Novel travelling wave structures: few-cycle-pulse solitons and soliton molecules
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Zitong Chen, Man Jia,School of Physical Science and Technology, Ningbo University, Ningbo 315211, China
First author contact:*Author to whom any correspondence should be addressed. Received:2020-08-1Revised:2020-11-13Accepted:2020-11-18Online:2021-01-25
Abstract We discuss a fifth order KdV (FOKdV) equation via a novel travelling wave method by introducing a background term. Results show that the background term plays an essential role in finding new abundant travelling wave structures, such as the soliton induced by negative background, the periodic travelling wave excited by the positive background, the few-cycle-pulse (FCP) solitons with and without background, the soliton molecules excited by the background. The FCP solitons are first obtained for the FOKdV equation. Keywords：New travelling method;a fifth order KdV equation;solitons induced by background;few-cycle-pulse solitons;soliton molecules
PDF (3864KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Zitong Chen, Man Jia. Novel travelling wave structures: few-cycle-pulse solitons and soliton molecules. Communications in Theoretical Physics, 2021, 73(2): 025003- doi:10.1088/1572-9494/abcfb4
1. Introduction
It has been realized that there are many different types of waves in numerous media based on large universal nonlinear equations, such as the KdV equation [1], the KP equation in water surface gravity waves and plasma [2], the mKdV equation in plasma and optics [3, 4], the NLS equation in nonlinear optics [5], the sine-Gordon equation in field theory [6] and biaxial ferromagnets [7], etc. Many interesting and effective methods and techniques, including the Lie point symmetry and group theory [8], the bilinear method [9], Darboux transformations and Bäcklund transformations, the Lax pair and the inverse scattering method [10], CRE [11] and CTE expand [12], etc, have been used to search for the exact solutions of the universal nonlinear equations. Among all the methods, travelling wave analysis is one of the basic and effective ways to construct the wave solutions of nonlinear systems since the first derivation of KdV equation in fluid mechanics.
Due to the fundamentality of the travelling wave method, it is necessary to develop the method to find new progresses. Recently, some new developments about the travelling wave analysis have been made. Some novel few-cycle-pulses (FCPs) solitons and soliton molecules solutions have been found for the famous Sawada-Kotera equation by using the travelling wave method [13].
FCPs are usually referred to the pulses containing only several oscillations of the electromagnetic field [14-16]. The spectrum of the FCPs is wide so that it is almost impossible to allocate the carrier frequency. Therefore, the FCPs are called the broadband pulses sometimes.
Theoretical studies show that FCPs solitons play an important role in extreme nonlinear optics and interaction of matter with strong optical fields, it is urgent and essential to construct models from the governing Maxwell-Bloch equation and the Schrödinger-vonNeumann equation and search for this kind of solution [16-18]. However, most of the results are related to breathers and solitons by numerical computations.
Experimental research with many phenomena of bounded solitons have been observed [19-21] in many optical systems and predicted in Bose-Einstein condensates [22]. The bounded states of solitons, or coherent structures of solitons, are also called soliton molecules have been investigated widely in many nonlinear systems. Even in fermions and magnetic flux [23], there also exists the kink bounded states. It has been pointed out a new mechanism to form soliton molecules has been found by introducing velocity resonance [24].
Because the new travelling method is effective to search for the FCPs solitons and soliton molecules which are significant to explain many physical phenomena, we will apply the new travelling wave method to a fifth order KdV equation (FOKdV) [25]$\begin{eqnarray}{u}_{t}+30{u}^{2}{u}_{x}+20{u}_{x}{u}_{{xx}}+10{{uu}}_{{xxx}}+{u}_{{xxxxx}}=0,\end{eqnarray}$which is first derived in fluid models and later has been applied in many nonlinear fields, such as plasma waves [26], ocean gravity waves, surface and internal waves [27], and electromagnetic waves [28], etc. The FOKdV equation equation (1) can be considered as a generalization of the well known fifth-order nonlinear system for some particular parameters selections [11, 29].
The FOKdV equation has been proved to possess many integrable properties, such as the bilinear form by introducing an independent auxiliary variable [30], Lax pair, etc. Studies show it also has periodic and solitary waves solutions. But by now, the FCP soliton solutions have not been found.
In our manuscript, we investigate the FOKdV equation via a new travelling wave method to search for its exact solutions, especially for the FCP solitons and soliton molecules solutions. The paper is arranged as follows. In section 2, the exact solutions of the FOKdV equation via the new travelling wave method are explored and four different kinds of travelling wave solutions are constructed. Section 3 is a short summary and discussion.
2. Novel travelling wave solutions to the FOKdV equation
In order to find novel possible travelling waves solutions, we start from a transformation related to the bilinear transformation of the FOKdV equation equation (1)$\begin{eqnarray}\begin{array}{rcl}u & = & {u}_{0}+2{\left(\mathrm{ln}F\right)}_{{xx}},\quad F=F({kx}+\omega t,\tau )\equiv F(z,\tau ),\\ z & = & {kx}+\omega t,\end{array}\end{eqnarray}$where τ is an independent auxiliary variable and can be treated as arbitrary phase shift finally. Substitution for u into equation (1) yields, after some manipulation and one integration in x, the bilinear equation for F is$\begin{eqnarray}\begin{array}{l}\left[\displaystyle \frac{{k}^{5}}{6}{D}_{z}^{6}+5{k}^{3}{u}_{0}{D}_{z}^{4}+\left(30{{ku}}_{0}^{2}+\omega \right){D}_{z}^{2}-\displaystyle \frac{5{k}^{2}}{6}{D}_{\tau }{D}_{z}^{3}\right]\\ \times \,F\cdot F=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\left({k}^{3}{D}_{z}^{4}+6{{ku}}_{0}{D}_{z}^{2}+{D}_{\tau }{D}_{z}\right)F\cdot F=0,\end{eqnarray}$where Dz, Dt and Dτ are the bilinear operators first defined by Hirota [31],$\begin{eqnarray*}\begin{array}{l}{D}_{x}^{m}{D}_{t}^{n}\left(a\cdot b\right)\\ =\,{\left.{\left(\displaystyle \frac{\partial }{\partial x}-\displaystyle \frac{\partial }{\partial x^{\prime} }\right)}^{m}{\left(\displaystyle \frac{\partial }{\partial t}-\displaystyle \frac{\partial }{\partial t^{\prime} }\right)}^{n}a\left(x,t\right)b\left(x^{\prime} ,t^{\prime} \right)\right|}_{x^{\prime} =x,t^{\prime} =t}.\end{array}\end{eqnarray*}$Equation (4) can be considered as the subsidiary condition that the independent auxiliary variable τ satisfies.
It is clear the bilinear forms equations (3)-(4) are different from the known results for the FOKdV equation because we have introduced a background u0. We will show the exitance of u0 make it possible for us to construct more interesting travelling wave structures.
Equations (3)-(4) have introduced the general travelling wave forms, so that one can easily find that the related travelling wave structures of the FOKdV equation. Once F is known by solving equations (3)-(4), the possible new solutions of the FOKdV equation can be found by the transformation equation (2).
Because the terms Dz6, Dz4 and Dz2 are not coupled with each other, we suppose F satisfy a separate form given by$\begin{eqnarray}F(z,\tau )=G(z,\tau )+H(z,\tau ),\end{eqnarray}$where G(z,τ)≡G and H(z,τ)≡H are restricted with$\begin{eqnarray}{G}_{z}^{2}={a}_{0}+{a}_{1}G+{a}_{2}{G}^{2},\end{eqnarray}$$\begin{eqnarray}{H}_{z}^{2}={b}_{0}+{b}_{1}H+{b}_{2}{H}^{2},\end{eqnarray}$with ai and bi, i=0,1,2 being arbitrary constants.
Substituting equations (6)-(7) into (3)-(4), vanishing all the coefficients of the differential power ploynomials G and H, we immediately obtain nine constraint equations reading as$\begin{eqnarray}\begin{array}{l}\omega +k(3{a}_{2}^{2}{k}^{4}+10{a}_{2}{b}_{2}{k}^{4}+3{b}_{2}^{2}{k}^{4}\\ \quad +\,20{a}_{2}{k}^{2}{u}_{0}+20{b}_{2}{k}^{2}{u}_{0}+30{u}_{0}^{2})=0,\end{array}\end{eqnarray}$$\begin{eqnarray}{a}_{1}{b}_{2}+{a}_{2}{b}_{1}=0,\end{eqnarray}$$\begin{eqnarray}(5{a}_{2}{k}^{2}+5{b}_{2}{k}^{2}+12{u}_{0})({a}_{1}{b}_{2}+{a}_{2}{b}_{1})=0,\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}4{a}_{0}{a}_{1}{a}_{2}+8{a}_{0}{a}_{2}{b}_{1}-4{a}_{0}{b}_{1}{b}_{2}-{a}_{1}^{3}\\ \quad -\,3{a}_{1}^{2}{b}_{1}-4{a}_{1}{a}_{2}{b}_{0}+8{a}_{1}{b}_{0}{b}_{2}-3{a}_{1}b{1}^{2}\\ \quad +\,4{b}_{0}{b}_{1}{b}_{2}-{b}_{1}^{3}=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}\omega ({a}_{1}-{b}_{1})+k({a}_{1}{a}_{2}^{2}{k}^{4}+15{a}_{1}{a}_{2}{b}_{2}{k}^{4}\\ \quad +\,15{a}_{1}{b}_{2}^{2}{k}^{4}-{b}_{1}{b}_{2}^{2}{k}^{4}+10{a}_{1}{a}_{2}{k}^{2}{u}_{0}\\ \quad +\,60{a}_{1}{b}_{2}{k}^{2}{u}_{0}-10{b}_{1}{b}_{2}{k}^{2}{u}_{0}+30{a}_{1}{u}_{0}^{2}-30{b}_{1}{u}_{0}^{2})=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}\omega ({a}_{2}+{b}_{2})+k({a}_{2}^{3}{k}^{4}+15{a}_{2}^{2}{b}_{2}{k}^{4}\\ \quad +\,15{a}_{2}{b}_{2}^{2}{k}^{4}+{b}_{2}^{3}{k}^{4}+10{a}_{2}^{2}{k}^{2}{u}_{0}\\ \quad +\,60{a}_{2}{b}_{2}{k}^{2}{u}_{0}+10{b}_{2}^{2}{k}^{2}{u}_{0}+30{a}_{2}{u}_{0}^{2}+30{b}_{2}{u}_{0}^{2})=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}\omega ({a}_{1}-{b}_{1})+k({a}_{1}{a}_{2}^{2}{k}^{4}+15{a}_{1}{a}_{2}{b}_{2}{k}^{4}\\ \quad +\,15{a}_{1}{b}_{2}^{2}{k}^{4}-{b}_{1}{b}_{2}^{2}{k}^{4}+10{a}_{1}{a}_{2}{k}^{2}{u}_{0}\\ \quad +\,60{a}_{1}{b}_{2}{k}^{2}{u}_{0}-10{b}_{1}{b}_{2}{k}^{2}{u}_{0}+30{a}_{1}{u}_{0}^{2}-30{b}_{1}{u}_{0}^{2})=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}4\omega ({a}_{0}+{b}_{0})+k(64{a}_{0}{a}_{2}^{2}{k}^{4}-15{a}_{1}^{2}{a}_{2}{k}^{4}\\ \quad -\,25{a}_{1}{a}_{2}{b}_{1}{k}^{4}-25{a}_{1}{b}_{1}{b}_{2}{k}^{4}\\ \quad -\,40{a}_{2}^{2}{b}_{0}{k}^{4}+80{a}_{2}{b}_{0}{b}_{2}{k}^{4}-30{a}_{2}{b}_{1}^{2}{k}^{4}\\ \quad +\,24{b}_{0}{b}_{2}^{2}{k}^{4}-5{b}_{1}^{2}{b}_{2}{k}^{4}\\ \quad +\,160{a}_{0}{a}_{2}{k}^{2}{u}_{0}-30{a}_{1}^{2}{k}^{2}{u}_{0}-60{a}_{1}{b}_{1}{k}^{2}{u}_{0}\\ \quad +\,160{b}_{0}{b}_{2}{k}^{2}{u}_{0}-30{b}_{1}^{2}{k}^{2}{u}_{0}\\ \quad +\,120{a}_{0}{u}_{0}^{2}+120{b}_{0}{u}_{0}^{2})=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}4\omega ({a}_{0}+{b}_{0})-k(24{a}_{0}{a}_{2}^{2}{k}^{4}+80{a}_{0}{a}_{2}{b}_{2}{k}^{4}\\ -\,40{a}_{0}{b}_{2}^{2}{k}^{4}-5{a}_{1}^{2}{a}_{2}{k}^{4}-30{a}_{1}^{2}{b}_{2}{k}^{4}\\ \quad -\,25{a}_{1}{a}_{2}{b}_{1}{k}^{4}-25{a}_{1}{b}_{1}{b}_{2}{k}^{4}+64{b}_{0}{b}_{2}^{2}{k}^{4}\\ \quad -\,15{b}_{1}^{2}{b}_{2}{k}^{4}+160{a}_{0}{a}_{2}{k}^{2}{u}_{0}\\ \quad -\,30{a}_{1}^{2}{k}^{2}{u}_{0}-60{a}_{1}{b}_{1}{k}^{2}{u}_{0}+160{b}_{0}{b}_{2}{k}^{2}{u}_{0}-30{b}_{1}^{2}{k}^{2}{u}_{0}\\ \quad +\,120{a}_{0}u{0}^{2}+120{b}_{0}{u}_{0}^{2})=0,\end{array}\end{eqnarray}$with nine parameters ω, k, u0, ai and bi, i=0,1,2 being determined.
Solving equations (8)-(16), some significant solutions to the fifth KdV equation are found.
Case 1. Solitons induced by the negative background.
In this case, the solutions to equations (8)-(16) are$\begin{eqnarray}\begin{array}{rcl}{a}_{1} & = & -{b}_{1},\qquad {a}_{2}={b}_{2}=-\displaystyle \frac{6{u}_{0}}{5{k}^{2}}=6a\gt 0,\\ \omega & = & -126{a}^{2}{k}^{5},\end{array}\end{eqnarray}$with the corresponding solution of G and H being$\begin{eqnarray}\begin{array}{rcl}H(z,y) & = & -\displaystyle \frac{{b}_{1}}{12a}\pm \displaystyle \frac{\sqrt{25({b}_{1}^{2}-24{{ab}}_{0})}}{60a}\cosh \left(\sqrt{6a}(z-{z}_{1})\right),\\ G(z,y) & = & \displaystyle \frac{{b}_{1}}{12a}\pm \displaystyle \frac{\sqrt{25({b}_{1}^{2}-24{{aa}}_{0})}}{60a}\cosh \left(\sqrt{6a}(z-{z}_{2})\right).\end{array}\end{eqnarray}$
After some redefinitions of the constants, the solution to the FOKdV equation now is$\begin{eqnarray}u=-5{{ak}}^{2}+12{{ak}}^{2}{{\rm{{\rm{sech}} }}}^{2}(\sqrt{6a}k(x-126{a}^{2}{k}^{4}t+{x}_{0})),\end{eqnarray}$where a>0, c, k and x0 are arbitrary constants. It is obvious the solution equation (19) is related to the background term −5ak2. If the background disappears (a = 0), the solution becomes a trivial vacuum solution u=0. This type of soliton is considered as the negative background induced solitons. It is interesting that the background induced soliton can only travel to the right because ω/k=−126a2k4<0.
Case 2. Periodic waves induced by positive background.$\begin{eqnarray}\begin{array}{rcl}{a}_{1} & = & -{b}_{1},\qquad {a}_{2}={b}_{2}=-\displaystyle \frac{6{u}_{0}}{5{k}^{2}}=-6a\lt 0,\\ \omega & = & -126{a}^{2}{k}^{5}.\end{array}\end{eqnarray}$The corresponding solutions of G and H now become$\begin{eqnarray}\begin{array}{rcl}H(z,y) & = & \displaystyle \frac{{b}_{1}}{12a}\pm \displaystyle \frac{\sqrt{25({b}_{1}^{2}+24{{ab}}_{0})}}{60a}\cos \left(\sqrt{6a}(z-{z}_{1})\right),\\ G(z,y) & = & -\displaystyle \frac{{b}_{1}}{12a}\pm \displaystyle \frac{\sqrt{25({b}_{1}^{2}+24{{aa}}_{0})}}{60a}\cos \left(\sqrt{6a}(z-{z}_{2})\right).\end{array}\end{eqnarray}$Thus the solution u to the FOKdV equation is expressed by$\begin{eqnarray}\begin{array}{rcl}u & = & 5{{ak}}^{2}+2\left[\mathrm{ln}\left(\sqrt{{k}^{2}({b}_{1}^{2}+24{{ab}}_{0})}\cos (\sqrt{6a}k\xi )\right.\right.\\ & & \pm \sqrt{{k}^{2}({b}_{1}^{2}+24{{aa}}_{0})}\times {\left.\left.\cos (\sqrt{6a}k\eta \right)\right]}_{{xx}},\\ \xi & = & x-126{a}^{2}{k}^{4}t-{\xi }_{0},\qquad \eta =x-126{a}^{2}{k}^{4}t-{\eta }_{0}.\end{array}\end{eqnarray}$After redefinitions of some arbitrary constants. From the expression equation (22), we immediately find the periodic wave solution equation (22) is also induced by the positive background.
The character of this solution is based on the analytical condition for$\begin{eqnarray}\begin{array}{l}\sqrt{{k}^{2}({b}_{1}^{2}+24{{ab}}_{0})}\cos (\sqrt{6a}k\xi )\\ \quad \pm \,\sqrt{{k}^{2}({b}_{1}^{2}+24{{aa}}_{0})}\cos (\sqrt{6a}k\eta )\gt 0,\end{array}\end{eqnarray}$for real a0, b0 and b1. Because the analytical condition is complicated, here we just show some significant results for some special conditions. For instance, if the parameters are selected to satisfy ${k}^{2}({b}_{1}^{2}+24{{ab}}_{0})\lt 0$, ${k}^{2}({b}_{1}^{2}\,+24{{aa}}_{0})\gt 0$ and a0> b0 with ‘+’ in equation (22), the solution will show different types of periodic waves. Figures 1-3 exhibit three types of periodic waves structures of the solution for the parameters selected as$\begin{eqnarray}a=\displaystyle \frac{1}{2},\qquad k=\displaystyle \frac{1}{2},\qquad {a}_{0}=-{b}_{0}=1,\qquad {b}_{1}=2,\end{eqnarray}$with different ξ0 and η0.
Figure 1.
New window|Download| PPT slide Figure 1.The periodic wave for the solution equation (22) where the parameters are selected as equation (24) with ξ0=−η0=2 for (a) the structure and (b) the shape for t=0.
Figure 2.
New window|Download| PPT slide Figure 2.The periodic wave described by equations (22) and (24) with ξ0=2, η0=−9/10 for (a) the structure and (b) the shape for t=0.
Figure 3.
New window|Download| PPT slide Figure 3.The periodic wave structure (a) and the shape (b) for t=0 described by equations (22) and (24) with ξ0=2, η0=−1.
For this type of periodic waves, the results suggest the parameters ξ0 and η0 denoting the initial positions of the two waves are important to form different periodic wave structures.
Case 3. Solitons with and without background.$\begin{eqnarray}\begin{array}{rcl} & & {a}_{1}=-{b}_{1},\qquad {a}_{2}={b}_{2},\\ & & \omega =-16{a}_{2}{k}^{5}-40{a}_{2}{k}^{3}{u}_{0}-30{{ku}}_{0}^{2},\\ & & H(z,y)=\displaystyle \frac{{a}_{1}}{2{a}_{2}}\pm \displaystyle \frac{\sqrt{{a}_{1}^{2}-4{a}_{2}{b}_{0}}}{2{a}_{2}}\cosh \left(\sqrt{{a}_{2}}(z-{z}_{1})\right),\\ & & G(z,y)=-\displaystyle \frac{{a}_{1}}{2{a}_{2}}\pm \displaystyle \frac{\sqrt{{a}_{1}^{2}-4{a}_{2}{a}_{0}}}{2{a}_{2}}\cosh \left(\sqrt{{a}_{2}}(z-{z}_{2})\right).\end{array}\end{eqnarray}$
After redefinitions of the arbitrary constants, the solution for u to the FOKdV equation is given by$\begin{eqnarray}u={u}_{0}+2{k}^{2}{\rm{{\rm{sech}} }}[{kx}-(16{k}^{5}+40{k}^{3}{u}_{0}+30{{ku}}_{0}^{2})t+{x}_{0}].\end{eqnarray}$This solution is just the usual known soliton solution with and without background.
Thus the solution u for the FOKdV equation is$\begin{eqnarray}\begin{array}{rcl}u & = & 2\left[\mathrm{ln}\left({b}_{1}\sqrt{{a}_{1}^{2}-4{a}_{0}{a}_{2}}\cosh (\sqrt{{a}_{2}}\xi )\right.\right.\\ & & \pm {a}_{1}\sqrt{\displaystyle \frac{{b}_{1}}{{a}_{1}}(4{a}_{0}{a}_{2}-{a}_{1}^{2})}\\ & & \qquad {\left.\left.\times \cos \left(\sqrt{\displaystyle \frac{{a}_{2}}{{a}_{1}}{b}_{1}}\eta \right)\right)\right]}_{{xx}}-\displaystyle \frac{{a}_{2}{k}^{2}({a}_{1}-{b}_{1})}{5{a}_{1}},\\ & & \xi ={kx}+\omega t-{\xi }_{0},\qquad \eta ={kx}+\omega t-{\eta }_{0},\\ & & \omega =-\displaystyle \frac{{k}^{2}{a}_{2}^{2}({a}_{1}^{2}-22{a}_{1}{b}_{1}+{b}_{1}^{2})}{5{a}_{1}^{2}}.\end{array}\end{eqnarray}$It is clear the analytical conditions for the solution equation (28) for real parameters a0, a1, a2 and b1 are$\begin{eqnarray}\left|{b}_{1}\sqrt{{a}_{1}^{2}-4{a}_{0}{a}_{2}}\right|\gt \left|{a}_{1}\sqrt{\displaystyle \frac{{b}_{1}}{{a}_{1}}(4{a}_{0}{a}_{2}-{a}_{1}^{2})}\right|,\qquad {a}_{1}\ne {b}_{1},\end{eqnarray}$$\begin{eqnarray}{\xi }_{0}-{\eta }_{0}\ne (2n+1)\pi ,n=0,\pm 1,\pm 2,\ldots ,{a}_{1}={b}_{1}.\end{eqnarray}$With the nonsingular conditions equation (29) or (30), the solution equation (28) represents the FCP soliton structures. Generally, with the condition equation (29), the solution gives a FCP soliton structure with nonzero background while the solution with condition equation (30) shows the FCP soliton without background.
Here we write down a special solution for a1=b1,$\begin{eqnarray}\begin{array}{rcl}u & = & 2\left[\mathrm{ln}\left(\sqrt{{b}_{1}^{2}-4{a}_{0}{a}_{2}}\cosh (\sqrt{{a}_{2}}\xi )\right.\right.\\ & & {\left.\left.+\sqrt{4{a}_{0}{a}_{2}-{b}_{1}^{2}}\cos (\sqrt{{a}_{2}}\eta \right)\right]}_{{xx}},\\ \xi & = & {kx}+4{a}_{2}^{2}{k}^{5}t-{\xi }_{0},\qquad \eta ={kx}+4{a}_{2}^{2}{k}^{5}t-{\eta }_{0}.\end{array}\end{eqnarray}$Some special FCP structures are exhibited in figures 4-7 with different parameters selections with (a) the structure and (b) the wave shape at t=0.
Figure 4.
New window|Download| PPT slide Figure 4.A kind of FCP soliton solution described by equation (31) with a2=1, $k=\tfrac{1}{2}$, ${\xi }_{0}=\tfrac{3}{4}$, η0=0, (a) the structure and (b) the plot for t=0.
Figure 5.
New window|Download| PPT slide Figure 5.Another kind of FCP soliton solution given by equation (31) with the parameters selected as a2=1, $k=\tfrac{1}{2}$, ξ0=η0=0, (a) the structure and (b) the wave shape at t=0.
Figure 6.
New window|Download| PPT slide Figure 6.The FCP soliton solution equation (31) with the choice of the parameters being a2=1, $k=\tfrac{1}{2}$, ξ0=0, ${\eta }_{0}=\tfrac{3}{4}$, (a) the structure and (b) the wave shape at t=0.
Figure 7.
New window|Download| PPT slide Figure 7.The profile of a FCP soliton solution structure given by equation (31) with a2=1, $k=\tfrac{1}{2}$, ξ0=0, ${\eta }_{0}=\tfrac{\pi }{2}$, (a) the structure and (b) the wave at t=0.
Another interesting solution occurs when b1<0. If b1<0, by introducing b1=−ca1, (c>0 and a1>0), the solution to the FOKdV equation now becomes$\begin{eqnarray}\begin{array}{rcl}u & = & -\displaystyle \frac{{a}_{2}{k}^{2}}{5}(c+1)+2[\mathrm{ln}({a}_{1}\sqrt{c}\cosh (\sqrt{{{ca}}_{2}}\eta )\\ & & {\left.\pm {{ca}}_{1}\cosh (\sqrt{{a}_{2}}\xi ))\right]}_{{xx}},\\ \xi & = & {kx}+\omega t-{\xi }_{0},\qquad \eta ={kx}+\omega t-{\eta }_{0},\\ \omega & = & -\displaystyle \frac{{a}_{2}^{2}{k}^{5}}{5}({c}^{2}+22c+1).\end{array}\end{eqnarray}$The result suggests this kind of solution is also induced by the nonzero background. If c=−1, the solution will reduced to the FCP solitons equation (31).
Furthermore, we can conclude from equation (32), the two-soliton solution admits the right travelling direction because the propagation speed of the two solitons is$\begin{eqnarray}\displaystyle \frac{\omega }{k}=-\displaystyle \frac{{a}_{2}^{2}{k}^{4}}{5}({c}^{2}+22c+1)\lt 0,\end{eqnarray}$for c>0. Equation (32) is a special two-soliton solution because the two solitons share the same velocity, which means the two-soliton forms a coherent structure or a bounded state named as soliton molecule. Soliton molecules are popular phenomenons in nonlinear systems [24, 32, 33], even in non-interagble system [34]. Figure 8 shows some particular bounded states of soliton molecules with the choices of the parameters: (a) {a1=1, a2=2, c=1/3, k=1/2,ξ=0, η=−5}, (b) {a1=1, a2=2, c=1/5, k=1/2, ξ=0, η=−5}, (c) {a1=1, a2=2, c=1/100, k=1/2,ξ= 0, η=−5} and (d) {a1=1, a2=2, c=1/500, k=1/2, ξ=0, η=−5}, respectively.
Figure 8.
New window|Download| PPT slide Figure 8.The profile of some different soliton molecules structures given by equation (32) with different parameters selections.
The profile illustrates that the parameter c is related to the shape of the soliton molecules. The wave heights of the two solitons become nearly the same when c is small enough.
3. Summary
In summary, we apply a new travelling wave method to an integrable FOKdV equation. The new travelling wave assumption is related to the famous bilinear form, but it is different from the bilinear form because we have introduced a background term. Our results show some special novel abundant travelling structures, such as the solitons induced by the negative background, the new periodic travelling wave excited by the positive background, etc, are constructed due to the background term.
Furthermore, the FCP solitons have also been found that have never been constructed except for the numerical method. Because the widely applications of the FOKdV equation, it may help us to understand more about the nonlinear phenomena in related fields and areas.
Another interesting founding is the soliton molecule solutions. The soliton molecules are obtained without introducing the generation mechanics for the system.
Though most physical systems are non-integrable, studies show that the non-integrable system may also possess the soliton molecules solutions. Because the FOKdV equation possesses both the soliton molecules solutions and FCP soliton solutions, it is interesting to search for the FCP soliton solutions for some non-integrable systems by applying the new travelling wave analysis.
The special soliton solutions with and without background are different to the normal solitons. It is important and necessary to study the stability of this kind of solitons so that we can find its applications in real physical systems.
Since the Lax hierarchy includes Kaup-Kupershmidt equation, seventh order SK equation, seventh order KdV equation, etc, we believe the equations mentioned above may also have the similar results by applying the travelling wave method in our manuscript.
Acknowledgments
The authors are grateful to Professor S Y Lou for his helpful discussions. The authors also acknowledge the support of NNSFC (Grant No. 11 675 084) and K C Wong Magna Fund in Ningbo University.
,School of Physical Science and Technology, 
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