Compound WKI-SP hierarchy and multiloop soliton solutions

Xiaorui Hu Tianle Xu Junyang Zhang Shoufeng Shen Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, Zhejiang, China

Abstract

The generalized hierarchies of compound WKI-SP (Wadati-Konno-Ichikawa and short pulse) equations are presented. The proposed integrable nonlinear equations include the WKI-type equations, the SP-type equations and the compound generalized WKI-SP equations. A chain of hodograph transformations are established to relate the compound WKI-SP equations with the MKdV-SG (modified Korteweg-de Vries and sine-Gordon) equations. As applications, the multiloop soliton solutions of one compound WKI-SP (named WKI-SP^(1,1)) equation are obtained. We emphasize on showing abundant solitonic behaviors of two loop solitons. The role of each parameter plays in the movement of two-loop solion are shown detailedly in a table.

keywords:

WKI-SP equation, MKdV-SG equation, hodograph transformation, loop soliton.

^†^†journal: XXX

1 Introduction

A remarkable development in our understanding of a certain class of nonlinear integrable equations known as the consistency condition for a system of linear differential equations has taken place in the past decade. This zero curvature equations on real Lie algebras such as sl(2, $\mathbb{R}$ )and so(3, $\mathbb{R}$ ) lay the foundation for constructing soliton hierarchies. The most famous one is the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, including the fundamental Korteweg-de Vries (KdV) equation, the the modified KdV (MKdV) equation, the sine-Gordon (SG) equation, the nonlinear Schrödinger (NLS) equation and so on.

In 1979, Wadati, Konno and Ichikawa (WKI) [1, 2] constructed a new series of integrable nonlinear evolution equations, such as

\displaystyle u_{t}+\bigg{[}\frac{u_{x}}{(1+u^{2})^{3/2}}\bigg{]}_{xx}=0.

(1)

The nonlinear terms of Eq.(1) have saturation effects. Eq.(1) describes nonlinear transverse oscillations of elastic beams under tension [3]. In the many past decades, lots of research have been done for this WKI equation (1). For example, an infinite number of conservation laws and the Hamiltonian form was found by Wadati et al [1]. The one-soliton and two-soliton solutions were obtained by using inverse scattering transform method in Ref.[4]. Qu and Zhang [5] derived the WKI model from motion of curves in E³ and gave the corresponding group-invariant solutions. The Darboux transformation is proposed by Zhang et. al. [6] and the direct scattering problem with box-like initial value was considered by Tu and Xu [7].

Recently, the systems of short pulse type have attracted considerable attention due to their important applications in physics. The very first such equation may be the short pulse (SP) equation

\displaystyle u_{xt}=u+\frac{1}{6}(u^{3})_{xx},

(2)

which possesses a Lax pair of the WKI type. The SP equation (2) is proposed by Schäfer and Wayne [8, 9] as an alternative (to the NLS equation) model for approximating the evolution of ultrashort intense infrared pulses in silica optics. It is shown in [9] by numerical simulations that the SP equation can be successfully used for describing pulses with broad spectrum. It turns out that the SP equation made its first appearance in Rabelo’s paper [10] in his study pseudospherical surfaces. The SP and two-component SP (real or complex) equations are proposed as special integrable cases in the negative WKI hierarchy for the first time in Refs.[11, 12]. Multi-soliton solutions and the Cauchy problem for a two-component SP system are given in Ref.[13]. The SP equation is proved integrable [15, 16] since it admits Lax pair, recursion operator, and bi-Hamiltonian structure and various exact solutions. Different approaches have been employed to construct solutions for the SP equation. In particular, Anton Sakovich and Sergei Sakovich [17] found an exact nonsingular solitary wave solution from the breather solution of the SG equation by means of a transformation between these two integrable equations. Kuetche et al [18] calculated the two-loop soliton solution by use of Hirota bilinear method and Hodnett-Moloney approach. Matsuno [19, 20] obtained the multiloop soliton, multibreather and the periodic solutions. Liu et al [21] constructed the $N$ -fold Darboux transformation given by determinants. The Riemann-Hilbert approach [22, 23] was applied to the SP equation and the long-time behavior of the solution was studied. Liu and Mao [24] established the $N$ -Bäcklund transformation and worked out various solutions including loop solitons, breather solutions, and their interaction solutions. Feng et al [25, 26] studied its integrable discretizations and both semidiscrete and full-discrete systems were obtained.

In this paper, we focus on the construction of the generalized hierarchies of compound WKI-SP equations from the consistency condition of two linear differential system. It is known that both the WKI equation and the SP equation admit loop soliton solutions [4, 27]. It is noted that they can be transformed to certain soliton equations which admit smooth soliton solutions through hodograph transformations[28]. For example, the WKI elastic beam equation is transformed to the potential mKdV equation. The SP equation can be transformed into the integrable SG equation. Hence we want to seek for a connection between the compound MKdV-SG equations (given by Gu in Ref.[29, 30]) and the WKI-SP equations. As applications, we will calculate multiloop soliton solutions for one WKI-SP equation.

2 Compound equations of the WKI-SP type

Many of the hardest problems and most interesting phenomena being studied by mathematicians, engineers and physicists are nonlinear in nature. Often, these phenomena can be modeled (and there is good reason to believe that these models are accurate) by nonlinear partial differential equations and to be sure, it will be many years to come before we have the mathematical sophistication to handle these equations completely. In the last decades, the inverse scattering transform (IST) has been employed to solve many physically significant equations. Due to the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems. Along with the IST, a systematic method has developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering. These evolution equations are expressed as the consistency condition

\displaystyle U_{t}-V_{x}+[U,V]=0,\quad[U,V]=UV-VU

(3)

for a system of linear differential equations

\displaystyle\Phi_{x}=U(x,t;\lambda)\Phi,\quad\quad\Phi_{t}=V(x,t;\lambda)\Phi,

(4)

where $U$ , $V$ and $\Phi$ are complex $N\times N$ matrixes. The matrixes $U$ and $V$ are usually rational functions of the parameter $\lambda$ .

2.1 AKNS-type and WKI-type Lax pair

To increase the readability of the article, we firstly recall the similar results for the known AKNS hierarchy and the WKI hierarchy.

In 1973, a wide class of nonlinear evolution equations were presented by Ablowitz, Kaup,Newell and Segur (AKNS) [31], where $U$ and $V$ read

\displaystyle U=\left(\begin{array}[]{cc}-i\lambda&q(x,t)\cr r(x,t)&i\lambda\end{array}\right),\quad V=\left(\begin{array}[]{cc}A(x,t;\lambda)&B(x,t;\lambda)\cr C(x,t;\lambda)&-A(x,t;\lambda)\end{array}\right).

(9)

The corresponding consistency condition (3) leads to the flowing equations

	$\displaystyle A_{x}=qc-rB,$		(10)
	$\displaystyle q_{t}-B_{x}-2i\lambda B-2qA=0,$		(11)
	$\displaystyle r_{t}-C_{x}+2i\lambda C+2rA=0.$		(12)

For example, AKNS set

\displaystyle A=\sum_{j=0}^{N}a_{j}\lambda^{N},\quad B=\sum_{j=0}^{N}b_{j}\lambda^{N},\quad C=\sum_{j=0}^{N}c_{j}\lambda^{N}.

(13)

To illustrate, take $N=3$ . By solving Eqs.(10), AKNS found

	$\displaystyle A=a_{3}\lambda^{3}+a_{2}\lambda^{2}+(\frac{1}{2}a_{3}qr+a_{1})\lambda+\frac{1}{2}a_{2}qr-\frac{i}{4}a_{3}(qr_{x}-q_{x}r)+a_{0},$		(14)
	$\displaystyle B=ia_{3}q\lambda^{2}+(ia_{2}q-\frac{1}{2}a_{3}q_{x})\lambda+(ia_{1}q+\frac{i}{2}a_{3}q^{2}r-\frac{1}{2}a_{2}q_{x}-\frac{i}{4}a_{3}q_{xx}),$		(15)
	$\displaystyle C=ia_{3}r\lambda^{2}+(ia_{2}r+\frac{1}{2}a_{3}r_{x})\lambda+(ia_{1}r+\frac{i}{2}a_{3}qr^{2}+\frac{1}{2}a_{2}r_{x}-\frac{i}{4}a_{3}r_{xx})$		(16)

together with the evolution equations

	$\displaystyle q_{t}+\frac{i}{4}a_{3}(r_{xxx}-6qrq_{x})+\frac{1}{2}a_{2}(q_{xx}-2q^{2}r)-ia_{1}q_{x}-2a_{0}q=0,$		(17)
	$\displaystyle r_{t}+\frac{i}{4}a_{3}(r_{xxx}-6qrr_{x})-\frac{1}{2}a_{2}(r_{xx}-2qr^{2})-ia_{1}r_{x}+2a_{0}r=0.$		(18)

Case I. As a special case, let $r=-q$ and

\displaystyle a_{0}=a_{1}=a_{2}=0,\quad a_{3}=-4i.

(19)

Then AKNS got the modified mKdV equation

\displaystyle q_{t}+6q^{2}q_{x}+q_{xxx}=0.

(20)

Case II. In the same way, by taking

\displaystyle A=\frac{a(x,t)}{\lambda},\quad B=\frac{b(x,t)}{\lambda},C=\frac{c(x,t)}{\lambda},

(21)

AKNS found

\displaystyle q_{xt}=-4iaq,\quad r_{xt}=-4iar,\quad a_{x}=\frac{i}{2}(qr)_{t}.

(22)

As a special but important case, we list the following SG equation

\displaystyle u_{xt}=\sin u

(23)

with

\displaystyle a=\frac{i}{4}\cos u,\quad b=c=\frac{i}{4}\sin u,\quad r=-q=\frac{1}{2}u_{x}.

(24)

In 1979, Wadati, Konno and Ichikawa [1, 2] proposed a generalization of the inverse scattering formalism (especially $U$ ) and found a new series of integrable nonlinear evolution equations. In fact, they considered

\displaystyle U=\left(\begin{array}[]{cc}-i\lambda&\lambda q\cr\lambda r&i\lambda\end{array}\right),\quad V=\left(\begin{array}[]{cc}A&B\cr C&-A\end{array}\right).

(29)

The corresponding consistency condition (3) leads to the flowing equations

	$\displaystyle A_{x}=\lambda(qC-rB),$		(30)
	$\displaystyle\lambda q_{t}-B_{x}-2i\lambda B-2\lambda qA=0,$		(31)
	$\displaystyle\lambda r_{t}-C_{x}+2i\lambda C+2\lambda rA=0.$		(32)

Case III. As a special case, by choosing

	$\displaystyle A=-\frac{4i}{\sqrt{1-rq}}\lambda^{3}+\frac{rq_{x}-qr_{x}}{(1-rq)^{3/2}}\lambda^{2},$		(33)
	$\displaystyle B=\frac{4q}{\sqrt{1-rq}}\lambda^{3}+\frac{2iq_{x}}{(1-rq)^{3/2}}\lambda^{2}-\bigg{(}\frac{q_{x}}{(1-rq)^{3/2}}\bigg{)}_{x}\lambda,$		(34)
	$\displaystyle C=\frac{4r}{\sqrt{1-rq}}\lambda^{3}-\frac{2ir_{x}}{(1-rq)^{3/2}}\lambda^{2}-\bigg{(}\frac{r_{x}}{(1-rq)^{3/2}}\bigg{)}_{x}\lambda,$		(35)

they obtained

	$\displaystyle q_{t}+\bigg{(}\frac{q_{x}}{(1-rq)^{3/2}}\bigg{)}_{xx}=0,$		(36)
	$\displaystyle r_{t}+\bigg{(}\frac{r_{x}}{(1-rq)^{3/2}}\bigg{)}_{xx}=0.$		(37)

For $r=-q$ , above equation is reduced to the typical WKI equation (1).

Case IV. Moreover, as another special case, by choosing $r=q=iu_{x}$ and

\displaystyle A=\frac{i\lambda}{2}u^{2}+\frac{1}{4i\lambda},\quad B=\frac{i\lambda}{2}u^{2}u_{x}-\frac{1}{2}u,\quad C=\frac{i\lambda}{2}u^{2}u_{x}+\frac{1}{2}u,

(38)

one can get the known SP equation (2). This Lax pair for (2) was firstly discovered by Sakovich A and Sakovich S [15] in 2005.

2.2 WKI-SP type integrable equations

In this paper, we focus on a series of compound equations of the WKI-SP type, which will be derived on the basis of the linear system (4). For convenience, we rewrite $U$ as

\displaystyle U=\left(\begin{array}[]{cc}\lambda&\lambda p(x,t)\cr\lambda p(x,t)&-\lambda\end{array}\right).

(41)

Eqs.(30)-(32) become

	$\displaystyle A_{x}=\lambda p(C-B),$		(42)
	$\displaystyle\lambda p_{t}-B_{x}+2\lambda B-2\lambda pA=0,$		(43)
	$\displaystyle\lambda p_{t}-C_{x}-2\lambda C+2\lambda pA=0.$		(44)

Consequently, we have

	$\displaystyle C-B=\frac{A_{x}}{\lambda p},$		(45)
	$\displaystyle C+B=-\frac{(C-B)_{x}}{2\lambda}+2pA,$		(46)
	$\displaystyle p_{t}=\frac{(C+B)_{x}}{2\lambda}+(C-B).$		(47)

Different from the previous references, we select $A$ as follows:

\displaystyle A=\sum_{j=0}^{n+m}a_{j}\lambda^{2n-2j+1},\quad\quad(n\geq 0,m\geq 0).

(48)

Substituting (45) and (46) into (47) and then expanding it into a power series of $\lambda$ , we obtain

\displaystyle p_{t}=-\frac{1}{4}\bigg{(}\frac{a_{n-1,x}}{p}\bigg{)}_{xx}+(pa_{n})_{x}+\frac{a_{nx}}{p}

(49)

together with the following two sets of recursive formulas

	$\displaystyle(pa_{0})_{x}+\frac{a_{0x}}{p}=0,$		(50)
	$\displaystyle(pa_{j})_{x}+\frac{a_{j,x}}{p}=\frac{1}{4}\bigg{(}\frac{a_{j-1,x}}{p}\bigg{)}_{xx},\quad(j=1,2,\ldots,n-2,n-1),$		(51)

and

	$\displaystyle-\frac{1}{4}\bigg{(}\frac{a_{n+m,x}}{p}\bigg{)}_{xx}=0,$		(52)
	$\displaystyle\frac{1}{4}\bigg{(}\frac{a_{j,x}}{p}\bigg{)}_{xx}=(pa_{j+1})_{x}+\frac{a_{j+1,x}}{p},\quad(j=n+m-1,\ldots,n+1,n).$		(53)

It is seen that $\{a_{0},a_{1},\ldots,a_{n-1}\}$ and $\{a_{n},a_{n+1},\ldots,a_{n+m-1},a_{n+m}\}$ are determined by Eqs.(50)-(51) and Eqs.(52)-(53), respectively. For convenience, we denote the defined equation in (49) as WKI-SP^(n,m) equation for the concrete $n$ and $m$ .

Furthermore, by solving Eqs.(50)-(51), we obtain

	$\displaystyle a_{0}=\frac{a_{0}^{0}(t)}{\sqrt{1+p^{2}}},$		(54)
	$\displaystyle a_{j}=\frac{1}{4\sqrt{1+p^{2}}}\partial^{-1}\bigg{(}\frac{p}{\sqrt{1+p^{2}}}\bigg{(}\frac{a_{j-1,x}}{p}\bigg{)}_{xx}\bigg{)},\quad(j=1,2,\ldots,n-1),$		(55)

which imply

\displaystyle a_{1}=-\frac{a_{0}^{0}(t)\biggl{[}(p^{3}+p)p_{xx}-\frac{1}{2}(6p^{2}+1)p_{x}^{2}\biggr{]}}{4(1+p^{2})^{\frac{7}{2}}},

(56)

\begin{split}&a_{2}=-\frac{a_{0}^{0}(t)}{32(1+p^{2})^{\frac{13}{2}}}\bigg{[}2p(1+p^{2})^{3}p_{xxxx}-2(1+p^{2})^{2}(15p^{2}+1)p_{x}p_{xxx}-(1+p^{2})^{2}(20p^{2}-1)p_{xx}^{2}\\ &+21p(1+p^{2})(10p^{2}-1)p_{x}^{2}p_{xx}-\frac{21}{4}(40p^{4}-16p^{2}-1)p_{x}^{4}\bigg{]},\\ &\quad\vdots\end{split}

By solving Eqs.(52)-(53), we have

	$\displaystyle a_{n+m}=a_{n+m}^{0}(t),$		(57)
	$\displaystyle a_{j}=4\partial^{-1}\bigg{(}p\partial^{-2}\bigg{(}\frac{a_{j+1,x}}{p}\bigg{)}\bigg{)}+4\partial^{-1}(p\partial^{-1}(pa_{j+1})),$		(58)
	$\displaystyle(j=n+m-1,n+m-2,\ldots,n),$

which imply

	$\displaystyle a_{n+m}=a_{n+m}^{0}(t),$		(59)
	$\displaystyle a_{n+m-1}=2a_{n+m}^{0}(t)(\partial^{-1}p)^{2},$		(60)
	$\displaystyle a_{n+m-2}=a_{n+m}^{0}(t)\bigg{[}\frac{2}{3}(\partial^{-1}p)^{4}+16(\partial^{-1}p)(\partial^{-3}p)-8(\partial^{-2}p)^{2}\bigg{]},$		(61)
	$\displaystyle\begin{split}&a_{n+m-3}=a_{n+m}^{0}(t)\bigg{\{}\frac{4}{45}(\partial^{-1}p)^{6}+\frac{32}{3}\biggl{[}(\partial^{-1}p)^{3}(\partial^{-3}p)+(\partial^{-1}p)\bigg{(}\partial^{-2}(\partial^{-1}p)^{3}\bigg{)}-(\partial^{-2}p)\bigg{(}\partial^{-1}(\partial^{-1}p)^{3}\bigg{)}\\ &-\frac{3}{2}\partial^{-1}\bigg{(}p(\partial^{-2}p)^{2}\bigg{)}\biggr{]}+64\bigg{[}(\partial^{-1}p)(\partial^{-5}p)-(\partial^{-2}p)(\partial^{-4}p)+\frac{1}{2}(\partial^{-3}p)^{2}\bigg{]}\bigg{\}},\end{split}$		(62)
	$\displaystyle\quad\vdots$

In order to express the fomulas clearly, we write $\partial^{-1}$ instead of $\int^{x}\cdot dx$ sometimes.

2.2.1 SP-hierarchy for $n=0$

If $n=0$ , (48) becomes

\displaystyle A=\sum_{j=0}^{m}a_{j}\lambda^{-2j+1}=a_{0}\lambda+a_{1}\lambda^{-1}+a_{2}\lambda^{-3}+\cdots+a_{m}\lambda^{-(2m-1)}.

(63)

Then we obtain the generalized hierarchy of SP equations

\displaystyle p_{t}=(pa_{0})_{x}+\frac{a_{0x}}{p}.

(64)

Here $a_{0}$ is determined by the recursive formulas (57) and (58) with $n=0$ . Now we give some special cases of the SP hierarchy.

Case 1. For $m=1$ and $p=u_{x}$ , we have

\displaystyle a_{1}=a^{0}_{1}(t),\quad a_{0}=2a_{1}^{0}(t)u^{2},

(65)

which exactly give rise to the short pulse equation

\displaystyle u_{xt}=4a_{1}^{0}(t)\bigg{(}\frac{1}{6}(u^{3})_{xx}+u\bigg{)}.

(66)

According to the convention above, we label Eq.(66) as WKI-SP^(0,1) equation.

Case 2. For $m=2$ and $p=u_{xxx}$ , we have

\displaystyle a_{2}=a^{0}_{2}(t),\quad a_{1}=2a_{2}^{0}(t)u^{2}_{xx},\quad a_{0}=\frac{2}{3}a_{2}^{0}(t)u_{xx}^{4}+16a_{2}^{0}(t)uu_{xx}-8a_{2}^{0}(t)u_{x}^{2},

(67)

which yields a high-order short pulse (i.e.WKI-SP^(0,2)) equation

\displaystyle u_{xxxt}=\frac{2}{15}a_{2}^{0}(t)(u_{xx}^{5})_{xx}+a_{2}^{0}(t)[u_{xxx}(16uu_{xx}-8u_{x}^{2})]_{x}+\frac{8}{3}a_{2}^{0}(t)u_{xx}^{3}+16a_{2}^{0}(t)u.

(68)

This equation is just the local form of the one given in Ref.[32].

Case 3. For $m=3$ and $p=u_{xxxxx}$ , we have

a_{3}=a^{0}_{3}(t),\quad a_{2}=2a_{3}^{0}(t)u_{xxxx}^{2},\quad a_{1}=\frac{2}{3}a_{3}^{0}(t)u_{xxxx}^{4}+16a_{3}^{0}(t)u_{xx}u_{xxxx}-8a_{3}^{0}(t)u_{xxx}^{2},

(69)

\begin{split}a_{0}=&\frac{4}{45}a_{3}^{0}(t)u_{xxxx}^{6}+32a_{3}^{0}(t)\bigg{[}\frac{1}{3}u_{xx}u_{xxxx}^{3}-\frac{1}{2}u_{xxx}^{2}u_{xxxx}^{2}+2uu_{xxxx}-2u_{x}u_{xxx}+u_{xx}^{2}+u_{xxxx}\partial^{-1}(u_{xxxx}^{2}u_{xxx})\\ &+\frac{32}{3}u_{xxxx}\partial^{-2}u_{xxxx}^{3}\bigg{]},\end{split}

(70)

which yields the WKI-SP^(0,3) equation

\begin{split}&u_{xxxxxt}=a_{3}^{0}(t)\bigg{\{}u_{xxxxxx}\bigg{[}\frac{4}{45}u_{xxxx}^{6}+32\bigg{(}\frac{1}{3}u_{xx}u_{xxxx}^{3}-\frac{1}{2}u_{xxx}^{2}u_{xxxx}^{2}+u_{xxxx}\partial^{-1}(u_{xxxx}^{2}u_{xxx})+\frac{32}{3}u_{xxxx}\partial^{-2}(u_{xxxx}^{3})\\ &+64\bigg{(}uu_{xxxx}-u_{x}u_{xxx}+\frac{1}{2}u_{xx}^{2}\bigg{)}\bigg{)}\bigg{]}+(u_{xxxxx}+{u^{-1}_{xxxxx}})\bigg{[}\frac{8}{15}u_{xxxx}^{5}u_{xxxxx}+32u_{xxxxx}\\ &\bigg{(}u_{xx}u_{xxxx}^{2}-u_{xxx}^{2}u_{xxxx}+\partial^{-1}(u_{xxxx}^{2}u_{xxx})+\frac{32}{3}u_{xxxxx}\partial^{-2}(u_{xxxx}^{3})\bigg{)}+64uu_{xxxxx}\bigg{]}\bigg{\}}.\end{split}

(71)

2.2.2 WKI-hierarchy for $m=0$

If $m=0$ , (48) is rewritten as

\displaystyle A=\sum_{j=0}^{n}a_{j}\lambda^{2n-2j+1}=a_{0}\lambda^{2n+1}+a_{1}\lambda^{2n-1}+a_{2}\lambda^{-3}+\cdots+a_{n}\lambda.

(72)

Then we obtain the generalized hierarchy of WKI equations

\displaystyle p_{t}=-\frac{1}{4}\bigg{(}\frac{a_{n-1,x}}{p}\bigg{)}_{xx}+a_{n}^{0}(t)p_{x},

(73)

with the recursive formulas (54) and (55). Some special equations in the WKI hierarchy are given in the follows.

Case 4. For $n=1$ and $p=u_{x}$ , Eq.(73) with (56) is exactly the WKI equation

\displaystyle u_{xt}=\frac{a_{0}^{0}(t)}{4}\left[\frac{u_{xx}}{(1+u_{x}^{2})^{\frac{3}{2}}}\right]_{xx}+a^{0}_{1}(t)u_{xx}.

(74)

Case 5. For $n=2$ and $p=u_{x}$ , Eq.(73) with (2.2) is the high-order WKI (i.e. WKI-SP^(2,0)) equation

\begin{split}&u_{xt}=\frac{a_{0}^{0}(t)}{16\left(1+u_{x}^{2}\right)^{\frac{13}{2}}}\Bigg{[}u_{xxxxxx}\left(1+u_{x}^{2}\right)^{4}-20u_{x}u_{xx}u_{xxxxx}\left(1+u_{x}^{2}\right)^{3}-35u_{xxxx}\Bigg{(}u_{xxx}\left(u_{x}^{3}+u_{x}\right)-\frac{39}{7}u_{xx}^{2}\\ &(u_{x}^{2}-\frac{1}{6})\Bigg{)}\left(1+u_{x}^{2}\right)^{2}+270u_{xx}\Bigg{(}\left(1+u_{x}^{2}\right)^{2}(u_{x}^{2}-\frac{1}{6})u_{xxx}^{2}-\frac{77}{18}u_{x}u_{xxx}u_{xx}^{2}(u_{x}^{2}-\frac{1}{2})\left(1+u_{x}^{2}\right)\\ &+\frac{28}{9}u_{xx}^{4}(u_{x}^{4}-u_{x}^{2}+\frac{1}{16})\Bigg{)}\Bigg{]}+\frac{1}{4}a_{1}^{0}\left(t\right)\left(\frac{u_{x}}{\sqrt{1+u_{x}^{2}}}\right)_{xxx}+a_{3}^{0}(t)u_{xx}.\end{split}

(75)

2.2.3 Compound WKI-SP hierarchy for $n\neq 0,m\neq 0$

For $n\neq 0$ and $m\neq 0$ , we have the generalized compound WKI-SP^(n,m) equations (49). Here, we give three examples.

Case 6. For $n=1,m=1$ and $p=u_{x}$ , Eq.(49) gives the WKI-SP^(1,1) equation

\displaystyle u_{xt}=\frac{a_{0}^{0}(t)}{4}\left[\frac{u_{xx}}{\bigg{(}1+u_{x}^{2}\bigg{)}^{\frac{3}{2}}}\right]_{xx}+a_{1}^{0}(t)u_{xx}+4a_{2}^{0}(t)\bigg{(}\frac{1}{6}(u^{3})_{xx}+u\bigg{)}.

(76)

Case 7. For $n=1,m=2$ and $p=u_{xxx}$ , we obtain the following WKI-SP^(1,2) equation

\displaystyle u_{xxxt}=\frac{a_{0}^{0}(t)}{4}\left[\frac{u_{xxxx}}{(1+u_{xxx}^{2})^{\frac{3}{2}}}\right]_{xx}+a_{2}^{0}(t)\bigg{(}\frac{2}{15}(u_{xx}^{5})_{xx}+[u_{xxx}(16uu_{xx}-8u_{x}^{2})]_{x}+\frac{8}{3}u_{xx}^{3}+16u\bigg{)}.

(77)

Case 8. For $n=2,m=1$ and $p=u_{x}$ , we obtain the following WKI-SP^(2,1) equation

\begin{split}&u_{xt}=\frac{a_{0}^{0}(t)}{16\left(1+u_{x}^{2}\right)^{\frac{13}{2}}}\Bigg{[}u_{xxxxxx}\left(1+u_{x}^{2}\right)^{4}-20u_{x}u_{xx}u_{xxxxx}\left(1+u_{x}^{2}\right)^{3}-35u_{xxxx}\Bigg{(}u_{xxx}\left(u_{x}^{3}+u_{x}\right)-\frac{39}{7}u_{xx}^{2}\\ &(u_{x}^{2}-\frac{1}{6})\Bigg{)}\left(1+u_{x}^{2}\right)^{2}+270u_{xx}\Bigg{(}\left(1+u_{x}^{2}\right)^{2}(u_{x}^{2}-\frac{1}{6})u_{xxx}^{2}-\frac{77}{18}u_{x}u_{xxx}u_{xx}^{2}\left(u_{x}^{2}-\frac{1}{2}\right)(1+u_{x}^{2})\\ &+\frac{28}{9}u_{xx}^{4}(u_{x}^{4}-u_{x}^{2}+\frac{1}{16})\Bigg{)}\Bigg{]}+\frac{1}{4}a_{1}^{0}\left(t\right)\left(\frac{u_{x}}{\sqrt{1+u_{x}^{2}}}\right)_{xxx}+4a_{3}^{0}(t)\left(u+\frac{1}{6}\left(u^{3}\right)_{xx}\right).\end{split}

(78)

3 Hodograph transformations between MKdV-SG and WKI-SP eqautions

In Ref.[29, 30], Gu obtained the compound equations of the MKdV-SG type

\displaystyle q_{t}-\frac{1}{4}\Bigg{[}4a_{n}q+\bigg{(}\frac{a_{nx}}{q}\bigg{)}_{x}\Bigg{]}_{x}+\frac{a_{n+1,x}}{q}=0

(79)

together with the following two sets of recursive formulas

	$\displaystyle a_{0x}=0,$		(80)
	$\displaystyle a_{hx}=\frac{1}{4}q\Bigg{[}4a_{h-1}q+\bigg{(}\frac{a_{h-1,x}}{q}\bigg{)}_{x}\Bigg{]}_{x},h=1,2,...\,,n,$		(81)

and

	$\displaystyle\Bigg{[}4a_{l+m}q+\bigg{(}\frac{a_{l+m,x}}{q}\bigg{)}_{x}\Bigg{]}_{x}=0,$		(82)
	$\displaystyle\Bigg{[}4a_{h-1}q+\bigg{(}\frac{a_{h-1,x}}{q}\bigg{)}_{x}\Bigg{]}_{x}=4\frac{a_{h,x}}{q},h=n+m,...\,,n+2.$		(83)

In this section, we would build the hodograph transformations between some members in the above MKdV-SG type equations and the WKI-SP^(n,m) equation. To see it clearly, we denote the concrete equation among (79) as MKdV-SG^(n,m) equation for specific $n$ and $m$ .

3.1 Hodograph transformation between MKdV-SG^(1,1) and WKI-SP^(1,1) equation

By taking $n=1,m=1$ and $q=-\frac{\theta_{s}}{2}$ in Eq.(79), the author gave the MKdV-SG^(1,1) equation

\displaystyle\theta_{ts}-\frac{1}{4}\alpha_{0}(t)\biggl{(}\frac{3}{2}\theta_{s}^{2}\theta_{ss}+\theta_{ssss}\biggr{)}-\alpha_{1}(t)\theta_{ss}=\alpha_{2}(t)\sin{\theta},

(84)

which is related to the motion of a nonlinear one-dimensional lattice of atoms.

Integrating (84) with respect to $s$ leads to

\displaystyle\theta_{t}-\frac{1}{4}\alpha_{0}(t)\Bigg{(}\frac{1}{2}\theta_{s}^{3}+\theta_{sss}\Bigg{)}-\alpha_{1}(t)\theta_{s}=\alpha_{2}(t)\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}.

(85)

A conservation law of Eq.(85) is given by

\displaystyle(\cos\theta)_{t}+\Bigg{[}\frac{1}{4}\alpha_{0}(t)\bigg{(}\theta_{ss}\sin{\theta}-\frac{1}{2}\theta_{s}^{2}\cos{\theta}\bigg{)}-\alpha_{1}(t)\cos{\theta}+\frac{1}{2}\alpha_{2}(t)\bigg{(}\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\bigg{)}^{2}\Bigg{]}_{s}=0.

(86)

Consider the hodograph transformation

\displaystyle x(s,t)=\int^{s}\cos\theta(s_{1},t){\rm{d}}s_{1}+x_{0},\quad\quad\quad\quad t^{\prime}(s,t)=t,

(87)

which leads to

\displaystyle\frac{\partial}{\partial s}=\cos\theta\frac{\partial}{\partial x},\quad\quad\frac{\partial}{\partial t}=\Bigg{[}\frac{1}{4}\alpha_{0}(t)\bigg{(}\frac{1}{2}\theta_{s}^{2}\cos{\theta}-\theta_{ss}\sin{\theta}\bigg{)}+\alpha_{1}(t)\cos{\theta}-\frac{1}{2}\alpha_{2}(t)\bigg{(}\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\bigg{)}^{2}\Bigg{]}\frac{\partial}{\partial x}+\frac{\partial}{\partial t^{\prime}}.

(88)

Applying (88) to (86), we obtain

\displaystyle(\tan\theta)_{t^{\prime}}-\frac{1}{4}\alpha_{0}(t)(\sin{\theta})_{xxx}-\frac{\alpha_{2}(t)}{2}\bigg{(}\frac{(\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1})^{2}}{\cos{\theta}}\bigg{)}_{x}\frac{1}{\sin{\theta}}=0,

(89)

with

\displaystyle\theta_{s}=(\cos\theta)\theta_{x}=(\sin\theta)_{x},\quad\theta_{ss}=(\cos\theta)(\theta_{s})_{x}=(\cos\theta)(\sin\theta)_{xx}.

(90)

Introduce a new dependent variable

\displaystyle u(s,t)=\int^{s}\sin\theta(s_{1},t){\rm{d}}s_{1},

(91)

which gives

\displaystyle u_{s}=\sin\theta.

(92)

Applying (88) to (91), we obtain

\displaystyle u_{s}=(\cos\theta)u_{x}.

(93)

Combing (92) with (93) gives

\displaystyle\tan\theta=u_{x},

(94)

then it follows

\displaystyle\sin\theta=\frac{u_{x}}{\sqrt{1+u_{x}^{2}}},\quad\quad\cos\theta=\frac{1}{\sqrt{1+u_{x}^{2}}}.

(95)

Substituting (94) and (95) into (89) yields

\displaystyle u_{xt^{\prime}}-\frac{1}{4}\alpha_{0}(t)\bigg{(}\frac{u_{x}}{\sqrt{1+u_{x}^{2}}}\bigg{)}_{xxx}-\alpha_{2}(t)\bigg{(}u+\frac{1}{6}(u^{3})_{xx}\bigg{)}=0,

(96)

which is just the WKI-SP^(1,1) equation given by (76) with $a^{0}_{1}(t)=0$ .

3.2 Hodograph transformation between MKdV-SG^(2,1) and WKI-SP^(2,1) equation

For $n=2,m=1$ , the MKdV-SG^(2,1) equation is given by

\displaystyle q_{t}-\frac{1}{4}\bigg{[}\bigg{(}\frac{a_{2,s}}{q}\bigg{)}_{s}+4a_{2}q\bigg{]}_{s}+\frac{a_{3,s}}{q}=0.

(97)

By solving (81), one obtain

	$\displaystyle a_{0}=\alpha_{0}(t),$		(98)
	$\displaystyle a_{1}=\frac{1}{2}\alpha_{0}(t)q^{2}+\alpha_{1}(t),$		(99)
	$\displaystyle a_{2}=\alpha_{0}(t)\bigg{(}\frac{1}{4}qq_{ss}-\frac{1}{8}q_{s}^{2}+\frac{3}{8}q^{4}\bigg{)}+\frac{1}{2}\alpha_{1}(t)q^{2}+\alpha_{2}(t).$		(100)

Let $q=-\frac{\theta_{s}}{2}$ and choose the following special solution for $a_{3}$ in (83)

\displaystyle a_{3}=\frac{1}{4}\alpha_{3}(t)\cos\theta.

(101)

The MKdV-SG^(2,1) equation (97) is expressed by

\begin{split}&\theta_{st}+\bigg{[}\frac{1}{128}\alpha_{0}(t)\bigg{(}-8\theta_{ssssss}-20\theta_{ss}^{3}-80\theta_{s}\theta_{ss}\theta_{sss}-20\theta_{s}^{2}\theta_{ssss}-15\theta_{s}^{4}\theta_{ss}\bigg{)}+\alpha_{1}(t)(-\frac{1}{4}\theta_{ssss}-\frac{3}{8}\theta_{s}^{2}\theta_{ss})-\alpha_{2}(t)\theta_{ss}\bigg{]}\\ &-\alpha_{3}(t)\sin\theta=0.\end{split}

(102)

Integrating Eq.(102) with respect to $s$ leads to

\displaystyle\theta_{t}-\bigg{[}\frac{1}{128}\alpha_{0}(t)\bigg{(}8\theta_{sssss}+20\theta_{s}\theta_{ss}^{2}+20\theta_{s}^{2}\theta_{sss}+3\theta_{s}^{5}\bigg{)}-\frac{1}{8}\alpha_{1}(t)\bigg{(}2\theta_{sss}+\theta_{s}^{3}\bigg{)}-\alpha_{2}(t)\theta_{s}\bigg{]}=\alpha_{3}(t)\int^{s}\sin\theta(s_{1},t)\mathrm{d}s_{1}.

(103)

Multiplying both sides of Eq.(103) by $-\sin\theta$ , one conservation law is given by

\displaystyle\begin{split}&(\cos\theta)_{t}+\bigg{[}\frac{1}{128}\alpha_{0}(t)\bigg{(}12\theta_{s}^{2}\theta_{ss}\sin\theta+8\theta_{ssss}\sin\theta+4\theta_{ss}^{2}\cos\theta-3\theta_{s}^{4}\cos\theta-8\theta_{s}\theta_{sss}\cos\theta\bigg{)}\\ &+\frac{1}{8}\alpha_{1}(t)\bigg{(}2\theta_{ss}\sin\theta-\theta_{s}^{2}\cos\theta\bigg{)}-\alpha_{2}(t)\cos\theta+\frac{1}{2}\alpha_{3}(t)\bigg{(}\int^{s}\sin\theta(s_{1},t)\mathrm{d}s_{1}\bigg{)}^{2}\bigg{]}_{s}=0.\end{split}

(104)

Consider the same hodograph transformation (87). It leads to

\displaystyle\frac{\partial}{\partial s}=\cos\theta\frac{\partial}{\partial x},

(105)

\displaystyle\begin{split}&\frac{\partial}{\partial t}=\bigg{[}-\frac{1}{128}\alpha_{0}(t)\bigg{(}12\theta_{s}^{2}\theta_{ss}\sin\theta+8\theta_{ssss}\sin\theta+4\theta_{ss}^{2}\cos\theta-3\theta_{s}^{4}\cos\theta-8\theta_{s}\theta_{sss}\cos\theta\bigg{)}\\ &\qquad-\frac{1}{8}\alpha_{1}(t)\bigg{(}2\theta_{ss}\sin\theta-\theta_{s}^{2}\cos\theta\bigg{)}+\alpha_{2}(t)\cos\theta-\frac{1}{2}\alpha_{3}(t)\bigg{(}\int^{s}\sin\theta(s_{1},t)\mathrm{d}s_{1}\bigg{)}^{2}\bigg{]}\frac{\partial}{\partial x}+\frac{\partial}{\partial t^{\prime}}.\end{split}

(106)

We have

	$\displaystyle\theta_{s}=(\cos\theta)\theta_{x}=(\sin\theta)_{x},\quad\theta_{ss}=\cos\theta(\theta_{s})_{x}=\cos\theta(\sin\theta)_{xx},$		(107)
	$\displaystyle\theta_{sss}=\cos^{2}\theta\sin\theta_{xxx}+{\cos\theta}(\cos{\theta})_{x}(\sin\theta)_{xx},$		(108)
	$\displaystyle\begin{split}&\theta_{ssss}=2\cos^{2}\theta(\cos\theta)_{x}(\sin\theta)_{xxx}+\cos^{3}\theta(\sin{\theta})_{xxxx}+\cos{\theta}(\cos\theta)_{x}^{2}(\sin\theta)_{xx}\\ &+\cos^{2}\theta(\cos\theta)_{xx}(\sin\theta)_{xx}+\cos^{2}\theta(\cos\theta)_{x}(\sin\theta)_{xxx}.\end{split}$		(109)

Applying (105) and (106) to Eq.(104), we obtain

\displaystyle\begin{split}&(\tan\theta)_{t^{\prime}}+\frac{\alpha_{0}(t)}{128}\bigg{[}24(\sin\theta)_{x}(\sin\theta)_{xx}^{2}+12(\sin\theta)_{x}^{2}(\sin\theta)_{xxx}+16(\sin\theta)_{xxx}(\cos\theta)_{x}^{2}+16(\sin\theta)_{xxx}\cos{\theta}(\cos\theta)_{xx}\\ &+32(\sin\theta)_{xxxx}\cos{\theta}(\cos\theta)_{x}+8(\sin\theta)_{xxxxx}\cos^{2}\theta+16(\sin\theta)_{xx}(\cos\theta)_{x}(\cos\theta)_{xx}+8(\sin\theta)_{xxx}(\cos\theta)_{x}^{2}\\ &+8(\sin\theta)_{xx}(\cos\theta)_{x}(\cos\theta)_{xx}+8(\sin\theta)_{xx}\cos\theta(\cos\theta)_{xxx}+8(\sin\theta)_{xxx}\cos\theta(\cos\theta)_{xx}+8(\sin\theta)_{xxx}(\cos\theta)_{x}^{2}\\ &+8(\sin\theta)_{xxx}(\cos\theta)_{x}^{2}+8(\sin\theta)_{xxx}\cos\theta(\cos\theta)_{xx}+8(\sin\theta)_{xxxx}\cos\theta(\cos\theta)_{x}\bigg{]}-\frac{1}{4}\alpha_{1}(t)(\sin\theta)_{xxx}\\ &-\frac{\alpha_{3}(t)}{2}\frac{1}{\sin\theta}\bigg{[}\frac{\bigg{(}\int^{s}\sin\theta(s_{1},t)\mathrm{d}s_{1}\bigg{)}^{2}}{\cos\theta}\bigg{]}_{x}=0.\end{split}

(110)

Introduce a new dependent variable $v(s,t)=\int^{s}\sin\theta(s_{1},t){\rm{d}}s_{1}$ (note $\tan\theta=v_{x}$ , $\sin\theta=\frac{v_{x}}{\sqrt{1+v_{x}^{2}}},\cos\theta=\frac{1}{\sqrt{1+v_{x}^{2}}}$ ). Then Eq.(110) is transformed to

\begin{split}&v_{xt^{\prime}}-\frac{\alpha_{0}\left(t\right)}{16\left(1+v_{x}^{2}\right)^{\frac{13}{2}}}\Bigg{[}v_{xxxxxx}\left(1+v_{x}^{2}\right)^{4}-20v_{x}v_{xx}v_{xxxxx}\left(1+v_{x}^{2}\right)^{3}-35v_{xxxx}\Bigg{(}v_{xxx}\left(v_{x}^{3}+v_{x}\right)-\frac{39}{7}v_{xx}^{2}\\ &(v_{x}^{2}-\frac{1}{6})\Bigg{)}\left(1+v_{x}^{2}\right)^{2}+270v_{xx}\Bigg{(}\left(1+v_{x}^{2}\right)^{2}(v_{x}^{2}-\frac{1}{6})v_{xxx}^{2}-\frac{77}{18}v_{x}v_{xxx}v_{xx}^{2}(v_{x}^{2}-\frac{1}{2})\left(1+v_{x}^{2}\right)\\ &+\frac{28}{9}v_{xx}^{4}\left(v_{x}^{4}-v_{x}^{2}+\frac{1}{16}\right)\Bigg{)}\Bigg{]}-\frac{1}{4}\alpha_{1}\left(t\right)\left(\frac{v_{x}}{\sqrt{1+v_{x}^{2}}}\right)_{xxx}-\alpha_{3}(t)\left(v+\frac{1}{6}\left(v^{3}\right)_{xx}\right)=0,\end{split}

(111)

which is just the WKI-SP^(2,1) equation (78).

3.3 Hodograph transformation between MKdV-SG^(1,2) and WKI-SP^(1,2) equation

For $n=1,m=2$ , the MKdV-SG^(1,2) equation is given by

\displaystyle q_{t}-\frac{1}{4}\bigg{[}\bigg{(}\frac{a_{1,s}}{q}\bigg{)}_{s}+4a_{1}q\bigg{]}_{s}+\frac{a_{2s}}{q}=0.

(112)

By solving the recursive formulas (81) and (83) with $q=-\frac{\theta_{s}}{2}$ , we get

\displaystyle a_{0}=\alpha_{0}(t),\quad a_{1}=\frac{1}{2}\alpha_{0}(t)q^{2}+\alpha_{1}(t),\quad a_{3}=\frac{1}{4}\alpha_{3}(t)\cos{\theta},\quad

(113)

and

\begin{split}&\frac{a_{2s}}{q}=C_{1}\sin{\theta}+C_{2}\cos{\theta}+2\sin{\theta}\int^{s}\sin\theta(s,t)\left(\int^{s_{1}}\sin\theta(s_{2},t)\mathrm{d}s_{2}\right)\mathrm{d}s_{1}\\ &+2\cos{\theta}\int^{s}\cos\theta(s_{1},t)\left(\int^{s_{1}}\sin\theta(s_{2},t)\mathrm{d}s_{2}\right)\mathrm{d}s_{1},\end{split}

(114)

with $q=-\frac{\theta_{s}}{2}$ .

Then the MKdV-SG^(1,2) equation (97) is rewritten as

\begin{split}&\theta_{t}=\frac{1}{4}\alpha_{0}(t)\theta_{sss}+\frac{1}{8}\alpha_{0}(t)\theta_{s}^{3}+\alpha_{1}(t)\theta_{s}+2C_{1}\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\\ &+2C_{2}\int^{s}\cos{\theta}(s_{1},t)\mathrm{d}s_{1}+\frac{2}{3}\biggl{(}\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\biggr{)}^{3}\\ &+4\int^{s}\cos{\theta}(s_{1},t)\biggl{[}\int^{s_{1}}\sin{\theta}(s_{2},t)\biggl{(}\int^{s_{2}}\sin{\theta}(s_{3},t)\mathrm{d}s_{3}\biggr{)}\mathrm{d}s_{2}\biggr{]}\mathrm{d}s_{1}.\end{split}

(115)

The corresponding conservation law of Eq.(115) is

\begin{split}&(\cos\theta)_{t}=\Bigg{[}-\frac{1}{8}\alpha_{0}(t)\left(2\theta_{ss}-\theta_{s}^{2}\cos{\theta}\right)+\alpha_{1}(t)\cos{\theta}-C_{1}\left(\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\right)^{2}\\ &-\frac{1}{6}\left(\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\right)-\int^{s}A\mathrm{d}s_{1}\Bigg{]}_{s}\end{split}

(116)

with

\begin{split}&A=2C_{2}\sin{\theta}\int^{s}\cos{\theta}(s_{1},t)\mathrm{d}s_{1}+4\sin{\theta}\int^{s}\cos{\theta}(s_{1},t)\left[\int^{s_{1}}\cos{\theta}(s_{2},t)\left(\int^{s_{2}}\cos{\theta}(s_{3},t)\mathrm{d}s_{3}\right)\mathrm{d}s_{2}\right]\mathrm{d}s_{1}.\end{split}

(117)

The same hodograph transformation (87) leads (116) to

\displaystyle\frac{\partial}{\partial s}=\cos\theta\frac{\partial}{\partial x},

(118)

\displaystyle\begin{split}&\frac{\partial}{\partial t}=\Biggl{[}-\frac{1}{8}\alpha_{0}(t)(2\theta_{ss}\sin{\theta}-\theta_{s}^{2}\cos{\theta})+\alpha_{1}(t)\cos{\theta}-C_{1}\biggl{(}\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\biggr{)}-\frac{1}{6}\biggl{(}\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\biggr{)}^{4}\\ &-\int^{s}A\mathrm{d}s\Biggr{]}\frac{\partial}{\partial x}+\frac{\partial}{\partial t^{\prime}}.\end{split}

(119)

Applying (118) and (119) to Eq.(116), we obtain

\displaystyle(\tan(\theta))_{t^{\prime}}-\frac{1}{4}(\sin{\theta})_{xxx}-\Biggl{[}\frac{C_{1}\left(\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\right)^{2}+\frac{1}{6}\left(\int^{s}\sin{\theta}(s_{1},t)\mathrm{d}s_{1}\right)^{4}-\int^{s}A\mathrm{d}s_{1}}{\cos{\theta}}\Biggr{]}_{x}\frac{1}{\sin{\theta}}=0.

(120)

\begin{split}&v_{xt^{\prime}}=\frac{1}{4}\alpha_{0}(t)\left(\frac{v_{x}}{\sqrt{1+v_{x}^{2}}}\right)_{xxx}+C_{1}v^{2}v_{xx}+2C_{2}xv_{xx}+2C_{2}v_{xx}\int^{x}v\mathrm{d}x^{\prime}+4v_{xx}\int^{x}v^{\prime}_{x}\left[\left(\int^{x^{\prime}}v\mathrm{d}x^{\prime\prime\prime}\right)\mathrm{d}x^{\prime\prime}\right]\mathrm{d}x^{\prime}\\ &+2C_{1}v(1+v_{x}^{2})+\frac{2}{3}v^{3}(1+v_{x}^{2})+\frac{1}{6}v^{4}v_{xx}+2C_{2}x(1+v_{x}^{2})+4(1+v_{x}^{2})\int^{x}\left(\int^{x^{\prime}}v\mathrm{d}x^{\prime\prime}\right)\mathrm{d}x^{\prime}.\end{split}

(121)

We set $v=u_{xx}$ with $C_{1}=C_{2}=0$ and obtain

\begin{split}&u_{xxxt^{\prime}}=\frac{1}{4}\alpha_{0}(t)\left(\frac{u_{xxx}}{\sqrt{1+u_{xxx}^{2}}}\right)_{xxx}+\frac{1}{30}(u_{xx}^{5})_{xx}+\left[(4uu_{xx}-2u_{x}^{2})u_{xxx}\right]_{x}+\frac{2}{3}u_{xx}^{3}+4u.\end{split}

(122)

Now Eq.(122) is just the WKI-SP^(1,2) equation.

Remark 1: We believe that the hodograph transformation defined by (87) and (91) is not only true of the three pairs of equations given above, but also valid for the the whole MKdV-SG and the WKI-SP hierarchy.

4 Loop soliton solutions of the WKI-SP^(1,1) equation

For simplicity, we rewrite the MKdV-SG^(1,1) equation and the WKI-SP^(1,1) equation as

\displaystyle\theta_{ts}+\alpha\bigg{[}\frac{3}{2}\theta_{s}^{2}\theta_{ss}+\theta_{ssss}\bigg{]}=\beta\sin(\theta)

(123)

and

\displaystyle u_{xt}+\alpha\bigg{(}\frac{u_{x}}{\sqrt{1+u_{x}^{2}}}\bigg{)}_{xxx}=\beta\bigg{(}u+\frac{1}{6}(u^{3})_{xx}\bigg{)}

(124)

with $\alpha$ and $\beta$ being arbitrary real constants.

In Ref.[33, 34], the authors presented the following $N$ -soliton solution for the MKdV-SG^(1,1) equation (123):

\displaystyle\theta=2i\ln\frac{{f}^{*}}{f}

(125)

with

	$\displaystyle f=f_{N}=\sum_{\mu=0,1}\exp\bigg{[}\sum_{j=1}^{N}\mu_{j}(\xi_{j}+\frac{\pi}{2}i)+\sum_{1\leq j<l\leq N}^{N}\mu_{j}\mu_{l}A_{jl}\bigg{]},$		(126)
	$\displaystyle\xi_{j}=k_{j}s+\omega_{j}t+\xi_{j0},\quad\quad\omega_{j}=\frac{\beta}{k_{j}}-\alpha k_{j}^{3},\quad\quad{\rm{e}}^{A_{jl}}=\frac{(k_{j}-k_{l})^{2}}{(k_{j}+k_{l})^{2}},\quad\quad k_{i}\neq\pm k_{j}.$		(127)

where the summation of $\mu\equiv\{\mu_{1},\mu_{2},\ldots,\mu_{N}\}$ should be done for all permutations of $\mu_{j}=0,1,j=1,2,\ldots,N$ . Here ${f}^{*}$ is the conjugate of $f$ .

Due to the hodograph transformation (87) and (91), the loop soliton solutions of the WKI-SP^(1,1) equation are represented by the parametric form as

	$\displaystyle u=\frac{1}{\beta}[\theta_{t}+\alpha(\frac{1}{2}\theta_{s}^{3}+\theta_{sss})],\quad(\theta=2i\ln\frac{\bar{f}}{f}),$		(128)
	$\displaystyle x=s-\frac{2}{\beta}\bigg{[}(\ln{f\bar{f}})_{t}+\alpha(\ln{f\bar{f}})_{sss}+6\alpha\int^{s}\bigg{(}(\ln f)^{2}_{ss}+(\ln\bar{f})^{2}_{ss}\bigg{)}\mathrm{d}s\bigg{]}+x_{0},$		(129)

where $f$ satisfies (126).

4.1 One-loop soliton solution

In the case of $N=1$ , we have

\displaystyle f=1+{\rm{ie}}^{\xi_{1}},\quad\xi_{1}=k_{1}s+\bigg{(}\frac{\beta}{k_{1}}-\alpha k_{1}^{3}\bigg{)}t+\xi_{10}.

(130)

Substituting (130) into (128) and (129) leads to

	$\displaystyle u(s,t)=\frac{2}{k_{1}}{\rm{sech}}\xi_{1},$		(131)
	$\displaystyle x(s,t)=s-\frac{2}{k_{1}}\tanh\xi_{1}+d+\frac{2}{k_{1}},$		(132)

which is the well-known one soliton solution and depicted in Figures 1 and 2. The positive and negative $k_{1}$ correspond to loop soliton and antiloop soliton, respectively. Most of the time, we call both of them loop soliton for convenience.

Refer to caption — Figure 1: a, b, and c are the profiles of x as a function of s, u as a function of s, and one-loop soliton solution of the WKI-SP^(1,1) equation with parameters $\alpha=2$ , $\beta=2$ , $k_{1}=1.5$ , $\xi_{10}=0$ and $d_{1}=-\frac{4}{3}$ respectively.

To clarify the motion of the loop in the $(x,t)$ coordinate system, we rewrite (132) into the form

\displaystyle x-c_{1}t+x_{10}=\frac{\xi_{1}}{k_{1}}-\frac{2}{k_{1}}\tanh\xi_{1}+d+\frac{2}{k_{1}},

(133)

with $c_{1}={\alpha}k_{1}^{2}-\frac{\beta}{k_{1}^{2}}$ and $x_{10}=\frac{\xi_{10}}{k_{1}}$ .

Here, $c_{1}$ denotes the velocity of the loop soliton. For the given nonzero $\alpha$ and $\beta$ , it shows that the loop soliton could propagate to either the left (i.e.,negative x direction) or the right (i.e., positive x direction). This is different from the SP equation and the WKI equation. The soliton of the SP equation and the WKI equation can only move in one direction of $x$ -axis. From (131), we see the amplitude (defined by $A_{1}$ ) of the loop soliton is $\frac{2}{k_{1}}$ . Then we have

\displaystyle c_{1}=\frac{4\alpha}{A_{1}^{2}}-\frac{{\beta}A_{1}^{2}}{4}.

(134)

Interestingly, the above expression shows the loop solitons of the WKI-SP^(1,1) equation have abundant solitonic behaviors. On the one hand, the large loop can move more rapidly than the small loop. On the other hand, the small loop can also move more rapidly than the large loop, which is different from the typical solitonic movement.

4.2 Interaction of two loop solitons

For the two-loop soliton solution, (126) gives

\displaystyle f=1+{\rm ie}^{\xi_{1}}+{\rm ie}^{\xi_{2}}-{\gamma}{\rm e}^{\xi_{1}+\xi_{2}}

(135)

with

\displaystyle\gamma=\bigg{(}\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\bigg{)}^{2}.

(136)

Due to (128) and (129), the parametric representation of the solution reads

	$\displaystyle u(s,t)=\frac{2\sqrt{\gamma}}{k_{1}k_{2}}\frac{(k_{1}+k_{2})\cosh\Psi_{1}\cosh\Psi_{2}+(k_{1}-k_{2})\sinh\Psi_{1}\sinh\Psi_{2}}{\cosh^{2}\Psi_{1}+{\gamma}\sinh^{2}\Psi_{2}},$		(137)
	$\displaystyle x(s,t)=s+\frac{1}{k_{1}k_{2}}\frac{(k_{1}-k_{2})\sinh 2\Psi_{1}-\gamma(k_{1}+k_{2})\sinh 2\Psi_{2}}{\cosh^{2}\Psi_{1}+\gamma\sinh^{2}\Psi_{2}}+\frac{2(k_{1}+k_{2})}{k_{1}k_{2}}+d,$		(138)

where we have put

\displaystyle\Psi_{1}=\frac{1}{2}(\xi_{1}-\xi_{2}),\quad\Psi_{2}=\frac{1}{2}(\xi_{1}+\xi_{2})+\frac{1}{2}\ln\gamma

(139)

for simplicity.

The two-loop soliton has abundant solitonic behaviors. Two loops can move in the same direction of the $x$ -axes while they could move towards each other in the opposite directions. It could be the fast small loop chasing the slow high one, and it might be the fast high loop chasing the slow small one. To figure out what each parameter does, we rewrite the amplitude and the velocity as follows

\displaystyle A_{i}=\frac{2}{k_{i}},\quad c_{i}={\alpha}k_{i}^{2}-\frac{\beta}{k_{i}^{2}},\quad(i=1,2).

(140)

A table is listed to intuitively show the chasing or the meeting movement. Since $|k_{1}|\neq|k_{2}|$ , we assume $|k_{1}|>|k_{2}|$ without generality. In the following table, $v_{low}$ represents the speed of the small loop soliton while $v_{high}$ represents the speed of the large loop soliton.

Table 1: The role of each parameter plays in the movement of two-loop soliton

		one loop chases another loop				two loops
		both move to the positive $x$ -axis	both move to the negative $x$ -axis	$v_{low}>v_{high}$	$v_{low}<v_{high}$	meet each other
	$\sqrt[4]{\frac{\beta}{\alpha}}<\|k_{2}\|<\|k_{1}\|$	$\checkmark$ $(\alpha>0,\beta>0)$	$\checkmark$ $(\alpha<0,\beta<0)$	$\checkmark$
$\alpha\beta>0$	$\|k_{2}\|<\|k_{1}\|<\sqrt[4]{\frac{\beta}{\alpha}}$	$\checkmark$ $(\alpha<0,\beta<0)$	$\checkmark$ $(\alpha>0,\beta>0)$		$\checkmark$
	$\|k_{2}\|<\sqrt[4]{\frac{\beta}{\alpha}}<\|k_{1}\|$					$\checkmark$
	$\frac{\sqrt{-\alpha\beta}}{\|\alpha\|\|k_{1}\|}<\|k_{2}\|<\|k_{1}\|$	$\checkmark$ $(\alpha>0,\beta<0)$	$\checkmark$ $(\alpha<0,\beta>0)$	$\checkmark$
$\alpha\beta<0$	$\|k_{2}\|<min\{\|k_{1}\|,\frac{\sqrt{-\alpha\beta}}{\|\alpha\|\|k_{1}\|}\}$	$\checkmark$ $(\alpha>0,\beta<0)$	$\checkmark$ $(\alpha<0,\beta>0)$		$\checkmark$

Some diverse figures are worked out to exhibit different solitonic movements. In Figures 3 and 4, both the two loop solitons are moving to the right. There are both a small but fast loop soliton chasing a larger and slower loop soliton. Two loop solitons with dissimilar amplitudes are shown in Figures 3, in which a small and fast loop soliton maybe observed traveling around a larger and slower loop soliton. Two loop solitons with similar amplitudes ( $k_{1}\approx k_{2}$ ) are shown in Figures 4, in which the loops do not over lap and they just seems to exchange their amplitudes during the period of the nonlinear interaction.

Figures 5 exhibit two loop solitons are moving to the left. Here a small and slower loop soliton catches up with a larger and fast loop soliton.

Figures 6 illustrate one loop soliton and one antiloop soliton move towards each other in the opposite directions. The antiloop travels to the right and the loop travels to the left. As time goes, two solitons merge and then they separate each other with leaving the original wave profiles.

4.3 Interaction of three loop solitons

For the three-loop soliton case, formulas (126) reduce to

\displaystyle f=1+{\rm ie}^{\xi_{1}}+{\rm ie}^{\xi_{2}}+{\rm ie}^{\xi_{3}}-{\gamma_{12}}{\rm e}^{\xi_{1}+\xi_{2}}-{\gamma_{13}}{\rm e}^{\xi_{1}+\xi_{3}}-{\gamma_{23}}{\rm e}^{\xi_{2}+\xi_{3}}-{\rm i}{\gamma_{123}}{\rm e}^{\xi_{1}+\xi_{2}+\xi_{3}}

(141)

with

\displaystyle\gamma_{ij}=\bigg{(}\frac{k_{i}-k_{j}}{k_{i}+k_{j}}\bigg{)}^{2},\quad\gamma_{123}=\gamma_{12}\gamma_{13}\gamma_{23}.

(142)

By virtue of (128) and (129), the three-loop soliton in the parametric form is provided as follows

	$\displaystyle u(s,t)=\frac{2\Xi_{1}}{\Xi},$		(143)
	$\displaystyle x(s,t)=s+\frac{2\Xi_{2}}{\Xi}+\frac{2(k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3})}{k_{1}k_{2}k_{3}}+d.$		(144)

where

\displaystyle\begin{split}\Xi&=k_{1}k_{2}k_{3}[2(\gamma_{12}\sinh^{2}\Psi_{3}+\gamma_{13}\sinh^{2}\Psi_{2}+\gamma_{23}\sinh^{2}\Psi_{1})+2(1-\gamma_{13})\sqrt{\gamma_{12}\gamma_{23}}\cosh(\Psi_{1}+\Psi_{3})\\ &+2(1-\gamma_{23})\sqrt{\gamma_{12}\gamma_{13}}\cosh(\Psi_{2}+\Psi_{3})+2(1-\gamma_{12})\sqrt{\gamma_{13}\gamma_{23}}\cosh(\Psi_{1}+\Psi_{2})\\ &+\gamma_{123}\cosh(2(\Psi_{1}+\Psi_{2}+\Psi_{3}))+(\gamma_{12}+\gamma_{13}+\gamma_{23})]\end{split}

(145)

\begin{split}\Xi_{1}=&\sum_{1\leqslant m<n\leqslant 3}\gamma_{m,n}\bigg{(}\sum_{1\leqslant i<j\leqslant 3}k_{i}k_{j}-2k_{m}k_{n}\bigg{)}-\gamma_{123}\sum_{1\leqslant i<j\leqslant 3}k_{i}k_{j}\\ &+2k_{1}k_{2}\sqrt{\gamma_{13}\gamma_{23}}\bigg{[}\gamma_{12}\cosh(\Psi_{1}+\Psi_{2}+2\Psi_{3})+\cosh(\Psi_{1}-\Psi_{2})\bigg{]}\\ &+2k_{1}k_{3}\sqrt{\gamma_{12}\gamma_{23}}\bigg{[}\gamma_{13}\cosh(\Psi_{1}+2\Psi_{2}+\Psi_{3})+\cosh(\Psi_{1}-\Psi_{3})\bigg{]}\\ &+2k_{2}k_{3}\sqrt{\gamma_{12}\gamma_{13}}\bigg{[}\gamma_{23}\cosh(2\Psi_{1}+\Psi_{2}+\Psi_{3})+\cosh(\Psi_{2}-\Psi_{3})\bigg{]}\end{split}

(146)

\begin{split}\Xi_{2}=&\gamma_{123}(k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3})\sinh(2(\Psi_{1}+\Psi_{2}+\Psi_{3}))+\gamma_{12}(-k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3})\sinh(2\Psi_{3})\\ &+\gamma_{13}(k_{1}k_{2}-k_{1}k_{3}+k_{2}k_{3})\sinh(2\Psi_{2})+\gamma_{23}(k_{1}k_{2}+k_{1}k_{3}-k_{2}k_{3})\sinh(2\Psi_{1})\\ &+k_{2}k_{3}(\gamma_{23}-1)\sqrt{\gamma_{12}\gamma_{13}}\sinh(\Psi_{2}+\Psi_{3})+k_{1}k_{3}(\gamma_{13}-1)\sqrt{\gamma_{12}\gamma_{23}}\sinh(\Psi_{1}+\Psi_{3})\\ &+k_{1}k_{2}(\gamma_{12}-1)\sqrt{\gamma_{13}\gamma_{23}}\sinh(\Psi_{1}+\Psi_{2})\end{split}

(147)

with

	$\displaystyle\Psi_{1}=\frac{1}{2}(-\xi_{1}+\xi_{2}+\xi_{3})+\frac{1}{2}\ln(\gamma_{23}),$		(148)
	$\displaystyle\Psi_{2}=\frac{1}{2}(\xi_{1}-\xi_{2}+\xi_{3})+\frac{1}{2}\ln(\gamma_{13}),$		(149)
	$\displaystyle\Psi_{3}=\frac{1}{2}(\xi_{1}+\xi_{2}-\xi_{3})+\frac{1}{2}\ln(\gamma_{12}).$		(150)

Two examples of such solution are plotted below (see Figures 7 and 8). Figures 7 describe three loop solitons with different speed propagating to the positive direction. It shows rich catch-up phenomenon. Figures 8 describes two loop solitons colliding with one antiloop soliton.

Remark 2. Due to the multiloop soliton solutions obtained in this section, the corresponding multibreather solutions and soliton molecules would be constructed directly. Under certain condition imposed on the parameters, the breather solution is shown to yield a nonsingular oscillating pulse solution of the WKI-SP^(1,1) equation, which we shall term the breather solution as well. In fact, the $M$ -breather solution of the WKI-SP^(1,1) equation can be constructed from the $M$ -breather solution of the MKdV-SG^(1,1) equation (125)-(127) with $N=2M$ by choosing the parameters appropriately. So we specify the parameters as

\displaystyle k_{2j-1}=k_{2j}^{*},\quad\xi_{2j-1,0}=\xi_{2j,0}^{*}

(151)

to satisfy

\displaystyle\xi_{2j-1}=\xi_{2j}^{*}.

(152)

The parametric solution (128) and (129) with (151) and (152) describes multiple collisions of $M$ breathers.

Soliton molecules [35, 36, 37, 38] can be formed in some possible mechanisms both theoretically and experimentally. Here, we can obtain loop soliton molecules of the WKI-SP^(1,1) equation by introducing the velocity resonance conditions[39]

\displaystyle\frac{k_{i}}{k_{j}}=\frac{\alpha k_{i}^{3}-\frac{\beta}{k_{i}}}{\alpha k_{j}^{3}-\frac{\beta}{k_{j}}}.

(153)

Two loop solitons are bounded to form a loop soliton molecule which are shown in Fig.9(a). One can see that the distance between two loops will not be changed as time goes on. Further, one can also immediately construct the breather molecules by using the velocity resonant mechanism. Two breathers constitute one breather molecule which is shown in Fig.9(b). It is worth mentioning that such loop-type soliton molecules and nonsingular oscillating pulse type breather molecules are firstly reported theoretically.

5 Conclusion

In this paper, we have constructed the generalized compound WKI-SP^(n,m) equations by virtue of two compatible linear spectral equations. By solving two sets of recursive equations, we showed three SP-type equations (named WKI-SP^(0,1), WKI-SP^(0,2) and WKI-SP^(0,3)), two WKI-type equations (named WKI-SP^(1,0) and WKI-SP^(2,0)) and three compound WKI-SP equations (called WKI-SP^(1,1), WKI-SP^(1,2) and WKI-SP^(2,1)). The classical WKI equation (1) is just the WKI-SP^(1,0) equation while the the SP equation equation (2) is the WKI-SP^(0,1) equation. The WKI-SP^(0,2) and WKI-SP^(0,3) equations are the high-order short pulse equations.

It is known that there is a novel hodograph between the SP equation (2) and the SG equation. And there also exits the hodograph transformation between the WKI equation (1) and the (potential) MKdV equation. In this paper, with the help of one conservation law, we have found one hodograph transformation which successfully transformed the compound WKI-SP^(1,1) equation into the compound MKdV-SG^(1,1) equation. Meanwhile, the same hodograph transformation could also enable us to convert the compound WKI-SP^(1,2) equation and the WKI-SP^(2,1) equation into the MKdV-SG^(1,2) equation and the MKdV-SG^(2,1) equation respectively.

We constructed multiloop soliton solutions for the WKI-SP^(1,1) equation as applications. The loop (antiloop) soliton solutions arise from the kink (antikink) solutions of the MKdV-SG^(1,1) equation. Let $m$ and $N-m$ be the number of positive and negative $k_{j}$ , respectively. Then, the corresponding soliton solution would include $m$ loop solitons and $N-m$ antiloop solitons. Here, we also call antiloop as loop for convenience. The analytic solutions for the one-loop, two-loop and three-loop solitons in the parametric forms were shown. Especially, we stressed the description of two-loop soliton, which exhibit abundant solitonic behaviors. On the one hand, two loop solitons can either move in the same direction or in the opposite direction of the $x$ -axis. It is worth mentioned that there could be a small loop with a fast speed chasing a large loop with a slow speed. Certainly, we also obtained the typical solitonic behavior that the large loop moves more rapidly than the small loop. On the other hand, there are two kinds of collision processes depending on the ratio of the eigenvalues involved. In the first case of two loop solitons with dissimilar amplitudes chasing each other, the small one would travel around the larger one. The phase shift of the small loop soliton can be seen to originate mainly from the delay caused by its travelling around the large loop soliton. In the second case of the collision process for two loop solitons with similar amplitudes, the slow loop soliton is pushed out forward with a considerable phase shift when compared with that of the fast loop. That is to say the slow loop soliton pass through the fast one.

It is also remarkable that all the models given by the WKI-SP^(n,m) hierarchy here could be with time-varying coefficients. Hence, their dispersion relations will have a time dependent velocity and the solitons will accelerate. These equations with variable coefficients may be nice candidates in applications having accelerated ultra-short optical pulses. The method and the results mentioned in this paper would be extended to the complex WKI-SP hierarchy, the nonlocal WKI-SP hierarchy, the semi-discrete and the discrete WKI-SP hierarchy, the multi-component WKI-SP hierarchy (and with their corresponding complex, nonlocal, semi-discrete and discrete forms). These problems are to be pursued in our near future work. The Darboux transformation, Bäcklund transformation and the Riemann-Hilbert problem for the compound WKI-SP equation are also in our near consideration.

Acknowledgments

This work is supported by the national natural science foundation of China (Grant No. 11771395 and No. 11871336) and the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY18A010034).

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Compound WKI-SP hierarchy and multiloop soliton solutions

Abstract

keywords:

1 Introduction

2 Compound equations of the WKI-SP type

2.1 AKNS-type and WKI-type Lax pair

2.2 WKI-SP type integrable equations

2.2.1 SP-hierarchy for n=0n=0

2.2.2 WKI-hierarchy for m=0m=0

2.2.3 Compound WKI-SP hierarchy for n≠0,m≠0n\neq 0,m\neq 0

3 Hodograph transformations between MKdV-SG and WKI-SP eqautions

3.1 Hodograph transformation between MKdV-SG(1,1) and WKI-SP(1,1) equation

3.2 Hodograph transformation between MKdV-SG(2,1) and WKI-SP(2,1) equation

3.3 Hodograph transformation between MKdV-SG(1,2) and WKI-SP(1,2) equation

4 Loop soliton solutions of the WKI-SP(1,1) equation

4.1 One-loop soliton solution

4.2 Interaction of two loop solitons

4.3 Interaction of three loop solitons

5 Conclusion

Acknowledgments

References

References

2.2.1 SP-hierarchy for $n=0$

2.2.2 WKI-hierarchy for $m=0$

2.2.3 Compound WKI-SP hierarchy for $n\neq 0,m\neq 0$

3.1 Hodograph transformation between MKdV-SG^(1,1) and WKI-SP^(1,1) equation

3.2 Hodograph transformation between MKdV-SG^(2,1) and WKI-SP^(2,1) equation

3.3 Hodograph transformation between MKdV-SG^(1,2) and WKI-SP^(1,2) equation

4 Loop soliton solutions of the WKI-SP^(1,1) equation