3 The proximate long water equations
The travelling wave variable is assumed that
(8)
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u ( x , t ) = u ( ξ ) , v ( x , t ) = v ( ξ ) ξ = μ x − ct 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 )
(9a)
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− cu ′ − μ uu ′ − μ v ′ + α μ 2 u ″ = 0 - cu' - \mu uu' - \mu v' + \alpha {\mu ^2}u'' = 0
(9b)
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− cv ′ − μ ( uv ) ′ − α μ 2 v ″ = 0 - {\rm{cv}}' - \mu ({\rm{uv}})' - \alpha {\mu ^2}v'' = 0
Integrating the ODEs above in regard to ξ yields
(10a)
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F 1 − cu − 1 2 μ u 2 − μ v + α μ 2 u ′ = 0 {F_1} - {\rm{cu}} - {1 \over 2}\mu {u^2} - \mu v + \alpha {\mu ^2}u' = 0
(10b)
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F 2 − cv − μ uv − α μ 2 v ′ = 0 {F_2} - cv - \mu uv - \alpha {\mu ^2}{v^\prime} = 0
F 1 and F 2 are integration constants to be determined later.
Eqs (9a) and (10a) can be described respectively in the following forms
(11a)
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v ′ = − c μ u ′ − uu ′ + α μ u ″ v' = - {c \over \mu }u' - uu' + \alpha \mu u''
(11b)
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v = F 1 μ − c μ u − 1 2 u 2 + α μ u ′ 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
(12)
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( F 2 − c μ F 1 ) + ( c 2 μ − F 1 ) u + 3 c 2 u 2 + 1 2 μ u 3 − α 2 μ 3 u ″ = 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 u 3 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.
(13)
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u ( ξ ) = c 0 + a 0 + a 1 ϕ ′ b 0 + b 1 ϕ , u(\xi ) = {c_0} + {{{a_0} + {a_1}\phi '} \over {{b_0} + {b_1}\phi }}
(14)
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ϕ ′ = ε e 0 + e 1 ϕ + e 2 ϕ 2 + e 3 ϕ 3 + e 4 ϕ 4 \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
ϕ i ∑ j = 0 4 e j ϕ j ( 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
ϕ i ∑ j = 0 4 e j ϕ j ( 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 a 1 = ±2αμb 1 ,
c 0 = − c μ {c_0} = - {c \over \mu }
, a 0 = 0, F 2 = 0,
b 0 = e 3 4 e 4 b 1 {b_0} = {{{e_3}} \over {4{e_4}}}{b_1}
,
e 1 = e 3 ( 4 e 2 e 4 − e 3 2 ) 8 e 4 2 {e_1} = {{{e_3}(4{e_2}{e_4} - e_3^2)} \over {8e_4^2}}
,
F 1 = − 3 α 2 μ 4 e 3 2 − 8 α 2 μ 4 e 2 e 4 + 2 c 2 e 4 4 μ e 4 {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 e 0 , e 2 , e 3 are arbitrary constants, and μ,c, e 4 and b 1 are arbitrary nonzero constants.
Case 2 a 1 = 0, b 0 = 0,
b 1 = a 0 2 α μ e 0 {b_1} = {{{a_0}} \over {2\alpha \mu \sqrt {{e_0}} }}
,
F 2 = α 3 μ 4 e 0 ( 2 e 3 e 0 − e 1 e 2 ) 2 e 0 {F_2} = {{{\alpha ^3}{\mu ^4}\sqrt {{e_0}} (2{e_3}{e_0} - {e_1}{e_2})} \over {2{e_0}}}
,
F 1 = μ ( ± 2 c 0 α μ e 1 e 0 − 2 c 0 2 e 0 − 4 α 2 μ 2 e 0 e 2 + α 2 μ 2 e 1 2 ) 4 e 0 {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 ( ± α μ e 1 − 2 c 0 e 0 ) 2 e 0 c = {{\mu ( \pm \alpha \mu {e_1} - 2{c_0}\sqrt {{e_0}} )} \over {2\sqrt {{e_0}} }}
, where c 0 ,e 1 ,e 2 ,e 3 ,a 0 are arbitrary constants, and μ ≠ 0,e 0 > 0,e 4 ≠ 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) e 0 = 1, e 2 = −(1 − k 2 ), e 4 = k 2 , e 1 = e 3 = 0,
u 1.1.1 = − c μ + 2 μ α ( 1 − k 2 ) cs ( ξ ) dn ( ξ ) {u_{1.1.1}} = - {c \over \mu } + 2\mu \alpha (1 - {k^2}){\rm{cs}}(\xi ){\rm{dn}}(\xi )
,
v 1.1.1 = − 4 α 2 μ 2 ns ( ξ ) 2 {v_{1.1.1}} = - 4{\alpha ^2}{\mu ^2}{\rm{ns}}{(\xi )^2}
;
u 1.1.2 = − c μ − 2 μ α ( 1 − k 2 ) cs ( ξ ) dn ( ξ ) {u_{1.1.2}} = - {c \over \mu } - 2\mu \alpha (1 - {k^2}){\rm{cs}}(\xi ){\rm{dn}}(\xi )
,
v 1.1.2 = − 4 α 2 μ 2 k 2 sn ( ξ ) 2 {v_{1.1.2}} = - 4{\alpha ^2}{\mu ^2}{k^2}{\rm{sn}}{(\xi )^2}
;
u 1.1.3 = − c μ − 2 μ α ( 1 − k 2 ) sc ( ξ ) nd ( ξ ) {u_{1.1.3}} = - {c \over \mu } - 2\mu \alpha (1 - {k^2}){\rm{sc}}(\xi ){\rm{nd}}(\xi )
,
v 1.1.3 = − 4 α 2 μ 2 dc ( ξ ) 2 {v_{1.1.3}} = - 4{\alpha ^2}{\mu ^2}{\rm{dc}}{(\xi )^2}
,
u 1.1.4 = − c μ + 2 μ α ( 1 − k 2 ) sc ( ξ ) nd ( ξ ) {u_{1.1.4}} = - {c \over \mu } + 2\mu \alpha (1 - {k^2}){\rm{sc}}(\xi ){\rm{nd}}(\xi )
,
v 1.1.4 = − 4 α 2 μ 2 k 2 cd ( ξ ) 2 {v_{1.1.4}} = - 4{\alpha ^2}{\mu ^2}{k^2}{\rm{cd}}{(\xi )^2}
,
1.2) e 0 = 1 − k 2 , e 2 = 2k 2 − 1, e 4 = −k 2 , e 1 = e 3 = 0,
u 1.2.1 = − c μ − 2 μ α sc ( ξ ) dn ( ξ ) {u_{1.2.1}} = - {c \over \mu } - 2\mu \alpha {\rm{sc}}(\xi ){\rm{dn}}(\xi )
,
v 1.2.1 = − 4 α 2 μ 2 ( 1 − k 2 ) nc ( ξ ) 2 {v_{1.2.1}} = - 4{\alpha ^2}{\mu ^2}({\rm{1}} - {k^2}){\rm{nc}}{(\xi )^2}
;
u 1.2.2 = − c μ + 2 μ α sc ( ξ ) dn ( ξ ) {u_{1.2.2}} = - {c \over \mu } + 2\mu \alpha {\rm{sc}}(\xi ){\rm{dn}}(\xi )
,
v 1.2.2 = 4 α 2 μ 2 k 2 cn ( ξ ) 2 {v_{1.2.2}} = 4{\alpha ^2}{\mu ^2}{k^2}{\rm{cn}}{(\xi )^2}
;
1.3) e 0 = k 2 − 1, e 2 = 2 − k 2 , e 4 = −1, e 1 = e 3 = 0
u 1.3.1 = − c μ − 2 μ α k 2 sd ( ξ ) cn ( ξ ) {u_{1.3.1}} = - {c \over \mu } - 2\mu \alpha {k^2}{\rm{sd}}(\xi ){\rm{cn}}(\xi )
,
v 1.3.1 = − 4 α 2 μ 2 ( k 2 − 1 ) nd ( ξ ) 2 {v_{1.3.1}} = - 4{\alpha ^2}{\mu ^2}({k^2} - {\rm{1}}){\rm{nd}}{(\xi )^2}
,
1.4) e 0 = k 2 , e 2 = −(1 + k 2 ), e 4 = 1, e 1 = e 3 = 0,
u 1.4.1 = − c μ − 2 μ α dn ( ξ ) cs ( ξ ) {u_{1.4.1}} = - {c \over \mu } - 2\mu \alpha {\rm{dn}}(\xi ){\rm{cs}}(\xi )
,
v 1.3.1 = − 4 α 2 μ 2 k 2 sn ( ξ ) 2 {v_{1.3.1}} = - 4{\alpha ^2}{\mu ^2}{k^2}{\rm{sn}}{(\xi )^2}
;
u 1.4.2 = − c μ + 2 μ α dn ( ξ ) cs ( ξ ) {u_{1.4.2}} = - {c \over \mu } + 2\mu \alpha {\rm{dn}}(\xi ){\rm{cs}}(\xi )
,
v 1.4.2 = − 4 α 2 μ 2 ns ( ξ ) 2 {v_{1.4.2}} = - 4{\alpha ^2}{\mu ^2}{\rm{ns}}{(\xi )^2}
,
u 1.4.4 = − c μ − 2 μ α ( 1 − k 2 ) nd ( ξ ) sc ( ξ ) {u_{1.4.4}} = - {c \over \mu } - 2\mu \alpha (1 - {k^2}){\rm{nd}}(\xi ){\rm{sc}}(\xi )
,
v 1.4.4 = − 4 α 2 μ 2 dc ( ξ ) 2 {v_{1.4.4}} = - 4{\alpha ^2}{\mu ^2}{\rm{dc}}{(\xi )^2}
,
1.5) e 0 = −k 2 , e 2 = 2k 2 − 1, e 4 = 1 − k 2 , e 1 = e 3 = 0,
u 1.5.1 = − c μ + 2 μ α sn ( ξ ) dc ( ξ ) {u_{1.5.1}} = - {c \over \mu } + 2\mu \alpha {\rm{sn}}(\xi ){\rm{dc}}(\xi )
,
v 1.5.1 = 4 α 2 μ 2 k 2 cn ( ξ ) 2 {v_{1.5.1}} = 4{\alpha ^2}{\mu ^2}{k^2}{\rm{cn}}{(\xi )^2}
;
u 1.5.2 = − c μ − 2 μ α sn ( ξ ) dc ( ξ ) {u_{1.5.2}} = - {c \over \mu } - 2\mu \alpha {\rm{sn}}(\xi ){\rm{dc}}(\xi )
,
v 1.5.2 = − 4 α 2 μ 2 ( 1 − k 2 ) nc ( ξ ) 2 {v_{1.5.2}} = - 4{\alpha ^2}{\mu ^2}(1 - {k^2}){\rm{nc}}{(\xi )^2}
,
1.6) e 0 = −1, e 2 = 2 − k 2 , e 4 = k 2 − 1, e 1 = e 3 = 0,
u 1.61 = − c μ + 2 μ α k 2 sn ( ξ ) cd ( ξ ) {u_{1.61}} = - {c \over \mu } + 2\mu \alpha {k^2}{\rm{sn}}(\xi ){\rm{cd}}(\xi )
,
v 1.61 = 4 α 2 μ 2 dn ( ξ ) 2 {v_{1.61}} = 4{\alpha ^2}{\mu ^2}{\rm{dn}}{(\xi )^2}
;
u 1.62 = − c μ − 2 μ α k 2 sn ( ξ ) cd ( ξ ) {u_{1.62}} = - {c \over \mu } - 2\mu \alpha {k^2}{\rm{sn}}(\xi ){\rm{cd}}(\xi )
,
v 1.62 = − 4 α 2 μ 2 ( k 2 − 1 ) nd ( ξ ) 2 {v_{1.62}} = - 4{\alpha ^2}{\mu ^2}({k^2} - 1){\rm{nd}}{(\xi )^2}
;
1.7) e 0 = 1 − k 2 , e 2 = 2 − k 2 , e 4 = 1, e 1 = e 3 = 0,
u 1.7.1 = − c μ − 2 μ α nc ( ξ ) ds ( ξ ) {u_{1.7.1}} = - {c \over \mu } - 2\mu \alpha {\rm{nc}}(\xi ){\rm{ds}}(\xi )
,
v 1.7.1 = − 4 α 2 μ 2 ( 1 − k 2 ) sc ( ξ ) 2 {v_{1.7.1}} = - 4{\alpha ^2}{\mu ^2}({\rm{1}} - {k^2}){\rm{sc}}{(\xi )^2}
;
u 1.7.2 = − c μ + 2 μ α nc ( ξ ) ds ( ξ ) {u_{1.7.2}} = - {c \over \mu } + 2\mu \alpha {\rm{nc}}(\xi ){\rm{ds}}(\xi )
,
v 1.7.2 = − 4 α 2 μ 2 cs ( ξ ) 2 {v_{1.7.2}} = - 4{\alpha ^2}{\mu ^2}{\rm{cs}}{(\xi )^2}
,
1.8) e 0 = 1, e 2 = 2 − k 2 , e 4 = 1 − k 2 , e 1 = e 3 = 0,
u 1.8.1 = − c μ + 2 μ α ns ( ξ ) dc ( ξ ) {u_{1.8.1}} = - {c \over \mu } + 2\mu \alpha {\rm{ns}}(\xi ){\rm{dc}}(\xi )
,
v 1.8.1 = − 4 α 2 μ 2 cs ( ξ ) 2 {v_{1.8.1}} = - 4{\alpha ^2}{\mu ^2}{\rm{cs}}{(\xi )^2}
;
u 1.8.2 = − c μ − 2 μ α ns ( ξ ) dc ( ξ ) {u_{1.8.2}} = - {c \over \mu } - 2\mu \alpha {\rm{ns}}(\xi ){\rm{dc}}(\xi )
,
v 1.8.2 = − 4 α 2 μ 2 ( 1 − k 2 ) sc ( ξ ) 2 {v_{1.8.2}} = - 4{\alpha ^2}{\mu ^2}(1 - {k^2}){\rm{sc}}{(\xi )^2}
,
1.9) e 0 = 1, e 2 = 2k 2 − 1, e 4 = k 2 (k 2 − 1), e 1 = e 3 = 0,
u 1.9.1 = − c μ + 2 μ α ns ( ξ ) cd ( ξ ) {u_{1.9.1}} = - {c \over \mu } + 2\mu \alpha {\rm{ns}}(\xi ){\rm{cd}}(\xi )
,
v 1.91 = − 4 α 2 μ 2 ds ( ξ ) 2 {v_{1.91}} = - 4{\alpha ^2}{\mu ^2}{\rm{ds}}{(\xi )^2}
;
u 1.9.2 = − c μ − 2 μ α ns ( ξ ) cd ( ξ ) {u_{1.9.2}} = - {c \over \mu } - 2\mu \alpha {\rm{ns}}(\xi ){\rm{cd}}(\xi )
,
v 1.9.2 = − 4 α 2 μ 2 k 2 ( k 2 − 1 ) sd ( ξ ) 2 {v_{1.9.2}} = - 4{\alpha ^2}{\mu ^2}{k^2}({k^2} - 1){\rm{sd}}{(\xi )^2}
,
1.10) e 0 = k 2 (k 2 − 1), e 2 = 2k 2 − 1, e 4 = 1, e 1 = e 3 = 0,
u 1.10.1 = − c μ − 2 μ α cd ( ξ ) ns ( ξ ) {u_{1.10.1}} = - {c \over \mu } - 2\mu \alpha {\rm{cd}}(\xi ){\rm{ns}}(\xi )
,
v 1.101 = − 4 α 2 μ 2 k 2 ( k 2 − 1 ) sd ( ξ ) 2 {v_{1.101}} = - 4{\alpha ^2}{\mu ^2}{k^2}({k^2} - 1){\rm{sd}}{(\xi )^2}
;
u 1.10.2 = − c μ − 2 μ α cd ( ξ ) ns ( ξ ) {u_{1.10.2}} = - {c \over \mu } - 2\mu \alpha {\rm{cd}}(\xi ){\rm{ns}}(\xi )
,
v 1.10.2 = − 4 α 2 μ 2 ds ( ξ ) 2 {v_{1.10.2}} = - 4{\alpha ^2}{\mu ^2}{\rm{ds}}{(\xi )^2}
,
1.11) e 0 = 0, e 2 = −(1 + k 2 ), e 4 = k 2 , e 1 = e 3 = 0,
u 1.11.1 = − c μ − 2 μ α cs ( ξ ) dn ( ξ ) {u_{1.11.1}} = - {c \over \mu } - 2\mu \alpha {\rm{cs}}(\xi ){\rm{dn}}(\xi )
,
v 1.11.1 = − 4 α 2 μ 2 k 2 sn ( ξ ) 2 {v_{1.11.1}} = - 4{\alpha ^2}{\mu ^2}{k^2}{\rm{sn}}{(\xi )^2}
;
1.12)
e 0 = k 4 4 , e 2 = 1 2 ( k 2 − 2 ) , e 4 = 1 4 , e 1 = e 3 = 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,
v 1.14.1 = − α 2 μ 2 k 4 ( 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}}}
;
v 1.14.2 = − α 2 μ 2 ( 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)
e 0 = k 2 4 , e 2 = ( k 2 − 2 ) 2 , e 4 = k 2 4 , e 1 = e 3 = 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,
u 1.13.1 = − c μ ∓ 2 i α μ 1 − k 2 nd ( ξ ) {u_{1.13.1}} = - {c \over \mu } \mp 2i\alpha \mu \sqrt {1 - {k^2}} {\rm{nd}}(\xi )
,
v 1.13.1 = − α 2 μ 2 k 2 ( i 1 − k 2 sn ( 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}}}
;
u 1.13.2 = − c μ ± 2 i α μ 1 − k 2 nd ( ξ ) {u_{1.13.2}} = - {c \over \mu } \pm 2i\alpha \mu \sqrt {1 - {k^2}} {\rm{nd}}(\xi )
,
v 1.13.2 = − α 2 μ 2 k 2 ( i 1 − k 2 sn ( 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}
;
u 1.13.3 = − c μ ∓ 2 i α μ dn ( ξ ) {u_{1.13.3}} = - {c \over \mu } \mp 2i\alpha \mu {\rm{dn}}(\xi )
,
v 1.13.3 = − α 2 μ 2 k 2 ( sn ( ξ ) ± i cn ( ξ ) ) 2 {v_{1.13.3}} = - {{{\alpha ^2}{\mu ^2}{k^2}} \over {{{\left( {{\rm{sn}}(\xi ) \pm i{\rm{cn}}(\xi )} \right)}^2}}}
;
u 1.13.4 = − c μ ± 2 i α μ dn ( ξ ) {u_{1.13.4}} = - {c \over \mu } \pm 2i\alpha \mu {\rm{dn}}(\xi )
,
v 1.13.3 = − α 2 μ 2 k 2 ( sn ( ξ ) ± i cn ( 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) e 0 = 1,e 2 = 2 − 4k 2 ,e 4 = 1,e 1 = e 3 = 0,
u 1.14.1 = − c μ + 2 α μ ( nc ( ξ ) ds ( ξ ) − k 2 sn ( 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)
,
v 1.14.1 = − 4 α 2 μ 2 sc ( ξ ) − 2 dn ( ξ ) − 2 {v_{1.14.1}} = - 4{\alpha ^2}{\mu ^2}{\rm{sc}}{(\xi )^{ - 2}}{\rm{dn}}{(\xi )^{ - 2}}
;
u 1.14.2 = − c μ − 2 α μ ( nc ( ξ ) ds ( ξ ) − k 2 sn ( 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)
,
v 1.14.2 = − 4 α 2 μ 2 sc ( ξ ) 2 dn ( ξ ) 2 {v_{1.14.2}} = - 4{\alpha ^2}{\mu ^2}{\rm{sc}}{(\xi )^2}{\rm{dn}}{(\xi )^2}
,
1.15)
e 0 = ( k − 1 ) 2 4 D 1 , e 2 = 1 2 ( 1 + 6 k + k 2 ) , e 4 = D 1 2 ( k − 1 ) 2 4 , e 1 = e 3 = 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,
u 1.15.1 = − c μ − 2 α μ ( sn ( ξ ) ( k 2 cn 2 ( ξ ) + dn 2 ( ξ ) ) ( 1 + ( k + 1 ) sn ( ξ ) + k sn 2 ( ξ ) ) + cn 2 ( ξ ) dn 2 ( ξ ) ( k + 1 + 2 k sn ( ξ ) ) ) ( 1 + ( k + 1 ) sn ( ξ ) + k sn 2 ( 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 )}}
,
v 1.15.1 = − α 2 μ 2 ( k − 1 ) 2 D 1 ( 1 + sn ( ξ ) ) 2 ( 1 + k sn ( ξ ) ) 2 dn ( ξ ) 2 cn ( 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}}}
u 1.15.2 = − c μ + 2 α μ ( sn ( ξ ) ( k 2 cn 2 ( ξ ) + dn 2 ( ξ ) ) ( 1 + ( k + 1 ) sn ( ξ ) + k sn 2 ( ξ ) ) + cn 2 ( ξ ) dn 2 ( ξ ) ( k + 1 + 2 k sn ( ξ ) ) ) ( 1 + ( k + 1 ) sn ( ξ ) + k sn 2 ( 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 )}}
,
v 1.15.2 = − α 2 μ 2 ( k − 1 ) 2 dn ( ξ ) ) 2 cn ( ξ ) ) 2 ( 1 + sn ( ξ ) ) 2 ( 1 + k sn ( 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)
e 0 = ( k + 1 ) 2 4 D 1 , e 2 = 1 2 ( 1 − 6 k + k 2 ) , e 4 = D 1 2 ( k + 1 ) 2 4 , e 1 = e 3 = 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,
u 1.16.1 = − c μ − 2 α μ ( sn ( ξ ) ( k 2 cn 2 ( ξ ) + dn 2 ( ξ ) ) ( 1 + ( − k + 1 ) sn ( ξ ) − k sn 2 ( ξ ) ) + cn 2 ( ξ ) dn 2 ( ξ ) ( − k + 1 − 2 k sn ( ξ ) ) ) ( 1 + ( − k + 1 ) sn ( ξ ) − k sn 2 ( 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 )}}
,
v 1.16.1 = − α 2 μ 2 ( k + 1 ) 2 D 1 ( 1 + sn ( ξ ) ) 2 ( 1 − k sn ( ξ ) ) 2 dn ( ξ ) 2 cn ( 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}}}
;
v 1.16.2 = − α 2 μ 2 ( k + 1 ) 2 dn ( ξ ) ) 2 cn ( ξ ) ) 2 ( 1 + sn ( ξ ) ) 2 ( 1 − k sn ( 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)
e 0 = k 4 − 2 k 3 + k 2 , e 2 = 6 k − k 2 − 1 , e 4 = − 4 k , e 1 = e 3 = 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,
u 1.17.1 = − c μ − 2 μ α ( ( k 2 cn ( ξ ) 2 + dn ( ξ ) 2 ) ( 1 + k sn 2 ( ξ ) ) + 2 k cn ( ξ ) 2 dn ( ξ ) 2 ) ( 1 + k sn ( 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 )}}
,
v 1.17.1 = − 4 α 2 μ 2 ( k 4 − 2 k 3 + k 2 ) ( 1 + k sn ( ξ ) 2 ) 2 k 2 cn ( ξ ) 2 dn ( 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}}}
;
u 1.17.2 = − c μ + 2 μ α ( ( k 2 cn ( ξ ) 2 + dn ( ξ ) 2 ) ( 1 + k sn 2 ( ξ ) ) + 2 k cn ( ξ ) 2 dn ( ξ ) 2 ) ( 1 + k sn ( 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 )}}
,
v 1.17.2 = 16 α 2 μ 2 k cn ( ξ ) 2 dn ( ξ ) ) 2 ( 1 + k sn ( 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)
e 0 = k 4 + 2 k 3 + k 2 , e 2 = − 6 k − k 2 − 1 , e 4 = 4 k , e 1 = e 3 = 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,
v 1.18.1 = − 4 α 2 μ 2 ( k 4 + 2 k 3 + k 2 ) ( − 1 + k sn ( ξ ) 2 ) 2 k 2 cn ( ξ ) 2 dn ( 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}}}
;
v 1.18.2 = − 16 α 2 μ 2 k cn ( ξ ) 2 dn ( ξ ) ) 2 ( − 1 + k sn ( 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) e 0 = 2 + 2k 1 − k 2 , e 2 = 6k 1 − k 2 + 2, e 4 = 4k 1 , e 1 = e 3 = 0,
u 1.19.1 = − c μ + 2 μ α ( ( cn ( ξ ) 2 − sn ( ξ ) 2 ) ( k 1 − dn ( ξ ) 2 ) − 2 k 2 sn ( ξ ) 2 cn ( ξ ) 2 ) ( k 1 − dn ( 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 )}}
,
v 1.19.1 = − 4 α 2 μ 2 ( 2 + 2 k 1 − k 2 ) ( k 1 − dn ( ξ ) 2 ) 2 k 4 sn ( ξ ) 2 cn ( 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}}}
;
v 1.19.2 = − 16 α 2 μ 2 k 1 k 4 sn ( ξ ) 2 cn ( ξ ) ) 2 ( k 1 + 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) e 0 = 2 − 2k 1 − k 2 , e 2 = −6k 1 − k 2 + 2, e 4 = −4k 1 , e 1 = e 3 = 0,
u 1.20.1 = − c μ − 2 μ α ( ( cn ( ξ ) 2 − sn ( ξ ) 2 ) ( k 1 + dn ( ξ ) 2 ) + 2 k 2 sn ( ξ ) 2 cn ( ξ ) 2 ) ( k 1 + 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 )}}
,
v 1.20.1 = − 4 α 2 μ 2 ( 2 − 2 k 1 − k 2 ) ( k 1 + dn ( ξ ) 2 ) 2 k 4 sn ( ξ ) 2 cn ( 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}}}
;
u 1.20.2 = − c μ + 2 μ α ( ( cn ( ξ ) 2 − sn ( ξ ) 2 ) ( k 1 + dn ( ξ ) 2 ) + 2 k 2 sn ( ξ ) 2 cn ( ξ ) 2 ) ( k 1 + 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 )}}
,
v 1.20.2 = 16 α 2 μ 2 k 1 k 4 sn ( ξ ) 2 cn ( ξ ) ) 2 ( k 1 − dn ( 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)
e 0 = k 2 − 1 4 ( D 3 2 + D 2 2 ) , e 2 = k 2 + 1 2 , e 4 = ( D 3 2 k 2 − D 2 2 ) ( k 2 − 1 ) 4 , e 1 = e 3 = 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,
u 1.21.1 = − c μ + 2 μ α ( D 2 dn ( ξ ) + D 3 cn ( ξ ) + D 2 D 2 2 − D 3 2 D 2 2 − D 3 2 k 2 sn ( ξ ) dn ( ξ ) + D 3 k 2 D 2 2 − D 3 2 D 2 2 − D 3 2 k 2 cn ( ξ ) sn ( ξ ) ) ( D 2 cn ( ξ ) + D 3 dn ( ξ ) ) ( D 2 2 − D 3 2 D 2 2 − D 3 2 k 2 + 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)}}
,
v 1.21.1 = − α 2 μ 2 ( k 2 − 1 ) ( D 2 cn ( ξ ) + D 3 dn ( ξ ) ) 2 ( D 3 2 + D 2 2 ) ( D 2 2 − D 3 2 D 2 2 − D 3 2 k 2 + 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}}}
;
u 1.21.2 = − c μ − 2 μ α ( D 2 dn ( ξ ) + D 3 cn ( ξ ) + D 2 D 2 2 − D 3 2 D 2 2 − D 3 2 k 2 sn ( ξ ) dn ( ξ ) + D 3 k 2 D 2 2 − D 3 2 D 2 2 − D 3 2 k 2 cn ( ξ ) sn ( ξ ) ) ( D 2 cn ( ξ ) + D 3 dn ( ξ ) ) ( D 2 2 − D 3 2 D 2 2 − D 3 2 k 2 + 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)}}
,
v 1.21.2 = − α 2 μ 2 ( D 3 2 k 2 − D 2 2 ) ( k 2 − 1 ) ( D 2 2 − D 3 2 D 2 2 − D 3 2 k 2 + sn ( ξ ) ) 2 ( D 2 cn ( ξ ) + D 3 dn ( 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)
e 0 = k 4 4 ( D 3 2 + D 2 2 ) , e 2 = k 2 2 − 1 , e 4 = D 3 2 + D 2 2 4 , e 1 = e 3 = 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,
u 1.22.1 = − c μ − 2 μ α ( k 2 sn ( ξ ) cn ( ξ ) ( D 2 sn ( ξ ) + D 3 cn ( ξ ) ) + dn ( ξ ) ( D 2 2 − ( k 2 − 1 ) D 3 2 D 2 2 + D 3 2 + dn ( ξ ) ) ( D 2 cn ( ξ ) − D 3 sn ( ξ ) ) ) ( D 2 sn ( ξ ) + D 3 cn ( ξ ) ) ( D 2 2 − ( k 2 − 1 ) D 3 2 D 2 2 + D 3 2 + 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)}}
,
v 1.22.1 = − α 2 μ 2 k 4 ( D 2 sn ( ξ ) + D 3 cn ( ξ ) ) 2 ( D 3 2 + D 2 2 ) ( D 2 2 − ( k 2 − 1 ) D 3 2 D 2 2 + D 3 2 + 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}}}
;
u 1.22.2 = − c μ + 2 μ α ( k 2 sn ( ξ ) cn ( ξ ) ( D 2 sn ( ξ ) + D 3 cn ( ξ ) ) + dn ( ξ ) ( D 2 2 − ( k 2 − 1 ) D 3 2 D 2 2 + D 3 2 + dn ( ξ ) ) ( D 2 cn ( ξ ) − D 3 sn ( ξ ) ) ) ( D 2 sn ( ξ ) + D 3 cn ( ξ ) ) ( D 2 2 − ( k 2 − 1 ) D 3 2 D 2 2 + D 3 2 + 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)}}
,
v 1.22.2 = − α 2 μ 2 ( D 3 2 + D 2 2 ) ( D 2 2 − ( k 2 − 1 ) D 3 2 D 2 2 + D 3 2 + dn ( ξ ) ) 2 ( D 2 sn ( ξ ) + D 3 cn ( 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)
e 0 = 2 k − k 2 − 1 D 2 2 , e 2 = 2 k 2 + 2 , e 4 = − D 2 2 ( k 2 + 2 k + 1 ) , e 1 = e 3 = 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,
u 1.23.1 = − c μ + 8 μ α k sn ( ξ ) cn ( ξ ) dn ( ξ ) k 2 sn ( ξ ) 4 − 1 {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}}
,
v 1.23.1 = − 4 α 2 μ 2 ( 2 k − k 2 − 1 ) ( k sn ( ξ ) 2 + 1 ) 2 ( k sn ( ξ ) 2 − 1 ) 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}}}
;
u 1.23.2 = − c μ − 8 μ α k sn ( ξ ) cn ( ξ ) dn ( ξ ) k 2 sn ( ξ ) 4 − 1 {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}}
,
v 1.23.2 = 4 α 2 μ 2 ( k 2 + 2 k + 1 ) ( k sn ( ξ ) 2 − 1 ) 2 ( k sn ( 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)
e 0 = − 2 k − k 2 − 1 D 2 2 , e 2 = 2 k 2 + 2 , e 4 = − D 2 2 ( k 2 + 2 k + 1 ) , e 1 = e 3 = 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,
u 1.24.2 = − c μ + 8 μ α k sn ( ξ ) cn ( ξ ) dn ( ξ ) k 2 sn ( ξ ) 4 − 1 {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}}
,
v 1.24.2 = 4 α 2 μ 2 ( k 2 + 2 k + 1 ) ( k sn ( ξ ) 2 + 1 ) 2 ( k sn ( ξ ) 2 − 1 ) 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)
e 0 = e 4 = 1 4 , e 2 = 1 − 2 k 2 2 , e 1 = e 3 = 0 , {e_0} = {e_4} = {1 \over 4},{e_2} = {{1 - 2{k^2}} \over 2},{e_1} = {e_3} = 0,
u 1.25.3 = − c μ + 2 μ α k cn ( ξ ) ( dn ( ξ ) ∓ iksn ( ξ ) ) ( k sn ( ξ ) ± i dn ( 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)}}
,
v 1.25.3 = − α 2 μ 2 ( k sn ( ξ ) ± i dn ( ξ ) ) 2 {v_{1.25.3}} = - {{{\alpha ^2}{\mu ^2}} \over {{{\left( {k{\rm{sn}}(\xi ) \pm i{\rm{dn}}(\xi )} \right)}^2}}}
;
u 1.25.4 = − c μ − 2 k cn ( ξ ) ( dn ( ξ ) ∓ iksn ( ξ ) ) ( k sn ( ξ ) ± i dn ( 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)}}
,
v 1.25.4 = − α 2 μ 2 ( k sn ( ξ ) ± i dn ( ξ ) ) 2 {v_{1.25.4}} = - {\alpha ^2}{\mu ^2}{\left( {k{\rm{sn}}(\xi ) \pm i{\rm{dn}}(\xi )} \right)^2};
;
u 1.25.7 = − c μ ± 2 μ α ds ( ξ ) {u_{1.25.7}} = - {c \over \mu } \pm 2\mu \alpha {\rm{ds}}(\xi )
,
v 1.25.7 = − α 2 μ 2 ns ( 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}
;
u 1.25.8 = − c μ ∓ 2 μ α ds ( ξ ) {u_{1.25.8}} = - {c \over \mu } \mp 2\mu \alpha {\rm{ds}}(\xi )
,
v 1.25.8 = − α 2 μ 2 sn ( 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)
e 0 = e 4 = k 2 − 1 4 , e 2 = k 2 + 1 2 , e 1 = e 3 = 0 , {e_0} = {e_4} = {{{k^2} - 1} \over 4},{e_2} = {{{k^2} + 1} \over 2},{e_1} = {e_3} = 0,
u 1.26.1 = − c μ ± 2 α μ k cd ( ξ ) {u_{1.26.1}} = - {c \over \mu } \pm 2\alpha \mu k{\rm{cd}}(\xi )
,
v 1.26.1 = − α 2 μ 2 ( k 2 − 1 ) ( ks d ( 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}}}
;
u 1.26.2 = − c μ ∓ 2 α μ k cd ( ξ ) {u_{1.26.2}} = - {c \over \mu } \mp 2\alpha \mu k{\rm{cd}}(\xi )
,
v 1.26.2 = − α 2 μ 2 ( k 2 − 1 ) ( ks d ( 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}
;
u 1.26.3 = − c μ ∓ 2 μ α k dc ( ξ ) {u_{1.26.3}} = - {c \over \mu } \mp 2\mu \alpha k{\rm{dc}}(\xi )
,
v 1.26.3 = − α 2 μ 2 ( k 2 − 1 ) ( 1 ± k sn ( ξ ) ) 2 dn ( 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}}}
;
u 1.26.4 = − c μ ± 2 μ α k dc ( ξ ) {u_{1.26.4}} = - {c \over \mu } \pm 2\mu \alpha k{\rm{dc}}(\xi )
,
v 1.26.4 = − α 2 μ 2 ( k 2 − 1 ) dn ( ξ ) 2 ( 1 ± k sn ( 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)
e 0 = − ( 1 − k 2 ) 2 4 , e 2 = k 2 + 1 2 , e 4 = − 1 4 , e 1 = e 3 = 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,
u 1.27.1 = − c μ ∓ μ α sn ( ξ ) {u_{1.27.1}} = - {c \over \mu } \mp \mu \alpha {\rm{sn}}(\xi )
,
v 1.27.1 = α 2 μ 2 ( 1 − k 2 ) 2 ( k cn ( 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}}
;
u 1.27.2 = − c μ ± μ α sn ( ξ ) {u_{1.27.2}} = - {c \over \mu } \pm \mu \alpha {\rm{sn}}(\xi )
,
v 1.27.2 = α 2 μ 2 ( k cn ( 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)
e 0 = 1 4 , e 2 = k 2 + 1 2 , e 4 = ( 1 − k 2 ) 2 4 , e 1 = e 3 = 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,
u 1.28.1 = − c μ ± 2 μ α ns ( ξ ) {u_{1.28.1}} = - {c \over \mu } \pm 2\mu \alpha {\rm{ns}}(\xi )
,
v 1.28.1 = − α 2 μ 2 ( dn ( ξ ) ± cn ( ξ ) ) 2 sn ( ξ ) − 2 {v_{1.28.1}} = - {\alpha ^2}{\mu ^2}{\left( {{\rm{dn}}(\xi ) \pm {\rm{cn}}(\xi )} \right)^2}{\rm{sn}}{(\xi )^{ - 2}}
;
u 1.28.2 = − c μ ∓ 2 μ α ns ( ξ ) {u_{1.28.2}} = - {c \over \mu } \mp 2\mu \alpha {\rm{ns}}(\xi )
,
v 1.28.2 = − α 2 μ 2 ( 1 − k 2 ) 2 sn ( 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)
e 0 = 1 4 , e 2 = ( k 2 − 2 ) 2 2 , e 4 = k 4 4 , e 1 = e 3 = 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,
u 1.29.1 = − c μ ∓ 2 μ α 1 − k 2 sc ( ξ ) {u_{1.29.1}} = - {c \over \mu } \mp 2\mu \alpha \sqrt {1 - {k^2}} {\rm{sc}}(\xi )
,
v 1.29.1 = − α 2 μ 2 ( 1 − k 2 ± dn ( ξ ) ) 2 cn ( 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}}}
;
u 1.29.2 = − c μ ± 2 μ α 1 − k 2 sc ( ξ ) {u_{1.29.2}} = - {c \over \mu } \pm 2\mu \alpha \sqrt {1 - {k^2}} {\rm{sc}}(\xi )
,
v 1.29.2 = − α 2 μ 2 k 4 cn ( ξ ) 2 ( 1 − k 2 ± 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) e 0 = 0, e 2 > 0, e 4 < 0, e 1 = e 3 = 0,
u 1.30.1 = − c μ + 2 α μ e 2 t an h ( e 2 ξ ) {u_{1.30.1}} = - {c \over \mu } + 2\alpha \mu \sqrt {{e_2}} t{\rm{an}}h(\sqrt {{e_2}} \xi )
,
v 1.30.1 = 4 α 2 μ 2 e 2 sec h 2 ( e 2 ξ ) {v_{1.30.1}} = 4{\alpha ^2}{\mu ^2}{e_2}\sec {h^2}(\sqrt {{e_2}} \xi )
,
1.31) e 0 = 0, e 2 > 0, e 4 > 0, e 1 = e 3 = 0,
u 1.31.1 = − c μ + 2 α μ e 2 cot h ( e 2 ξ ) {u_{1.31.1}} = - {c \over \mu } + 2\alpha \mu \sqrt {{e_2}} \cot h(\sqrt {{e_2}} \xi )
,
v 1.31.1 = − 4 α 2 μ 2 e 2 csc h 2 ( e 2 ξ ) {v_{1.31.1}} = - 4{\alpha ^2}{\mu ^2}{e_2}\csc {h^2}(\sqrt {{e_2}} \xi )
,
1.32)
e 0 = e 2 2 4 e 4 , e 2 < 0 , e 4 > 0 , e 1 = e 3 = 0 , {e_0} = {{e_2^2} \over {4{e_4}}},{e_2} < 0,{e_4} > 0,{e_1} = {e_3} = 0,
u 1.32.1 = − c μ + 2 α μ − e 2 2 sec h ( − e 2 2 ξ ) csch ( − e 2 2 ξ ) {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 )
,
v 1.32.1 = 2 α 2 μ 2 e 2 tanh − 2 ( − e 2 2 ξ ) {v_{1.32.1}} = 2{\alpha ^2}{\mu ^2}{e_2}{\rm{tan}}{{\rm{h}}^{ - 2}}(\sqrt {{{ - {e_2}} \over 2}} \xi )
;
v 1.32.2 = 2 α 2 μ 2 e 2 tanh 2 ( − e 2 2 ξ ) {v_{1.32.2}} = 2{\alpha ^2}{\mu ^2}{e_2}{\rm{tan}}{{\rm{h}}^2}(\sqrt {{{ - {e_2}} \over 2}} \xi )
,
1.33)
e 0 = 0 , e 2 = 1 , e 4 = 1 2 , e 1 = e 3 = 0 , {e_0} = 0,{e_2} = 1,{e_4} = {1 \over 2},{e_1} = {e_3} = 0,
u 1.33.1 = − c μ − 2 2 μ α cot h ( D − ξ ) {u_{1.33.1}} = - {c \over \mu } - 2\sqrt 2 \mu \alpha \cot h(D - \xi )
,
v 1.33.1 = − 2 α 2 μ 2 tan h 2 ( D − ξ ) ( 2 − 2 tan h 2 ( D − ξ ) ) − 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) e 0 = 0, e 2 < 0, e 4 > 0, e 1 = e 3 = 0,
u 1.34.1 = − c μ − 2 α μ − e 2 tan ( − e 2 ξ ) {u_{1.34.1}} = - {c \over \mu } - 2\alpha \mu \sqrt { - {e_2}} \tan (\sqrt { - {e_2}} \xi )
,
v 1.34.1 = 4 α 2 μ 2 e 2 sec 2 ( − e 2 ξ ) {v_{1.34.1}} = 4{\alpha ^2}{\mu ^2}{e_2}\mathop {\sec }\nolimits^2 (\sqrt { - {e_2}} \xi )
;
u 1.34.2 = − c μ + 2 α μ − e 2 cot ( − e 2 ξ ) {u_{1.34.2}} = - {c \over \mu } + 2\alpha \mu \sqrt { - {e_2}} \cot (\sqrt { - {e_2}} \xi )
,
v 1.34.2 = 4 α 2 μ 2 e 2 c sc 2 ( − e 2 ξ ) {v_{1.34.2}} = 4{\alpha ^2}{\mu ^2}{e_2}c{\rm{s}}{{\rm{c}}^2}(\sqrt { - {e_2}} \xi )
;
1.35)
e 0 = e 2 2 4 e 4 , e 2 < 0 , e 4 > 0 , e 1 = e 3 = 0 , {e_0} = {{e_2^2} \over {4{e_4}}},{e_2} < 0,{e_4} > 0,{e_1} = {e_3} = 0,
u 1.36.1 = − c μ + 2 2 e 2 μ α csc ( 2 e 2 ξ ) {u_{1.36.1}} = - {c \over \mu } + 2\sqrt {2{e_2}} \mu \alpha \csc (\sqrt {2{e_2}} \xi )
,
v 1.36.1 = − 2 α 2 μ 2 e 2 cot 2 ( e 2 2 ξ ) {v_{1.36.1}} = - 2{\alpha ^2}{\mu ^2}{e_2}\mathop {\cot }\nolimits^2 (\sqrt {{{{e_2}} \over 2}} \xi )
;
u 1.36.2 = − c μ − 2 2 e 2 μ α csc ( 2 e 2 ξ ) {u_{1.36.2}} = - {c \over \mu } - 2\sqrt {2{e_2}} \mu \alpha \csc (\sqrt {2{e_2}} \xi )
,
v 1.36.2 = − 2 α 2 μ 2 e 2 tan 2 ( e 2 2 ξ ) {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) e 0 = 0, e 2 = 0, e 4 > 0, e 1 = e 3 = 0,
u 1.36.1 = − c μ − 2 α μ e 4 e 4 ξ + ξ 0 {u_{1.36.1}} = - {c \over \mu } - {{2\alpha \mu \sqrt {{e_4}} } \over {\sqrt {{e_4}} \xi + {\xi _0}}}
,
v 1.36.1 = − 4 α 2 μ 2 e 4 ( e 4 ξ + ξ 0 ) 2 {v_{1.36.1}} = - {{4{\alpha ^2}{\mu ^2}{e_4}} \over {{{\left( {\sqrt {{e_4}} \xi + {\xi _0}} \right)}^2}}}
,
1.37) e 0 = k , e 1 = −4k , e 2 = −1 + 6k − k 2 , e 3 = 2(k − 1)2 , e 4 = −(k − 1)2 ,
u 1.37.1 = − c μ − 4 α μ k cn ( ξ ) dn ( ξ ) 1 − k sn ( 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}}}
,
v 1.37.1 = 2 α 2 μ 2 ( k 2 + 6 k + 1 ) − 16 α 2 μ 2 k cn ( ξ ) 2 dn ( ξ ) 2 ( 1 − k sn ( ξ ) 2 ) 2 + 4 α 2 μ 2 ( ( − 1 − k 2 + 6 k ) ( k sn ( ξ ) + 1 ) 2 − 2 ( k − 1 ) 2 ) ( k sn ( ξ ) + 1 ) 2 ( 1 − k sn ( 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)}};
;
u 1.37.2 = − c μ + 4 α μ k cn ( ξ ) dn ( ξ ) 1 − k sn ( 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}}}
,
v 1.37.2 = 2 α 2 μ 2 ( k 2 + 6 k + 1 ) − 4 α 2 μ 2 ( ( − 1 − k 2 + 6 k ) ( k sn ( ξ ) + 1 ) 2 − 2 ( k − 1 ) 2 ) ( k sn ( ξ ) + 1 ) 2 ( 1 − k sn ( 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)}}
;
u 1.37.3 = − c μ − 4 α μ k 3 k cn ( ξ ) dn ( ξ ) sn 2 ( ξ ) k 3 sn ( ξ ) 2 − ( 1 − dn ( 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}}}
,
v 1.37.3 = 2 α 2 μ 2 ( k 2 + 6 k + 1 ) − 16 α 2 μ 2 k 7 cn 2 ( ξ ) dn 2 ( ξ ) sn 4 ( ξ ) ( k 3 sn ( ξ ) 2 − ( 1 − dn ( ξ ) 2 ) 2 ) 2 + 4 α 2 μ 2 ( ( − 1 − k 2 + 6 k ) ( k k sn ( ξ ) − dn ( ξ ) 2 + 1 ) 2 − 2 ( k − 1 ) 2 k 7 sn ( ξ ) 2 ) k k sn ( ξ ) ( k k sn ( ξ ) − dn ( ξ ) 2 + 1 ) ( k 3 sn ( ξ ) 2 − ( 1 − dn ( 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)}}
;
u 1.37.4 = − c μ + 4 α μ k 3 k cn ( ξ ) dn ( ξ ) sn 2 ( ξ ) k 3 sn ( ξ ) 2 − ( 1 − dn ( 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}}}
,
v 1.37.4 = 2 α 2 μ 2 ( k 2 + 6 k + 1 ) − 4 α 2 μ 2 ( ( − 1 − k 2 + 6 k ) ( k k sn ( ξ ) − dn ( ξ ) 2 + 1 ) 2 − 2 ( k − 1 ) 2 k 7 sn ( ξ ) 2 ) k k sn ( ξ ) ( k k sn ( ξ ) − dn ( ξ ) 2 + 1 ) ( k 3 sn ( ξ ) 2 − ( 1 − dn ( 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)
e 0 = − e 1 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 ) , {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}),
u 1.38.1 = − c μ − 4 μ α − e 1 dn ( − e 1 ξ ) ( 1 + cn ( − e 1 ξ ) ) ( 1 + cn ( − e 1 ξ ) ) 2 − sn ( − e 1 ξ ) 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}}}
,
v 1.38.1 = − α 2 μ 2 e 1 ( k 2 + 1 ) + 16 α 2 μ 2 e 1 dn ( − e 1 ξ ) 2 ( cn ( − e 1 ξ ) + 1 ) 2 ( ( cn ( − e 1 ξ ) + 1 ) 2 − sn ( − e 1 ξ ) 2 ) 2 + 4 α 2 μ 2 e 1 ( ( k 2 − 2 ) ( sn ( − e 1 ξ ) + cn ( − e 1 ξ ) + 1 ) 2 − 2 ( 1 − k 2 ) sn ( − e 1 ξ ) 2 ) sn ( − e 1 ξ ) 2 ( sn ( − e 1 ξ ) + cn ( − e 1 ξ ) + 1 ) ( ( cn ( − e 1 ξ ) + 1 ) 2 − sn ( − e 1 ξ ) 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)}}
;
u 1.38.2 = − c μ + 4 μ α − e 1 dn ( − e 1 ξ ) ( 1 + cn ( − e 1 ξ ) ) ( 1 + cn ( − e 1 ξ ) ) 2 − sn ( − e 1 ξ ) 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}}}
,
v 1.38.2 = − α 2 μ 2 e 1 ( k 2 + 1 ) − 4 α 2 μ 2 e 1 ( ( k 2 − 2 ) ( sn ( − e 1 ξ ) + cn ( − e 1 ξ ) + 1 ) 2 − 2 ( 1 − k 2 ) sn ( − e 1 ξ ) 2 ) sn ( − e 1 ξ ) 2 ( sn ( − e 1 ξ ) + cn ( − e 1 ξ ) + 1 ) ( ( cn ( − e 1 ξ ) + 1 ) 2 − sn ( − e 1 ξ ) 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)}}
;
u 1.38.3 = − c μ + 4 μ α − e 1 dn ( − e 1 ξ ) ( cn ( − e 1 ξ ) + 1 ) sn ( − e 1 ξ ) 2 − ( cn ( − e 1 ξ ) + 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}}}
,
v 1.38.3 = − α 2 μ 2 e 1 ( k 2 + 1 ) + 16 α 2 μ 2 e 1 dn ( − e 1 ξ ) 2 ( cn ( − e 1 ξ ) + 1 ) 2 ( sn ( − e 1 ξ ) 2 − ( cn ( − e 1 ξ ) + 1 ) 2 ) 2 + 4 α 2 μ 2 e 1 ( ( k 2 − 2 ) ( sn ( − e 1 ξ ) + cn ( − e 1 ξ ) + 1 ) 2 − 2 ( 1 − k 2 ) ( cn ( − e 1 ξ ) + 1 ) 2 ) ( cn ( − e 1 ξ ) + 1 ) ( sn ( − e 1 ξ ) + cn ( − e 1 ξ ) + 1 ) ( ( cn ( − e 1 ξ ) + 1 ) 2 − sn ( − e 1 ξ ) 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)}}
;
u 1.38.4 = − c μ − 4 μ α − e 1 dn ( − e 1 ξ ) ( cn ( − e 1 ξ ) + 1 ) sn ( − e 1 ξ ) 2 − ( cn ( − e 1 ξ ) + 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}}}
,
v 1.38.4 = − α 2 μ 2 e 1 ( k 2 + 1 ) − 4 α 2 μ 2 e 1 ( ( k 2 − 2 ) ( sn ( − e 1 ξ ) + cn ( − e 1 ξ ) + 1 ) 2 − 2 ( 1 − k 2 ) ( cn ( − e 1 ξ ) + 1 ) 2 ) ( cn ( − e 1 ξ ) + 1 ) ( sn ( − e 1 ξ ) + cn ( − e 1 ξ ) + 1 ) ( ( cn ( − e 1 ξ ) + 1 ) 2 − sn ( − e 1 ξ ) 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) e 0 > 0, e 1 = −4e 0 , e 2 = 4e 0 (2 − k 2 ), e 3 = −8e 0 (1 − k 2 ), e 4 = 4e 0 (1 − k 2 ),
u 1.39.1 = − c μ − 8 μ α e 0 ( 1 − k 2 ) ( 1 − k 2 sn ( 2 e 0 ξ ) + dn ( 2 e 0 ξ ) ) cn ( 2 e 0 ξ ) 2 − ( 1 − k 2 sn ( 2 e 0 ξ ) + dn ( 2 e 0 ξ ) ) 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}}}
,
v 1.39.1 = − 4 α 2 μ 2 e 0 ( k 2 + 1 ) − 2 9 α 2 μ 2 e 0 ( 1 − k 2 ) ( 1 − k 2 sn ( 2 e 0 ξ ) + dn ( 2 e 0 ξ ) ) 2 ( cn ( 2 e 0 ξ ) 2 − ( 1 − k 2 sn ( 2 e 0 ξ ) + dn ( 2 e 0 ξ ) ) 2 ) 2 + 16 α 2 μ 2 e 0 ( 2 − k 2 ) ( ( 1 − k 2 sn ( 2 e 0 ξ ) + cn ( 2 e 0 ξ ) + dn ( 2 e 0 ξ ) ) 2 + cn ( 2 e 0 ξ ) 2 ) cn ( 2 e 0 ξ ) ( 1 − k 2 sn ( 2 e 0 ξ ) + cn ( 2 e 0 ξ ) + dn ( 2 e 0 ξ ) ) 2 ( − 1 − k 2 sn ( 2 e 0 ξ ) + cn ( 2 e 0 ξ ) − dn ( 2 e 0 ξ ) ) {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)}}
.
u 1.39.2 = − c μ + 8 μ α e 0 ( 1 − k 2 ) ( 1 − k 2 sn ( 2 e 0 ξ ) + dn ( 2 e 0 ξ ) ) cn ( 2 e 0 ξ ) 2 − ( 1 − k 2 sn ( 2 e 0 ξ ) + dn ( 2 e 0 ξ ) ) 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}}}
,
v 1.39.2 = − 4 α 2 μ 2 e 0 ( k 2 + 1 ) − 16 α 2 μ 2 e 0 ( 2 − k 2 ) ( ( 1 − k 2 sn ( 2 e 0 ξ ) + cn ( 2 e 0 ξ ) + dn ( 2 e 0 ξ ) ) 2 + cn ( 2 e 0 ξ ) 2 ) cn ( 2 e 0 ξ ) ( 1 − k 2 sn ( 2 e 0 ξ ) + cn ( 2 e 0 ξ ) + dn ( 2 e 0 ξ ) ) 2 ( − 1 − k 2 sn ( 2 e 0 ξ ) + cn ( 2 e 0 ξ ) − dn ( 2 e 0 ξ ) ) {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)
e 0 = 0 , e 1 = 0 , e 2 > 0 , e 3 2 = 4 e 4 e 2 ≠ 0 , , {e_0} = 0,{e_1} = 0,{e_2} > 0,e_3^2 = 4{e_4}{e_2} \ne 0,
u 1.40.1 = − c μ ± 8 α μ e 4 e 2 e 2 sec h 2 ( e 2 2 ξ ) ( e 3 2 tan h ( e 2 2 ξ ) + e 4 e 2 ( 1 − tan h ( e 2 2 ξ ) ) 2 ) ( e 3 2 − e 4 e 2 ( 1 − tan h ( e 2 2 ξ ) ) 2 ) ( e 3 2 − e 4 e 2 ( 3 tan h 2 ( e 2 2 ξ ) + 2 tan h ( e 2 2 ξ ) − 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)}}
,
v 1.40.1 = α 2 μ 2 ( 3 e 3 2 − 8 e 4 e 2 ) 4 e 4 − 32 α 2 μ 2 e 4 2 e 2 3 e 3 sec h 4 ( e 2 2 ξ ) ( e 3 2 tanh ( e 2 2 ξ ) + e 4 e 2 ( 1 − tanh ( e 2 2 ξ ) 2 ) 2 ( e 3 2 + e 4 e 2 ( 1 − tanh ( e 2 2 ξ ) ) 2 ) 2 ( e 3 2 − e 4 e 2 ( 3 tanh 2 ( e 2 2 ξ ) + 2 tanh ( e 2 2 ξ ) − 5 ) ) 2 − 8 α 2 μ 2 e 4 e 2 2 e 3 ( e 3 2 − e 4 e 2 ( 1 − tanh 2 ( e 2 2 ξ ) ) 2 − 2 e 4 e 2 e 3 2 sech 4 ( e 2 2 ξ ) ] sech 2 ( e 2 2 ξ ) ( e 3 2 − e 4 e 2 ( 1 − tanh ( e 2 2 ξ ) 2 ) 2 ( e 3 2 − e 4 e 2 ( 1 − tanh ( e 2 2 ξ ) ) 2 − 2 e 4 e 2 sech 2 ( e 2 2 ξ ) {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,
k 1 = 1 − k 2 {k_1} = \sqrt {1 - {k^2}}
, D 1 ,D 2 , D 3 (D 1 D 2 D 3 ≠ 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) e 1 = e 3 = 0,e 0 = k 2 ,e 2 = −(1 + k 2 ),e 4 = 1, ξ = μx + c 0 μ t
u 2.1.1 = c 0 + 2 α μ k sn ( ξ ) {u_{2.1.1}} = {c_0} + {{2\alpha \mu } \over k}{\rm{sn}}(\xi )
,
v 2.1.1 = α 2 μ 2 ( 1 + k 2 ) − 2 α 2 μ 2 ( 1 − k ds ( ξ ) cs ( ξ ) ) k 2 ns ( 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}}}
;
u 2.1.2 = c 0 + 2 α μ k cd ( ξ ) {u_{2.1.2}} = {c_0} + {{2\alpha \mu } \over k}{\rm{cd}}(\xi )
,
v 2.1.2 = α 2 μ 2 ( 1 + k 2 ) − 2 α 2 μ 2 ( 1 + k ( 1 − k 2 ) nc ( ξ ) sc ( ξ ) ) k 2 dc ( 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)
e 1 = e 3 = 0 , e 0 = k 4 4 , e 2 = 1 2 ( k 2 − 2 ) , e 4 = 1 4 , ξ = μ x + c 0 μ t {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
u 2.2.1 = c 0 + 4 α μ k 2 ( 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)}}
,
v 2.2.1 = − α 2 μ 2 ( k 2 − 2 ) − 4 α 2 μ 2 ( 2 − k 2 cs ( ξ ) ( ds ( ξ ) ± ns ( ξ ) ) ) k 4 ( 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)
e 1 = e 3 = 0 , e 0 = k 2 4 , e 2 = ( k 2 − 2 ) 2 , e 4 = k 2 4 , ξ = μ x + c 0 μ t {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
u 2.3.1 = c 0 + 4 α μ k ( sn ( ξ ) ± i cn ( ξ ) ) {u_{2.3.1}} = {c_0} + {{4\alpha \mu } \over {k\left( {{\rm{sn}}(\xi ) \pm i{\rm{cn}}(\xi )} \right)}}
,
v 2.3.1 = − 1 2 α 2 μ 2 ( k 2 − 2 ) − 4 α 2 μ 2 ( 2 − ikdn ( ξ ) ( i cn ( ξ ) ± sn ( ξ ) ) ) k 2 ( sn ( ξ ) ± i cn ( 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)
e 0 = k , e 1 = − 4 k , e 2 = − 1 + 6 k − k 2 , e 3 = 2 ( k − 1 ) 2 , e 4 = − ( k − 1 ) 2 , ξ = μ x + μ ( c 0 ± 2 α μ k ) 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
u 2.4.1 = c 0 + 2 α μ ( k sn ( ξ ) + 1 ) k {u_{2.4.1}} = {c_0} + {{2\alpha \mu \left( {\sqrt k {\rm{sn}}(\xi ) + 1} \right)} \over {\sqrt k }}
,
v 2.4.1 = − 2 α 2 μ 2 ( k − 1 ) 2 + 4 α 2 μ 2 k ± 4 α 2 μ 2 k ( k sn ( ξ ) + 1 ) − 2 α 2 μ 2 ( ( k sn ( ξ ) + 1 ) 2 − k dn ( 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)
e 0 = − e 1 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 ) , ξ = μ x + μ ( c 0 ± α μ − e 1 ) 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
u 2.5.1 = c 0 + 4 α μ − e 1 ( ns ( − e 1 ξ ) + cs ( − e 1 ξ ) + 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)
,
v 2.5.1 = − α 2 μ 2 e 1 k 2 − 14 α 2 μ 2 e 1 ± 4 α 2 μ 2 ( ns ( − e 1 ξ ) + cs ( − e 1 ξ ) + 1 ) + 4 α 2 μ 2 e 1 − 2 ( 2 ( ns ( − e 1 ξ ) + cs ( − e 1 ξ ) + 1 ) 2 − e 1 ds ( − e 1 ξ ) 2 ( 1 + cn ( − e 1 ξ ) ) 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)
;
u 2.5.2 = c 0 + 4 α μ ( sn ( − e 1 ξ ) + cn ( − e 1 ξ ) + 1 ) − e 1 ( cn ( − e 1 ξ ) + 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)}}
,
v 2.5.2 = − α 2 μ 2 e 1 k 2 − 14 α 2 μ 2 e 1 ± 4 α 2 μ 2 ( sn ( − e 1 ξ ) + cn ( − e 1 ξ ) + 1 ) e 1 ( cn ( − e 1 ξ ) + 1 ) + 4 α 2 μ 2 ( 2 ( sn ( − e 1 ξ ) + cn ( − e 1 ξ ) + 1 ) 2 + e 1 dn ( − e 1 ξ ) 2 ( 1 + cn ( − e 1 ξ ) ) 2 ) e 1 2 dn ( − e 1 ξ ) 2 ( 1 + cn ( − e 1 ξ ) ) 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)
e 0 = − e 1 4 ≠ 0 , e 1 = − ( β ξ ) 2 , e 2 = − ( 2 e 1 + β 2 ) , e 3 = 2 ( e 1 + β 2 ) , e 4 = − ( e 1 + β 2 ) , ξ = μ x + μ ( c 0 ± α μ k β ) 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
u 2.6.1 = c 0 + 4 α μ − e 1 k ( k + ds ( β ξ ) − ns ( β ξ ) ) , {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),
,
v 2.6.1 = α 2 μ 2 ( 2 e 1 + β 2 ) + α 2 μ 2 k 2 β 2 ± 4 α 2 μ 2 k ( k − ns ( β ξ ) + ds ( β ξ ) ) − 4 α 2 μ 2 ( 2 ( k sn ( β ξ ) + dn ( β ξ ) − 1 ) 2 + − e 1 k β ( 1 − cn ( β ξ ) ) ) − e 1 k 2 sn ( β ξ ) 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}}}
,
u 2.6.2 = c 0 + 4 α μ ( k sn ( β ξ ) − dn ( β ξ ) − 1 ) − e 1 ( 1 + dn ( β ξ ) ) {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)}}
,
v 2.6.2 = α 2 μ 2 ( 2 e 1 + β 2 ) + α 2 μ 2 k 2 β 2 ± 4 α 2 μ 2 ( k sn ( β ξ ) − dn ( β ξ ) − 1 ) ( 1 + dn ( β ξ ) ) + 4 α 2 μ 2 ( 2 ( k sn ( β ξ ) − dn ( β ξ ) − 1 ) 2 + − e 1 k 2 β ( 1 + dn ( β ξ ) ) ) e 1 ( 1 + dn ( β ξ ) ) 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)
e 0 = − e 2 2 4 e 4 , e 2 < 0 , e 4 > 0 , e 1 = e 3 = 0 , ξ = μ x + c 0 μ t {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
u 2.7.2 = c 0 − 4 α μ e 4 e 2 − e 2 2 tanh − e 2 2 ( 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 )}}
,
v 2.7.2 = − α 2 μ 2 e 2 + 4 α 2 μ 2 e 4 e 2 3 ( 4 e 4 + e 2 2 csc h ( − e 2 2 ξ ) sech ( − e 2 2 ξ ) ) coth 2 − e 2 2 ( 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)
e 0 = − e 2 2 4 e 4 , e 2 > 0 , e 4 > 0 , e 1 = e 3 = 0 , ξ = μ x + c 0 μ t {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, i 2 = −1, , c 0 , 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 ) = c 0 + a 0 + a 1 ϕ ′ b 0 + b 1 ϕ u(\xi ) = {c_0} + {{{a_0} + {a_1}\phi '} \over {{b_0} + {b_1}\phi }}
,
v ( ξ ) = d 0 + f 10 + f 11 ϕ ′ g 10 + g 11 ϕ + ( f 20 + f 21 ϕ ′ g 20 + g 21 ϕ ) 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 ].
