Soliton molecules and the CRE method in the extended mKdV equation
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Bo Ren,1,4, Ji Lin,2,4, Ping Liu31Institute of Nonlinear Science, Shaoxing University, Shaoxing, 312000, China 2Department of Physics, Zhejiang Normal University, Jinhua, 321004, China 3College of Electron and Information Engineering, University of Electronic Science and Technology of China Zhongshan Institute, Zhongshan, 528402, China
First author contact:4 Authors to whom any correspondence should be addressed. Received:2020-01-31Revised:2020-02-21Accepted:2020-03-9Online:2020-04-22
Abstract The soliton molecules of the (1+1)-dimensional extended modified Korteweg–de Vries (mKdV) system are obtained by a new resonance condition, which is called velocity resonance. One soliton molecule and interaction between a soliton molecule and one-soliton are displayed by selecting suitable parameters. The soliton molecules including the bright and bright soliton, the dark and bright soliton, and the dark and dark soliton are exhibited in figures 1–3, respectively. Meanwhile, the nonlocal symmetry of the extended mKdV equation is derived by the truncated Painlevé method. The consistent Riccati expansion (CRE) method is applied to the extended mKdV equation. It demonstrates that the extended mKdV equation is a CRE solvable system. A nonauto-Bäcklund theorem and interaction between one-soliton and cnoidal waves are generated by the CRE method. Keywords：extended mKdV equation;soliton molecule;CRE method;nonlocal symmetry
PDF (2995KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Bo Ren, Ji Lin, Ping Liu. Soliton molecules and the CRE method in the extended mKdV equation. Communications in Theoretical Physics, 2020, 72(5): 055005- doi:10.1088/1572-9494/ab7ed6
1. Introduction
The features of resonance phenomena are an important problem, both experimentally and theoretically. For the integrable systems, resonance in solitons may lead to various types of new excitations such as the breathers [1], the soliton fissions, the soliton fusions [2], the rational-exponential waves [3] and so on. The soliton molecule, which can be treated as the soliton bound state, has attracted considerable attention. The soliton molecules were first predicted theoretically in the framework of the nonlinear Schrödinger-Ginzburg-Landau equation [4] and the coupled nonlinear Schrödinger equations [5]. Recently, a new velocity resonance mechanism is introduced to form soliton molecules [6, 7]. For velocity resonance, high-order dispersive terms play a key role in the nonlinear integrable systems [6]. Based on the velocity resonance, the soliton molecules of the (2+1)-dimensional fifth-order Korteweg–de Vries (KdV) equation [8], the complex modified KdV equation [9] and the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation [10] are constructed by using the Darboux transformation and the variable separation method. Meanwhile, some novel interactions between soliton molecules and breather solutions, and between soliton molecules and dromions, are presented by the velocity resonance mechanism [8–11]. The interaction between solitons and other nonlinear excitations plays an important topic [12–20]. The consistent Riccati expansion (CRE) or the consistent tanh expansion (CTE) method can be applied to find these types of solutions [21]. The method has been applied to various nonlinear systems, including the modified Kadomtsev-Petviashvili (KP) equation [22], the modified KdV-Calogero-Bogoyavlenkskii-Schiff equation [23], the supersymmetric integrable systems [24] and the non-integrable cubic generalised KP equation [25]. In this paper, the main purpose of our work is to construct soliton molecules, interaction between a soliton molecule and one-soliton, and interaction between solitons and cnoidal periodic wave solutions for an extended modified KdV (mKdV) equation.
In the sense of Marchant et al [26, 27], one can get the extended mKdV equation. The extended mKdV equation reads$\begin{eqnarray}\begin{array}{l}{u}_{t}+\alpha ({u}_{{xxx}}+6\sigma {u}^{2}{u}_{x})\\ \quad +\,\beta {\left(6{u}^{5}+10\sigma {{uu}}_{x}^{2}+10\sigma {u}^{2}{u}_{{xx}}+{u}_{{xxxx}}\right)}_{x}=0,\end{array}\end{eqnarray}$where α and β are the third-order and fifth-order dispersion coefficients matching with the relevant nonlinear terms. The extended mKdV equation (1) describes the evolution of steeper waves with shorter wavelengths than in the mKdV equation [28]. The extended mKdV equation (1) becomes the focusing form and defocusing form with σ=+1 and σ=−1. The equation (1) gives rise to multiple soliton solutions and multiple singular soliton solutions for σ=+1 and σ=−1, respectively [28]. The extended mKdV equation (1) for the focusing form, i.e. σ=+1 reads$\begin{eqnarray}\begin{array}{l}{u}_{t}+\alpha ({u}_{{xxx}}+6{u}^{2}{u}_{x})\\ \ \ \ \ +\beta {\left(6{u}^{5}+10{{uu}}_{x}^{2}+10{u}^{2}{u}_{{xx}}+{u}_{{xxxx}}\right)}_{x}=0.\end{array}\end{eqnarray}$The extended mKdV equation of the focusing form (2) possesses the Lax pair, the infinitely conservation laws, the Darboux transformation, the Painlevé property and the multi-soliton solution [29]. The long-time asymptotic behavior for the extended mKdV equation (2) has been analyzed recently [30].
This paper is organized as follows. In section 2, the soliton molecules and interaction between a soliton molecule and one-soliton of the extended mKdV equation are obtained by a new resonance condition. In section 3, the nonlocal symmetry and its application of the extended mKdV equation can be constructed by the truncated Painlevé method. In section 4, the extended mKdV equation is used in the CRE method. The interacted one-soliton with periodic waves can be derived by using a nonauto-Bäcklund transformation theorem. Some concluding remarks will be made in the last section.
2. Soliton molecules and interaction between a soliton molecule and one-soliton for the extended mKdV equation
To determine the multi-soliton solution of the extended mKdV equation (2), the dependent variable transformation reads [29]$\begin{eqnarray}u=2{\left(\arctan \displaystyle \frac{F}{G}\right)}_{x}.\end{eqnarray}$The auxiliary functions of F(x, t) and G(x, t) for the three-soliton solution are selected as$\begin{eqnarray}\begin{array}{rcl}F(x,t) & = & 1+\exp ({\theta }_{1})+\exp ({\theta }_{2})+\exp ({\theta }_{3})\\ & & +\,{a}_{12}\exp ({\theta }_{1}+{\theta }_{2})+{a}_{13}\exp ({\theta }_{1}+{\theta }_{3})\\ & & +\,{a}_{23}\exp ({\theta }_{2}+{\theta }_{3})+{b}_{123}\exp ({\theta }_{1}+{\theta }_{2}+{\theta }_{3}),\\ G(x,t) & = & 1-\exp ({\theta }_{1})-\exp ({\theta }_{2})-\exp ({\theta }_{3})\\ & & +\,{a}_{12}\exp ({\theta }_{1}+{\theta }_{2})+{a}_{13}\exp ({\theta }_{1}+{\theta }_{3})\\ & & +\,{a}_{23}\exp ({\theta }_{2}+{\theta }_{3})-{b}_{123}\exp ({\theta }_{1}+{\theta }_{2}+{\theta }_{3}),\end{array}\end{eqnarray}$with θi=kix+ωit+ci. By substituting (4) and (3) into (2), the phases shift aij and b123 read$\begin{eqnarray}\begin{array}{rcl}{a}_{{ij}} & = & -\displaystyle \frac{{\left({k}_{i}-{k}_{j}\right)}^{2}}{{\left({k}_{i}+{k}_{j}\right)}^{2}},\quad 1\leqslant i\leqslant j\leqslant 3,\\ {b}_{123} & = & {a}_{12}{a}_{13}{a}_{23},\end{array}\end{eqnarray}$and the dispersion relation$\begin{eqnarray}\alpha {k}_{i}^{3}+\beta {k}_{i}^{5}+{\omega }_{i}=0.\end{eqnarray}$If the parameters a13, a23, b123 and $\exp ({\theta }_{3})$ in (4) select zero, the three-soliton solution will reduce to the two-soliton solution.
We shall study the soliton molecule with a new resonance condition. The new condition $({k}_{i}\ne {k}_{j})$ of velocity resonance reads$\begin{eqnarray}\displaystyle \frac{{k}_{i}}{{k}_{j}}=\displaystyle \frac{{\omega }_{i}}{{\omega }_{j}}=\displaystyle \frac{\alpha {k}_{i}^{3}+\beta {k}_{i}^{5}}{\alpha {k}_{j}^{3}+\beta {k}_{j}^{5}}.\end{eqnarray}$The ith and jth solitons of resonant condition (7) are bounded to form a soliton molecule or an asymmetric soliton by selecting the parameters in (7). We take the two-soliton and the new resonance condition to describe the soliton molecule in figures 1–3. For the figure 1, the parameters are selected as$\begin{eqnarray}\begin{array}{rcl}{k}_{1} & = & \displaystyle \frac{1}{2},\quad {k}_{2}=1,\quad \alpha =1,\\ \beta & = & -0.8,\quad {c}_{1}=0,\quad {c}_{2}=10.\end{array}\end{eqnarray}$For the figure 2, the parameters are selected as$\begin{eqnarray}\begin{array}{rcl}{k}_{1} & = & \displaystyle \frac{1}{2},\quad {k}_{2}=-1,\quad \alpha =1,\\ \beta & = & -0.8,\quad {c}_{1}=0,\quad {c}_{2}=10.\end{array}\end{eqnarray}$The parameters are selected as the following form in figure 3$\begin{eqnarray}\begin{array}{rcl}{k}_{1} & = & -\displaystyle \frac{1}{2},\quad {k}_{2}=-1,\quad \alpha =1,\\ \beta & = & -0.8,\quad {c}_{1}=0,\quad {c}_{2}=10.\end{array}\end{eqnarray}$From the figures 1–3, the soliton molecule can be classified into three cases, i.e., k1>0, k2>0; k1k2<0 and k1<0, k2<0. It represents that the soliton molecule can consist of the bright and bright soliton, the dark and bright soliton, and the dark and dark soliton, respectively. The velocities of two solitons in the molecule are the same value with different amplitude.
Figure 1.
New window|Download| PPT slide Figure 1.(a) Soliton molecule structure described by (3) with the parameter selections (8). (b) Density plot of the corresponding soliton molecule.
Figure 2.
New window|Download| PPT slide Figure 2.(a) Soliton molecule structure described by (3) with the parameter selections (9). (b) Density plot of the corresponding soliton molecule.
Figure 3.
New window|Download| PPT slide Figure 3.(a) Soliton molecule structure described by (3) with the parameter selections (10). (b) Density plot of the corresponding soliton molecule.
To study the interaction between a soliton molecule and one-soliton, we use the three-soliton. We describe this phenomenon in figure 4. For figure 4, the parameters are selected as$\begin{eqnarray}\begin{array}{rcl}{k}_{1} & = & \displaystyle \frac{1}{2},\quad {k}_{2}=1,\quad {k}_{3}=0.8,\quad \alpha =1,\\ \beta & = & -0.8,\quad {c}_{1}=0,\quad {c}_{2}=15,\quad {c}_{3}=35.\end{array}\end{eqnarray}$The interaction between a soliton molecule and one-soliton is elastic from figure 4.
Figure 4.
New window|Download| PPT slide Figure 4.(a) The interaction between a soliton molecule and one-soliton with the parameters selections (11). (b) The corresponding density plot.
3. Nonlocal symmetry and its application of the extended mKdV equation
The nonlocal symmetry for the extended mKdV equation can be constructed with the truncated Painlevé method [31–36]. The Laurent series of the extended mKdV equation (2) reads$\begin{eqnarray}u=\displaystyle \frac{{u}_{0}}{f}+{u}_{1},\end{eqnarray}$where the function f(x, t)=0 is the equation of singularity manifold. The functions of u0 and u1 are functions of x and t. By balancing the coefficients of each power of f independently, we get$\begin{eqnarray}{u}_{0}={{if}}_{x},\end{eqnarray}$and the Schwarzian extended mKdV form$\begin{eqnarray}K+\alpha C+\displaystyle \frac{3\beta }{2}{C}^{2}+\beta {C}_{{xx}}=0,\end{eqnarray}$where K and the Schwarzian derivative C satisfy$\begin{eqnarray}K(x,t)=\displaystyle \frac{{f}_{t}}{{f}_{x}},\ \ \ C(x,t)=\{f;x\}=\displaystyle \frac{\partial }{{\partial }_{x}}\displaystyle \frac{{f}_{{xx}}}{{f}_{x}}-\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{f}_{{xx}}}{{f}_{x}}\right)}^{2}.\end{eqnarray}$By the truncated Painlevé method, the nonlocal symmetry of the extended mKdV equation reads$\begin{eqnarray}{\sigma }^{u}={{if}}_{x}.\end{eqnarray}$The nonlocal symmetry (16) is the residual of the singularity manifold f. To solve the initial value problem of the Lie’s first principle of the nonlocal symmetry, we localize the nonlocal symmetry to the Lie point symmetry by introducing a new field$\begin{eqnarray}{f}_{x}=g.\end{eqnarray}$The initial value problem related by the nonlocal symmetry (17) becomes$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{d\bar{u}}{d\epsilon } & = & {ig},\quad \bar{u}{| }_{\epsilon =0}=u,\\ \displaystyle \frac{d\bar{f}}{d\epsilon } & = & -{f}^{2},\quad \bar{f}{| }_{\epsilon =0}=f,\\ \displaystyle \frac{d\bar{g}}{d\epsilon } & = & -2{gf},\quad \bar{g}{| }_{\epsilon =0}=g.\end{array}\end{eqnarray}$By solving the above initial value problem, the Bäcklund transformation (BT) theorem of the enlarged systems (2), (14) and (17) is obtained.
If f, g and u is a solution of the enlarged systems (2), (14) and (17), $\bar{f},\bar{g}$ and $\bar{u}$ is also a solution of the enlarged systems$\begin{eqnarray}\bar{f}=\displaystyle \frac{f}{1+\epsilon f},\quad \bar{g}=\displaystyle \frac{g}{{\left(1+\epsilon f\right)}^{2}},\quad \bar{u}=u+\displaystyle \frac{i\epsilon g}{1+\epsilon f},\end{eqnarray}$with an arbitrary group parameter ε.
4. CRE method and soliton-cnoidal wave solutions of the extended mKdV equation
The results presented in this paper can be also applied to the defocusing form of the extended mKdV equation. The extended mKdV equation (1) becomes the defocusing form, i.e. σ=−1. In this part, we shall use the CRE/CTE method to study the defocusing form of the extended mKdV equation. Based on the CRE/CTE method [21], the truncated solution of the defocusing form of the extended mKdV equation reads as the following form by using leading order analysis$\begin{eqnarray}u={u}_{0}R(f)+{u}_{1},\end{eqnarray}$where u0, u1 and f are arbitrary functions and R satisfies the Riccati equation$\begin{eqnarray}{R}_{f}={a}_{0}+{a}_{1}R+{a}_{2}{R}^{2},\end{eqnarray}$with a0, a1 and a2 being arbitrary constants. Substituting (20) into (2) and balancing the coefficients of R lead to$\begin{eqnarray}{u}_{0}={f}_{x},\quad {u}_{1}=-\displaystyle \frac{{f}_{{xx}}}{2{f}_{x}},\end{eqnarray}$and$\begin{eqnarray}\begin{array}{l}K+\alpha (C-2{f}_{x}^{2})\\ \quad +\,\beta \left(\displaystyle \frac{3}{2}{C}^{2}+{C}_{{xx}}-10{f}_{x}^{2}C+6{f}_{x}^{4}-10{f}_{{xx}}^{2}\right)=0,\end{array}\end{eqnarray}$where K(x, t) and C(x, t) satisfy (15). The CRE will be transformed to the CTE with a0=1, a1=0 and a2=−1. The special solution of the Riccati equation reads$\begin{eqnarray}R=\tanh (f).\end{eqnarray}$It demonstrates that a CRE solvable model would be a CTE solvable model, and vice versa. We can construct the theorem, which is treated as a nonauto-BT theorem between the solution of f and u.
If f is a solution of the equation (23), then u given by$\begin{eqnarray}u={f}_{x}\tanh (f)-\displaystyle \frac{{f}_{{xx}}}{2{f}_{x}},\end{eqnarray}$will be a solution of the extended mKdV equation with the defocusing form.
The solution of the extended mKdV equation with the defocusing form can be obtained by using the above theorem. We shall list two cases by means of the theorem.
Case I. A trivial solution of the defocusing form is$\begin{eqnarray}f={kx}+\omega t,\quad \omega =2\alpha {k}^{3}-6\beta {k}^{5},\end{eqnarray}$where k is an arbitrary constant and ω is determined by the dispersion relation. The one-soliton solution of the defocusing form reads as the following form by using the nonauto-BT theorem$\begin{eqnarray}u=k\tanh \left({kx}+(2\alpha {k}^{3}-6\beta {k}^{5})t\right).\end{eqnarray}$
Case II. In order to obtain the interaction between one-soliton and other nonlinear excitations of the defocusing form, this type solution can be determined by a sum of a trivial solution and an arbitrary function$\begin{eqnarray}f={kx}+\omega t+F(X),\quad X={k}_{1}x+{\omega }_{1}t,\end{eqnarray}$where k, ω, k1 and ω1 are arbitrary constants. The elliptic function equation can be obtained by substituting (28) into (23)$\begin{eqnarray}\begin{array}{rcl}{F}_{1}{\left(X\right)}_{X}^{2} & = & 4{F}_{1}{\left(X\right)}^{4}+{C}_{3}{F}_{1}{\left(X\right)}^{3}+{C}_{2}{F}_{1}{\left(X\right)}^{2}\\ & & +\,{C}_{1}{F}_{1}(X)+{C}_{0},\quad {F}_{1}(X)=F(X),\end{array}\end{eqnarray}$with$\begin{eqnarray*}\begin{array}{rcl}{C}_{0} & = & \displaystyle \frac{78{\alpha }^{3}{k}^{3}}{125{\beta }^{2}{k}_{1}^{3}({k}_{1}\omega -k{\omega }_{1})}+\displaystyle \frac{2\alpha {k}^{2}(7k{\omega }_{1}-{k}_{1}\omega )}{5\beta {k}_{1}^{4}({k}_{1}\omega -k{\omega }_{1})}\\ & & +\,\displaystyle \frac{5k({k}_{1}\omega -k{\omega }_{1})}{6\alpha {k}_{1}^{5}}+\displaystyle \frac{4{k}^{4}}{{k}_{1}^{4}},\\ {C}_{1} & = & \displaystyle \frac{234{\alpha }^{3}{k}^{2}}{125{\beta }^{2}{k}_{1}^{2}({k}_{1}\omega -k{\omega }_{1})}+\displaystyle \frac{4\alpha k(10k{\omega }_{1}-{k}_{1}\omega )}{5\beta {k}_{1}^{3}({k}_{1}\omega -k{\omega }_{1})}\\ & & +\,\displaystyle \frac{5({k}_{1}\omega -k{\omega }_{1})}{6\alpha {k}_{1}^{4}}+\displaystyle \frac{16{k}^{3}}{{k}_{1}^{3}},\\ {C}_{2} & = & \displaystyle \frac{234{\alpha }^{3}k}{125{\beta }^{2}{k}_{1}({k}_{1}\omega -k{\omega }_{1})}+\displaystyle \frac{2\alpha (19k{\omega }_{1}-{k}_{1}\omega )}{5\beta {k}_{1}^{2}({k}_{1}\omega -k{\omega }_{1})}+\displaystyle \frac{24{k}^{2}}{{k}_{1}^{2}},\\ {C}_{3} & = & \displaystyle \frac{78{\alpha }^{3}}{125{\beta }^{2}({k}_{1}\omega -k{\omega }_{1})}+\displaystyle \frac{12\alpha {\omega }_{1}}{5\beta {k}_{1}({k}_{1}\omega -k{\omega }_{1})}+\displaystyle \frac{16k}{{k}_{1}}.\end{array}\end{eqnarray*}$By the nonauto-BT, the interaction between one-soliton and cnoidal waves of the defocusing form reads as$\begin{eqnarray}\begin{array}{rcl}u & = & \left(k+{k}_{1}F{\left(X\right)}_{X}\right)\tanh \left({kx}+\omega t+F(X\right)\\ & & -\,\displaystyle \frac{{k}_{1}^{2}F{\left(X\right)}_{{XX}}}{2\left(k+{k}_{1}F{\left(X\right)}_{X}\right)}.\end{array}\end{eqnarray}$If we get the solution of the elliptic function equation (29), the explicit interaction between one-soliton and cnoidal periodic wave solutions for the defocusing form will be obtained by the expression (30).
5. Conclusion
In summary, the soliton molecules and some novel interaction solutions of the extended mKdV equation were studied by the velocity resonance mechanism and the CRE method. The soliton molecule and the interaction between a soliton molecule and one-soliton were constructed by the new resonance condition of velocity resonance. The interaction between a soliton molecule and one-soliton is elastic by some detail analysis. The nonlocal symmetry and the BT theorem of the extended mKdV equation were derived by the truncated Painlevé method. Based on the CRE/CTE method, the extended mKdV equation is a CRE/CTE solvable system. A nonauto-BT theorem was constructed by using the CRE/CTE method. The interaction between one-soliton and cnoidal periodic waves was given by the nonauto-BT theorem.
Acknowledgments
This work is supported by the National Natural Science Foundation of China Grant Nos. 11775146, 11305106, 11975156, 11775047 and 11835011.
,1,4, Ji Lin
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