_{y}+2{\lambda }_{1}\sigma {\sigma }_{y},\end{eqnarray}$
$\begin{eqnarray}{\partial }_{x}^{-1}{\left(\frac{{\sigma }_{x}}{\sigma }\right)}_{y}=\frac{{\sigma }_{y}}{\sigma },\quad \end{eqnarray}$
$\begin{eqnarray}{\left(\frac{{\sigma }_{x}}{\sigma }\right)}_{y}={\lambda }_{1}{\sigma }_{y}+2\gamma {v}_{y}+\frac{{\sigma }_{y}}{{\sigma }^{2}}.\end{eqnarray}$
By using ( $\begin{eqnarray}\begin{array}{rcl} & & ({\bar{A}}_{1}-{A}_{1})+\lambda \sigma ({\bar{B}}_{1}-{B}_{1})-\sigma {A}_{1,x}\\ & & \quad =4\lambda {\sigma }_{x}{\sigma }^{-1}(1-2\gamma \sigma v-{\lambda }_{1}{\sigma }^{2}+{\sigma }_{x})=0,\end{array}\end{eqnarray}$
which is equivalent to $\begin{eqnarray}{A}_{1}+\sigma {A}_{1,x}+\lambda \sigma {B}_{1}={\bar{A}}_{1}+\lambda \sigma {\bar{B}}_{1},\end{eqnarray}$
with A1, B1 and ${\bar{A}}_{1},{\bar{B}}_{1}$ being given by ( $\begin{eqnarray}{A}_{2}+\sigma {A}_{2,x}+\lambda \sigma {B}_{2}={\bar{A}}_{2}+\lambda \sigma {\bar{B}}_{2},\end{eqnarray}$
we next only need to verify $\begin{eqnarray}({\bar{A}}_{2}-{A}_{2})+\lambda \sigma ({\bar{B}}_{2}-{B}_{2})-\sigma {A}_{2,x}=0,\end{eqnarray}$
with A2, B2 and ${\bar{A}}_{2},{\bar{B}}_{2}$ being given by ( $\begin{eqnarray}\begin{array}{l}({\bar{A}}_{2}-{A}_{2})+\lambda \sigma ({\bar{B}}_{2}-{B}_{2})-\sigma {A}_{2,x}\\ \quad =\lambda \left[\frac{2{\sigma }_{y}}{\sigma }+4\sigma {\partial }_{x}^{-1}{\left(\frac{{\sigma }_{x}^{2}}{2{\sigma }^{2}}-\frac{\gamma {\sigma }_{x}}{\sigma }v\right)}_{y}-2{\lambda }_{1}\sigma {\sigma }_{y}\right].\end{array}\end{eqnarray}$
So we need to show $\begin{eqnarray}{\partial }_{x}^{-1}{\left(\frac{{\sigma }_{x}^{2}}{2{\sigma }^{2}}-\frac{\gamma {\sigma }_{x}}{\sigma }v\right)}_{y}=\frac{{\lambda }_{1}{\sigma }_{y}}{2}-\frac{{\sigma }_{y}}{2{\sigma }^{2}},\end{eqnarray}$
that is to say $\begin{eqnarray}{\left(\frac{{\sigma }_{x}^{2}}{2{\sigma }^{2}}-\frac{\gamma {\sigma }_{x}}{\sigma }v\right)}_{y}=\frac{{\lambda }_{1}{\sigma }_{{xy}}}{2}-{\left(\frac{{\sigma }_{y}}{2{\sigma }^{2}}\right)}_{x}.\end{eqnarray}$
On the one hand, again by using ( $\begin{eqnarray}\begin{array}{rcl}{\left(\displaystyle \frac{{\sigma }_{x}^{2}}{2{\sigma }^{2}}-\displaystyle \frac{\gamma {\sigma }_{x}}{\sigma }v\right)}_{y} & = & \left({\lambda }_{1}{\sigma }_{y}+2\gamma {v}_{y}+\displaystyle \frac{{\sigma }_{y}}{{\sigma }^{2}}\right)\left(-\gamma v+\displaystyle \frac{{\sigma }_{x}}{\sigma }\right)\\ & & -\,\displaystyle \frac{\gamma {\sigma }_{x}}{\sigma }{v}_{y}\\ & = & -\,\gamma {\lambda }_{1}{\sigma }_{y}v-2{\gamma }^{2}{{vv}}_{y}-\gamma \displaystyle \frac{{\sigma }_{y}}{{\sigma }^{2}}v+{\lambda }_{1}\displaystyle \frac{{\sigma }_{x}{\sigma }_{y}}{\sigma }\\ & & +\,\gamma \displaystyle \frac{{\sigma }_{x}}{\sigma }{v}_{y}+\displaystyle \frac{{\sigma }_{x}{\sigma }_{y}}{{\sigma }^{3}}\\ & = & \gamma {\lambda }_{1}{\sigma }_{y}v-\gamma \displaystyle \frac{{\sigma }_{y}}{{\sigma }^{2}}v-{\lambda }_{1}\displaystyle \frac{{\sigma }_{y}}{\sigma }\\ & & +\,{\lambda }_{1}^{2}\sigma {\sigma }_{y}-\displaystyle \frac{\gamma }{\sigma }{v}_{y}+\gamma {\lambda }_{1}\sigma {v}_{y}+\displaystyle \frac{{\sigma }_{x}{\sigma }_{y}}{{\sigma }^{3}}.\end{array}\end{eqnarray}$
On the other hand, by applying ( $\begin{eqnarray}\begin{array}{rcl}\frac{{\lambda }_{1}{\sigma }_{{xy}}}{2}-{\left(\frac{{\sigma }_{y}}{2{\sigma }^{2}}\right)}_{x} & = & (\gamma {\sigma }_{y}v+\gamma \sigma {v}_{y}+{\lambda }_{1}\sigma {\sigma }_{y})\left({\lambda }_{1}-\frac{1}{{\sigma }^{2}}\right)\\ & & +\,\frac{{\sigma }_{x}{\sigma }_{y}}{{\sigma }^{3}}\\ & = & \gamma {\lambda }_{1}{\sigma }_{y}v+\gamma {\lambda }_{1}\sigma {v}_{y}+{\lambda }_{1}^{2}\sigma {\sigma }_{y}\\ & & -\,\gamma \frac{{\sigma }_{y}}{{\sigma }^{2}}v-\frac{\gamma }{\sigma }{v}_{y}-{\lambda }_{1}\frac{{\sigma }_{y}}{\sigma }+\frac{{\sigma }_{x}{\sigma }_{y}}{{\sigma }^{3}}.\end{array}\end{eqnarray}$
This shows that (Let us secondly verify that the coefficients of ψx in (A1 ) and (A3 ) are also equal. By employing (2.18 ) and (2.20 ), we getA18 ), (A5 ), (2.14 ) and (2.19 ), a direct calculation yieldsA9 ) associated with the spectral parameter λ1 , the final expression in (A20 ) is equal to zero. Directly we can also compute thatA5 ), (A7 ) and (A11 ) associated with the spectral parameter λ1 have been applied. Therefore, based on (A20 ) and (A21 ), we show that equality (A19 ) holds. This completes the proof.
$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{t} & = & \frac{a}{c}\left[{\sigma }^{2}{A}_{1,x}({\lambda }_{1})+{\sigma }_{x}{B}_{1}({\lambda }_{1})-\sigma {B}_{1,x}({\lambda }_{1})\right]\\ & & +\frac{b}{c}\left[{\sigma }^{2}{A}_{2,x}({\lambda }_{1})+{\sigma }_{x}{B}_{2}-\sigma {B}_{2,x}-4{\lambda }_{1}{\sigma }_{y}\right].\end{array}\end{eqnarray}$
We now verify the following equality: $\begin{eqnarray}\begin{array}{rcl} & & \frac{a}{c}(\sigma {A}_{1}+{B}_{1}+\sigma {B}_{1,x}-2\gamma \sigma {{vB}}_{1})\\ & & \quad +\frac{b}{c}(\sigma {A}_{2}+{B}_{2}+\sigma {B}_{2,x}-2\gamma \sigma {{vB}}_{2})+{\sigma }_{t}\\ & & \quad =\frac{a}{c}\left[\sigma {\bar{A}}_{1}+{\bar{B}}_{1}(1+{\sigma }_{x}-2\gamma \sigma v)\right]\\ & & \quad +\frac{b}{c}\left[\sigma {\bar{A}}_{2}+{\bar{B}}_{2}(1+{\sigma }_{x}-2\gamma \sigma v)-4\lambda {\sigma }_{y}\right].\end{array}\end{eqnarray}$
Via ( $\begin{eqnarray}\begin{array}{rcl} & & \sigma {\bar{A}}_{1}+(1+{\sigma }_{x}-2\gamma \sigma v){\bar{B}}_{1}\\ & & \,-(\sigma {A}_{1}+{B}_{1}+\sigma {B}_{1,x}-2\gamma \sigma {{vB}}_{1})\\ & & \,-\left[{\sigma }^{2}{A}_{1,x}\left({\lambda }_{1})+{\sigma }_{x}{B}_{1}({\lambda }_{1})-\sigma {B}_{1,x}({\lambda }_{1}\right)\right]\\ & & \,=\sigma ({\bar{A}}_{1}-{A}_{1})+{\lambda }_{1}{\sigma }^{2}({\bar{B}}_{1}-{B}_{1})-{\sigma }^{2}{A}_{1,x}({\lambda }_{1})\\ & & \,+{\sigma }_{x}\left({B}_{1}-{B}_{1}\left({\lambda }_{1}\right)\right)\\ & & \,=4\lambda {\sigma }_{x}+{\lambda }_{1}{\sigma }^{2}({\bar{B}}_{1}-{B}_{1})-{\sigma }^{2}{A}_{1,x}({\lambda }_{1})\\ & & \,+{\sigma }_{x}\left(-4\lambda +4{\lambda }_{1}\right)\\ & & \,=\sigma [{\bar{A}}_{1}({\lambda }_{1})-{A}_{1}({\lambda }_{1})]+{\lambda }_{1}{\sigma }^{2}[{\bar{B}}_{1}({\lambda }_{1})-{B}_{1}({\lambda }_{1})]\\ & & \,-{\sigma }^{2}{A}_{1,x}({\lambda }_{1}).\end{array}\end{eqnarray}$
By virtue of the formula ( $\begin{eqnarray}\begin{array}{rcl} & & \sigma {\bar{A}}_{2}+(1+{\sigma }_{x}-2\gamma \sigma v){\bar{B}}_{2}-4\lambda {\sigma }_{y}\\ & & \,-\,(\sigma {A}_{2}+{B}_{2}+\sigma {B}_{2,x}-2\gamma \sigma {{vB}}_{2})\\ & & \,-\left[{\sigma }^{2}{A}_{2,x}({\lambda }_{1})+{\sigma }_{x}{B}_{2}-\sigma {B}_{2,x}-4{\lambda }_{1}{\sigma }_{y}\right]\\ & & \,=\sigma ({\bar{A}}_{2}-{A}_{2})+{\lambda }_{1}{\sigma }^{2}({\bar{B}}_{2}-{B}_{2})-4\lambda {\sigma }_{y}-{\sigma }^{2}{A}_{2,x}({\lambda }_{1})\\ & & \,+\,4{\lambda }_{1}{\sigma }_{y}\\ & & \,=4\lambda {\sigma }_{y}+{\lambda }_{1}{\sigma }^{2}({\bar{B}}_{2}-{B}_{2})-4\lambda {\sigma }_{y}-{\sigma }^{2}{A}_{2,x}({\lambda }_{1})+4{\lambda }_{1}{\sigma }_{y}\\ & & \,=\sigma [{\bar{A}}_{2}({\lambda }_{1})-{A}_{2}({\lambda }_{1})]+{\lambda }_{1}{\sigma }^{2}({\bar{B}}_{2}-{B}_{2})\\ & & \,-\,{\sigma }^{2}{A}_{2,x}({\lambda }_{1})=0,\end{array}\end{eqnarray}$
where the formulas (