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The construction of solutions to Zakharov–Kuznetsov equation with fractional power nonlinear terms

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Abstract

In the paper, we study a plasma fluid physical model, namely the Zakharov–Kuznetsov (ZK, for simplicity) equation with fractional power nonlinear terms by the complete discrimination system for polynomial method, and give a detailed construction of all its single traveling wave solutions. The results show abundant traveling wave patterns of the ZK equation.

Introduction

Due to the role of nonlinear differential equations in modeling physical phenomena, an important task in nonlinear science is to find the exact solutions to those equations. Correspondingly, a lot of useful methods have been proposed to solve nonlinear differential equations, including direct expansion methods and ansatz methods (see, for example [1,2,3,4,5] and the references therein). In addition, some further arguments on L-integrability and Liouville theorem in infinite-dimensional Hamiltonian systems have been given [6]. However, for those direct expansion methods, a weakness is that one cannot obtain complete results though the considered equations can be reduced to integral forms. In order to overcome this weakness, Liu proposed the complete discrimination system for polynomial method to give the complete classifications of all single traveling wave solutions for such nonlinear differential equations [7,8,9,10,11]. In Liu’s method, we firstly reduce the considered partial differential equation to the integral form under the traveling wave transformation, then we can use the complete discrimination system for polynomial to give the factorization when the integrand is determined by a polynomial, and finally we classify the solutions of the corresponding integral. If the equation cannot be directly reduced to an integral form, we can use Liu’s trial equation method to get an integrable sub-equation, from which we obtain exact solutions by using the complete discrimination system for polynomial [12,13,14,15,16,17]. Liu’s methods have been further applied and developed extensively to solve more nonlinear problems (see, for example, [17,18,19,20,21,22,23] and the references therein). For exact solutions of nonlinear differential equations, there are a large number of results (see, for example, Seadawy et al.’s interesting works [24,25,26,27,28,29,30,31,32,33]). Recently, Ma et al. proposed a powerful method, namely the transformed rational function method, for generating traveling wave solutions and studied deeply and extensively another kind of exact solutions called lump solutions [34,35,36,37,38,39,40,41].

In the paper, we consider a kind of the Zakharov–Kuznetsov (ZK, for simplicity) equations with fractional power nonlinear terms. The ZK equation was first derived to describe the behavior of weakly nonlinear ion-acoustic waves in plasma which comprises cold ions and hot isothermal electrons when there is a uniform magnetic field [42,43,44]. Moreover, it also governs two-dimensional modulations of a KdV soliton equation in fluid mechanics. The usual ZK equation reads

$$ u_{t}+auu_{x}+(u_{xx}+u_{yy})_{x}=0, $$
(1)

which is more difficult to study than the KP equation because it is not integrable under the meaning of the inverse scattering transform method. In addition, we know that the solitary-wave solutions of the ZK equation are inelastic.

Furthermore, to describe ion-acoustic waves in cold-ion plasma when the behavior of electrons is not isothermal, Schamel [45] derived the new ZK equations with fractional power nonlinear terms as follows:

$$ u_{t}+u^{\frac{1}{2}}u_{x}+au_{xxx}=0 $$
(2)

and

$$ u_{t}+\bigl(1+bu^{\frac{1}{2}}\bigr)u_{x}+ \frac{1}{2}u_{xxx}=0. $$
(3)

Monro and Parkes also obtained a different fractional power form of the ZK equation

$$ 16u_{t}+20\bigl(u^{\frac{3}{2}}\bigr)_{x}+u_{xxx}=0. $$
(4)

Wazwaz obtained some special forms of exact solutions to a fractional ZK equation by the sine-cosine ansatz method [46]. Other related studies on the equation can be seen in [47, 48]. However, to our knowledge, all single traveling wave solutions to the ZK equations have not been given [43,44,45,46,47,48].

In the paper, we use the complete discrimination system for polynomial to give a complete classification of all single traveling wave solutions for two fractional power ZK equations. From a practical point of view, the classification of traveling wave solutions is very convenient since we can determine the type of solutions when the conditions of concrete parameters are given.

The paper is organized as follows. In Sect. 2, we give the classification for \(\gamma =\frac{1}{2}\). In Sect. 3, we give the classification for \(\gamma =\frac{3}{2}\). In the last section, we give a short discussion and conclusion.

Classification of solutions: first case

We consider a general form of the fractional power ZK equation

$$ au_{t}+b\bigl(u^{\gamma }\bigr)_{x}+u_{xxx}=0, $$
(5)

where a and b are coefficients, and γ in general is a fractional number such as \(\gamma =\frac{1}{2}\) and \(\gamma = \frac{3}{2}\) and so on. It is easy to see that this ZK equation includes the previous ZK equations as special examples.

Substituting traveling wave transformation

$$ u(x,t)=u(\xi ),\quad \xi =x-\omega t $$
(6)

into Eq. (5), we get the ODE as follows:

$$ -a\omega u'+b\bigl(u^{\gamma }\bigr)'+u'''=0. $$
(7)

Integrating the above equation twice yields

$$ \bigl(u'\bigr)^{2}=b_{0}+b_{1}u+a \omega u^{2}-\frac{2b}{\gamma +1}u^{\gamma +1}, $$
(8)

where \(b_{0}\) and \(b_{1}\) are two integral constants. If γ is a rational number, we can write it as \(\gamma =\frac{n}{m}\), and then we have

$$ \bigl(u'\bigr)^{2}=b_{0}+b_{1}u+a \omega u^{2}-\frac{2bm}{n+m}u^{\frac{n+m}{m}}. $$
(9)

Further, we take \(u=v^{m}\) to get

$$ m^{2}v^{2m-2}\bigl(v'\bigr)^{2}=b_{0}+b_{1}v^{m}+a \omega v^{2m}-\frac{2bm}{n+m}v ^{n+m}. $$
(10)

Writing it as an integral form, we have

$$ \int \frac{v^{m-1}\,\mathrm{d}v}{\sqrt{b_{0}+b_{1}v^{m}+a\omega v^{2m}- \frac{2bm}{n+m}v^{n+m}}}=\pm \frac{1}{m}(\xi -\xi _{0}), $$
(11)

where \(\xi _{0}\) is an integral constant. This integral form is our starting point of solving exact solutions. If we denote \(F(v)=b_{0}+b _{1}v^{m}+a\omega v^{2m}-\frac{2bm}{n+m}v^{n+m}\), it is easy to see that the solutions of integral depend on the roots of \(F(v)\) completely. A powerful mathematical tool, namely a complete discrimination system for polynomial, can be used to solve the problem. Firstly, we take \(\gamma =\frac{1}{2}\) respectively to give all solutions to the corresponding integral

$$ \int \frac{v\,\mathrm{d}v}{\sqrt{b_{0}+b_{1}v^{2}-\frac{4b}{3}v^{3}+a \omega v^{4}}}=\pm \frac{1}{2}(\xi -\xi _{0}). $$
(12)

We consider two cases \(b_{0}=0\) and \(b_{0}\neq 0\) to give solutions respectively.

Case 1. \(b_{0}=0\). Then the integral form becomes

$$ \int \frac{\mathrm{d}v}{\sqrt{b_{1}-\frac{4b}{3}v+a\omega v^{2}}}= \pm \frac{1}{2}(\xi -\xi _{0}). $$
(13)

Denote \(\Delta =b^{2}_{3}-4b_{1}a\omega \) as the discrimination of \(f(v)=b_{1}-\frac{4b}{3}v+a\omega v^{2}\). According to the values of Δ, we obtain the following solutions.

Family 2.1.1. \(\Delta =0\). Then we have \(f(v)=a\omega (v- \alpha )^{2}\), where \(\alpha =\sqrt{\frac{b_{1}}{a\omega }}\). We get the solution

$$ u=\biggl\{ \alpha \pm \exp \biggl(\pm \frac{\sqrt{a\omega }}{2}(\xi -\xi _{0})\biggr)\biggr\} ^{2}. $$
(14)

Family 2.1.2. \(\Delta >0\). Then we have \(f(v)=a\omega (v- \alpha )(v-\beta )\), where α and β are two distinct real roots of \(f=0\). We get the solution

$$ u=\biggl\{ \frac{\alpha -\beta (1\pm \exp (\pm \frac{\sqrt{a\omega }}{2}( \xi -\xi _{0})))}{\pm \exp (\pm \frac{\sqrt{a\omega }}{2}(\xi -\xi _{0}))}\biggr\} ^{2}. $$
(15)

Family 2.1.3. \(\Delta <0\). Then we have \(f(v)=a\omega ((v- \alpha )^{2}+\beta ^{2})\), where \(\alpha =\frac{2b}{3a\omega }\), \(\beta =\sqrt{\frac{b_{1}}{a\omega }-\frac{4b^{2}}{9a^{2}\omega ^{2}}}\). We get the solution

$$ u=\biggl\{ \alpha \pm \beta \coth \biggl(\pm \frac{\sqrt{a\omega }}{2}(\xi -\xi _{0})\biggr)\biggr\} ^{2}. $$
(16)

Case 2. \(b_{0}\neq 0\). Without loss of generality, assume \(a\omega >0\) and denote

$$ F(v)=v^{4}+a_{3}v^{3}+a_{2}v^{2}+a_{1}x+a_{0}, $$
(17)

where \(a_{3}=-\frac{4b}{3a\omega }\), \(a_{2}=\frac{b_{1}}{a\omega }\), \(a _{0}=\frac{b_{0}}{a\omega }\). The complete discrimination system for the fourth order polynomial \(F(v)\) is given by [11]

$$\begin{aligned}& \begin{aligned} &D_{1}=4,\qquad D_{2}=-p,\qquad D_{3}=8rp-2p^{3}-9q^{2}, \\ &D_{4}=4p^{4}r-p^{3}q^{2}+36prq^{2}-32r^{2}p^{2} -\frac{27}{4}q^{4}+64r ^{3}, \\ &F_{2}=9q^{2}-32pr, \end{aligned} \end{aligned}$$
(18)

where \(p=a_{2}\), \(q=\frac{a_{3}^{3}}{8}-\frac{a_{2}a_{3}}{2}\), \(r=a _{0}+\frac{a_{2}a_{3}^{2}}{16}-\frac{3a_{3}^{4}}{256} \). There are the following cases to be discussed.

Family 2.2.1. \(D_{4}=0\), \(D_{3}=0\), \(D_{2}<0 \). Then we have

$$ F(v)=v^{2}+lv+s^{2}, $$
(19)

where l, s are real numbers, and \(l^{2}-4s^{2}<0 \). We have the solution represented by the implicit function u on ξ

$$\begin{aligned}& \pm \frac{\sqrt{a\omega }}{2}(\xi -\xi _{0})= \frac{1}{2\sqrt{2s-l}}\ln \frac{\sqrt{u}\mp \sqrt{2s-l} u^{ \frac{1}{4}}+s}{\sqrt{u}\pm \sqrt{2s-l} u^{\frac{1}{4}}+s} \\& \quad {}+\frac{1}{\sqrt{2s+l}}\biggl\{ \arctan \frac{4u^{\frac{1}{4}}\pm \sqrt{2s-l}}{2\sqrt{2s+l}}+\arctan \frac{4u^{\frac{1}{4}}\mp \sqrt{2s-l}}{2\sqrt{2s+l}} \biggr\} . \end{aligned}$$
(20)

Family 2.2.2. \(D_{4}=0\), \(D_{3}=0\), \(D_{2}=0 \). Then we have

$$ F(v)=(v-\alpha )^{4}. $$
(21)

If \(\alpha >0\), we have

$$\begin{aligned} \pm \frac{\sqrt{a\omega }}{2}(\xi -\xi _{0})=\frac{1}{2\sqrt{ \alpha }}\ln \biggl\vert \frac{u^{\frac{1}{4}} -\sqrt{\alpha }}{u^{\frac{1}{4}}+\sqrt{ \alpha }} \biggr\vert -\frac{u^{\frac{1}{4}}}{\sqrt{u}-\alpha }. \end{aligned}$$
(22)

If \(\alpha <0\), we have

$$\begin{aligned} \pm \frac{\sqrt{a\omega }}{2}(\xi -\xi _{0})=\frac{1}{\sqrt{- \alpha }}\arctan \frac{u^{\frac{1}{4}}}{\sqrt{-\alpha }} -\frac{u ^{\frac{1}{4}}}{\sqrt{u}-\alpha }. \end{aligned}$$
(23)

Family 2.2.3. \(D_{4}=0\), \(D_{3}=0\), \(D_{2}>0\), \(E_{2}=0 \). Then we have

$$ F(v)=(v-\alpha )^{2}(u-\beta )^{2}, $$
(24)

where α, β are real numbers, and \(\alpha >\beta \). If \(\alpha >\beta >0\), we have

$$\begin{aligned} \pm \frac{\sqrt{a\omega }}{2}(\alpha +\beta ) (\xi -\xi _{0})=\sqrt{ \alpha }\ln \biggl\vert \frac{u^{\frac{1}{4}}-\sqrt{\alpha }}{u^{\frac{1}{4}} +\sqrt{ \alpha }} \biggr\vert -\sqrt{\beta }\ln \biggl\vert \frac{u^{\frac{1}{4}}-\sqrt{\beta }}{u ^{\frac{1}{4}}+\sqrt{\beta }} \biggr\vert . \end{aligned}$$
(25)

If \(0>\alpha >\beta \), we have

$$\begin{aligned} \pm \frac{\sqrt{a\omega }}{2}(\alpha +\beta ) (\xi -\xi _{0})=2\sqrt{- \alpha }\arctan \frac{u^{\frac{1}{4}}}{\sqrt{-\alpha }}-2\sqrt{- \beta }\arctan \frac{u^{\frac{1}{4}}}{\sqrt{-\beta }}. \end{aligned}$$
(26)

If \(\alpha >0>\beta \), we have

$$\begin{aligned} \pm \frac{\sqrt{a\omega }}{2}(\alpha +\beta ) (\xi -\xi _{0})=\sqrt{ \alpha }\ln \biggl\vert \frac{u^{\frac{1}{4}}-\sqrt{\alpha }}{u^{\frac{1}{4}}+\sqrt{ \alpha }} \biggr\vert -2\sqrt{-\beta }\arctan \frac{u^{\frac{1}{4}}}{\sqrt{- \beta }}. \end{aligned}$$
(27)

Family 2.2.4. In all other cases such as \(F(v)=(v-\alpha )^{2}(v- \beta )(v-\gamma )\) and so on, the corresponding solutions can be expressed in terms of elliptic functions and hyper-elliptic functions. We omit them for brevity.

Classification of solutions: second case

Now we consider the second case \(\gamma =\frac{3}{2}\). Then we have

$$ \int \frac{v\,\mathrm{d}v}{\sqrt{b_{0}+b_{1}v^{2}+a\omega v^{4}- \frac{4b}{5}v^{5}}}=\pm \frac{1}{2}(\xi -\xi _{0}). $$
(28)

By taking the transformation

$$ v=\biggl(-\frac{5}{4b}\biggr)^{\frac{1}{5}}w-\frac{a\omega }{4b}, $$
(29)

the above equation becomes

$$ \int \frac{(w-s_{0})\,\mathrm{d}w}{\sqrt{F(w)}}=\pm \biggl(-\frac{4b}{5} \biggr)^{ \frac{2}{5}}\frac{1}{2}(\xi -\xi _{0}), $$
(30)

where

$$ F(w)=w^{5}+pw^{3}+qw^{2}+rw+s, $$
(31)

and

$$\begin{aligned}& \begin{aligned}&p=10d^{2}-4c_{4}d,\qquad q=-10d^{3}+6c_{4}d^{2}+c_{2},\qquad r=5d^{4}-4c_{4}d^{3}-2c _{2}d, \\ &s=b_{0}-d^{5}+c_{4}d^{4}+c_{2}d^{2},\qquad c_{4}=a\omega \biggl(-\frac{5}{4b}\biggr)^{ \frac{4}{5}}, \\ &c_{2}=\biggl(b-\frac{5}{4b}\biggr)^{\frac{2}{5}},\qquad s_{0}= \frac{a \omega }{4b}\biggl(-\frac{4b}{5}\biggr)^{\frac{1}{5}}. \end{aligned} \end{aligned}$$
(32)

We write its complete discrimination system of \(F(w)\) as follows (see [7, 10]):

$$\begin{aligned}& \begin{aligned} &D_{2}=-p,\qquad D_{3}=40rp-12p^{3}-45q^{2}, \\ &D_{4}=12p^{4}r-4p^{3}q^{2}+117prq^{2}-88r^{2}p^{2} \\ &\hphantom{D_{4}=}{}-40qsp^{2}-27q^{4}-300qrs+160r^{3}, \\ &D_{5}=-1600qsr^{3}-3750pqs^{3}+2000ps^{2}r^{2}-4p^{3}q^{2}r^{2}+16p ^{3}q^{3}s \\ &\hphantom{D_{5}=}{}-900rs^{2}p^{3} +825p^{2}q^{2}s^{2}+144pq^{2}r^{3}+2250rq^{2}s^{2}+16p ^{4}r^{3} \\ &\hphantom{D_{5}=}{}+108p^{5}s^{2}-128r^{4}p^{2} -27r^{2}q^{4}+108sq^{5}+256r^{5}+3125s ^{4} \\ &\hphantom{D_{5}=}{}-72rsqp^{4}+560sqr^{2}p^{2}-630prsq^{3}, \\ & E_{2}=160r^{2}p^{3}+900q^{2}r^{2}-48rp^{5}+60rP^{2}q^{2}+1500pqrs+16q ^{2}p^{4} \\ &\hphantom{E_{2}=}{}-1100qsp^{3} +625s^{2}p^{2}-3375sq^{3}, \\ &F_{2}=3q^{2}-8rp.\end{aligned} \end{aligned}$$
(33)

According to the above complete discrimination system, we list the following eleven cases to discuss. Among these, in first five cases, the solutions can be represented in terms of elementary functions, while in other cases the solutions are given by elliptic functions or elliptic integrals.

Family 3.1. \(D_{5}=0\), \(D_{4}=0\), \(D_{3}>0\), \(E_{2}\neq 0 \). Then we have

$$ F(w)=(w-\alpha )^{2}(w-\beta )^{2}(w-\gamma ), $$
(34)

where α, β, γ are real numbers, and \(\gamma \neq \alpha >\beta \neq \gamma \). When \(w>\gamma \), we have

$$\begin{aligned}& \pm (\xi -\xi _{0})=\frac{2(\alpha -s_{0})}{(\alpha -\beta ) \sqrt{ \gamma -\alpha }}\arctan {\frac{\sqrt{w-\gamma }}{\sqrt{\gamma - \alpha }}} \\& \hphantom{\pm (\xi -\xi _{0})=}{}-\frac{2(s_{0}-\beta )}{(\alpha -\beta ) \sqrt{\gamma -\beta }} \arctan {\frac{\sqrt{w-\gamma }}{\sqrt{\gamma -\beta }}}, \quad \gamma > \alpha , \end{aligned}$$
(35)
$$\begin{aligned}& \pm (\xi -\xi _{0})=-\frac{2(s_{0}-\beta )}{(\alpha -\beta )\sqrt{ \gamma -\beta }} \arctan {\frac{\sqrt{((-\frac{5}{4b})^{1/5}- \frac{a}{4b})^{-1}v-\gamma }}{\sqrt{\gamma -\beta }}} \\& \hphantom{\pm (\xi -\xi _{0})=}{}+\frac{(\alpha -s_{0})}{(\alpha -\beta ) \sqrt{\alpha -\gamma }} \\& \hphantom{\pm (\xi -\xi _{0})=}{}\times \ln { \biggl\vert \frac{ \sqrt{((-\frac{5}{4b})^{1/5}-\frac{a}{4b})^{-1}v- \gamma }-\sqrt{\alpha -\gamma }}{ \sqrt{((-\frac{5}{4b})^{1/5}- \frac{a}{4b})^{-1}v-\gamma }+\sqrt{\alpha -\gamma }} \biggr\vert },\quad \beta < \gamma < \alpha , \end{aligned}$$
(36)

or

$$\begin{aligned} \pm (\xi -\xi _{0}) =&\frac{\alpha -s_{0}}{(\alpha -\beta )\sqrt{ \alpha -\gamma }} \ln { \biggl\vert \frac{\sqrt{((-\frac{5}{4b})^{1/5}- \frac{a}{4b})^{-1}v-\gamma }- \sqrt{\alpha -\gamma }}{\sqrt{((- \frac{5}{4b})^{1/5}-\frac{a}{4b})^{-1}v-\gamma }+\sqrt{\alpha - \gamma }} \biggr\vert } \\ &{}-\frac{s_{0}-\beta }{(\alpha -\beta )\sqrt{\beta -\gamma }}\ln { \biggl\vert \frac{\sqrt{((-\frac{5}{4b})^{1/5}-\frac{a}{4b})^{-1}v-\gamma } -\sqrt{ \beta -\gamma }}{\sqrt{((-\frac{5}{4b})^{1/5}-\frac{a}{4b})^{-1}v- \gamma }+\sqrt{\beta -\gamma }} \biggr\vert },\quad \gamma < \beta . \end{aligned}$$
(37)

Family 3.2. \(D_{5}=0\), \(D_{4}=0\), \(D_{3}=0\), \(D_{2}\neq 0\), \(F_{2} \neq 0 \). Then we have

$$ F(w)=(w-\alpha )^{3}(w-\beta )^{2}, $$
(38)

where α, β are real numbers, and \(\alpha \neq \beta \). When \(w>\alpha \), we have

$$\begin{aligned} \pm \frac{\alpha -\beta }{2}(\xi -\xi _{0}) =&\frac{\beta -s_{0}}{\sqrt{((- \frac{5}{4b})^{1/5}-\frac{a}{4b})^{-1}v-\alpha }} \\ &{}+(\alpha -s_{0})\sqrt{\alpha -\beta }\arctan {\frac{\sqrt{((- \frac{5}{4b})^{1/5}-\frac{a}{4b})^{-1}v- \alpha }}{\sqrt{\alpha - \beta }}},\quad \alpha >\beta , \end{aligned}$$
(39)

or

$$\begin{aligned} \pm \frac{\alpha -\beta }{2}(\xi -\xi _{0}) =&\frac{\beta -s_{0}}{\sqrt{((- \frac{5}{4b})^{1/5}-\frac{a}{4b})^{-1}v-\alpha }} \\ &{}-\frac{\alpha -s_{0}}{2\sqrt{\beta -\alpha }}\ln { \biggl\vert \frac{\sqrt{((- \frac{5}{4b})^{1/5}-\frac{a}{4b})^{-1}v-\alpha }-\sqrt{\beta - \alpha }}{ \sqrt{((-\frac{5}{4b})^{1/5}-\frac{a}{4b})^{-1}v-\alpha }+\sqrt{\beta -\alpha }} \biggr\vert },\quad \alpha < \beta . \end{aligned}$$
(40)

Family 3.3. \(D_{5}=0\), \(D_{4}=0\), \(D_{3}=0\), \(D_{2}\neq 0\), \(F_{2}=0 \). Then we have

$$ F(w)=(w-\alpha )^{4}(w-\beta ), $$
(41)

where α, β, are real numbers, and \(\alpha \neq \beta \). When \(w>\alpha \), we have

$$\begin{aligned} \pm (\xi -\xi _{0}) =&\frac{\alpha -s_{0}}{\alpha -\beta }\frac{\sqrt{((- \frac{5}{4b})^{1/5}-\frac{a}{4b})^{-1}v-\beta }}{((-\frac{5}{4b})^{1/5}- \frac{a}{4b})^{-1}v-\alpha } \\ &{}+\biggl\{ \frac{\alpha -s_{0}}{2(\alpha -\beta )^{\frac{3}{2}}}-\frac{1}{\sqrt{ \alpha -\beta }}\biggr\} \arctan { \frac{ \sqrt{((-\frac{5}{4b})^{1/5}- \frac{a}{4b})^{-1}v-\beta }}{\sqrt{\beta -\alpha }}},\quad \alpha < \beta , \end{aligned}$$
(42)

or

$$\begin{aligned} \pm (\xi -\xi _{0}) =&\frac{\alpha -s_{0}}{\beta -\alpha }\frac{\sqrt{(- \frac{5}{4b})^{1/5}-\frac{a}{4b})^{-1}v-\beta }}{((-\frac{5}{4b})^{1/5}- \frac{a}{4b})^{-1}v-\alpha } \\ &{}+\biggl\{ \frac{\alpha -s_{0}}{(\beta -\alpha )^{\frac{3}{2}}}+\frac{1}{\sqrt{ \beta -\alpha }}\biggr\} \\ &{}\times \ln { \biggl\vert \frac{\sqrt{((-\frac{5}{4b})^{1/5}- \frac{a}{4b})^{-1}v-\beta }-\sqrt{\beta -\alpha }}{ \sqrt{((- \frac{5}{4b})^{1/5}-\frac{a}{4b})^{-1}v-\beta }+\sqrt{\beta -\alpha }} \biggr\vert }, \quad \alpha >\beta . \end{aligned}$$
(43)

Family 3.4. \(D_{5}=0\), \(D_{4}=0\), \(D_{3}=0\), \(D_{2}=0 \). Then we have

$$ F(w)=(w-\alpha )^{5}, $$
(44)

where α is real number. When \(w>\alpha \), we have

$$\begin{aligned} \pm (\xi -\xi _{0}) =&-2\biggl(\biggl(\biggl(-\frac{5}{4b} \biggr)^{1/5}-\frac{a}{4b}\biggr)^{-1}v- \alpha \biggr)^{-\frac{1}{2}} \\ &{}-\frac{2(\alpha -s_{0})}{3} \biggl(\biggl(\biggl(-\frac{5}{4b} \biggr)^{1/5}- \frac{a}{4b}\biggr)^{-1}v-\alpha \biggr)^{-\frac{2}{3}}. \end{aligned}$$
(45)

Family 3.5. \(D_{5}=0\), \(D_{4}=0\), \(D_{3}<0\), \(E_{2}\neq 0 \). Then we have

$$ F(w)=(w-\alpha ) \bigl(w^{2}+rw+s\bigr)^{2}, $$
(46)

where α is real number, and \(r^{2}-4s<0 \). When \(w>\alpha \), we have

$$\begin{aligned} \pm (\xi -\xi _{0}) =&\frac{b-\alpha +s_{0}}{4ab}\ln \biggl(\biggl(\biggl(- \frac{5}{4b}\biggr)^{1/5}- \frac{a}{4b}\biggr)^{-2} \biggl(\biggl(-\frac{5}{4b}\biggr)^{1/5}-\frac{a}{4b} \biggr)^{-2}v^{2} \\ &{}+r\biggl(\biggl(-\frac{5}{4b}\biggr)^{1/5}-\frac{a}{4b} \biggr)^{-1}v+s\biggr) \\ &{}+ \frac{7b-\alpha +s _{0}}{2b\sqrt{4b-a^{2}}}\arctan \frac{2(-\frac{5}{4b})^{1/5}- \frac{a}{4b})^{-1}v-a}{4b-a^{2}}, \end{aligned}$$
(47)

where

$$ b=\sqrt{\alpha ^{2}+r\alpha +s},\qquad a=\sqrt{2b-r-2\alpha }. $$
(48)

Family 3.6. \(D_{5}=0\), \(D_{4}>0 \). Then we have

$$ F(w)=(w-\alpha )^{2}(w-\alpha _{1}) (w-\alpha _{2}) (w-\alpha _{3}), $$
(49)

where α, \(\alpha _{1}\), \(\alpha _{2}\), \(\alpha _{3}\) are real numbers, and \(\alpha _{1}>\alpha _{2}>\alpha _{3} \). We have

$$\begin{aligned} \pm (\xi -\xi _{0}) =& \int \frac{\mathrm{d}w}{\sqrt{(w-\alpha _{1})(w- \alpha _{2})(w-\alpha _{3})}} \\ &{}-\frac{2(\alpha -s_{0})}{(\alpha -\alpha _{2})\sqrt{\alpha _{2}-\alpha _{3}}}\biggl\{ \operatorname {F}(\varphi ,k) -\frac{\alpha _{1}-\alpha _{2}}{\alpha _{1}- \alpha }\varPi \biggl( \varphi ,\frac{\alpha _{1}-\alpha _{2}}{\alpha _{1}-\alpha },k\biggr)\biggr\} , \end{aligned}$$
(50)

where

$$\begin{aligned}& \operatorname {F}(\varphi ,k)= \int _{0}^{\varphi }\frac{\mathrm{d}\varphi }{\sqrt{1-k ^{2}\sin ^{2}\varphi }}, \end{aligned}$$
(51)
$$\begin{aligned}& \varPi (\varphi ,h,k)= \int _{0}^{\varphi }\frac{\mathrm{d}\varphi }{(1+h \sin ^{2}\varphi )\sqrt{1-k^{2}\sin ^{2}\varphi }}. \end{aligned}$$
(52)

Family 3.7. \(D_{5}=0\), \(D_{4}=0\), \(D_{3}<0\), \(E_{2}=0 \). Then we have

$$ F(w)=(w-\alpha )^{3}\bigl((w-l_{1})^{2}+s_{1}^{2} \bigr), $$
(53)

where α, \(l_{1}\), and \(s_{1} \) are real numbers. When \(w> \alpha \) and \(\alpha \neq l_{1}+s_{1}\), we have

$$\begin{aligned} \pm (\xi -\xi _{0}) =& \int \frac{\mathrm{d}w}{\sqrt{(w-\alpha )((w-l _{1})^{2}+s_{1}^{2})}} \\ &{}+(\alpha -s_{0})\biggl\{ \frac{\tan \theta +\cot \theta }{2(s_{1}\tan \theta -l_{1}-\alpha )\sqrt{\frac{s_{1}}{\sin ^{3}{2\theta }}}}\operatorname {F}(\varphi ,k)- \frac{s_{1}\tan \theta +s_{1}\cot \theta }{s_{1}\cot \theta +l _{1}+\alpha } \\ &{}\times \biggl\{ (\frac{\tan \theta +l_{1}+\alpha }{(s_{1}\cot \theta +l_{1}- \alpha )\sin \varphi }\sqrt{1-k^{2} \sin ^{2}\varphi } +\operatorname {F}(\varphi ,k)-\operatorname {E}(\varphi ,k)\biggr\} \biggr\} , \end{aligned}$$
(54)

where

$$ E(\varphi , k)= \int _{0}^{\varphi }\sqrt{1-k^{2}\sin ^{\theta }} \,\mathrm{d}\theta . $$
(55)

Family 3.8. \(D_{5}=0\), \(D_{4}<0 \). Then we have

$$ F(w)=(w-\alpha )^{2}(w-\beta ) \bigl((w-l_{1})^{2}+s_{1}^{2} \bigr), $$
(56)

where α, \(l_{1}\), and \(s_{1} \) are real numbers. The solution is represented by

$$\begin{aligned} \pm (\xi -\xi _{0}) =& \int \frac{\mathrm{d}w}{\sqrt{(w-\beta )((w-l _{1})^{2}+s_{1}^{2})}} \\ &{}+(\alpha -s_{0})\biggl\{ \frac{\tan \theta +\cot \theta }{2(s_{1}\tan \theta -l_{1}-\alpha )\sqrt{\frac{s}{\sin ^{3}{2\theta }}}}\operatorname {F}(\varphi ,k)- \frac{s_{1}\tan \theta +s\cot \theta }{s_{1}\cot \theta +l_{1}+\alpha } \\ &{}\times \biggl\{ (\frac{\tan \theta +l_{1}+\alpha }{(s\cot \theta +l_{1}- \alpha )\sin \varphi }\sqrt{1-k^{2} \sin ^{2}\varphi } +\operatorname {F}(\varphi ,k)-\operatorname {E}(\varphi ,k)\biggr\} \biggr\} . \end{aligned}$$
(57)

Family 3.9. \(D_{5}=0\), \(D_{4}=0\), \(D_{3}>0\), \(E_{2}=0 \). Then we have

$$ F(w)=(w-\alpha )^{3}(w-\beta ) (w-\gamma ), $$
(58)

where α, β, and γ are real numbers. The solution is represented by

$$\begin{aligned} \pm (\xi -\xi _{0}) =& \int \frac{\mathrm{d}w}{\sqrt{(w-\alpha )(w- \beta )(w-\gamma )}} \\ &{}+\frac{(\alpha -\beta )(\alpha -s_{0})}{2\sqrt{\alpha -\gamma }}\operatorname {E}\biggl( \arcsin \sqrt{\frac{\alpha -\gamma }{w-\gamma }},\sqrt{ \frac{ \beta -\gamma }{\alpha -\gamma }}\biggr) -\sqrt{\frac{w-\beta }{(w-\gamma )(w-\alpha )}}. \end{aligned}$$
(59)

In other cases, we can give the corresponding solutions similarly. We omit them for simplicity.

Family 3.10. In the following three cases: \(D_{5}>0\), \(D_{4}>0\), \(D_{3}>0\), \(D_{2}>0\) or \(D_{5}<0\) or \(D_{5}>0\wedge (D_{4}\leq 0 \vee D_{3}\leq 0\vee D_{2}\leq 0) \), where means “and”, means “or”, we have respectively

$$\begin{aligned}& F(w)=(w-\alpha _{1}) (w-\alpha _{2}) (w-\alpha _{3}) (w-\alpha _{4}) (w-\alpha _{5}), \end{aligned}$$
(60)
$$\begin{aligned}& F(w)=(w-\alpha _{1}) (w-\alpha _{2}) (w-\alpha _{3}) \bigl((w-l)^{2}+s^{2}\bigr), \end{aligned}$$
(61)

or

$$ F(w)=(w-\alpha ) \bigl((w-l_{1})^{2}+s_{1}^{2} \bigr)) \bigl((w-l_{2})^{2}+s_{2}^{2} \bigr). $$
(62)

Then the corresponding solutions can be expressed by hyper-elliptic functions or hyper-elliptic integral. We omit them for brevity.

Conclusion

By the complete discrimination system for polynomial method, we give the complete classification of the traveling wave solutions to a fractional power ZK equation under \(\gamma =\frac{1}{2}\) and \(\gamma = \frac{3}{2}\). The last case is more difficult than the first one since we need the complete discrimination system for fifth order polynomial. These results mean that there are rich traveling wave patterns for the ZK equation. Our results provide a complete classification of all single traveling wave solutions to two fractional power ZK equations. If we take the concrete parameters in a real model, we can give the corresponding representation of the solution. Therefore, it is rather convenient for practice. On the other hand, most of the solutions have implicit function forms if we consider w or v as the function of ξ. But, inversely, if we take ξ as the function of w or v, the solutions will become explicit functions. In general, these implicit function solutions cannot be obtained by the direct expansion methods and ansatz methods in [46, 47].

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Acknowledgements

The authors would like to thank to reviewers for their helpful suggestions.

Funding

This project is supported by the National Natural Science Foundation of China under Grant No. 51674086, 51534004.

Author information

YL gave the main ideas and computations including those in Sect. 1, Sect. 3, and Sect. 4. XW gave the computation of Sect. 2. Both authors read and approved the final manuscript.

Correspondence to Yang Liu.

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Keywords

  • The complete discrimination system for polynomial
  • Exact solution
  • Traveling wave solution
  • Zakharov–Kuznetsov equation