In the last few decades, the research works on soliton molecules have received considerable interest because of their versatile applications in optics [1–4], Bose-Einstein condensates [5, 6], fluid physics [7], and so on. The soliton molecules in the nonlinear Schrödinger-Ginzburg-Landau equation [8] and coupled nonlinear Schrödinger equations [9] have been predicted theoretically in the early twentieth century. In 2005, soliton molecules were predicted numerically and observed experimentally in dispersion-managed fiber [10]. The author of a previous study [11] presented the first experimental results on the polarization dynamics of vector soliton molecules with periodic polarization switching. The real-time evolution of femtosecond soliton molecules in the cavity of a mode-locked laser was observed using the time-stretch technique [1, 12]. In another study [13], the first observation of the entire buildup process of soliton molecules has been reported. A new kind of soliton molecule with nanosecond soliton separation was demonstrated in 2019 [14]. Moreover, the scattering of a two-soliton molecule on the modified Pöschl-Teller potential well has been studied by means of a collective coordinate approach [15]. The numerical and theoretical research works on the dynamics of the matter-wave soliton molecules in multicomponent immiscible bulk Bose-Einstein condensates were forwarded in an earlier study [16].
In addition, much more theoretical research results of soliton molecules for various nonlinear partial differential equations (PDEs) can also been discovered in various studies [17–25] and references therein. Lou [17] introduced the velocity resonance mechanism to three fifth-order systems, and the soliton molecules of them were constructed based on the Hirota bilinear method. By means of the mechanism of velocity resonance, the molecule consisting of two identical solitons was obtained for the modified KdV equation [18] in the frame of Darboux transformation. Later, besides the bell-shaped soliton molecule, the soliton molecules such as kink molecule [19], dromion molecule [20], breather molecule [21, 22], and half periodic kink molecule [23] for PDEs were also presented, and the characteristics and propagation states of these soliton molecules were revealed in depth [24, 25].
So far, most of the studies have focused on the bright soliton molecules, the dark soliton molecules have been less investigated theoretically and have not yet been experimentally observed. Based on the integrable three-level coupled Maxwell-Bloch equations with the mixed focusing-defocusing case, the two- and three-dark-soliton molecules were analytically demonstrated [26]. In the wick-type nonlinear Schrödinger equation and the integrable higher-order nonlinear Schrödinger equation, the interactions between dark soliton molecules have been discussed under the velocity resonance condition [27, 28].
However, to our knowledge, there is a paucity of investigations on the soliton molecules in the mKdV-type bilinear equation with higher-order nonlinear terms and dispersions. In contrast, whether this model holds dark solitons and breathers in the form of a bound state has not yet been fully discussed either. Motivated by the above two observations, we plan to examine the soliton molecules and breathers in the following combined mKdV-type bilinear equation:1 ) are defined as [29]1 ) can be written as:4 ) covers some specific equations. When α1 ≠ 0, and the other ${\alpha }_{n}^{\prime} s$ are zeros, equation (4 ) becomes the celebrated mKdV (mKdV-3) equation:4 ) reduces to the fifth-order mKdV (mKdV-5) equation:4 ) accompanying with α3 ≠ 0. By taking α1, α2 ≠ 0, equation (4 ) turns into the third fifth-order mKdV (mKdV-35) equation. The multi-soliton solutions, periodic wave solutions, and rational solutions for the mKdV-35 equation have been solved in previous studies [44, 48, 49].
$\begin{eqnarray}\left({D}_{t}+\displaystyle \sum _{n=1}^{N}{\alpha }_{n}{D}_{x}^{2n+1}\right){f}^{* }\cdot f=0,\end{eqnarray}$
$\begin{eqnarray}{D}_{x}^{2}{f}^{* }\cdot f=0,\end{eqnarray}$
where the asterisk denotes the complex conjugate, and the arbitrary constants ${\alpha }_{n}^{{\prime} }s(n=1,2,3,\cdots ,N)$ represent the (2n + 1) th-order dispersion coefficients matching with the relevant nonlinear terms. The bilinear derivative operators Dx and Dt in equation ( $\begin{eqnarray}\begin{array}{l}{D}_{x}^{{m}_{1}}{D}_{t}^{{m}_{2}}(a\cdot b)={\left(\displaystyle \frac{\partial }{\partial x}-\displaystyle \frac{\partial }{\partial x^{\prime} }\right)}^{{m}_{1}}\\ \quad \times \,{\left.{\left(\displaystyle \frac{\partial }{\partial t}-\displaystyle \frac{\partial }{\partial t^{\prime} }\right)}^{{m}_{2}}a(x,t)\cdot b\left(x^{\prime} ,t^{\prime} \right)\right|}_{x=x^{\prime} ,t=t^{\prime} }.\end{array}\end{eqnarray}$
By making use of the transformation [30] $\begin{eqnarray}v=-{\rm{i}}{\left(\mathrm{ln}\displaystyle \frac{{f}^{* }}{f}\right)}_{x},\end{eqnarray}$
the usual form of the nonlinear evolution equation for equation ( $\begin{eqnarray}\begin{array}{l}{v}_{t}+\left[{\alpha }_{1}(2{v}^{3}+{v}_{2x})+{\alpha }_{2}(6{v}^{5}+10{{vv}}_{x}^{2}+10{v}^{2}{v}_{2x}+{v}_{4x})\right.\\ \quad +{\alpha }_{3}(20{v}^{7}+70{v}^{4}{v}_{2x}\\ \quad +140{v}^{3}{v}_{x}^{2}+14{v}^{2}{v}_{4x}+42{{vv}}_{2x}^{2}\\ \quad {\left.+56{{vv}}_{x}{v}_{3x}+70{v}_{x}^{2}{v}_{2x}+14{v}_{6x})\right]}_{x}+\cdots \ =\ 0\end{array}\end{eqnarray}$
with vnx ≡ ∂nv/∂xn. As an extended version of the mKdV-type bilinear equation [31], equation ( $\begin{eqnarray}{v}_{t}+6{v}^{2}{v}_{x}+{v}_{3x}=0.\end{eqnarray}$
It has been applied for representing nonlinear phenomena such as transmission lines in Schottky barrier [32], acoustic waves [33], models of traffic congestion [34], Alfvén waves [35], and so on. Moreover, lots of effective mathematical techniques have been used to seek its exact solutions including soliton solutions, rational solutions, interaction solutions between cnoidal waves and kink solitary waves, among others [36–41]. While α2 ≠ 0, equation ( $\begin{eqnarray}{v}_{t}+{\alpha }_{2}{\left[6{v}^{5}+10({{vv}}_{x}^{2}+{v}^{2}{v}_{2x})+{v}_{4x}\right]}_{x}=0,\end{eqnarray}$
which has been used to describe the propagation of pulse wave in a deformable elastic vessel filled with inviscid blood [42]. The Riemann-Hilbert problems, periodic wave solutions, soliton solutions, rational solutions, etc., to mKdV-5 equation have been obtained in different studies [43–46]. Using the Jacobi elliptic function expansion method and Hirota's method, Parkers and Wazwaz et al [45–47] considered the periodic wave solutions and multi-soliton solutions for the seventh-order mKdV (mKdV-7) equation: $\begin{eqnarray}\begin{array}{l}{v}_{t}+{\alpha }_{3}\left[20{v}^{7}+70({v}^{4}{v}_{2x}+2{v}^{3}{v}_{x}^{2})+14({v}^{2}{v}_{4x}\right.\\ \quad {\left.+3{{vv}}_{2x}^{2}+4{{vv}}_{x}{v}_{3x}+5{v}_{x}^{2}{v}_{2x})+{v}_{6x}\right]}_{x}=0,\end{array}\end{eqnarray}$
which is exactly equation (The structure of this paper is arranged as follows: In sections 2 and 3 , by means of the velocity resonance restriction, the two-soliton molecules for the mKdV-35 equation and the three-soliton molecules for mKdV-357 equation are, respectively, constructed. The evolution behaviors for these soliton molecules are analyzed and exhibited in graphical ways at the same time. In section 4 , we continue our discussions on the existence of the multi-soliton molecules for the linear combination system of mKdV-type bilinear equation with more higher dispersion and nonlinear terms. In the last section, some conclusions are provided.
1. Soliton molecules and breathers for mKdV-35 equation
First, we would like to consider the soliton molecule structures for the following mKdV-35-type bilinear equation:3 ), the nonlinear evolution form for equation (8 ) can be expressed as follows:8 ):
$\begin{eqnarray}({D}_{t}+{\alpha }_{1}{D}_{x}^{3}+{\alpha }_{2}{D}_{x}^{5}){f}^{* }\cdot f=0,\end{eqnarray}$
$\begin{eqnarray}{D}_{x}^{2}{f}^{* }\cdot f=0.\end{eqnarray}$
Under the dependent variable transformation ( $\begin{eqnarray}\begin{array}{l}{v}_{t}+\left[{\alpha }_{1}(2{v}^{3}+{v}_{2x})+{\alpha }_{2}\right.\\ \quad {\left.\times (6{v}^{5}+10{{vv}}_{x}^{2}+10{v}^{2}{v}_{2x}+{v}_{4x})\right]}_{x}=0.\end{array}\end{eqnarray}$
Taking advantage of the Hirota bilinear method directly, we get the multi-soliton solutions for equation ( $\begin{eqnarray}f=\displaystyle \sum _{\nu =0,1}\exp \left[\displaystyle \sum _{m\lt n}^{N}{A}_{{mn}}{\nu }_{m}{\nu }_{n}+\displaystyle \sum _{n=1}^{N}{\nu }_{n}\left({\xi }_{n}+\displaystyle \frac{\pi }{2}{\rm{i}}\right)\right],\end{eqnarray}$
$\begin{eqnarray}{\xi }_{n}={k}_{n}x+{\omega }_{n}t+{\xi }_{n0},\end{eqnarray}$
$\begin{eqnarray}{\omega }_{n}=-{\alpha }_{1}{k}_{n}^{3}-{\alpha }_{2}{k}_{n}^{5},\end{eqnarray}$
$\begin{eqnarray}\exp \,{A}_{{mn}}=\displaystyle \frac{{\left({k}_{m}-{k}_{n}\right)}^{2}}{{\left({k}_{m}+{k}_{n}\right)}^{2}}.\end{eqnarray}$
1.1. One-soliton solution
To make it more concrete, the one-soliton solution of the mKdV-35 equation (9 ) can be written as:11 ), it is obvious that the wave number k1 is not only related to the phase velocity of the soliton as $\tfrac{{\omega }_{1}}{{k}_{1}}\,=-{\alpha }_{1}{k}_{1}^{2}-{\alpha }_{2}{k}_{1}^{4}$, but also determines the amplitude of the soliton. Besides, because ${\rm{sech}} {\xi }_{1}$ is an even function, both the bright and dark soliton solutions for equation (9 ) can be produced by taking k1 as a negative and a positive constant respectively, see figure 1, which displays the dark and bright soliton solutions for equation (9 ) with parameters being chosen as (a) k1 = 1, (b) k1 = −1, and other constants α1 = α2 = 1, ξ10 = 0. As time increases, the dark and bright solitons propagate along the line x = 2t at the same speed and height, respectively.
$\begin{eqnarray}v=-{k}_{1}{\rm{sech}} {\xi }_{1},\ {\xi }_{1}={k}_{1}x-({\alpha }_{1}{k}_{1}^{3}+{\alpha }_{2}{k}_{1}^{5})t+{\xi }_{10}\end{eqnarray}$
with number wave k1 and phase ξ10 being arbitrary constants. From solution (
Figure 1. The transmission states of (a) the dark soliton (k1 = 1) and (b) the bright soliton (k1 = −1) for mKdV-35 equation ( |
1.2. Two-soliton molecules and breather solution
The two-soliton solution for equation (9 ) possesses the form:12 ), one may find that the suitable choices of wave numbers yield the following three different types of two-soliton profiles: (i) bright-bright soliton, k1, k2 > 0; (ii) bright-dark soliton, k1k2 < 0; (iii) dark-dark soliton, k1, k2 < 0. To identify the certain effects of wave numbers on the shapes of the two-soliton solutions, we display the intensity profiles of the two-soliton solutions in figure 2 by changing only the values of k1 and k2, leaving the other parameters unchanged. Figure 2 confirms that the intensity or energy of the soliton in every case propagates without change after the collision with the other soliton except for a phase shift.
$\begin{eqnarray}v=-{\rm{i}}{\left(\mathrm{ln}\displaystyle \frac{1+{{\rm{e}}}^{{\xi }_{1}-\tfrac{{\rm{i}}\pi }{2}}+{{\rm{e}}}^{{\xi }_{2}-\tfrac{{\rm{i}}\pi }{2}}+{a}_{12}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}-{\rm{i}}\pi }}{1+{{\rm{e}}}^{{\xi }_{1}+\tfrac{{\rm{i}}\pi }{2}}+{{\rm{e}}}^{{\xi }_{2}+\tfrac{{\rm{i}}\pi }{2}}+{a}_{12}{e}^{{\xi }_{1}+{\xi }_{2}+{\rm{i}}\pi }}\right)}_{x},\end{eqnarray}$
where ${a}_{12}=\tfrac{{\left({k}_{1}-{k}_{2}\right)}^{2}}{{\left({k}_{1}+{k}_{2}\right)}^{2}}$ and the variables ξj = kjx + ωjt + ξj0(j = 1, 2) with dispersion relations $\begin{eqnarray}{\omega }_{j}=-{\alpha }_{1}{k}_{j}^{3}-{\alpha }_{2}{k}_{j}^{5}.\end{eqnarray}$
According to the two-soliton solution (
Figure 2. The propagation states of the two-soliton solutions for the mKdV-35 equation: (a) bright-bright soliton (k1 = −1, k2 = −0.5), (b) bright-dark soliton (k1 = −1, k2 = 0.5) and (c) dark-dark soliton (k1 = 1, k2 = 0.5) with other parameter selections α1 = α2 = 1 and ξ10 = ξ20 = 0. |
To seek possible soliton molecule solutions for equation (9 ), one may bring into the velocity resonance mechanism:12 ). By taking a simple calculation and eliminating the case k1 = ±k2, we have:15 ), the two solitons in solution (12 ) share the same velocity $-{k}_{1}^{2}({\alpha }_{1}+{\alpha }_{2}{k}_{1}^{2})$, implying that they are bounded to form a two-soliton molecule. Not unexpectedly, following the basis for the classification of two-soliton solutions, there are also three different types of two-soliton molecules: (i) bright-bright soliton molecule; (ii) bright-dark soliton molecule; (iii) dark-dark soliton molecule. Figure 3 shows the evolutions of different types of two-soliton molecules with the wave numbers being selected as $\left\{{k}_{1}=-0.8,{k}_{2}=-\tfrac{\sqrt{34}}{5}\right\}$, $\left\{{k}_{1}=0.8,{k}_{2}=-\tfrac{\sqrt{34}}{5}\right\}$ and $\left\{{k}_{1}=0.8,{k}_{2}=\tfrac{\sqrt{34}}{5}\right\}$, and other parameters α1 = 1, α2 = −0.5, ξ10 = 0, ξ20 = 10. Under the circumstances, the velocities of the soliton molecules are all the same, that is vsm = −0.435.
$\begin{eqnarray}\displaystyle \frac{{\omega }_{1}}{{k}_{1}}=\displaystyle \frac{{\omega }_{2}}{{k}_{2}}=-{\alpha }_{1}{k}_{1}^{2}-{\alpha }_{2}{k}_{1}^{4}=-{\alpha }_{1}{k}_{2}^{2}-{\alpha }_{2}{k}_{2}^{4}\end{eqnarray}$
to the two-soliton solution ( $\begin{eqnarray}{k}_{2}=\pm \displaystyle \frac{\sqrt{-{\alpha }_{2}({\alpha }_{1}+{\alpha }_{2}{k}_{1}^{2})}}{{\alpha }_{2}}.\end{eqnarray}$
Under the resonance restriction (
Figure 3. The evolution profiles of the two-soliton molecules of the mKdV-35 equation: (a) bright-bright soliton molecule (k1 = −0.8, ${k}_{2}=-\tfrac{\sqrt{34}}{5}$), (b) bright-dark soliton molecule (k1 = 0.8, ${k}_{2}=-\tfrac{\sqrt{34}}{5}$) and (c) dark-dark soliton molecule (k1 = 0.8, ${k}_{2}=\tfrac{\sqrt{34}}{5}$). |
As a matter of fact, the two-soliton solution (12 ) contains not only the two-soliton molecule but also a breather, which can be acquired by making the wave number of one of the solitons as the complex conjugate of the other, that is ${k}_{2}={k}_{1}^{* }$. Figure 4 displays the density map of the breather solution of mKdV-35 equation by choosing the parameters as α1 = 1, α2 = −0.5, k1 = 0.2 + 0.8i and ${\xi }_{10}={\xi }_{20}^{* }=1-{\rm{i}}$, and the speed of breather in this case is vb = −2.777. If different wave numbers that are complex conjugate to each other are taken, the two-soliton solution (12 ) transforms into the forms of the breathers with different transmission directions and periods.
Figure 4. The density plot of breather solution for equation ( |
2. Three-soliton molecules and soliton-brether molecules for the mKdV-357 equation
In this section, we will concentrate on extending the process used above to mKdV-357-type bilinear equation3 ), the usual form of the nonlinear evolution equation for equation (16 ) is yielded:17 ), the breather solution and the two-soliton molecules of the mKdV-357 equation can also be obtained. Here we omit the repetitive expressions. Sequentially, according to the Hirota bilinear technique, the three-soliton solution to the mKdV-357 equation takes the form:18 ) satisfy the velocity resonance condition:20 ) produces:21 ) to make kj ≠ ks(s, j = 1, 2, 3). For instance, after selecting the appropriate parameters, especially the wave numbers, as21 ) not only does not change the velocity of the soliton molecule, but also the total energy.
$\begin{eqnarray}({D}_{t}+{\alpha }_{1}{D}_{x}^{3}+{\alpha }_{2}{D}_{x}^{5}+{\alpha }_{3}{D}_{x}^{7}){f}^{* }\cdot f=0,\end{eqnarray}$
$\begin{eqnarray}{D}_{x}^{2}{f}^{* }\cdot f=0.\end{eqnarray}$
Upon using the transformation ( $\begin{eqnarray}\begin{array}{l}{v}_{t}+\left[{\alpha }_{1}(2{v}^{3}+{v}_{2x})+{\alpha }_{2}(6{v}^{5}+10{{vv}}_{x}^{2}\right.\\ \Space{0ex}{1.54em}{0ex}\quad +10{v}^{2}{v}_{2x}+{v}_{4x})+{\alpha }_{3}(20{v}^{7}+70{v}^{4}{v}_{2x}\\ \Space{0ex}{1.54em}{0ex}\quad +140{v}^{3}{v}_{x}^{2}+14{v}^{2}{v}_{4x}+42{{vv}}_{2x}^{2}\\ \Space{0ex}{1.54em}{0ex}\quad {\left.+56{{vv}}_{x}{v}_{3x}+70{v}_{x}^{2}{v}_{2x}+14{v}_{6x})\right]}_{x}=0.\end{array}\end{eqnarray}$
It is worth noting that, by performing the same steps in the previous section on equation ( $\begin{eqnarray}v=-{\rm{i}}{\left[\mathrm{ln}\displaystyle \frac{1-{\mathrm{ie}}^{{\xi }_{1}}-{\mathrm{ie}}^{{\xi }_{2}}-{\mathrm{ie}}^{{\xi }_{3}}-{a}_{12}{e}^{{\xi }_{1}+{\xi }_{2}}-{a}_{13}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{3}}-{a}_{23}{{\rm{e}}}^{{\xi }_{2}+{\xi }_{3}}+{\rm{i}}{a}_{123}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}+{\xi }_{3}}}{1+{\mathrm{ie}}^{{\xi }_{1}}+{\mathrm{ie}}^{{\xi }_{2}}+{\mathrm{ie}}^{{\xi }_{3}}-{a}_{12}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}}-{a}_{13}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{3}}-{a}_{23}{{\rm{e}}}^{{\xi }_{2}+{\xi }_{3}}-{\rm{i}}{a}_{123}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}+{\xi }_{3}}}\right]}_{x}\end{eqnarray}$
with $\begin{eqnarray}{\xi }_{j}={k}_{j}x+{\omega }_{j}t+{\xi }_{j0},\ \ \mathrm{for}\ j=1,2,3\end{eqnarray}$
$\begin{eqnarray}{\omega }_{j}=-{\alpha }_{1}{k}_{j}^{3}-{\alpha }_{2}{k}_{j}^{5}-{\alpha }_{3}{k}_{j}^{7},\ \ \mathrm{for}\ j=1,2,3\end{eqnarray}$
$\begin{eqnarray}{a}_{{mn}}=\displaystyle \frac{{\left({k}_{m}-{k}_{n}\right)}^{2}}{{\left({k}_{m}+{k}_{n}\right)}^{2}},\ \ \mathrm{for}\ 1\leqslant m\lt n\leqslant 3\end{eqnarray}$
$\begin{eqnarray}{a}_{123}={a}_{12}{a}_{13}{a}_{23}.\ \end{eqnarray}$
However, the existence of soliton molecules requires the solitons that make up bound-state solitons have the same velocity, meaning that the velocities of the three solitons in solution ( $\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\omega }_{1}}{{k}_{1}} & = & \displaystyle \frac{{\omega }_{2}}{{k}_{2}}=\displaystyle \frac{{\omega }_{3}}{{k}_{3}}={\alpha }_{1}{k}_{1}^{2}+{\alpha }_{2}{k}_{1}^{4}+{\alpha }_{3}{k}_{1}^{6}={\alpha }_{1}{k}_{2}^{2}\\ & & +{\alpha }_{2}{k}_{2}^{4}+{\alpha }_{3}{k}_{2}^{6}={\alpha }_{1}{k}_{3}^{2}+{\alpha }_{2}{k}_{3}^{4}+{\alpha }_{3}{k}_{3}^{6}.\end{array}\end{eqnarray}$
The direct computation of equation ( $\begin{eqnarray}{k}_{2},{k}_{3}=\pm \displaystyle \frac{\sqrt{2{\alpha }_{3}(-{\alpha }_{3}{k}_{1}^{2}-{\alpha }_{2}\pm \sqrt{-3{\alpha }_{3}^{2}{k}_{1}^{4}-2{\alpha }_{2}{\alpha }_{3}{k}_{1}^{2}-4{\alpha }_{1}{\alpha }_{3}+{\alpha }_{2}^{2}})}}{2{\alpha }_{3}}.\end{eqnarray}$
It is known that two solitons will be reduced to one when their wave numbers are the same. Hence, to avoid three solitons degenerating into two solitons or even one soliton, one can adopt different plus and minus signs in equation ( $\begin{eqnarray}\begin{array}{rcl}{\alpha }_{1} & = & 2,{\alpha }_{2}=-3,{\alpha }_{3}=1,{\xi }_{10}=0,\\ {\xi }_{20} & = & 30,{\xi }_{30}=10,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{k}_{1} & = & 0.5,\ \ {k}_{2}=\sqrt{\displaystyle \frac{11+\sqrt{37}}{8}},\\ {k}_{3} & = & \sqrt{\displaystyle \frac{11-\sqrt{37}}{8}},\ \ \mathrm{for}\ \mathrm{figure}\ 5({\rm{a}})\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{k}_{1} & = & 0.5,\ \ {k}_{2}=\sqrt{\displaystyle \frac{11+\sqrt{37}}{8}},\\ {k}_{3} & = & -\sqrt{\displaystyle \frac{11-\sqrt{37}}{8}},\mathrm{for}\ \mathrm{figure}\ 5({\rm{b}})\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{k}_{1} & = & 0.5,\ \ {k}_{2}=-\sqrt{\displaystyle \frac{11+\sqrt{37}}{8}},\\ {k}_{3} & = & -\sqrt{\displaystyle \frac{11-\sqrt{37}}{8}},\mathrm{for}\ \mathrm{figure}\ 5({\rm{c}})\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{k}_{1} & = & -0.5,{k}_{2}=-\sqrt{\displaystyle \frac{11+\sqrt{37}}{8}},\\ {k}_{3} & = & -\sqrt{\displaystyle \frac{11-\sqrt{37}}{8}},\mathrm{for}\ \mathrm{figure}\ 5({\rm{d}})\end{array}\end{eqnarray}$
four different kinds of three-soliton molecules of the mKdV-357 equation, including (i) three-bright-soliton molecule (3BSM); (ii) one-bright-two-dark-soliton molecule (1B2DSM); (iii) two-bright-one-dark-soliton molecule (2B1DSM); (iv) three-dark-soliton molecule (3DSM), are depicted in figure 5, and they travel at the same speed vsm = −0.328. Furthermore, by computing the extreme values of the amplitudes of the three-soliton molecules at t = 0, it is discovered that the first maximum, second maximum and third maximum of the absolute value of the amplitude of each soliton molecule are equal, namely, $| v{| }_{\max [1]}^{3{BSM}}=| v{| }_{\max [1]}^{1B2{DSM}}=| v{| }_{\max [1]}^{2B1{DSM}}=| v{| }_{\max [1]}^{3{DSM}}=1.46$, $| v{| }_{\max [2]}^{3{BSM}}=| v{| }_{\max [2]}^{1B2{DSM}}=| v{| }_{\max [2]}^{2B1{DSM}}=| v{| }_{\max [2]}^{3{DSM}}=0.78$ and $| v{| }_{\max [3]}^{3{BSM}}=| v{| }_{\max [3]}^{1B2{DSM}}=| v{| }_{\max [3]}^{2B1{DSM}}=| v{| }_{\max [3]}^{3{DSM}}=0.50$. Hence, the total energy $(| v{| }_{\max [1]}^{2}+| v{| }_{\max [2]}^{2}+| v{| }_{\max [3]}^{2})$ of each three-soliton molecule in figure 5 is equal. Therefore, it can be concluded that a set of wave numbers {∣k1∣, ∣k2∣, ∣k3∣} satisfying resonance condition (
Figure 5. The maps of (a) three-dark-soliton molecule, (b) one-bright-two-dark-soliton molecule, (c) two-bright-one-dark-soliton molecule and (d) three-bright-soliton molecule for mKdV-357 equation with parameters ( |
Similar to the situation stated above, the three-soliton solution (18 ) also describes the interaction between a breather and a soliton as soon as any two wave numbers that are complex conjugate to each other are selected. Furthermore, if we employ the velocity resonance mechanism to ensure that the velocity of the breather is resonant with the soliton, that is:18 ) turns out to be the soliton-breather molecule. The density plots of an elastic collision between a breather and a soliton and a soliton-breather molecule are shown in figure 6, where the wave parameters are:
$\begin{eqnarray}\begin{array}{l}{v}_{\mathrm{soliton}}=-{k}_{2}^{2}({\alpha }_{1}+{\alpha }_{2}{k}_{2}^{2}+{\alpha }_{3}{k}_{2}^{4})\\ \quad ={v}_{\mathrm{breather}}={\alpha }_{1}(3{k}_{1i}^{2}-{k}_{1r}^{2})\\ \quad -{\alpha }_{2}(5{k}_{1i}^{4}-10{k}_{1i}^{2}{k}_{1r}^{2}+{k}_{1r}^{4})\\ \quad +{\alpha }_{3}(7{k}_{1i}^{6}-35{k}_{1i}^{4}{k}_{1r}^{2}+21{k}_{1i}^{2}{k}_{1r}^{4}-{k}_{1r}^{6}),\end{array}\end{eqnarray}$
where ${k}_{1}={k}_{3}^{* }={k}_{1r}+{\rm{i}}{k}_{1i}$, the three-soliton solution ( $\begin{eqnarray}\begin{array}{ccc}{\alpha }_{1} & = & -{\alpha }_{2}={\alpha }_{3}={k}_{3}=1,{k}_{1}={k}_{2}^{\ast }=1+0.5{\rm{i}},\\ \Space{0ex}{1.54em}{0ex}{\xi }_{10} & = & {\xi }_{20}^{\ast }=1+i,{\xi }_{30}=0,{\unicode{x000A0}}{\rm{f}}{\rm{o}}{\rm{r}}{\unicode{x000A0}}{\rm{f}}{\rm{i}}{\rm{g}}{\rm{u}}{\rm{r}}{\rm{e}}{\unicode{x000A0}}6({\rm{a}})\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\Space{0ex}{1.54em}{0ex}{\alpha }_{1} & = & -{\alpha }_{2}=2,{\alpha }_{3}=1,{k}_{1}={k}_{3}^{* }=1+0.5{\rm{i}},\\ {\xi }_{10} & = & {\xi }_{30}^{* }=1+i,{\xi }_{20}=10.\ \ \ \mathrm{for}\ \mathrm{figure}\ 6({\rm{b}})\end{array}\end{eqnarray}$
Figure 6. The density plots of (a) the breather and soliton interaction solution and (b) the soliton-breather molecule solution for equation ( |
In fact, the profiles of the soliton-breather molecule are represented by two arbitrary complex parameters k1, ξ10 and four random constants α1, α2, α3, and ξ30. These wave parameters of different values control the velocities, amplitudes, transmission directions of solitons and breathers, and the relative distances between solitons and breathers.
3. Discussions
When we plan to apply the same procedure used above to the combined mKdV-type bilinear equation (1 ) with more higher-order bilinear terms to construct its multi-soliton molecule solutions, it is found that the soliton molecules comprising more than four solitons can not be realized. The reason for this is that when n ≥ 4, the single mKdV-type bilinear equation:6 ) and mKdV-7 equation (7 ) can be derived from the functional derivative of the higher-order conserved quantities of the mKdV equation with:5 )26 ) with n = 4) does not correspond to the higher-order mKdV equation that derived from the higher-order conserved quantities of the mKdV equation. According to Ref.[30], the ninth-order mKdV equation obtained from the fifth conserved quantity of the mKdV equation (mKdV-9' equation) has the following bilinear form:30 ) owns four-soliton solution, for example, a three-bright-one-dark-soliton solution of the mKdV-9' equation with parameters α = 1, k1 = −0.87, k2 = 0.3, k3 = −0.6, k4 = −0.5, ξ10 = −15, ξ20 = 15, ξ30 = −5 and ξ40 = 10 is shown in figure 7(a). Observing from figure 7(b), which displays the corresponding two-dimensional plots of four-soliton solutions at t = −200, −100, 200, respectively, it is not difficult to discover that the collisions between bright solitons are elastic, whereas the collisions between bright and dark solitons are inelastic, and every collision between two solitons is accompanied with a phase shift.
$\begin{eqnarray}({D}_{t}+{D}_{x}^{2n+1}){f}^{* }\cdot f=0,\end{eqnarray}$
$\begin{eqnarray}{D}_{x}^{2}{f}^{* }\cdot f=0\end{eqnarray}$
does not admit multi-soliton structures with more than four solitons. However, from [50], it is known that mKdV-5 equation ( $\begin{eqnarray}{v}_{t}=-\displaystyle \frac{\partial }{\partial x}\displaystyle \frac{\delta {I}_{j}}{\delta v},(j=3,4)\end{eqnarray}$
where Ij is the j-th conserved quantity of equation ( $\begin{eqnarray}{I}_{3}={\int }_{-\infty }^{\infty }\left({v}^{6}-5{v}^{2}{v}_{x}^{2}+\displaystyle \frac{1}{2}{v}_{2x}^{2}\right){\rm{d}}x,\end{eqnarray}$
$\begin{eqnarray}{I}_{4}={\int }_{-\infty }^{\infty }\left(\displaystyle \frac{5}{2}{v}^{8}-35{v}^{4}{v}_{x}^{2}+7{v}^{2}{v}_{2x}^{2}-\displaystyle \frac{7}{2}{v}_{x}^{4}-\displaystyle \frac{1}{2}{v}_{3x}^{2}\right){\rm{d}}x.\end{eqnarray}$
But the mKdV-9 equation (i.e., Equation ( $\begin{eqnarray}\begin{array}{l}\left[{D}_{t}+{D}_{x}^{9}+\alpha {D}_{x}^{6}{D}_{{t}_{1}}+3(\alpha +7){D}_{x}^{4}{D}_{{t}_{2}}\right.\\ \quad \left.-(4\alpha +21){D}_{x}^{2}{D}_{{t}_{3}}\right]{f}^{* }\cdot f=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}{D}_{x}^{2}{f}^{* }\cdot f=0,\end{eqnarray}$
$\begin{eqnarray}({D}_{{t}_{1}}+{D}_{x}^{3}){f}^{* }\cdot f=0,\end{eqnarray}$
$\begin{eqnarray}({D}_{{t}_{2}}+{D}_{x}^{5}){f}^{* }\cdot f=0,\end{eqnarray}$
$\begin{eqnarray}({D}_{{t}_{3}}+{D}_{x}^{7}){f}^{* }\cdot f=0\end{eqnarray}$
with α being an arbitrary constant and t1, t2, and t3 being three auxiliary variables. It can be verified that mKdV-9' bilinear equation (
Figure 7. (a) The profiles of the four-soliton solution and (b) the corresponding two-dimensional plot of four-soliton solutions at t = −200, −100, 200 for equation ( |
Furthermore, when we attempt to find the four-soliton molecule for the mKdV-3579' bilinear equation of the following form:17 ) and the mKdV-9' equation (30 ), it is proven that we can only get the two-soliton-one-breather molecule and the two-breather molecule, as plotted in figure 8, but can not receive four-soliton molecule. The wave parameters in figure 8(a) are
$\begin{eqnarray}\begin{array}{l}\left[{D}_{t}+{\alpha }_{1}{D}_{x}^{3}+{\alpha }_{2}{D}_{x}^{5}+{\alpha }_{3}{D}_{x}^{7}+{\alpha }_{4}({D}_{x}^{9}+\alpha {D}_{x}^{6}{D}_{{t}_{1}}\right.\\ \left.+3(\alpha +7){D}_{x}^{4}{D}_{{t}_{2}}-(4\alpha +21){D}_{x}^{2}{D}_{{t}_{3}})\right]{f}^{* }\cdot f=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}{D}_{x}^{2}{f}^{* }\cdot f=0,\end{eqnarray}$
$\begin{eqnarray}({D}_{{t}_{1}}+{D}_{x}^{3}){f}^{* }\cdot f=0,\end{eqnarray}$
$\begin{eqnarray}({D}_{{t}_{2}}+{D}_{x}^{5}){f}^{* }\cdot f=0,\end{eqnarray}$
$\begin{eqnarray}({D}_{{t}_{3}}+{D}_{x}^{7}){f}^{* }\cdot f=0,\end{eqnarray}$
which is the linear combination of the mKdV-357 equation ( $\begin{eqnarray}\alpha ={\alpha }_{1}=-{\alpha }_{2}={\alpha }_{3}=1,{k}_{3}={k}_{4}^{* }=0.6+0.2{\rm{i}},\end{eqnarray}$
$\begin{eqnarray}{\alpha }_{4}=-1.5,{\xi }_{10}=5,{\xi }_{20}=20,{\xi }_{30}={\xi }_{40}^{* }=1+i\end{eqnarray}$
with kj, ωj(j = 1, 2) satisfying the velocity resonance condition $\tfrac{{\omega }_{1}}{{k}_{1}}=\tfrac{{\omega }_{2}}{{k}_{2}}({k}_{1}\ne \pm {k}_{2})$ and the velocities of two solitons the same as that of the breather $\begin{eqnarray}\begin{array}{l}{v}_{s}=-{k}_{1}^{2}({\alpha }_{1}+{\alpha }_{2}{k}_{1}^{2}+{\alpha }_{3}{k}_{1}^{4}+{\alpha }_{4}{k}_{1}^{6})\\ \Space{0ex}{1.34em}{0ex}\quad =\,{v}_{b}={\alpha }_{1}(3{k}_{3i}^{2}-{k}_{3r}^{2})+{\alpha }_{2}(-5{k}_{3i}^{4}\\ \Space{0ex}{1.34em}{0ex}\quad +10{k}_{3i}^{2}{k}_{3r}^{2}-{k}_{3r}^{4})\\ \Space{0ex}{1.34em}{0ex}\quad +{\alpha }_{3}(7{k}_{3i}^{6}-35{k}_{3i}^{4}{k}_{3r}^{2}+21{k}_{3i}^{2}{k}_{3r}^{4}-{k}_{3r}^{6})\\ \Space{0ex}{1.34em}{0ex}\quad -{\alpha }_{4}(3{k}_{3i}^{2}-{k}_{3r}^{2})(3{k}_{3i}^{6}-27{k}_{3i}^{4}{k}_{3r}^{2}+33{k}_{3i}^{2}{k}_{3r}^{4}-{k}_{3r}^{6}),\\ \Space{0ex}{1.34em}{0ex}\quad \mathrm{where}\ {k}_{3}={k}_{3r}+{\rm{i}}{k}_{3i}\end{array}\end{eqnarray}$
and the wave constants in figure 8(b) are taken as $\begin{eqnarray}\alpha ={\alpha }_{1}={\alpha }_{2}={\alpha }_{3}=-{\alpha }_{4}=1,{k}_{3}={k}_{4}^{* }=0.4+0.5{\rm{i}},\end{eqnarray}$
$\begin{eqnarray}{k}_{1i}=0.6,{\xi }_{10}={\xi }_{20}^{* }=10+2i,{\xi }_{30}={\xi }_{40}^{* }=2+i\end{eqnarray}$
with two breathers holding the same velocity $\begin{eqnarray}\begin{array}{l}{v}_{{b}_{1}}={\alpha }_{1}(3{k}_{1i}^{2}-{k}_{1r}^{2})+{\alpha }_{2}(-5{k}_{1i}^{4}+10{k}_{1i}^{2}{k}_{1r}^{2}-{k}_{1r}^{4})\\ \quad +{\alpha }_{3}(7{k}_{1i}^{6}-35{k}_{1i}^{4}{k}_{1r}^{2}+21{k}_{1i}^{2}{k}_{1r}^{4}-{k}_{1r}^{6})\\ \quad -\ {\alpha }_{4}(3{k}_{1i}^{2}-{k}_{1r}^{2})(3{k}_{1i}^{6}-27{k}_{1i}^{4}{k}_{1r}^{2}\\ \quad +\ 33{k}_{1i}^{2}{k}_{1r}^{4}-{k}_{1r}^{6})\\ \quad =\ {v}_{{b}_{2}}={\alpha }_{1}(3{k}_{3i}^{2}-{k}_{3r}^{2})+{\alpha }_{2}(-5{k}_{3i}^{4}\\ \quad +\ 10{k}_{3i}^{2}{k}_{3r}^{2}-{k}_{3r}^{4})\\ \quad +\ {\alpha }_{3}(7{k}_{3i}^{6}-35{k}_{3i}^{4}{k}_{3r}^{2}+21{k}_{3i}^{2}{k}_{3r}^{4}-{k}_{3r}^{6})\\ \quad -\ {\alpha }_{4}(3{k}_{3i}^{2}-{k}_{3r}^{2})\\ \quad \times \ (3{k}_{3i}^{6}-27{k}_{3i}^{4}{k}_{3r}^{2}+33{k}_{3i}^{2}{k}_{3r}^{4}-{k}_{3r}^{6}).\\ \quad \mathrm{where}\ {k}_{j}={k}_{{jr}}+{\rm{i}}{k}_{{ji}}(j=1,3)\end{array}\end{eqnarray}$
The reason why we can not get the four-soliton molecule for the combined mKdV-3579' equation is that the real wave numbers that accord with the velocity resonance condition $\tfrac{{\omega }_{1}}{{k}_{1}}=\tfrac{{\omega }_{2}}{{k}_{2}}=\tfrac{{\omega }_{3}}{{k}_{3}}=\tfrac{{\omega }_{4}}{{k}_{4}}({k}_{m}\ne \pm {k}_{n},m,n=1,2,3,4)$ cannot be achieved.
Figure 8. The pictures of (a) the two-soliton-one-breather molecule and (b) the two-breather molecule for equation ( |
4. Conclusions
In this article, proceeding from the multi-soliton solutions obtained by the Hirota bilinear method, we constructed the soliton molecule solutions for the combined mKdV-type bilinear equation with higher order nonlinear and dispersion terms by velocity resonance mechanism. The results indicated that, because the wave numbers not only determine the propagation velocities of soliton molecules, but also affect the amplitudes of soliton molecules, besides common bright soliton molecules, the dark soliton molecules and mixed bright-dark-soliton molecules could also be obtained for combined mKdV-type bilinear equations by choosing different positive and negative wave numbers. To be specific, three types of two-soliton molecules including the bright-bright-soliton molecule, the bright-dark-soliton molecule, and the dark-dark-soliton molecule for the mKdV-35 equation, and four types of three-soliton molecules such as the three-bright-soliton molecule, the one-bright-two-dark-soliton molecule, the two-bright-one-dark-soliton molecule and the three-dark-soliton molecule for the mKdV-357 equation have been acquired. In addition, by taking the wave number of one of the solitons as the complex conjugate of the other, the breather solution, the soliton-breather molecule and the two-breather molecule solutions have also been designed for the mKdV-35 equation and mKdV-357 equation, respectively. To illustrate the soliton molecules in more detail, we gave some evolution plots of these molecules. And the characteristics and propagation states of these soliton molecules were well analyzed at the same time. Especially, by calculating the extreme values of the amplitudes of the three-soliton molecules of the mKdV-357 equation, it was discovered that a set of wave numbers {∣k1∣, ∣k2∣, ∣k3∣} satisfying the velocity resonance condition not only did not change the velocity of the soliton molecule, but also the total energy. In addition, the existence of multi-soliton molecules for combined mKdV-type bilinear equation (1 ) with more higher order bilinear terms have also been discussed. However, due to the fact that the single mKdV-type bilinear equation (26 ) with n ≥ 4 does not possess multi-soliton solutions comprising more than four solitons, the combined mKdV equation (4 ) no longer holds soliton molecules constituted by more than four solitons. Considering that the ninth-order mKdV equation derived from the fifth conserved quantity of the mKdV equation (mKdV-9' equation) owns a four-soliton solution, we further discussed the existence of the four-soliton molecule for mKdV-3579' equation. However, due to the inexistence of the real wave numbers that satisfy the velocity resonance restriction, it was verified that the mKdV-3579' equation did not possess the four-soliton molecule, except for the two-soliton-breather molecule and the two-breather molecule.
In fact, the soliton molecule solutions for more integrable or even nonintegrable nonlinear partial differential equations can also be established by the Darboux transformation, the variable separation method, etc. But most importantly, we hope the results obtained here may raise the possibility of relative experiments and potential applications.