1. bookVolume 6 (2021): Issue 1 (January 2021)
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Travelling wave solutions to the proximate equations for LWSW

Published Online: 25 May 2021
Page range: 335 - 346
Received: 02 Jan 2021
Accepted: 26 Feb 2021
Journal Details
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Journal
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01 Jan 2016
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Abstract

By feat of Maple 17 and a subsidiary ordinary differential equation, a new extension algebraic method is chosen to construct the travelling wave solutions to the proximate equation set involving an arbitrary parameter for long waves over shallow-water. Multiple triangle periodic solutions and new Jacobi elliptic function solutions are obtained. This procedure is applicable to other nonlinear partial differential equations as well.

Keywords

Introduction

Investigating exact solutions of nonlinear partial differential equations (PDEs) has always been a key subject in mathematical physics. Some direct ways are presented, for example, the tanh-function means [1,2,3], Jacobi elliptic function expansion way [4,5,6], the F-expansion [7], (G/G) - expansion way [8,9,10,11,12,13] and the homogeneous balance means [14, 15]. In this paper, a kind of new extension algebraic means is presented to construct the explicit solutions of the following proximate long water wave equations in shallow water [12] utuuxvx+αuxx=0 {u_t} - u{u_x} - {v_x} + \alpha {u_{{\rm{xx}}}} = 0 vt(uv)xαvxx=0 {v_t} - {({\rm{uv}})_x} - \alpha {v_{{\rm{xx}}}} = 0 where α ≠ 0 is an arbitrary constant.

Eqs (1a) and (1b) express the interaction of long water waves in time and one spatial dimension. The special case of Eqs (1a) and (1b) utuuxvx+12uxx=0 {u_t} - u{u_x} - {v_x} + {1 \over 2}{u_{{\rm{xx}}}} = 0 vt(uv)x12vxx=0 {v_t} - {(uv)_x} - {1 \over 2}{v_{{\rm{xx}}}} = 0 was discussed by Broer [16] and Whitham [17, 18]; Broer first discovered that Eqs (2a) and (2b) are equivalent to Boussinesq equations for two-way dispersive long waves in shallow water by an appropriate variational principles. Kupersh- midt [19] studied the symmetry and conservation law of Eq. (2). Furthermore, Wang and Zhou [20], Zhang [21], Yan [22] and Manafiana et al. [23] obtained abundant periodic solutions, soliton solutions and soliton-like solutions of Eq. (2) by using the homogeneous balance and extended homogeneous balance., Yan et al. [20, 21] derived some hyperbolic function solutions and periodic solutions of Eq. (2) by deleting tail auxiliary function method. Eqs (1a) and (1b) are the generalised case of Eq. (2), which involves an arbitrary constant. Wang et al. [12] obtained many hyperbolic solutions and trigonometric period solutions of Eqs (1a) and (1b) by using(G/G) -expansion means. This paper tries to seek some important and undiscovered travelling wave solutions, especially Jacobi elliptic function solutions of Eqs (1a) and (1b) by using an auxiliary ODE (ordinary differential equation).

The new extension algebraic means

We’d like to provide the main steps of a new extension algebraic means in this section based on (G/G)-expansion way in literatures [8,9,10,11,12,13] and the auxiliary equation methods in Refs [25,26,27,28,29,30,31,32]. Provided that a general nonlinear PDE with respect to two independent variables x and t expressed as F(ut,ux,utt,uxt,...)=0 F({u_t},{u_x},{u_{{\rm{tt}}}},{u_{{\rm{xt}}}},...) = 0 where u = u(x,t) is an unknown smooth function, F(ut,ux,utt,uxt,...) is a polynomial with respect to u(x,t) and its various partial derivatives.

where is an unknown smooth function, is a polynomial with respect toand its various partial derivatives.

We assume the travelling wave transformations u(x,t)=u(ξ),ξ=μxct u(x,t) = u(\xi ),\;\xi = \mu x - {\rm{ct}} reduces Eq. (3) to an ODE for u(ξ) as follows H(u,u,u,u,...)=0 H(u,u',u'',u''',...) = 0 where and in Eq. (4) are arbitrary constants to be determined later and the superscripts indicate the ordinary derivatives in regard to variable where μ and c in Eq. (4) are arbitrary constants to be determined later and the superscripts indicate the ordinary derivatives in regard to variable ξ

If Eq. (5) is integrable, then integrating term by term one or more times yields constant(s) of integration.

Assume that the solution of Eq. (5) can be shown by the following form u=c0+i=1n(ai0+ai1ϕbi0+bi1ϕ)i,αn1bn10, u = {c_0} + \sum\nolimits_{i = 1}^n {\left( {{{{a_{i0}} + {a_{i1}}\phi '} \over {{b_{i0}} + {b_{i1}}\phi }}} \right)^i},\;{\alpha _{n1}}{b_{n1}} \ne 0, where c0,ai0,ai1,bi0,bi1(i = 1,2,...,n) are arbitrary constants, n is a positive integer and ϕ = ϕ(ξ) satisfies the following assistant ODE ϕ=εj=0mejϕj,ε=±1 \phi ' = \varepsilon \sqrt {\sum\nolimits_{j = 0}^m {e_j}{\phi ^j}} ,\;\;\;\varepsilon = \pm 1 where ej, j = 0,1,2,3,...,m are arbitrary constants and m is a positive integer. The solutions of Eq. (7) are given by Table 1 in the appendix.

The integer m and n can be obtained through using the homogeneous balance between the highest-order nonlinear terms and the highest-order derivatives in Eq. (5). Substituting Eq. (6) along with Eq. (7) into Eq. (5) with the aim of Maple17 and equating the coefficients of ϕij=0mejϕj(i=0,1,2,3,4...) {\phi ^i}\sqrt {\sum\nolimits_{j = 0}^m {e_j}{\phi ^j}} (i = 0,1,2,3,4...) to zero forms an algebraic equation set for μ,c,c0,ai0,bi0(i = 0,1,2,...,n). Suppose the numerical values of the foregoing constants can be gotten by solving the algebraic equations set, now that the solutions of Eq. (7) have been known, substituting the numerical values of μ,c,c0,ai0,bi0(i = 0,1,2,...,n), ej(j = 0,1,2,...,m) and the solutions of Eq. (7) into Eq. (6), so the travelling wave solutions of Eq. (3) will be found.

In Section 3, we obtain new and multiple travelling wave solutions of the proximate long water Eq. (1) by using the proposed means.

The proximate long water equations

The travelling wave variable is assumed that u(x,t)=u(ξ),v(x,t)=v(ξ)ξ=μxct u(x,t) = u(\xi ),v(x,t) = v(\xi )\quad \xi = \mu x - ct

The travelling wave transformations Eq. (8) can convert Eq. (1) into the following ODEs for u = u(ξ) and v = v(ξ) cuμuuμv+αμ2u=0 - cu' - \mu uu' - \mu v' + \alpha {\mu ^2}u'' = 0 cvμ(uv)αμ2v=0 - {\rm{cv}}' - \mu ({\rm{uv}})' - \alpha {\mu ^2}v'' = 0

Integrating the ODEs above in regard to ξ yields F1cu12μu2μv+αμ2u=0 {F_1} - {\rm{cu}} - {1 \over 2}\mu {u^2} - \mu v + \alpha {\mu ^2}u' = 0 F2cvμuvαμ2v=0 {F_2} - cv - \mu uv - \alpha {\mu ^2}{v^\prime} = 0 where F1 and F2 are integration constants to be determined later.

Eqs (9a) and (10a) can be described respectively in the following forms v=cμuuu+αμu v' = - {c \over \mu }u' - uu' + \alpha \mu u'' v=F1μcμu12u2+αμu v = {{{F_1}} \over \mu } - {c \over \mu }u - {1 \over 2}{u^2} + \alpha \mu u'

Substituting Eq. (11) into Eq. (10b) and simplifying it, Eqs (1a) and (1b) are reduced to an ODE as follows (F2cμF1)+(c2μF1)u+3c2u2+12μu3α2μ3u=0 ({F_2} - {c \over \mu }{F_1}) + ({{{c^2}} \over \mu } - {F_1})u + {{3c} \over 2}{u^2} + {1 \over 2}\mu {u^3} - {\alpha ^2}{\mu ^3}u'' = 0

Making full use of the homogeneous balance between the highest derivative u and the highest nonlinear term u3 in Eq. (12) yields that m is an arbitrary positive integer and n = 1. We only analyse one case in the following discussion.

Let n = 1, m = 4,

In this situation, Eqs (6) and (7) are transformed into the following equations, respectively. u(ξ)=c0+a0+a1ϕb0+b1ϕ, u(\xi ) = {c_0} + {{{a_0} + {a_1}\phi '} \over {{b_0} + {b_1}\phi }} ϕ=εe0+e1ϕ+e2ϕ2+e3ϕ3+e4ϕ4 \phi ' = \varepsilon \sqrt {{e_0} + {e_1}\phi + {e_2}{\phi ^2} + {e_3}{\phi ^3} + {e_4}{\phi ^4}}

By substituting Eq. (13) into Eq. (12) with Eq. (14), the left-hand side of Eq. (12) is converted into another polynomial of ϕi(i = 0,1,2,3,4,...,8) and ϕij=04ejϕj(i=0,1,2,3,4,...,7) {\phi ^i}\sqrt {\sum\nolimits_{j = 0}^4 {e_j}{\phi ^j}} (i = 0,1,2,3,4,...,7) . And setting the coefficients of ϕi(i = 0,1,2,...,) and ϕij=04ejϕj(i=0,1,2,3,4,...,6) {\phi ^i}\sqrt {\sum\nolimits_{j = 0}^4 {e_j}{\phi ^j}} (i = 0,1,2,3,4,...,6) to zero gives rise to an algebraic equation set. Solving the algebraic equation set with Maple 17 yields the followings:

Case1 a1 = ±2αμb1, c0=cμ {c_0} = - {c \over \mu } , a0 = 0, F2 = 0, b0=e34e4b1 {b_0} = {{{e_3}} \over {4{e_4}}}{b_1} , e1=e3(4e2e4e32)8e42 {e_1} = {{{e_3}(4{e_2}{e_4} - e_3^2)} \over {8e_4^2}} , F1=3α2μ4e328α2μ4e2e4+2c2e44μe4 {F_1} = - {{3{\alpha ^2}{\mu ^4}e_3^2 - 8{\alpha ^2}{\mu ^4}{e_2}{e_4} + 2{c^2}{e_4}} \over {4\mu {e_4}}} .

Where e0, e2, e3 are arbitrary constants, and μ,c, e4 and b1 are arbitrary nonzero constants.

Case 2 a1 = 0, b0 = 0, b1=a02αμe0 {b_1} = {{{a_0}} \over {2\alpha \mu \sqrt {{e_0}} }} , F2=α3μ4e0(2e3e0e1e2)2e0 {F_2} = {{{\alpha ^3}{\mu ^4}\sqrt {{e_0}} (2{e_3}{e_0} - {e_1}{e_2})} \over {2{e_0}}} , F1=μ(±2c0αμe1e02c02e04α2μ2e0e2+α2μ2e12)4e0 {F_1} = {{\mu ( \pm 2{c_0}{\,{\alpha}{\mu}\,}{e_1}\sqrt {{e_0}} - 2c_0^2{e_0} - 4{\alpha ^2}{\mu ^2}{e_0}{e_2} + {\alpha ^2}{\mu ^2}e_1^2)} \over {4{e_0}}} , c=μ(±αμe12c0e0)2e0 c = {{\mu ( \pm \alpha \mu {e_1} - 2{c_0}\sqrt {{e_0}} )} \over {2\sqrt {{e_0}} }} , where c0,e1,e2,e3,a0 are arbitrary constants, and μ ≠ 0,e0 > 0,e4 ≠ 0.

Choosing different values of ej(j = 0, 1, 2, 3, 4), Eq. (11) has many different fundamental solutions, substituting all solutions of Eq. (15) into Eq. (13), and considering Eq. (11b) yields multiple Jaccobi elliptic and solution-like solutions of Eq. (1) and are listed as follows.

For case1, Jaccobi elliptic function solutions have

1.1) e0 = 1, e2 = −(1 − k2), e4 = k2, e1 = e3 = 0,

u1.1.1=cμ+2μα(1k2)cs(ξ)dn(ξ) {u_{1.1.1}} = - {c \over \mu } + 2\mu \alpha (1 - {k^2}){\rm{cs}}(\xi ){\rm{dn}}(\xi ) , v1.1.1=4α2μ2ns(ξ)2 {v_{1.1.1}} = - 4{\alpha ^2}{\mu ^2}{\rm{ns}}{(\xi )^2} ; u1.1.2=cμ2μα(1k2)cs(ξ)dn(ξ) {u_{1.1.2}} = - {c \over \mu } - 2\mu \alpha (1 - {k^2}){\rm{cs}}(\xi ){\rm{dn}}(\xi ) , v1.1.2=4α2μ2k2sn(ξ)2 {v_{1.1.2}} = - 4{\alpha ^2}{\mu ^2}{k^2}{\rm{sn}}{(\xi )^2} ;

u1.1.3=cμ2μα(1k2)sc(ξ)nd(ξ) {u_{1.1.3}} = - {c \over \mu } - 2\mu \alpha (1 - {k^2}){\rm{sc}}(\xi ){\rm{nd}}(\xi ) , v1.1.3=4α2μ2dc(ξ)2 {v_{1.1.3}} = - 4{\alpha ^2}{\mu ^2}{\rm{dc}}{(\xi )^2} , u1.1.4=cμ+2μα(1k2)sc(ξ)nd(ξ) {u_{1.1.4}} = - {c \over \mu } + 2\mu \alpha (1 - {k^2}){\rm{sc}}(\xi ){\rm{nd}}(\xi ) , v1.1.4=4α2μ2k2cd(ξ)2 {v_{1.1.4}} = - 4{\alpha ^2}{\mu ^2}{k^2}{\rm{cd}}{(\xi )^2} ,

1.2) e0 = 1 − k2, e2 = 2k2 − 1, e4 = −k2, e1 = e3 = 0,

u1.2.1=cμ2μαsc(ξ)dn(ξ) {u_{1.2.1}} = - {c \over \mu } - 2\mu \alpha {\rm{sc}}(\xi ){\rm{dn}}(\xi ) , v1.2.1=4α2μ2(1k2)nc(ξ)2 {v_{1.2.1}} = - 4{\alpha ^2}{\mu ^2}({\rm{1}} - {k^2}){\rm{nc}}{(\xi )^2} ; u1.2.2=cμ+2μαsc(ξ)dn(ξ) {u_{1.2.2}} = - {c \over \mu } + 2\mu \alpha {\rm{sc}}(\xi ){\rm{dn}}(\xi ) , v1.2.2=4α2μ2k2cn(ξ)2 {v_{1.2.2}} = 4{\alpha ^2}{\mu ^2}{k^2}{\rm{cn}}{(\xi )^2} ;

1.3) e0 = k2 − 1, e2 = 2 − k2, e4 = −1, e1 = e3 = 0

u1.3.1=cμ2μαk2sd(ξ)cn(ξ) {u_{1.3.1}} = - {c \over \mu } - 2\mu \alpha {k^2}{\rm{sd}}(\xi ){\rm{cn}}(\xi ) , v1.3.1=4α2μ2(k21)nd(ξ)2 {v_{1.3.1}} = - 4{\alpha ^2}{\mu ^2}({k^2} - {\rm{1}}){\rm{nd}}{(\xi )^2} ,

1.4) e0 = k2, e2 = −(1 + k2), e4 = 1, e1 = e3 = 0,

u1.4.1=cμ2μαdn(ξ)cs(ξ) {u_{1.4.1}} = - {c \over \mu } - 2\mu \alpha {\rm{dn}}(\xi ){\rm{cs}}(\xi ) , v1.3.1=4α2μ2k2sn(ξ)2 {v_{1.3.1}} = - 4{\alpha ^2}{\mu ^2}{k^2}{\rm{sn}}{(\xi )^2} ; u1.4.2=cμ+2μαdn(ξ)cs(ξ) {u_{1.4.2}} = - {c \over \mu } + 2\mu \alpha {\rm{dn}}(\xi ){\rm{cs}}(\xi ) , v1.4.2=4α2μ2ns(ξ)2 {v_{1.4.2}} = - 4{\alpha ^2}{\mu ^2}{\rm{ns}}{(\xi )^2} , u1.4.4=cμ2μα(1k2)nd(ξ)sc(ξ) {u_{1.4.4}} = - {c \over \mu } - 2\mu \alpha (1 - {k^2}){\rm{nd}}(\xi ){\rm{sc}}(\xi ) , v1.4.4=4α2μ2dc(ξ)2 {v_{1.4.4}} = - 4{\alpha ^2}{\mu ^2}{\rm{dc}}{(\xi )^2} ,

1.5) e0 = −k2, e2 = 2k2 − 1, e4 = 1 − k2, e1 = e3 = 0,

u1.5.1=cμ+2μαsn(ξ)dc(ξ) {u_{1.5.1}} = - {c \over \mu } + 2\mu \alpha {\rm{sn}}(\xi ){\rm{dc}}(\xi ) , v1.5.1=4α2μ2k2cn(ξ)2 {v_{1.5.1}} = 4{\alpha ^2}{\mu ^2}{k^2}{\rm{cn}}{(\xi )^2} ; u1.5.2=cμ2μαsn(ξ)dc(ξ) {u_{1.5.2}} = - {c \over \mu } - 2\mu \alpha {\rm{sn}}(\xi ){\rm{dc}}(\xi ) , v1.5.2=4α2μ2(1k2)nc(ξ)2 {v_{1.5.2}} = - 4{\alpha ^2}{\mu ^2}(1 - {k^2}){\rm{nc}}{(\xi )^2} ,

1.6) e0 = −1, e2 = 2 − k2, e4 = k2 − 1, e1 = e3 = 0,

u1.61=cμ+2μαk2sn(ξ)cd(ξ) {u_{1.61}} = - {c \over \mu } + 2\mu \alpha {k^2}{\rm{sn}}(\xi ){\rm{cd}}(\xi ) , v1.61=4α2μ2dn(ξ)2 {v_{1.61}} = 4{\alpha ^2}{\mu ^2}{\rm{dn}}{(\xi )^2} ; u1.62=cμ2μαk2sn(ξ)cd(ξ) {u_{1.62}} = - {c \over \mu } - 2\mu \alpha {k^2}{\rm{sn}}(\xi ){\rm{cd}}(\xi ) , v1.62=4α2μ2(k21)nd(ξ)2 {v_{1.62}} = - 4{\alpha ^2}{\mu ^2}({k^2} - 1){\rm{nd}}{(\xi )^2} ;

1.7) e0 = 1 − k2, e2 = 2 − k2, e4 = 1, e1 = e3 = 0,

u1.7.1=cμ2μαnc(ξ)ds(ξ) {u_{1.7.1}} = - {c \over \mu } - 2\mu \alpha {\rm{nc}}(\xi ){\rm{ds}}(\xi ) , v1.7.1=4α2μ2(1k2)sc(ξ)2 {v_{1.7.1}} = - 4{\alpha ^2}{\mu ^2}({\rm{1}} - {k^2}){\rm{sc}}{(\xi )^2} ; u1.7.2=cμ+2μαnc(ξ)ds(ξ) {u_{1.7.2}} = - {c \over \mu } + 2\mu \alpha {\rm{nc}}(\xi ){\rm{ds}}(\xi ) , v1.7.2=4α2μ2cs(ξ)2 {v_{1.7.2}} = - 4{\alpha ^2}{\mu ^2}{\rm{cs}}{(\xi )^2} ,

1.8) e0 = 1, e2 = 2 − k2, e4 = 1 − k2, e1 = e3 = 0,

u1.8.1=cμ+2μαns(ξ)dc(ξ) {u_{1.8.1}} = - {c \over \mu } + 2\mu \alpha {\rm{ns}}(\xi ){\rm{dc}}(\xi ) , v1.8.1=4α2μ2cs(ξ)2 {v_{1.8.1}} = - 4{\alpha ^2}{\mu ^2}{\rm{cs}}{(\xi )^2} ; u1.8.2=cμ2μαns(ξ)dc(ξ) {u_{1.8.2}} = - {c \over \mu } - 2\mu \alpha {\rm{ns}}(\xi ){\rm{dc}}(\xi ) , v1.8.2=4α2μ2(1k2)sc(ξ)2 {v_{1.8.2}} = - 4{\alpha ^2}{\mu ^2}(1 - {k^2}){\rm{sc}}{(\xi )^2} ,

1.9) e0 = 1, e2 = 2k2 − 1, e4 = k2(k2 − 1), e1 = e3 = 0,

u1.9.1=cμ+2μαns(ξ)cd(ξ) {u_{1.9.1}} = - {c \over \mu } + 2\mu \alpha {\rm{ns}}(\xi ){\rm{cd}}(\xi ) , v1.91=4α2μ2ds(ξ)2 {v_{1.91}} = - 4{\alpha ^2}{\mu ^2}{\rm{ds}}{(\xi )^2} ; u1.9.2=cμ2μαns(ξ)cd(ξ) {u_{1.9.2}} = - {c \over \mu } - 2\mu \alpha {\rm{ns}}(\xi ){\rm{cd}}(\xi ) , v1.9.2=4α2μ2k2(k21)sd(ξ)2 {v_{1.9.2}} = - 4{\alpha ^2}{\mu ^2}{k^2}({k^2} - 1){\rm{sd}}{(\xi )^2} ,

1.10) e0 = k2(k2 − 1), e2 = 2k2 − 1, e4 = 1, e1 = e3 = 0,

u1.10.1=cμ2μαcd(ξ)ns(ξ) {u_{1.10.1}} = - {c \over \mu } - 2\mu \alpha {\rm{cd}}(\xi ){\rm{ns}}(\xi ) , v1.101=4α2μ2k2(k21)sd(ξ)2 {v_{1.101}} = - 4{\alpha ^2}{\mu ^2}{k^2}({k^2} - 1){\rm{sd}}{(\xi )^2} ; u1.10.2=cμ2μαcd(ξ)ns(ξ) {u_{1.10.2}} = - {c \over \mu } - 2\mu \alpha {\rm{cd}}(\xi ){\rm{ns}}(\xi ) , v1.10.2=4α2μ2ds(ξ)2 {v_{1.10.2}} = - 4{\alpha ^2}{\mu ^2}{\rm{ds}}{(\xi )^2} ,

1.11) e0 = 0, e2 = −(1 + k2), e4 = k2, e1 = e3 = 0,

u1.11.1=cμ2μαcs(ξ)dn(ξ) {u_{1.11.1}} = - {c \over \mu } - 2\mu \alpha {\rm{cs}}(\xi ){\rm{dn}}(\xi ) , v1.11.1=4α2μ2k2sn(ξ)2 {v_{1.11.1}} = - 4{\alpha ^2}{\mu ^2}{k^2}{\rm{sn}}{(\xi )^2} ;

1.12) e0=k44,e2=12(k22),e4=14,e1=e3=0, {e_0} = {{{k^4}} \over 4},{e_2} = {1 \over 2}({k^2} - 2),{e_4} = {1 \over 4},{e_1} = {e_3} = 0,

v1.14.1=α2μ2k4(ns(ξ)±ds(ξ))2 {v_{1.14.1}} = - {{{\alpha ^2}{\mu ^2}{k^4}} \over {{{\left( {{\rm{ns}}(\xi ) \pm {\rm{ds}}(\xi )} \right)}^2}}} ; v1.14.2=α2μ2(ns(ξ)±ds(ξ))2 {v_{1.14.2}} = - {\alpha ^2}{\mu ^2}{\left( {{\rm{ns}}(\xi ) \pm {\rm{ds}}(\xi )} \right)^2} ,

1.13) e0=k24,e2=(k22)2,e4=k24,e1=e3=0, {e_0} = {{{k^2}} \over 4},{e_2} = {{({k^2} - 2)} \over 2},{e_4} = {{{k^2}} \over 4},{e_1} = {e_3} = 0,

u1.13.1=cμ2iαμ1k2nd(ξ) {u_{1.13.1}} = - {c \over \mu } \mp 2i\alpha \mu \sqrt {1 - {k^2}} {\rm{nd}}(\xi ) , v1.13.1=α2μ2k2(i1k2sn(ξ)±cn(ξ))2 {v_{1.13.1}} = - {{{\alpha ^2}{\mu ^2}{k^2}} \over {{{\left( {i\sqrt {1 - {k^2}} {\rm{sn}}(\xi ) \pm {\rm{cn}}(\xi )} \right)}^2}}} ; u1.13.2=cμ±2iαμ1k2nd(ξ) {u_{1.13.2}} = - {c \over \mu } \pm 2i\alpha \mu \sqrt {1 - {k^2}} {\rm{nd}}(\xi ) , v1.13.2=α2μ2k2(i1k2sn(ξ)±cn(ξ))2 {v_{1.13.2}} = - {\alpha ^2}{\mu ^2}{k^2}{\left( {i\sqrt {1 - {k^2}} {\rm{sn}}(\xi ) \pm {\rm{cn}}(\xi )} \right)^2} ; u1.13.3=cμ2iαμdn(ξ) {u_{1.13.3}} = - {c \over \mu } \mp 2i\alpha \mu {\rm{dn}}(\xi ) , v1.13.3=α2μ2k2(sn(ξ)±icn(ξ))2 {v_{1.13.3}} = - {{{\alpha ^2}{\mu ^2}{k^2}} \over {{{\left( {{\rm{sn}}(\xi ) \pm i{\rm{cn}}(\xi )} \right)}^2}}} ; u1.13.4=cμ±2iαμdn(ξ) {u_{1.13.4}} = - {c \over \mu } \pm 2i\alpha \mu {\rm{dn}}(\xi ) , v1.13.3=α2μ2k2(sn(ξ)±icn(ξ))2 {v_{1.13.3}} = - {\alpha ^2}{\mu ^2}{k^2}{\left( {{\rm{sn}}(\xi ) \pm i{\rm{cn}}(\xi )} \right)^2} ,

1.14) e0 = 1,e2 = 2 − 4k2,e4 = 1,e1 = e3 = 0,

u1.14.1=cμ+2αμ(nc(ξ)ds(ξ)k2sn(ξ))cd(ξ)) {u_{1.14.1}} = - {c \over \mu } + 2\alpha \mu \left( {{\rm{nc}}(\xi ){\rm{ds}}(\xi ) - {k^2}{\rm{sn}}(\xi )){\rm{cd}}(\xi )} \right) , v1.14.1=4α2μ2sc(ξ)2dn(ξ)2 {v_{1.14.1}} = - 4{\alpha ^2}{\mu ^2}{\rm{sc}}{(\xi )^{ - 2}}{\rm{dn}}{(\xi )^{ - 2}} ; u1.14.2=cμ2αμ(nc(ξ)ds(ξ)k2sn(ξ))cd(ξ)) {u_{1.14.2}} = - {c \over \mu } - 2\alpha \mu \left( {{\rm{nc}}(\xi ){\rm{ds}}(\xi ) - {k^2}{\rm{sn}}(\xi )){\rm{cd}}(\xi )} \right) , v1.14.2=4α2μ2sc(ξ)2dn(ξ)2 {v_{1.14.2}} = - 4{\alpha ^2}{\mu ^2}{\rm{sc}}{(\xi )^2}{\rm{dn}}{(\xi )^2} ,

1.15) e0=(k1)24D1,e2=12(1+6k+k2),e4=D12(k1)24,e1=e3=0, {e_0} = {{{{(k - 1)}^2}} \over {4{D_1}}},{e_2} = {1 \over 2}(1 + 6k + {k^2}),{e_4} = {{D_1^2{{(k - 1)}^2}} \over 4},{e_1} = {e_3} = 0,

u1.15.1=cμ2αμ(sn(ξ)(k2cn2(ξ)+dn2(ξ))(1+(k+1)sn(ξ)+ksn2(ξ))+cn2(ξ)dn2(ξ)(k+1+2ksn(ξ)))(1+(k+1)sn(ξ)+ksn2(ξ))dn(ξ)cn(ξ) {u_{1.15.1}} = - {c \over \mu } - {{2\alpha \mu \left( {{\rm sn}(\xi )\left( {{k^2}{{\rm cn}^2}(\xi ) + {{\rm dn}^2}(\xi )} \right)\left( {1 + (k + 1){\rm sn}(\xi ) + k{{\rm sn}^2}(\xi )} \right) + {{\rm cn}^2}(\xi ){{\rm dn}^2}(\xi )(k + 1 + 2k{\rm sn}(\xi ))} \right)} \over {\left( {1 + (k + 1){\rm sn}(\xi ) + k{{\rm sn}^2}(\xi )} \right){\rm dn}(\xi ){\rm cn}(\xi )}} , v1.15.1=α2μ2(k1)2D1(1+sn(ξ))2(1+ksn(ξ))2dn(ξ)2cn(ξ)2 {v_{1.15.1}} = - {{{\alpha ^2}{\mu ^2}{{(k - 1)}^2}{D_1}{{(1 + {\rm sn}(\xi ))}^2}{{(1 + k{\rm sn}(\xi ))}^2}} \over {{\rm dn}{{(\xi )}^2}{\rm cn}{{(\xi )}^2}}}

u1.15.2=cμ+2αμ(sn(ξ)(k2cn2(ξ)+dn2(ξ))(1+(k+1)sn(ξ)+ksn2(ξ))+cn2(ξ)dn2(ξ)(k+1+2ksn(ξ)))(1+(k+1)sn(ξ)+ksn2(ξ))dn(ξ)cn(ξ) {u_{1.15.2}} = - {c \over \mu } + {{2\alpha \mu \left( {{\rm{sn}}(\xi )\left( {{k^2}{\rm{c}}{{\rm{n}}^2}(\xi ) + {\rm{d}}{{\rm{n}}^2}(\xi )} \right)\left( {1 + (k + 1){\rm{sn}}(\xi ) + k{\rm{s}}{{\rm{n}}^2}(\xi )} \right) + {\rm{c}}{{\rm{n}}^2}(\xi ){\rm{d}}{{\rm{n}}^2}(\xi )\left( {k + 1 + 2k{\rm{sn}}(\xi )} \right)} \right)} \over {\left( {1 + (k + 1){\rm{sn}}(\xi ) + k{\rm{s}}{{\rm{n}}^2}(\xi )} \right){\rm{dn}}(\xi ){\rm{cn}}(\xi )}} , v1.15.2=α2μ2(k1)2dn(ξ))2cn(ξ))2(1+sn(ξ))2(1+ksn(ξ))2 {v_{1.15.2}} = - {{{\alpha ^2}{\mu ^2}{{(k - 1)}^2}{\rm{dn}}(\xi {{))}^2}{\rm{cn}}(\xi {{))}^2}} \over {{{\left( {1 + {\rm{sn}}(\xi {{))}^2}(1 + k{\rm{sn}}(\xi )} \right)}^2}}} ,

1.16) e0=(k+1)24D1,e2=12(16k+k2),e4=D12(k+1)24,e1=e3=0, {e_0} = {{{{(k + 1)}^2}} \over {4{D_1}}},{e_2} = {1 \over 2}(1 - 6k + {k^2}),{e_4} = {{D_1^2{{(k + 1)}^2}} \over 4},{e_1} = {e_3} = 0,

u1.16.1=cμ2αμ(sn(ξ)(k2cn2(ξ)+dn2(ξ))(1+(k+1)sn(ξ)ksn2(ξ))+cn2(ξ)dn2(ξ)(k+12ksn(ξ)))(1+(k+1)sn(ξ)ksn2(ξ))dn(ξ)cn(ξ) {u_{1.16.1}} = - {c \over \mu } - {{2\alpha \mu \left( {{\rm{sn}}(\xi )\left( {{k^2}{\rm{c}}{{\rm{n}}^2}(\xi ) + {\rm{d}}{{\rm{n}}^2}(\xi )} \right)\left( {1 + ( - k + 1){\rm{sn}}(\xi ) - k{\rm{s}}{{\rm{n}}^2}(\xi )} \right) + {\rm{c}}{{\rm{n}}^2}(\xi ){\rm{d}}{{\rm{n}}^2}(\xi )\left( { - k + 1 - 2k{\rm{sn}}(\xi )} \right)} \right)} \over {\left( {1 + ( - k + 1){\rm{sn}}(\xi ) - k{\rm{s}}{{\rm{n}}^2}(\xi )} \right){\rm{dn}}(\xi ){\rm{cn}}(\xi )}} , v1.16.1=α2μ2(k+1)2D1(1+sn(ξ))2(1ksn(ξ))2dn(ξ)2cn(ξ)2 {v_{1.16.1}} = - {{{\alpha ^2}{\mu ^2}{{(k + 1)}^2}{D_1}{{\left( {1 + {\rm{sn}}(\xi {{))}^2}(1 - k{\rm{sn}}(\xi )} \right)}^2}} \over {{\rm{dn}}{{(\xi )}^2}{\rm{cn}}{{(\xi )}^2}}} ; v1.16.2=α2μ2(k+1)2dn(ξ))2cn(ξ))2(1+sn(ξ))2(1ksn(ξ))2 {v_{1.16.2}} = - {{{\alpha ^2}{\mu ^2}{{(k + 1)}^2}{\rm{dn}}(\xi {{))}^2}{\rm{cn}}(\xi {{))}^2}} \over {{{\left( {1 + {\rm{sn}}(\xi {{))}^2}(1 - k{\rm{sn}}(\xi )} \right)}^2}}} ,

1.17) e0=k42k3+k2,e2=6kk21,e4=4k,e1=e3=0, {e_0} = {k^4} - 2{k^3} + {k^2},{e_2} = 6k - {k^2} - 1,{e_4} = - {4 \over k},{e_1} = {e_3} = 0,

u1.17.1=cμ2μα((k2cn(ξ)2+dn(ξ)2)(1+ksn2(ξ))+2kcn(ξ)2dn(ξ)2)(1+ksn(ξ)2)dn(ξ)cs(ξ) {u_{1.17.1}} = - {c \over \mu } - {{2\mu \alpha \left( {({k^2}{\rm{cn}}{{(\xi )}^2} + {\rm{dn}}{{(\xi )}^2})(1 + k{\rm{s}}{{\rm{n}}^2}(\xi )) + 2k{\rm{cn}}{{(\xi )}^2}{\rm{dn}}{{(\xi )}^2}} \right)} \over {\left( {1 + k{\rm{sn}}{{(\xi )}^2}} \right){\rm{dn}}(\xi ){\rm{cs}}(\xi )}} , v1.17.1=4α2μ2(k42k3+k2)(1+ksn(ξ)2)2k2cn(ξ)2dn(ξ))2 {v_{1.17.1}} = - {{4{\alpha ^2}{\mu ^2}({k^4} - 2{k^3} + {k^2}){{\left( {1 + k{\rm{sn}}{{(\xi )}^2}} \right)}^2}} \over {{k^2}{\rm{cn}}{{(\xi )}^2}{\rm{dn}}(\xi {{))}^2}}} ; u1.17.2=cμ+2μα((k2cn(ξ)2+dn(ξ)2)(1+ksn2(ξ))+2kcn(ξ)2dn(ξ)2)(1+ksn(ξ)2)dn(ξ)cs(ξ) {u_{1.17.2}} = - {c \over \mu } + {{2\mu \alpha \left( {({k^2}{\rm{cn}}{{(\xi )}^2} + {\rm{dn}}{{(\xi )}^2})(1 + k{\rm{s}}{{\rm{n}}^2}(\xi )) + 2k{\rm{cn}}{{(\xi )}^2}{\rm{dn}}{{(\xi )}^2}} \right)} \over {\left( {1 + k{\rm{sn}}{{(\xi )}^2}} \right){\rm{dn}}(\xi ){\rm{cs}}(\xi )}} , v1.17.2=16α2μ2kcn(ξ)2dn(ξ))2(1+ksn(ξ)2)2 {v_{1.17.2}} = {{16{\alpha ^2}{\mu ^2}k{\rm{cn}}{{(\xi )}^2}{\rm{dn}}(\xi {{))}^2}} \over {{{\left( {1 + k{\rm{sn}}{{(\xi )}^2}} \right)}^2}}} ,

1.18) e0=k4+2k3+k2,e2=6kk21,e4=4k,e1=e3=0, {e_0} = {k^4} + 2{k^3} + {k^2},{e_2} = - 6k - {k^2} - 1,{e_4} = {4 \over k},{e_1} = {e_3} = 0,

v1.18.1=4α2μ2(k4+2k3+k2)(1+ksn(ξ)2)2k2cn(ξ)2dn(ξ))2 {v_{1.18.1}} = - {{4{\alpha ^2}{\mu ^2}({k^4} + 2{k^3} + {k^2}){{\left( { - 1 + k{\rm{sn}}{{(\xi )}^2}} \right)}^2}} \over {{k^2}{\rm{cn}}{{(\xi )}^2}{\rm{dn}}(\xi {{))}^2}}} ; v1.18.2=16α2μ2kcn(ξ)2dn(ξ))2(1+ksn(ξ)2)2 {v_{1.18.2}} = - {{16{\alpha ^2}{\mu ^2}k{\rm{cn}}{{(\xi )}^2}{\rm{dn}}(\xi {{))}^2}} \over {{{\left( { - 1 + k{\rm{sn}}{{(\xi )}^2}} \right)}^2}}} ,

1.19) e0 = 2 + 2k1k2, e2 = 6k1k2 + 2, e4 = 4k1, e1 = e3 = 0,

u1.19.1=cμ+2μα((cn(ξ)2sn(ξ)2)(k1dn(ξ)2)2k2sn(ξ)2cn(ξ)2)(k1dn(ξ)2)sn(ξ)cd(ξ) {u_{1.19.1}} = - {c \over \mu } + {{2\mu \alpha \left( {\left( {{\rm{cn}}{{(\xi )}^2} - {\rm{sn}}{{(\xi )}^2}} \right)\left( {{k_1} - {\rm{dn}}{{(\xi )}^2}} \right) - 2{k^2}{\rm{sn}}{{(\xi )}^2}{\rm{cn}}{{(\xi )}^2}} \right)} \over {\left( {{k_1} - {\rm{dn}}{{(\xi )}^2}} \right){\rm{sn}}(\xi ){\rm{cd}}(\xi )}} , v1.19.1=4α2μ2(2+2k1k2)(k1dn(ξ)2)2k4sn(ξ)2cn(ξ))2 {v_{1.19.1}} = - {{4{\alpha ^2}{\mu ^2}(2 + 2{k_1} - {k^2}){{\left( {{k_1} - {\rm{dn}}{{(\xi )}^2}} \right)}^2}} \over {{k^4}{\rm{sn}}{{(\xi )}^2}{\rm{cn}}(\xi {{))}^2}}} ; v1.19.2=16α2μ2k1k4sn(ξ)2cn(ξ))2(k1+dn(ξ)2)2 {v_{1.19.2}} = - {{16{\alpha ^2}{\mu ^2}{k_1}{k^4}{\rm{sn}}{{(\xi )}^2}{\rm{cn}}(\xi {{))}^2}} \over {{{\left( {{k_1} + {\rm{dn}}{{(\xi )}^2}} \right)}^2}}} ,

1.20) e0 = 2 − 2k1k2, e2 = −6k1k2 + 2, e4 = −4k1, e1 = e3 = 0,

u1.20.1=cμ2μα((cn(ξ)2sn(ξ)2)(k1+dn(ξ)2)+2k2sn(ξ)2cn(ξ)2)(k1+dn(ξ)2)sn(ξ)cd(ξ) {u_{1.20.1}} = - {c \over \mu } - {{2\mu \alpha \left( {\left( {{\rm{cn}}{{(\xi )}^2} - {\rm{sn}}{{(\xi )}^2}} \right)\left( {{k_1} + {\rm{dn}}{{(\xi )}^2}} \right) + 2{k^2}{\rm{sn}}{{(\xi )}^2}{\rm{cn}}{{(\xi )}^2}} \right)} \over {\left( {{k_1} + {\rm{dn}}{{(\xi )}^2}} \right){\rm{sn}}(\xi ){\rm{cd}}(\xi )}} , v1.20.1=4α2μ2(22k1k2)(k1+dn(ξ)2)2k4sn(ξ)2cn(ξ)2 {v_{1.20.1}} = - {{4{\alpha ^2}{\mu ^2}(2 - 2{k_1} - {k^2}){{\left( {{k_1} + {\rm{dn}}{{(\xi )}^2}} \right)}^2}} \over {{k^4}{\rm{sn}}{{(\xi )}^2}{\rm{cn}}{{(\xi )}^2}}} ; u1.20.2=cμ+2μα((cn(ξ)2sn(ξ)2)(k1+dn(ξ)2)+2k2sn(ξ)2cn(ξ)2)(k1+dn(ξ)2)sn(ξ)cd(ξ) {u_{1.20.2}} = - {c \over \mu } + {{2\mu \alpha \left( {\left( {{\rm{cn}}{{(\xi )}^2} - {\rm{sn}}{{(\xi )}^2}} \right)\left( {{k_1} + {\rm{dn}}{{(\xi )}^2}} \right) + 2{k^2}{\rm{sn}}{{(\xi )}^2}{\rm{cn}}{{(\xi )}^2}} \right)} \over {\left( {{k_1} + {\rm{dn}}{{(\xi )}^2}} \right){\rm{sn}}(\xi ){\rm{cd}}(\xi )}} , v1.20.2=16α2μ2k1k4sn(ξ)2cn(ξ))2(k1dn(ξ)2)2 {v_{1.20.2}} = {{16{\alpha ^2}{\mu ^2}{k_1}{k^4}{\rm{sn}}{{(\xi )}^2}{\rm{cn}}(\xi {{))}^2}} \over {{{\left( {{k_1} - {\rm{dn}}{{(\xi )}^2}} \right)}^2}}} ,

1.21) e0=k214(D32+D22),e2=k2+12,e4=(D32k2D22)(k21)4,e1=e3=0, {e_0} = {{{k^2} - 1} \over {4(D_3^2 + D_2^2)}},{e_2} = {{{k^2} + 1} \over 2},{e_4} = {{\left( {D_3^2{k^2} - D_2^2} \right)\left( {{k^2} - 1} \right)} \over 4},{e_1} = {e_3} = 0,

u1.21.1=cμ+2μα(D2dn(ξ)+D3cn(ξ)+D2D22D32D22D32k2sn(ξ)dn(ξ)+D3k2D22D32D22D32k2cn(ξ)sn(ξ))(D2cn(ξ)+D3dn(ξ))(D22D32D22D32k2+sn(ξ)) {u_{1.21.1}} = - {c \over \mu } + {{2\mu \alpha \left( {{D_2}{\rm{dn}}(\xi ) + {D_3}{\rm{cn}}(\xi ) + {D_2}\sqrt {{{D_2^2 - D_3^2} \over {D_2^2 - D_3^2{k^2}}}} {\rm{sn}}(\xi ){\rm{dn}}(\xi ) + {D_3}{k^2}\sqrt {{{D_2^2 - D_3^2} \over {D_2^2 - D_3^2{k^2}}}} {\rm{cn}}(\xi ){\rm{sn}}(\xi )} \right)} \over {\left( {{D_2}{\rm{cn}}(\xi ) + {D_3}{\rm{dn}}(\xi )} \right)\left( {\sqrt {{{D_2^2 - D_3^2} \over {D_2^2 - D_3^2{k^2}}}} + {\rm{sn}}(\xi )} \right)}} , v1.21.1=α2μ2(k21)(D2cn(ξ)+D3dn(ξ))2(D32+D22)(D22D32D22D32k2+sn(ξ))2 {v_{1.21.1}} = - {{{\alpha ^2}{\mu ^2}\left( {{k^2} - 1} \right){{\left( {{D_2}{\rm{cn}}(\xi ) + {D_3}{\rm{dn}}(\xi )} \right)}^2}} \over {(D_3^2 + D_2^2){{\left( {\sqrt {{{D_2^2 - D_3^2} \over {D_2^2 - D_3^2{k^2}}}} + {\rm{sn}}(\xi )} \right)}^2}}} ; u1.21.2=cμ2μα(D2dn(ξ)+D3cn(ξ)+D2D22D32D22D32k2sn(ξ)dn(ξ)+D3k2D22D32D22D32k2cn(ξ)sn(ξ))(D2cn(ξ)+D3dn(ξ))(D22D32D22D32k2+sn(ξ)) {u_{1.21.2}} = - {c \over \mu } - {{2\mu \alpha \left( {{D_2}{\rm{dn}}(\xi ) + {D_3}{\rm{cn}}(\xi ) + {D_2}\sqrt {{{D_2^2 - D_3^2} \over {D_2^2 - D_3^2{k^2}}}} {\rm{sn}}(\xi ){\rm{dn}}(\xi ) + {D_3}{k^2}\sqrt {{{D_2^2 - D_3^2} \over {D_2^2 - D_3^2{k^2}}}} {\rm{cn}}(\xi ){\rm{sn}}(\xi )} \right)} \over {\left( {{D_2}{\rm{cn}}(\xi ) + {D_3}{\rm{dn}}(\xi )} \right)\left( {\sqrt {{{D_2^2 - D_3^2} \over {D_2^2 - D_3^2{k^2}}}} + {\rm{sn}}(\xi )} \right)}} , v1.21.2=α2μ2(D32k2D22)(k21)(D22D32D22D32k2+sn(ξ))2(D2cn(ξ)+D3dn(ξ))2 {v_{1.21.2}} = - {{{\alpha ^2}{\mu ^2}\left( {D_3^2{k^2} - D_2^2} \right)\left( {{k^2} - 1} \right){{\left( {\sqrt {{{D_2^2 - D_3^2} \over {D_2^2 - D_3^2{k^2}}}} + {\rm{sn}}(\xi )} \right)}^2}} \over {{{\left( {{D_2}{\rm{cn}}(\xi ) + {D_3}{\rm{dn}}(\xi )} \right)}^2}}} ,

1.22) e0=k44(D32+D22),e2=k221,e4=D32+D224,e1=e3=0, {e_0} = {{{k^4}} \over {4(D_3^2 + D_2^2)}},{e_2} = {{{k^2}} \over 2} - 1,{e_4} = {{D_3^2 + D_2^2} \over 4},{e_1} = {e_3} = 0,

u1.22.1=cμ2μα(k2sn(ξ)cn(ξ)(D2sn(ξ)+D3cn(ξ))+dn(ξ)(D22(k21)D32D22+D32+dn(ξ))(D2cn(ξ)D3sn(ξ)))(D2sn(ξ)+D3cn(ξ))(D22(k21)D32D22+D32+dn(ξ)) {u_{1.22.1}} = - {c \over \mu } - {{2\mu \alpha \left( {{k^2}{\rm{sn}}(\xi ){\rm{cn}}(\xi )\left( {{D_2}{\rm{sn}}(\xi ) + {D_3}{\rm{cn}}(\xi )} \right) + {\rm{dn}}(\xi )\left( {\sqrt {{{D_2^2 - ({k^2} - 1)D_3^2} \over {D_2^2 + D_3^2}}} + {\rm{dn}}(\xi )} \right)\left( {{D_2}{\rm{cn}}(\xi ) - {D_3}{\rm{sn}}(\xi )} \right)} \right)} \over {\left( {{D_2}{\rm{sn}}(\xi ) + {D_3}{\rm{cn}}(\xi )} \right)\left( {\sqrt {{{D_2^2 - ({k^2} - 1)D_3^2} \over {D_2^2 + D_3^2}}} + {\rm{dn}}(\xi )} \right)}} , v1.22.1=α2μ2k4(D2sn(ξ)+D3cn(ξ))2(D32+D22)(D22(k21)D32D22+D32+dn(ξ))2 {v_{1.22.1}} = - {{{\alpha ^2}{\mu ^2}{k^4}{{\left( {{D_2}{\rm{sn}}(\xi ) + {D_3}{\rm{cn}}(\xi )} \right)}^2}} \over {(D_3^2 + D_2^2){{\left( {\sqrt {{{D_2^2 - ({k^2} - 1)D_3^2} \over {D_2^2 + D_3^2}}} + {\rm{dn}}(\xi )} \right)}^2}}} ; u1.22.2=cμ+2μα(k2sn(ξ)cn(ξ)(D2sn(ξ)+D3cn(ξ))+dn(ξ)(D22(k21)D32D22+D32+dn(ξ))(D2cn(ξ)D3sn(ξ)))(D2sn(ξ)+D3cn(ξ))(D22(k21)D32D22+D32+dn(ξ)) {u_{1.22.2}} = - {c \over \mu } + {{2\mu \alpha \left( {{k^2}{\rm{sn}}(\xi ){\rm{cn}}(\xi )\left( {{D_2}{\rm{sn}}(\xi ) + {D_3}{\rm{cn}}(\xi )} \right) + {\rm{dn}}(\xi )\left( {\sqrt {{{D_2^2 - ({k^2} - 1)D_3^2} \over {D_2^2 + D_3^2}}} + {\rm{dn}}(\xi )} \right)\left( {{D_2}{\rm{cn}}(\xi ) - {D_3}{\rm{sn}}(\xi )} \right)} \right)} \over {\left( {{D_2}{\rm{sn}}(\xi ) + {D_3}{\rm{cn}}(\xi )} \right)\left( {\sqrt {{{D_2^2 - ({k^2} - 1)D_3^2} \over {D_2^2 + D_3^2}}} + {\rm{dn}}(\xi )} \right)}} , v1.22.2=α2μ2(D32+D22)(D22(k21)D32D22+D32+dn(ξ))2(D2sn(ξ)+D3cn(ξ))2 {v_{1.22.2}} = - {{{\alpha ^2}{\mu ^2}\left( {D_3^2 + D_2^2} \right){{\left( {\sqrt {{{D_2^2 - ({k^2} - 1)D_3^2} \over {D_2^2 + D_3^2}}} + {\rm{dn}}(\xi )} \right)}^2}} \over {{{\left( {{D_2}{\rm{sn}}(\xi ) + {D_3}{\rm{cn}}(\xi )} \right)}^2}}} ,

1.23) e0=2kk21D22,e2=2k2+2,e4=D22(k2+2k+1),e1=e3=0, {e_0} = {{2k - {k^2} - 1} \over {D_2^2}},{e_2} = 2{k^2} + 2,{e_4} = - D_2^2({k^2} + 2k + 1),{e_1} = {e_3} = 0,

u1.23.1=cμ+8μαksn(ξ)cn(ξ)dn(ξ)k2sn(ξ)41 {u_{1.23.1}} = - {c \over \mu } + {{8\mu \alpha k{\rm{sn}}(\xi ){\rm{cn}}(\xi ){\rm{dn}}(\xi )} \over {{k^2}{\rm{sn}}{{(\xi )}^4} - 1}} , v1.23.1=4α2μ2(2kk21)(ksn(ξ)2+1)2(ksn(ξ)21)2 {v_{1.23.1}} = - {{4{\alpha ^2}{\mu ^2}\left( {2k - {k^2} - 1} \right){{\left( {k{\rm{sn}}{{(\xi )}^2} + 1} \right)}^2}} \over {{{\left( {k{\rm{sn}}{{(\xi )}^2} - 1} \right)}^2}}} ; u1.23.2=cμ8μαksn(ξ)cn(ξ)dn(ξ)k2sn(ξ)41 {u_{1.23.2}} = - {c \over \mu } - {{8\mu \alpha k{\rm{sn}}(\xi ){\rm{cn}}(\xi ){\rm{dn}}(\xi )} \over {{k^2}{\rm{sn}}{{(\xi )}^4} - 1}} , v1.23.2=4α2μ2(k2+2k+1)(ksn(ξ)21)2(ksn(ξ)2+1)2 {v_{1.23.2}} = {{4{\alpha ^2}{\mu ^2}({k^2} + 2k + 1){{\left( {k{\rm{sn}}{{(\xi )}^2} - 1} \right)}^2}} \over {{{\left( {k{\rm{sn}}{{(\xi )}^2} + 1} \right)}^2}}} ,

1.24) e0=2kk21D22,e2=2k2+2,e4=D22(k2+2k+1),e1=e3=0, {e_0} = {{ - 2k - {k^2} - 1} \over {D_2^2}},{e_2} = 2{k^2} + 2,{e_4} = - D_2^2({k^2} + 2k + 1),{e_1} = {e_3} = 0,

u1.24.2=cμ+8μαksn(ξ)cn(ξ)dn(ξ)k2sn(ξ)41 {u_{1.24.2}} = - {c \over \mu } + {{8\mu \alpha k{\rm{sn}}(\xi ){\rm{cn}}(\xi ){\rm{dn}}(\xi )} \over {{k^2}{\rm{sn}}{{(\xi )}^4} - 1}} , v1.24.2=4α2μ2(k2+2k+1)(ksn(ξ)2+1)2(ksn(ξ)21)2 {v_{1.24.2}} = {{4{\alpha ^2}{\mu ^2}({k^2} + 2k + 1){{\left( {k{\rm{sn}}{{(\xi )}^2} + 1} \right)}^2}} \over {{{\left( {k{\rm{sn}}{{(\xi )}^2} - 1} \right)}^2}}} ,

1.25) e0=e4=14,e2=12k22,e1=e3=0, {e_0} = {e_4} = {1 \over 4},{e_2} = {{1 - 2{k^2}} \over 2},{e_1} = {e_3} = 0,

u1.25.3=cμ+2μαkcn(ξ)(dn(ξ)iksn(ξ))(ksn(ξ)±idn(ξ)) {u_{1.25.3}} = - {c \over \mu } + {{2\mu \alpha k{\rm{cn}}(\xi )\left( {{\rm{dn}}(\xi ) \mp {\rm{iksn}}(\xi )} \right)} \over {\left( {k{\rm{sn}}(\xi ) \pm i{\rm{dn}}(\xi )} \right)}} , v1.25.3=α2μ2(ksn(ξ)±idn(ξ))2 {v_{1.25.3}} = - {{{\alpha ^2}{\mu ^2}} \over {{{\left( {k{\rm{sn}}(\xi ) \pm i{\rm{dn}}(\xi )} \right)}^2}}} ; u1.25.4=cμ2kcn(ξ)(dn(ξ)iksn(ξ))(ksn(ξ)±idn(ξ)) {u_{1.25.4}} = - {c \over \mu } - {{2k{\rm{cn}}(\xi )\left( {{\rm{dn}}(\xi ) \mp {\rm{iksn}}(\xi )} \right)} \over {\left( {k{\rm{sn}}(\xi ) \pm i{\rm{dn}}(\xi )} \right)}} , v1.25.4=α2μ2(ksn(ξ)±idn(ξ))2 {v_{1.25.4}} = - {\alpha ^2}{\mu ^2}{\left( {k{\rm{sn}}(\xi ) \pm i{\rm{dn}}(\xi )} \right)^2}; ; u1.25.7=cμ±2μαds(ξ) {u_{1.25.7}} = - {c \over \mu } \pm 2\mu \alpha {\rm{ds}}(\xi ) , v1.25.7=α2μ2ns(ξ)2(1±cn(ξ))2 {v_{1.25.7}} = - {\alpha ^2}{\mu ^2}{\rm{ns}}{(\xi )^2}{\left( {1 \pm {\rm{cn}}(\xi )} \right)^2} ; u1.25.8=cμ2μαds(ξ) {u_{1.25.8}} = - {c \over \mu } \mp 2\mu \alpha {\rm{ds}}(\xi ) , v1.25.8=α2μ2sn(ξ)2(1±cn(ξ))2 {v_{1.25.8}} = - {\alpha ^2}{\mu ^2}{\rm{sn}}{(\xi )^2}{\left( {1 \pm {\rm{cn}}(\xi )} \right)^{ - 2}} ,

1.26) e0=e4=k214,e2=k2+12,e1=e3=0, {e_0} = {e_4} = {{{k^2} - 1} \over 4},{e_2} = {{{k^2} + 1} \over 2},{e_1} = {e_3} = 0,

u1.26.1=cμ±2αμkcd(ξ) {u_{1.26.1}} = - {c \over \mu } \pm 2\alpha \mu k{\rm{cd}}(\xi ) , v1.26.1=α2μ2(k21)(ksd(ξ)±nd(ξ))2 {v_{1.26.1}} = - {{{\alpha ^2}{\mu ^2}\left( {{k^2} - 1} \right)} \over {{{\left( {{\rm{ks}}d(\xi ) \pm {\rm{nd}}(\xi )} \right)}^2}}} ;

u1.26.2=cμ2αμkcd(ξ) {u_{1.26.2}} = - {c \over \mu } \mp 2\alpha \mu k{\rm{cd}}(\xi ) , v1.26.2=α2μ2(k21)(ksd(ξ)±nd(ξ))2 {v_{1.26.2}} = - {\alpha ^2}{\mu ^2}\left( {{k^2} - 1} \right){\left( {{\rm{ks}}d(\xi ) \pm {\rm{nd}}(\xi )} \right)^2} ;

u1.26.3=cμ2μαkdc(ξ) {u_{1.26.3}} = - {c \over \mu } \mp 2\mu \alpha k{\rm{dc}}(\xi ) , v1.26.3=α2μ2(k21)(1±ksn(ξ))2dn(ξ)2 {v_{1.26.3}} = - {{{\alpha ^2}{\mu ^2}\left( {{k^2} - 1} \right){{\left( {1 \pm k{\rm{sn}}(\xi )} \right)}^2}} \over {{\rm{dn}}{{(\xi )}^2}}} ;

u1.26.4=cμ±2μαkdc(ξ) {u_{1.26.4}} = - {c \over \mu } \pm 2\mu \alpha k{\rm{dc}}(\xi ) , v1.26.4=α2μ2(k21)dn(ξ)2(1±ksn(ξ))2 {v_{1.26.4}} = - {{{\alpha ^2}{\mu ^2}\left( {{k^2} - 1} \right){\rm{dn}}{{(\xi )}^2}} \over {{{\left( {1 \pm k{\rm{sn}}(\xi )} \right)}^2}}} ,

1.27) e0=(1k2)24,e2=k2+12,e4=14,e1=e3=0, {e_0} = - {{{{\left( {1 - {k^2}} \right)}^2}} \over 4},{e_2} = {{{k^2} + 1} \over 2},{e_4} = - {1 \over 4},{e_1} = {e_3} = 0,

u1.27.1=cμμαsn(ξ) {u_{1.27.1}} = - {c \over \mu } \mp \mu \alpha {\rm{sn}}(\xi ) , v1.27.1=α2μ2(1k2)2(kcn(ξ)±dn(ξ))2 {v_{1.27.1}} = {\alpha ^2}{\mu ^2}{\left( {1 - {k^2}} \right)^2}{\left( {k{\rm{cn}}(\xi ) \pm {\rm{dn}}(\xi )} \right)^{ - 2}} ; u1.27.2=cμ±μαsn(ξ) {u_{1.27.2}} = - {c \over \mu } \pm \mu \alpha {\rm{sn}}(\xi ) , v1.27.2=α2μ2(kcn(ξ)±dn(ξ))2 {v_{1.27.2}} = {\alpha ^2}{\mu ^2}{\left( {k{\rm{cn}}(\xi ) \pm {\rm{dn}}(\xi )} \right)^2} ,

1.28) e0=14,e2=k2+12,e4=(1k2)24,e1=e3=0, {e_0} = {1 \over 4},{e_2} = {{{k^2} + 1} \over 2},{e_4} = {{{{\left( {1 - {k^2}} \right)}^2}} \over 4},{e_1} = {e_3} = 0,

u1.28.1=cμ±2μαns(ξ) {u_{1.28.1}} = - {c \over \mu } \pm 2\mu \alpha {\rm{ns}}(\xi ) , v1.28.1=α2μ2(dn(ξ)±cn(ξ))2sn(ξ)2 {v_{1.28.1}} = - {\alpha ^2}{\mu ^2}{\left( {{\rm{dn}}(\xi ) \pm {\rm{cn}}(\xi )} \right)^2}{\rm{sn}}{(\xi )^{ - 2}} ; u1.28.2=cμ2μαns(ξ) {u_{1.28.2}} = - {c \over \mu } \mp 2\mu \alpha {\rm{ns}}(\xi ) , v1.28.2=α2μ2(1k2)2sn(ξ)2(dn(ξ)±cn(ξ))2 {v_{1.28.2}} = - {\alpha ^2}{\mu ^2}{\left( {1 - {k^2}} \right)^2}{\rm{sn}}{(\xi )^2}{\left( {{\rm{dn}}(\xi ) \pm {\rm{cn}}(\xi )} \right)^{ - 2}} ,

1.29) e0=14,e2=(k22)22,e4=k44,e1=e3=0, {e_0} = {1 \over 4},{e_2} = {{{{\left( {{k^2} - 2} \right)}^2}} \over 2},{e_4} = {{{k^4}} \over 4},{e_1} = {e_3} = 0,

u1.29.1=cμ2μα1k2sc(ξ) {u_{1.29.1}} = - {c \over \mu } \mp 2\mu \alpha \sqrt {1 - {k^2}} {\rm{sc}}(\xi ) , v1.29.1=α2μ2(1k2±dn(ξ))2cn(ξ)2 {v_{1.29.1}} = - {{{\alpha ^2}{\mu ^2}{{\left( {\sqrt {1 - {k^2}} \pm {\rm{dn}}(\xi )} \right)}^2}} \over {{\rm{cn}}{{(\xi )}^2}}} ; u1.29.2=cμ±2μα1k2sc(ξ) {u_{1.29.2}} = - {c \over \mu } \pm 2\mu \alpha \sqrt {1 - {k^2}} {\rm{sc}}(\xi ) , v1.29.2=α2μ2k4cn(ξ)2(1k2±dn(ξ))2 {v_{1.29.2}} = - {{{\alpha ^2}{\mu ^2}{k^4}{\rm{cn}}{{(\xi )}^2}} \over {{{\left( {\sqrt {1 - {k^2}} \pm {\rm{dn}}(\xi )} \right)}^2}}} .

The solutions of hyperbolic function have

1.30) e0 = 0, e2 > 0, e4 < 0, e1 = e3 = 0,

u1.30.1=cμ+2αμe2tanh(e2ξ) {u_{1.30.1}} = - {c \over \mu } + 2\alpha \mu \sqrt {{e_2}} t{\rm{an}}h(\sqrt {{e_2}} \xi ) , v1.30.1=4α2μ2e2sech2(e2ξ) {v_{1.30.1}} = 4{\alpha ^2}{\mu ^2}{e_2}\sec {h^2}(\sqrt {{e_2}} \xi ) ,

1.31) e0 = 0, e2 > 0, e4 > 0, e1 = e3 = 0,

u1.31.1=cμ+2αμe2coth(e2ξ) {u_{1.31.1}} = - {c \over \mu } + 2\alpha \mu \sqrt {{e_2}} \cot h(\sqrt {{e_2}} \xi ) , v1.31.1=4α2μ2e2csch2(e2ξ) {v_{1.31.1}} = - 4{\alpha ^2}{\mu ^2}{e_2}\csc {h^2}(\sqrt {{e_2}} \xi ) ,

1.32) e0=e224e4,e2<0,e4>0,e1=e3=0, {e_0} = {{e_2^2} \over {4{e_4}}},{e_2} < 0,{e_4} > 0,{e_1} = {e_3} = 0,

u1.32.1=cμ+2αμe22sech(e22ξ)csch(e22ξ) {u_{1.32.1}} = - {c \over \mu } + 2\alpha \mu \sqrt {{{ - {e_2}} \over 2}} \sec h(\sqrt {{{ - {e_2}} \over 2}} \xi ){\rm{csch}}(\sqrt {{{ - {e_2}} \over 2}} \xi ) , v1.32.1=2α2μ2e2tanh2(e22ξ) {v_{1.32.1}} = 2{\alpha ^2}{\mu ^2}{e_2}{\rm{tan}}{{\rm{h}}^{ - 2}}(\sqrt {{{ - {e_2}} \over 2}} \xi ) ; v1.32.2=2α2μ2e2tanh2(e22ξ) {v_{1.32.2}} = 2{\alpha ^2}{\mu ^2}{e_2}{\rm{tan}}{{\rm{h}}^2}(\sqrt {{{ - {e_2}} \over 2}} \xi ) ,

1.33) e0=0,e2=1,e4=12,e1=e3=0, {e_0} = 0,{e_2} = 1,{e_4} = {1 \over 2},{e_1} = {e_3} = 0,

u1.33.1=cμ22μαcoth(Dξ) {u_{1.33.1}} = - {c \over \mu } - 2\sqrt 2 \mu \alpha \cot h(D - \xi ) , v1.33.1=2α2μ2tanh2(Dξ)(22tanh2(Dξ))1 {v_{1.33.1}} = - 2{\alpha ^2}{\mu ^2}\tan {h^2}(D - \xi ){\left( {2 - 2\tan {h^2}(D - \xi )} \right)^{ - 1}} ,

Triangular periodic solutions have

1.34) e0 = 0, e2 < 0, e4 > 0, e1 = e3 = 0,

u1.34.1=cμ2αμe2tan(e2ξ) {u_{1.34.1}} = - {c \over \mu } - 2\alpha \mu \sqrt { - {e_2}} \tan (\sqrt { - {e_2}} \xi ) , v1.34.1=4α2μ2e2sec2(e2ξ) {v_{1.34.1}} = 4{\alpha ^2}{\mu ^2}{e_2}\mathop {\sec }\nolimits^2 (\sqrt { - {e_2}} \xi ) ;

u1.34.2=cμ+2αμe2cot(e2ξ) {u_{1.34.2}} = - {c \over \mu } + 2\alpha \mu \sqrt { - {e_2}} \cot (\sqrt { - {e_2}} \xi ) , v1.34.2=4α2μ2e2csc2(e2ξ) {v_{1.34.2}} = 4{\alpha ^2}{\mu ^2}{e_2}c{\rm{s}}{{\rm{c}}^2}(\sqrt { - {e_2}} \xi ) ;

1.35) e0=e224e4,e2<0,e4>0,e1=e3=0, {e_0} = {{e_2^2} \over {4{e_4}}},{e_2} < 0,{e_4} > 0,{e_1} = {e_3} = 0,

u1.36.1=cμ+22e2μαcsc(2e2ξ) {u_{1.36.1}} = - {c \over \mu } + 2\sqrt {2{e_2}} \mu \alpha \csc (\sqrt {2{e_2}} \xi ) , v1.36.1=2α2μ2e2cot2(e22ξ) {v_{1.36.1}} = - 2{\alpha ^2}{\mu ^2}{e_2}\mathop {\cot }\nolimits^2 (\sqrt {{{{e_2}} \over 2}} \xi ) ;

u1.36.2=cμ22e2μαcsc(2e2ξ) {u_{1.36.2}} = - {c \over \mu } - 2\sqrt {2{e_2}} \mu \alpha \csc (\sqrt {2{e_2}} \xi ) , v1.36.2=2α2μ2e2tan2(e22ξ) {v_{1.36.2}} = - 2{\alpha ^2}{\mu ^2}{e_2}{\rm{ta}}{{\rm{n}}^2}(\sqrt {{{{e_2}} \over 2}} \xi ) ,

Rational solutions have

1.36) e0 = 0, e2 = 0, e4 > 0, e1 = e3 = 0,

u1.36.1=cμ2αμe4e4ξ+ξ0 {u_{1.36.1}} = - {c \over \mu } - {{2\alpha \mu \sqrt {{e_4}} } \over {\sqrt {{e_4}} \xi + {\xi _0}}} , v1.36.1=4α2μ2e4(e4ξ+ξ0)2 {v_{1.36.1}} = - {{4{\alpha ^2}{\mu ^2}{e_4}} \over {{{\left( {\sqrt {{e_4}} \xi + {\xi _0}} \right)}^2}}} ,

1.37) e0 = k, e1 = −4k, e2 = −1 + 6kk2, e3 = 2(k − 1)2, e4 = −(k − 1)2,

u1.37.1=cμ4αμkcn(ξ)dn(ξ)1ksn(ξ)2 {u_{1.37.1}} = - {c \over \mu } - {{4\alpha \mu \sqrt k cn(\xi ){\rm{dn}}(\xi )} \over {1 - k{\rm{sn}}{{(\xi )}^2}}} , v1.37.1=2α2μ2(k2+6k+1)16α2μ2kcn(ξ)2dn(ξ)2(1ksn(ξ)2)2+4α2μ2((1k2+6k)(ksn(ξ)+1)22(k1)2)(ksn(ξ)+1)2(1ksn(ξ)) {v_{1.37.1}} = 2{\alpha ^2}{\mu ^2}({k^2} + 6k + 1) - {{16{\alpha ^2}{\mu ^2}k{\rm{cn}}{{(\xi )}^2}{\rm{dn}}{{(\xi )}^2}\;\;} \over {\;{{\left( {1 - k{\rm{sn}}{{(\xi )}^2}} \right)}^2}}} + {{4{\alpha ^2}{\mu ^2}\left( {\left( { - 1 - {k^2} + 6k} \right){{\left( {\sqrt k {\rm{sn}}(\xi ) + 1} \right)}^2} - 2(k - {{1)}^2}} \right)} \over {{{\left( {\sqrt k {\rm{sn}}(\xi ) + 1} \right)}^2}\left( {1 - \sqrt k {\rm{sn}}(\xi )} \right)}}; ; u1.37.2=cμ+4αμkcn(ξ)dn(ξ)1ksn(ξ)2 {u_{1.37.2}} = - {c \over \mu } + {{4\alpha \mu \sqrt k cn(\xi ){\rm{dn}}(\xi )} \over {1 - k{\rm{sn}}{{(\xi )}^2}}} , v1.37.2=2α2μ2(k2+6k+1)4α2μ2((1k2+6k)(ksn(ξ)+1)22(k1)2)(ksn(ξ)+1)2(1ksn(ξ)) {v_{1.37.2}} = 2{\alpha ^2}{\mu ^2}({k^2} + 6k + 1) - {{4{\alpha ^2}{\mu ^2}\left( {\left( { - 1 - {k^2} + 6k} \right){{\left( {\sqrt k {\rm{sn}}(\xi ) + 1} \right)}^2} - 2(k - {{1)}^2}} \right)} \over {{{\left( {\sqrt k {\rm{sn}}(\xi ) + 1} \right)}^2}\left( {1 - \sqrt k {\rm{sn}}(\xi )} \right)}} ; u1.37.3=cμ4αμk3kcn(ξ)dn(ξ)sn2(ξ)k3sn(ξ)2(1dn(ξ)2)2 {u_{1.37.3}} = - {c \over \mu } - {{4\alpha \mu {k^3}\sqrt k cn(\xi ){\rm{dn}}(\xi ){\rm{s}}{{\rm{n}}^2}(\xi )} \over {{k^3}{\rm{sn}}{{(\xi )}^2} - {{\left( {1 - {\rm{dn}}{{(\xi )}^2}} \right)}^2}}} , v1.37.3=2α2μ2(k2+6k+1)16α2μ2k7cn2(ξ)dn2(ξ)sn4(ξ)(k3sn(ξ)2(1dn(ξ)2)2)2+4α2μ2((1k2+6k)(kksn(ξ)dn(ξ)2+1)22(k1)2k7sn(ξ)2)kksn(ξ)(kksn(ξ)dn(ξ)2+1)(k3sn(ξ)2(1dn(ξ)2)2) {v_{1.37.3}} = 2{\alpha ^2}{\mu ^2}({k^2} + 6k + 1) - {{16{\alpha ^2}{\mu ^2}{k^7}c{n^2}(\xi ){\rm{d}}{{\rm{n}}^2}(\xi ){\rm{s}}{{\rm{n}}^4}(\xi )} \over {{{\left( {{k^3}{\rm{sn}}{{(\xi )}^2} - {{\left( {1 - {\rm{dn}}{{(\xi )}^2}} \right)}^2}} \right)}^2}}} + {{4{\alpha ^2}{\mu ^2}\left( {\left( { - 1 - {k^2} + 6k} \right){{\left( {k\sqrt k {\rm{sn}}(\xi ) - {\rm{dn}}{{(\xi )}^2} + 1} \right)}^2} - 2(k - {{1)}^2}{k^7}{\rm{sn}}{{(\xi )}^2}} \right)k\sqrt k {\rm{sn}}(\xi )} \over {\left( {k\sqrt k {\rm{sn}}(\xi ) - {\rm{dn}}{{(\xi )}^2} + 1} \right)\left( {{k^3}{\rm{sn}}{{(\xi )}^2} - {{\left( {1 - {\rm{dn}}{{(\xi )}^2}} \right)}^2}} \right)}} ; u1.37.4=cμ+4αμk3kcn(ξ)dn(ξ)sn2(ξ)k3sn(ξ)2(1dn(ξ)2)2 {u_{1.37.4}} = - {c \over \mu } + {{4\alpha \mu {k^3}\sqrt k cn(\xi ){\rm{dn}}(\xi ){\rm{s}}{{\rm{n}}^2}(\xi )} \over {{k^3}{\rm{sn}}{{(\xi )}^2} - {{\left( {1 - {\rm{dn}}{{(\xi )}^2}} \right)}^2}}} , v1.37.4=2α2μ2(k2+6k+1)4α2μ2((1k2+6k)(kksn(ξ)dn(ξ)2+1)22(k1)2k7sn(ξ)2)kksn(ξ)(kksn(ξ)dn(ξ)2+1)(k3sn(ξ)2(1dn(ξ)2)2) {v_{1.37.4}} = 2{\alpha ^2}{\mu ^2}({k^2} + 6k + 1) - {{4{\alpha ^2}{\mu ^2}\left( {\left( { - 1 - {k^2} + 6k} \right){{\left( {k\sqrt k {\rm{sn}}(\xi ) - {\rm{dn}}{{(\xi )}^2} + 1} \right)}^2} - 2(k - {{1)}^2}{k^7}{\rm{sn}}{{(\xi )}^2}} \right)k\sqrt k {\rm{sn}}(\xi )} \over {\left( {k\sqrt k {\rm{sn}}(\xi ) - {\rm{dn}}{{(\xi )}^2} + 1} \right)\left( {{k^3}{\rm{sn}}{{(\xi )}^2} - {{\left( {1 - {\rm{dn}}{{(\xi )}^2}} \right)}^2}} \right)}} ;

1.38) e0=e14,e1<0,e2=e1(2k2),e3=2e1(1k2),e4=e1(1k2), {e_0} = - {{{e_1}} \over 4},{e_1} < 0,{e_2} = - {e_1}(2 - {k^2}),{e_3} = 2{e_1}(1 - {k^2}),{e_4} = - {e_1}(1 - {k^2}),

u1.38.1=cμ4μαe1dn(e1ξ)(1+cn(e1ξ))(1+cn(e1ξ))2sn(e1ξ)2 {u_{1.38.1}} = - {c \over \mu } - {{4\mu \alpha \sqrt { - {e_1}} {\rm{dn}}(\sqrt { - {e_1}} \xi )\left( {1 + {\rm{cn}}(\sqrt { - {e_1}} \xi )} \right)} \over {{{\left( {{\rm{1}} + {\rm{cn}}(\sqrt { - {e_1}} \xi )} \right)}^2} - {\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2}}} , v1.38.1=α2μ2e1(k2+1)+16α2μ2e1dn(e1ξ)2(cn(e1ξ)+1)2((cn(e1ξ)+1)2sn(e1ξ)2)2+4α2μ2e1((k22)(sn(e1ξ)+cn(e1ξ)+1)22(1k2)sn(e1ξ)2)sn(e1ξ)2(sn(e1ξ)+cn(e1ξ)+1)((cn(e1ξ)+1)2sn(e1ξ)2) {v_{1.38.1}} = - {\alpha ^2}{\mu ^2}{e_1}({k^2} + 1) + {{16{\alpha ^2}{\mu ^2}{e_1}{\rm{dn}}{{(\sqrt { - {e_1}} \xi )}^2}{{\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2}} \over {{{\left( {{{\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2} - {\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2}} \right)}^2}}} + {{4{\alpha ^2}{\mu ^2}{e_1}\left( {\left( {{k^2} - 2} \right){{\left( {{\rm{sn}}(\sqrt { - {e_1}} \xi ) + {\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2} - 2\left( {1 - {k^2}} \right){\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2}} \right){\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2}} \over {\left( {{\rm{sn}}(\sqrt { - {e_1}} \xi ) + {\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)\left( {{{\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2} - {\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2}} \right)}} ; u1.38.2=cμ+4μαe1dn(e1ξ)(1+cn(e1ξ))(1+cn(e1ξ))2sn(e1ξ)2 {u_{1.38.2}} = - {c \over \mu } + {{4\mu \alpha \sqrt { - {e_1}} {\rm{dn}}(\sqrt { - {e_1}} \xi )\left( {1 + {\rm{cn}}(\sqrt { - {e_1}} \xi )} \right)} \over {{{\left( {{\rm{1}} + {\rm{cn}}(\sqrt { - {e_1}} \xi )} \right)}^2} - {\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2}}} , v1.38.2=α2μ2e1(k2+1)4α2μ2e1((k22)(sn(e1ξ)+cn(e1ξ)+1)22(1k2)sn(e1ξ)2)sn(e1ξ)2(sn(e1ξ)+cn(e1ξ)+1)((cn(e1ξ)+1)2sn(e1ξ)2) {{v_{1.38.2}} = - {\alpha ^2}{\mu ^2}{e_1}({k^2} + 1) - {4{\alpha ^2}{\mu ^2}{e_1}\left( {\left( {{k^2} - 2} \right){{\left( {{\rm{sn}}(\sqrt { - {e_1}} \xi ) + {\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2} - 2\left( {1 - {k^2}} \right){\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2}} \right){\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2}} \over {\left( {{\rm{sn}}(\sqrt { - {e_1}} \xi ) + {\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)\left( {{{\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2} - {\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2}} \right)}} ; u1.38.3=cμ+4μαe1dn(e1ξ)(cn(e1ξ)+1)sn(e1ξ)2(cn(e1ξ)+1)2 {u_{1.38.3}} = - {c \over \mu } + {{4\mu \alpha \sqrt { - {e_1}} {\rm{dn}}(\sqrt { - {e_1}} \xi )\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)} \over {{\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2} - {{\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2}}} , v1.38.3=α2μ2e1(k2+1)+16α2μ2e1dn(e1ξ)2(cn(e1ξ)+1)2(sn(e1ξ)2(cn(e1ξ)+1)2)2+4α2μ2e1((k22)(sn(e1ξ)+cn(e1ξ)+1)22(1k2)(cn(e1ξ)+1)2)(cn(e1ξ)+1)(sn(e1ξ)+cn(e1ξ)+1)((cn(e1ξ)+1)2sn(e1ξ)2) {v_{1.38.3}} = - {\alpha ^2}{\mu ^2}{e_1}({k^2} + 1) + {{16{\alpha ^2}{\mu ^2}{e_1}{\rm{dn}}{{(\sqrt { - {e_1}} \xi )}^2}{{\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2}} \over {{{\left( {{\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2} - {{\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2}} \right)}^2}}} + {{4{\alpha ^2}{\mu ^2}{e_1}\left( {\left( {{k^2} - 2} \right){{\left( {{\rm{sn}}(\sqrt { - {e_1}} \xi ) + {\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2} - 2\left( {1 - {k^2}} \right){{\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2}} \right)\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)} \over {\left( {{\rm{sn}}(\sqrt { - {e_1}} \xi ) + {\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)\left( {{{\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2} - {\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2}} \right)}} ; u1.38.4=cμ4μαe1dn(e1ξ)(cn(e1ξ)+1)sn(e1ξ)2(cn(e1ξ)+1)2 {u_{1.38.4}} = - {c \over \mu } - {{4\mu \alpha \sqrt { - {e_1}} {\rm{dn}}(\sqrt { - {e_1}} \xi )\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)} \over {{\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2} - {{\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2}}} , v1.38.4=α2μ2e1(k2+1)4α2μ2e1((k22)(sn(e1ξ)+cn(e1ξ)+1)22(1k2)(cn(e1ξ)+1)2)(cn(e1ξ)+1)(sn(e1ξ)+cn(e1ξ)+1)((cn(e1ξ)+1)2sn(e1ξ)2) {v_{1.38.4}} = - {\alpha ^2}{\mu ^2}{e_1}({k^2} + 1) - {{4{\alpha ^2}{\mu ^2}{e_1}\left( {\left( {{k^2} - 2} \right){{\left( {{\rm{sn}}(\sqrt { - {e_1}} \xi ) + {\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2} - 2\left( {1 - {k^2}} \right){{\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2}} \right)\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)} \over {\left( {{\rm{sn}}(\sqrt { - {e_1}} \xi ) + {\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)\left( {{{\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2} - {\rm{sn}}{{(\sqrt { - {e_1}} \xi )}^2}} \right)}} ,

1.39) e0 > 0, e1 = −4e0, e2 = 4e0(2 − k2), e3 = −8e0(1 − k2), e4 = 4e0(1 − k2),

u1.39.1=cμ8μαe0(1k2)(1k2sn(2e0ξ)+dn(2e0ξ))cn(2e0ξ)2(1k2sn(2e0ξ)+dn(2e0ξ))2 {u_{1.39.1}} = - {c \over \mu } - {{8\mu \alpha \sqrt {{e_0}(1 - {k^2})} \left( {\sqrt {1 - {k^2}} {\rm{sn}}(2\sqrt {{e_0}} \xi ) + {\rm{dn}}(2\sqrt {{e_0}} \xi )} \right)} \over {{\rm{cn}}{{(2\sqrt {{e_0}} \xi )}^2} - {{\left( {\sqrt {1 - {k^2}} {\rm{sn}}(2\sqrt {{e_0}} \xi ) + {\rm{dn}}(2\sqrt {{e_0}} \xi )} \right)}^2}}} , v1.39.1=4α2μ2e0(k2+1)29α2μ2e0(1k2)(1k2sn(2e0ξ)+dn(2e0ξ))2(cn(2e0ξ)2(1k2sn(2e0ξ)+dn(2e0ξ))2)2+16α2μ2e0(2k2)((1k2sn(2e0ξ)+cn(2e0ξ)+dn(2e0ξ))2+cn(2e0ξ)2)cn(2e0ξ)(1k2sn(2e0ξ)+cn(2e0ξ)+dn(2e0ξ))2(1k2sn(2e0ξ)+cn(2e0ξ)dn(2e0ξ)) {v_{1.39.1}} = - 4{\alpha ^2}{\mu ^2}{e_0}({k^2} + 1) - {{{2^9}{\alpha ^2}{\mu ^2}{e_0}(1 - {k^2}){{\left( {\sqrt {1 - {k^2}} {\rm{sn}}(2\sqrt {{e_0}} \xi ) + {\rm{dn}}(2\sqrt {{e_0}} \xi )} \right)}^2}} \over {{{\left( {{\rm{cn}}{{(2\sqrt {{e_0}} \xi )}^2} - {{\left( {\sqrt {1 - {k^2}} {\rm{sn}}(2\sqrt {{e_0}} \xi ) + {\rm{dn}}(2\sqrt {{e_0}} \xi )} \right)}^2}} \right)}^2}}} + {{16{\alpha ^2}{\mu ^2}{e_0}\left( {2 - {k^2}} \right)\left( {{{\left( {\sqrt {1 - {k^2}} {\rm{sn}}(2\sqrt {{e_0}} \xi ) + {\rm{cn}}(2\sqrt {{e_0}} \xi ) + {\rm{dn}}(2\sqrt {{e_0}} \xi )} \right)}^2} + {\rm{cn}}{{(2\sqrt {{e_0}} \xi )}^2}} \right){\rm{cn}}(2\sqrt {{e_0}} \xi )} \over {{{\left( {\sqrt {1 - {k^2}} {\rm{sn}}(2\sqrt {{e_0}} \xi ) + {\rm{cn}}(2\sqrt {{e_0}} \xi ) + {\rm{dn}}(2\sqrt {{e_0}} \xi )} \right)}^2}\left( { - \sqrt {1 - {k^2}} {\rm{sn}}(2\sqrt {{e_0}} \xi ) + {\rm{cn}}(2\sqrt {{e_0}} \xi ) - {\rm{dn}}(2\sqrt {{e_0}} \xi )} \right)}} . u1.39.2=cμ+8μαe0(1k2)(1k2sn(2e0ξ)+dn(2e0ξ))cn(2e0ξ)2(1k2sn(2e0ξ)+dn(2e0ξ))2 {u_{1.39.2}} = - {c \over \mu } + {{8\mu \alpha \sqrt {{e_0}(1 - {k^2})} \left( {\sqrt {1 - {k^2}} {\rm{sn}}(2\sqrt {{e_0}} \xi ) + {\rm{dn}}(2\sqrt {{e_0}} \xi )} \right)} \over {{\rm{cn}}{{(2\sqrt {{e_0}} \xi )}^2} - {{\left( {\sqrt {1 - {k^2}} {\rm{sn}}(2\sqrt {{e_0}} \xi ) + {\rm{dn}}(2\sqrt {{e_0}} \xi )} \right)}^2}}} , v1.39.2=4α2μ2e0(k2+1)16α2μ2e0(2k2)((1k2sn(2e0ξ)+cn(2e0ξ)+dn(2e0ξ))2+cn(2e0ξ)2)cn(2e0ξ)(1k2sn(2e0ξ)+cn(2e0ξ)+dn(2e0ξ))2(1k2sn(2e0ξ)+cn(2e0ξ)dn(2e0ξ)) {v_{1.39.2}} = - 4{\alpha ^2}{\mu ^2}{e_0}({k^2} + 1) - {{16{\alpha ^2}{\mu ^2}{e_0}\left( {2 - {k^2}} \right)\left( {{{\left( {\sqrt {1 - {k^2}} {\rm{sn}}(2\sqrt {{e_0}} \xi ) + {\rm{cn}}(2\sqrt {{e_0}} \xi ) + {\rm{dn}}(2\sqrt {{e_0}} \xi )} \right)}^2} + {\rm{cn}}{{(2\sqrt {{e_0}} \xi )}^2}} \right){\rm{cn}}(2\sqrt {{e_0}} \xi )} \over {{{\left( {\sqrt {1 - {k^2}} {\rm{sn}}(2\sqrt {{e_0}} \xi ) + {\rm{cn}}(2\sqrt {{e_0}} \xi ) + {\rm{dn}}(2\sqrt {{e_0}} \xi )} \right)}^2}\left( { - \sqrt {1 - {k^2}} {\rm{sn}}(2\sqrt {{e_0}} \xi ) + {\rm{cn}}(2\sqrt {{e_0}} \xi ) - {\rm{dn}}(2\sqrt {{e_0}} \xi )} \right)}} .

1.40) e0=0,e1=0,e2>0,e32=4e4e20,, {e_0} = 0,{e_1} = 0,{e_2} > 0,e_3^2 = 4{e_4}{e_2} \ne 0,

u1.40.1=cμ±8αμe4e2e2sech2(e22ξ)(e32tanh(e22ξ)+e4e2(1tanh(e22ξ))2)(e32e4e2(1tanh(e22ξ))2)(e32e4e2(3tanh2(e22ξ)+2tanh(e22ξ)5)) {u_{1.40.1}} = - {c \over \mu } \pm {{8\alpha \mu {e_4}{e_2}\sqrt {{e_2}} \sec {h^2}({{\sqrt {{e_2}} } \over 2}\xi )\left( {e_3^2\tan h({{\sqrt {{e_2}} } \over 2}\xi ) + {e_4}{e_2}{{\left( {1 - \tan h({{\sqrt {{e_2}} } \over 2}\xi )} \right)}^2}} \right)} \over {\left( {e_3^2 - {e_4}{e_2}{{\left( {1 - \tan h({{\sqrt {{e_2}} } \over 2}\xi )} \right)}^2}} \right)\left( {e_3^2 - {e_4}{e_2}\left( {3\tan {h^2}({{\sqrt {{e_2}} } \over 2}\xi ) + 2\tan h({{\sqrt {{e_2}} } \over 2}\xi ) - 5} \right)} \right)}} , v1.40.1=α2μ2(3e328e4e2)4e432α2μ2e42e23e3sech4(e22ξ)(e32tanh(e22ξ)+e4e2(1tanh(e22ξ)2)2(e32+e4e2(1tanh(e22ξ))2)2(e32e4e2(3tanh2(e22ξ)+2tanh(e22ξ)5))28α2μ2e4e22e3(e32e4e2(1tanh2(e22ξ))22e4e2e32sech4(e22ξ)]sech2(e22ξ)(e32e4e2(1tanh(e22ξ)2)2(e32e4e2(1tanh(e22ξ))22e4e2sech2(e22ξ) {v_{1.40.1}} = {{{\alpha ^2}{\mu ^2}\left( {3e_3^2 - 8{e_4}{e_2}} \right)} \over {4{e_4}}} - {{32{\alpha ^2}{\mu ^2}e_4^2e_2^3{e_3}sec{h^4}\left( {{{\sqrt {{e_2}} } \over 2}\xi } \right)\left( {e_3^2\tanh \left( {{{\sqrt {{e_2}} } \over 2}\xi } \right) + {e_4}{e_2}{{\left( {1 - \tanh {{\left( {{{\sqrt {{e_2}} } \over 2}\xi } \right)}^2}} \right)}^2}} \right.} \over {{{\left( {e_3^2 + {e_4}{e_2}{{\left( {1 - \tanh \left( {{{\sqrt {{e_2}} } \over 2}\xi } \right)} \right)}^2}} \right)}^2}{{\left( {e_3^2 - {e_4}{e_2}\left( {3\mathop {\tanh }\nolimits^2 \left( {{{\sqrt {{e_2}} } \over 2}\xi } \right) + 2\tanh \left( {{{\sqrt {{e_2}} } \over 2}\xi } \right) - 5} \right)} \right)}^2}}} - {{8{\alpha ^2}{\mu ^2}{e_4}e_2^2{e_3}\left( {e_3^2 - {e_4}{e_2}{{\left( {1 - \mathop {\tanh }\nolimits^2 \left( {{{\sqrt {{e_2}} } \over 2}\xi } \right)} \right)}^2} - 2{e_4}{e_2}e_3^2sec{h^4}\left( {{{\sqrt {{e_2}} } \over 2}\xi } \right)} \right]sec{h^2}\left( {{{\sqrt {{e_2}} } \over 2}\xi } \right)} \over {\left( {e_3^2 - {e_4}{e_2}{{\left( {1 - \tanh {{\left( {{{\sqrt {{e_2}} } \over 2}\xi } \right)}^2}} \right)}^2}\left( {e_3^2 - {e_4}{e_2}{{\left( {1 - \tanh \left( {{{\sqrt {{e_2}} } \over 2}\xi } \right)} \right)}^2} - 2{e_4}{e_2}sec{h^2}\left( {{{\sqrt {{e_2}} } \over 2}\xi } \right)} \right.} \right.}} .

where k(0 < k < 1) denotes the modulus of the Jacobi elliptic function, k1=1k2 {k_1} = \sqrt {1 - {k^2}} , D1,D2, D3(D1D2D3 ≠ 0, μ and c(μc ≠ 0) are arbitrary constants, and ξ = μx − ct.

For case 2, we can obtain a lot of travelling wave solutions using the same process as that in case 1. For the sake of simplicity, only interesting ones are listed,

2.1) e1 = e3 = 0,e0 = k2,e2 = −(1 + k2),e4 = 1, ξ = μx + c0μ t

u2.1.1=c0+2αμksn(ξ) {u_{2.1.1}} = {c_0} + {{2\alpha \mu } \over k}{\rm{sn}}(\xi ) , v2.1.1=α2μ2(1+k2)2α2μ2(1kds(ξ)cs(ξ))k2ns(ξ)2 {v_{2.1.1}} = {\alpha ^2}{\mu ^2}\left( {1 + {k^2}} \right) - {{2{\alpha ^2}{\mu ^2}\left( {1 - k{\rm{ds}}(\xi ){\rm{cs}}(\xi )} \right)} \over {{k^2}{\rm{ns}}{{(\xi )}^2}}} ; u2.1.2=c0+2αμkcd(ξ) {u_{2.1.2}} = {c_0} + {{2\alpha \mu } \over k}{\rm{cd}}(\xi ) , v2.1.2=α2μ2(1+k2)2α2μ2(1+k(1k2)nc(ξ)sc(ξ))k2dc(ξ)2 {v_{2.1.2}} = {\alpha ^2}{\mu ^2}\left( {1 + {k^2}} \right) - {{2{\alpha ^2}{\mu ^2}\left( {1 + k\left( {1 - {k^2}} \right){\rm{nc}}(\xi ){\rm{sc}}(\xi )} \right)} \over {{k^2}{\rm{dc}}{{(\xi )}^2}}} ;

2.2) e1=e3=0,e0=k44,e2=12(k22),e4=14,ξ=μx+c0μt {e_1} = {e_3} = 0,{e_0} = {{{k^4}} \over 4},{e_2} = {1 \over 2}({k^2} - 2),{e_4} = {1 \over 4},\xi = \mu x + {c_0}\mu \;t

u2.2.1=c0+4αμk2(ns(ξ)±ds(ξ)) {u_{2.2.1}} = {c_0} + {{4\alpha \mu } \over {{k^2}\left( {{\rm{ns}}(\xi ) \pm {\rm{ds}}(\xi )} \right)}} , v2.2.1=α2μ2(k22)4α2μ2(2k2cs(ξ)(ds(ξ)±ns(ξ)))k4(ns(ξ)±ds(ξ))2 {v_{2.2.1}} = - {\alpha ^2}{\mu ^2}\left( {{k^2} - 2} \right) - {{4{\alpha ^2}{\mu ^2}\left( {2 - {k^2}{\rm{cs}}(\xi )\left( {{\rm{ds}}(\xi ) \pm {\rm{ns}}(\xi )} \right)} \right)} \over {{k^4}{{\left( {{\rm{ns}}(\xi ) \pm {\rm{ds}}(\xi )} \right)}^2}}} ,

2.3) e1=e3=0,e0=k24,e2=(k22)2,e4=k24,ξ=μx+c0μt {e_1} = {e_3} = 0,{e_0} = {{{k^2}} \over 4},{e_2} = {{({k^2} - 2)} \over 2},{e_4} = {{{k^2}} \over 4},\xi = \mu x + {c_0}\mu \;t

u2.3.1=c0+4αμk(sn(ξ)±icn(ξ)) {u_{2.3.1}} = {c_0} + {{4\alpha \mu } \over {k\left( {{\rm{sn}}(\xi ) \pm i{\rm{cn}}(\xi )} \right)}} , v2.3.1=12α2μ2(k22)4α2μ2(2ikdn(ξ)(icn(ξ)±sn(ξ)))k2(sn(ξ)±icn(ξ))2 {v_{2.3.1}} = - {1 \over 2}{\alpha ^2}{\mu ^2}\left( {{k^2} - 2} \right) - {{4{\alpha ^2}{\mu ^2}\left( {2 - {\rm{ikdn}}(\xi )\left( {i{\rm{cn}}(\xi ) \pm {\rm{sn}}(\xi )} \right)} \right)} \over {{k^2}{{\left( {{\rm{sn}}(\xi ) \pm i{\rm{cn}}(\xi )} \right)}^2}}} ,

2.4) e0=k,e1=4k,e2=1+6kk2,e3=2(k1)2,e4=(k1)2,ξ=μx+μ(c0±2αμk)t {e_0} = k,{e_1} = - 4k,{e_2} = - 1 + 6k - {k^2},{e_3} = 2(k - {1)^2},{e_4} = - {(k - 1)^2},\xi = \mu x + \mu ({c_0} \pm 2\alpha \mu \sqrt k )t

u2.4.1=c0+2αμ(ksn(ξ)+1)k {u_{2.4.1}} = {c_0} + {{2\alpha \mu \left( {\sqrt k {\rm{sn}}(\xi ) + 1} \right)} \over {\sqrt k }} , v2.4.1=2α2μ2(k1)2+4α2μ2k±4α2μ2k(ksn(ξ)+1)2α2μ2((ksn(ξ)+1)2kdn(ξ)cd(ξ))k {v_{2.4.1}} = - 2{\alpha ^2}{\mu ^2}{(k - 1)^2} + 4{\alpha ^2}{\mu ^2}k \pm 4{\alpha ^2}{\mu ^2}k\left( {\sqrt k {\rm{sn}}(\xi ) + 1} \right) - {{2{\alpha ^2}{\mu ^2}\left( {{{\left( {\sqrt k {\rm{sn}}(\xi ) + 1} \right)}^2} - k{\rm{dn}}(\xi ){\rm{cd}}(\xi )} \right)} \over k} ,

2.5) e0=e14,e1<0,e2=e1(2k2),e3=2e1(1k2),e4=e1(1k2),ξ=μx+μ(c0±αμe1)t {e_0} = - {{{e_1}} \over 4},{e_1} < 0,{e_2} = - {e_1}(2 - {k^2}),{e_3} = 2{e_1}(1 - {k^2}),{e_4} = - {e_1}(1 - {k^2}),\xi = \mu x + \mu ({c_0} \pm \alpha \mu \sqrt { - {e_1}} )t

u2.5.1=c0+4αμe1(ns(e1ξ)+cs(e1ξ)+1) {u_{2.5.1}} = {c_0} + {{4\alpha \mu } \over {\sqrt { - {e_1}} }}\left( {{\rm{ns}}(\sqrt { - {e_1}} \xi ) + {\rm{cs}}(\sqrt { - {e_1}} \xi ) + 1} \right) , v2.5.1=α2μ2e1k214α2μ2e1±4α2μ2(ns(e1ξ)+cs(e1ξ)+1)+4α2μ2e12(2(ns(e1ξ)+cs(e1ξ)+1)2e1ds(e1ξ)2(1+cn(e1ξ))2) {v_{2.5.1}} = - {\alpha ^2}{\mu ^2}{e_1}{k^2} - 14{\alpha ^2}{\mu ^2}{e_1} \pm 4{\alpha ^2}{\mu ^2}\left( {{\rm{ns}}(\sqrt { - {e_1}} \xi ) + {\rm{cs}}(\sqrt { - {e_1}} \xi ) + 1} \right) + 4{\alpha ^2}{\mu ^2}{e_1}^{ - 2}\left( {2{{\left( {{\rm{ns}}(\sqrt { - {e_1}} \xi ) + {\rm{cs}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2} - {e_1}{\rm{ds}}{{(\sqrt { - {e_1}} \xi )}^2}{{\left( {1 + {\rm{cn}}(\sqrt { - {e_1}} \xi )} \right)}^2}} \right) ; u2.5.2=c0+4αμ(sn(e1ξ)+cn(e1ξ)+1)e1(cn(e1ξ)+1) {u_{2.5.2}} = {c_0} + {{4\alpha \mu \left( {{\rm{sn}}(\sqrt { - {e_1}} \xi ) + {\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)} \over {\sqrt { - {e_1}} \left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}} , v2.5.2=α2μ2e1k214α2μ2e1±4α2μ2(sn(e1ξ)+cn(e1ξ)+1)e1(cn(e1ξ)+1)+4α2μ2(2(sn(e1ξ)+cn(e1ξ)+1)2+e1dn(e1ξ)2(1+cn(e1ξ))2)e12dn(e1ξ)2(1+cn(e1ξ))2 {v_{2.5.2}} = - {\alpha ^2}{\mu ^2}{e_1}{k^2} - 14{\alpha ^2}{\mu ^2}{e_1} \pm {{4{\alpha ^2}{\mu ^2}\left( {{\rm{sn}}(\sqrt { - {e_1}} \xi ) + {\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)} \over {{e_1}\left( {{\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}} + {{4{\alpha ^2}{\mu ^2}\left( {2{{\left( {{\rm{sn}}(\sqrt { - {e_1}} \xi ) + {\rm{cn}}(\sqrt { - {e_1}} \xi ) + 1} \right)}^2} + {e_1}{\rm{dn}}{{(\sqrt { - {e_1}} \xi )}^2}{{\left( {1 + {\rm{cn}}(\sqrt { - {e_1}} \xi )} \right)}^2}} \right)} \over {{e_1}^2{\rm{dn}}{{(\sqrt { - {e_1}} \xi )}^2}{{\left( {1 + {\rm{cn}}(\sqrt { - {e_1}} \xi )} \right)}^2}}} ,

2.6) e0=e140,e1=(βξ)2,e2=(2e1+β2),e3=2(e1+β2),e4=(e1+β2),ξ=μx+μ(c0±αμkβ)t {e_0} = - {{{e_1}} \over 4} \ne 0,{e_1} = - {\left( {\beta \xi } \right)^2},{e_2} = - (2{e_1} + {\beta ^2}),{e_3} = 2({e_1} + {\beta ^2}),{e_4} = - ({e_1} + {\beta ^2}),\xi = \mu x + \mu ({c_0} \pm \alpha \mu k\beta )t

u2.6.1=c0+4αμe1k(k+ds(βξ)ns(βξ)), {u_{2.6.1}} = {c_0} + {{4\alpha \mu } \over {\sqrt { - {e_1}} k}}\left( {k + {\rm{ds}}(\beta \xi ) - {\rm{ns}}(\beta \xi )} \right), , v2.6.1=α2μ2(2e1+β2)+α2μ2k2β2±4α2μ2k(kns(βξ)+ds(βξ))4α2μ2(2(ksn(βξ)+dn(βξ)1)2+e1kβ(1cn(βξ)))e1k2sn(βξ)2 {v_{2.6.1}} = {\alpha ^2}{\mu ^2}\left( {2{e_1} + {\beta ^2}} \right) + {\alpha ^2}{\mu ^2}{k^2}{\beta ^2} \pm {{4{\alpha ^2}{\mu ^2}} \over k}\left( {k - {\rm{ns}}(\beta \xi ) + {\rm{ds}}(\beta \xi )} \right) - {{4{\alpha ^2}{\mu ^2}\left( {2{{\left( {k{\rm{sn}}(\beta \xi ) + {\rm{dn}}(\beta \xi ) - 1} \right)}^2} + \sqrt { - {e_1}} k\beta \left( {1 - {\rm{cn}}(\beta \xi )} \right)} \right)} \over { - {e_1}{k^2}{\rm{sn}}{{(\beta \xi )}^2}}} , u2.6.2=c0+4αμ(ksn(βξ)dn(βξ)1)e1(1+dn(βξ)) {u_{2.6.2}} = {c_0} + {{4\alpha \mu \left( {k{\rm{sn}}(\beta \xi ) - {\rm{dn}}(\beta \xi ) - 1} \right)} \over {\sqrt { - {e_1}} \left( {1 + {\rm{dn}}(\beta \xi )} \right)}} , v2.6.2=α2μ2(2e1+β2)+α2μ2k2β2±4α2μ2(ksn(βξ)dn(βξ)1)(1+dn(βξ))+4α2μ2(2(ksn(βξ)dn(βξ)1)2+e1k2β(1+dn(βξ)))e1(1+dn(βξ))2 {v_{2.6.2}} = {\alpha ^2}{\mu ^2}\left( {2{e_1} + {\beta ^2}} \right) + {\alpha ^2}{\mu ^2}{k^2}{\beta ^2} \pm {{4{\alpha ^2}{\mu ^2}\left( {k{\rm{sn}}(\beta \xi ) - {\rm{dn}}(\beta \xi ) - 1} \right)} \over {\left( {1 + {\rm{dn}}(\beta \xi )} \right)}} + {{4{\alpha ^2}{\mu ^2}\left( {2{{\left( {k{\rm{sn}}(\beta \xi ) - {\rm{dn}}(\beta \xi ) - 1} \right)}^2} + \sqrt { - {e_1}} {k^2}\beta \left( {1 + {\rm{dn}}(\beta \xi )} \right)} \right)} \over {{e_1}{{\left( {1 + {\rm{dn}}(\beta \xi )} \right)}^2}}} ,

2.7) e0=e224e4,e2<0,e4>0,e1=e3=0,ξ=μx+c0μt {e_0} = - {{e_2^2} \over {4{e_4}}},{e_2} < 0,{e_4} > 0,{e_1} = {e_3} = 0,\xi = \mu x + {c_0}\mu \;t

u2.7.2=c04αμe4e2e22tanhe22(ξ) {u_{2.7.2}} = {c_0} - {{4\alpha \mu {e_4}} \over {{e_2}\sqrt {{{ - {e_2}} \over 2}} {\rm{tanh}}\sqrt {{{ - {e_2}} \over 2}} (\xi )}} , v2.7.2=α2μ2e2+4α2μ2e4e23(4e4+e22csch(e22ξ)sech(e22ξ))coth2e22(ξ) {v_{2.7.2}} = - {\alpha ^2}{\mu ^2}{e_2} + {{4{\alpha ^2}{\mu ^2}{e_4}} \over {{e_2}^3}}\left( {4{e_4} + {e_2}^2\csc h(\sqrt {{{ - {e_2}} \over 2}} \xi )\sec h(\sqrt {{{ - {e_2}} \over 2}} \xi )} \right){\rm{cot}}{{\rm{h}}^2}\sqrt {{{ - {e_2}} \over 2}} (\xi ) ,

2.8) e0=e224e4,e2>0,e4>0,e1=e3=0,ξ=μx+c0μt {e_0} = - {{e_2^2} \over {4{e_4}}},{e_2} > 0,{e_4} > 0,{e_1} = {e_3} = 0,\xi = \mu x + {c_0}\mu \;t

where k(0 < k < 1) expresses the modulus of the Jacobi elliptic function, i2 = −1, , c0, μ,β and c(β μc ≠ 0) are arbitrary constants.

Remark 1

All the solutions to Eqs (1a) and (1b) obtained in this paper are checked by Maple17.

Remark 2

We also can suppose that u(ξ)=c0+a0+a1ϕb0+b1ϕ u(\xi ) = {c_0} + {{{a_0} + {a_1}\phi '} \over {{b_0} + {b_1}\phi }} , v(ξ)=d0+f10+f11ϕg10+g11ϕ+(f20+f21ϕg20+g21ϕ)2 v(\xi ) = {d_0} + {{{f_{10}} + {f_{11}}\phi '} \over {{g_{10}} + {g_{11}}\phi }} + {\left( {{{{f_{20}} + {f_{21}}\phi '} \over {{g_{20}} + {g_{21}}\phi }}} \right)^2} , which are substituted into Eq. (10) instead of Eq. (12). The same solutions of Eqs (1a) and (1b) can be obtained.

Remark 3

We only consider one special case to Eqs (1a) and (1b). We are able to get more solutions to Eqs (1a) and (1b) for other conditions such as other positive integers of n,m by the same process as that in the above. The solutions for Eq. (7) are listed in literature [33].

Conclusions

Making full use of solutions of one nonlinear ODE, many varieties of explicit and exact travelling wave solutions for Eq. (1) are found, which include Jacobi elliptic function solutions and combined Jacobi elliptic function solutions, hyperbolic function solutions, triangular periodic solutions and rational solutions. As far as our information goes, some solutions to Eqs (1a) and (1b) obtained in this paper are not proposed in any literature existing until now. The above procedures can be extended to many other nonlinear PDEs.

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