Applications of the extended rational sine-cosine and sinh-cosh techniques to some nonlinear complex models arising in mathematical physics

Suppose that the solution of Eq. (5) can be written in the following forms: (6) Θ(η)=a0sin(μη)a2+a1cos(μη), cos(μη)≠−a2a1, \Theta (\eta ) = {{{a_0}\sin (\mu \eta )} \over {{a_2} + {a_1}\cos (\mu \eta )}},\quad \cos (\mu \eta ) \ne - {{{a_2}} \over {{a_1}}}, (7) Θ(η)=a0cos(μη)a2+a1sin(μη), sin(μη)≠−a2a1, \Theta (\eta ) = {{{a_0}\cos (\mu \eta )} \over {{a_2} + {a_1}\sin (\mu \eta )}},\quad \sin (\mu \eta ) \ne - {{{a_2}} \over {{a_1}}}, where a₀, a₁ and a₂ are parameters to be found in terms of the other parameters. The non-zero constant μ is the wave number. The derivatives of the predicted solutions are (8) Θ′(η)=a0μ[cos(μη)a2+a1][a2+a1sin(μη)]2, \Theta '(\eta ) = {{{a_0}\mu \left[ {\cos (\mu \eta ){a_2} + {a_1}} \right]} \over {{{\left[ {{a_2} + {a_1}\sin (\mu \eta )} \right]}^2}}}, (9) \ (10) Θ″(η)=a0μ2sin(μη)[2a12+a1cos(μη)a2−a22][a2+a1cos(μη)]3, \Theta ''(\eta ) = {{{a_0}{\mu ^2}\sin (\mu \eta )\left[ {2a_1^2 + {a_1}\cos (\mu \eta ){a_2} - a_2^2} \right]} \over {{{\left[ {{a_2} + {a_1}\cos (\mu \eta )} \right]}^3}}}, in the first form and (11) Θ′(η)=−a0μ[sin(μη)a2+a1][a2+a1sin(μη)]2, \Theta '(\eta ) = - {{{a_0}\mu \left[ {\sin (\mu \eta ){a_2} + {a_1}} \right]} \over {{{\left[ {{a_2} + {a_1}\sin (\mu \eta )} \right]}^2}}}, (12) \ (13) Θ″(η)=a0μ2cos(ηξ)[2a12+a1sin(ηξ)a2−a22][a2+a1sin(ηξ)]3. \Theta ''(\eta ) = {{{a_0}{\mu ^2}\cos (\eta \xi )\left[ {2a_1^2 + {a_1}\sin (\eta \xi ){a_2} - a_2^2} \right]} \over {{{\left[ {{a_2} + {a_1}\sin (\eta \xi )} \right]}^3}}}. in the second form.

We substitute Eqs. (8) or (11) into Eq. (5) and get a polynomial in trigonometric functions. Collecting the coefficients of the same power of cos^m(μη) or sin^m(μη) and equating each summation to zero, we get a set of algebraic equations. We solve the system of equations to get the solutions of the equation into consideration.

Assume that the solutions of Eq. (5) may be written in the forms (14) Θ(η)=a0sinh(μη)a2+a1cosh(μη), cosh(μη)≠−a2a1, \Theta (\eta ) = {{{a_0}\sinh (\mu \eta )} \over {{a_2} + {a_1}\cosh (\mu \eta )}},\quad \cosh (\mu \eta ) \ne - {{{a_2}} \over {{a_1}}}, (15) Θ(η)=a0cosh(μη)a2+a1sinh(μη), sinh(μη)≠−a2a1, \Theta (\eta ) = {{{a_0}\cosh (\mu \eta )} \over {{a_2} + {a_1}\sinh (\mu \eta )}},\quad \sinh (\mu \eta ) \ne - {{{a_2}} \over {{a_1}}}, where a₀, a₁ and a₂ are parameters to be found in terms of the other parameters. The non-zero constant μ is the wave number. The derivatives of the predicted solutions are (16) Θ′(η)=a0μ[cosh(μη)a2+a1][a2+a1sinh(μη)]2, \Theta '(\eta ) = {{{a_0}\mu \left[ {\cosh (\mu \eta ){a_2} + {a_1}} \right]} \over {{{\left[ {{a_2} + {a_1}\sinh (\mu \eta )} \right]}^2}}}, (17) \ (18) Θ″(η)=−a0μ2sinh(μη)[2a12+a1cosh(μη)a2−a22][a2+a1cosh(μη)]3, \Theta ''(\eta ) = - {{{a_0}{\mu ^2}\sinh (\mu \eta )\left[ {2a_1^2 + {a_1}\cosh (\mu \eta ){a_2} - a_2^2} \right]} \over {{{\left[ {{a_2} + {a_1}\cosh (\mu \eta )} \right]}^3}}}, in the first form and (19) Θ′(η)=a0μ[sinh(μη)a2−a1][a2+a1sinh(μη)]2, \Theta '(\eta ) = {{{a_0}\mu \left[ {\sinh (\mu \eta ){a_2} - {a_1}} \right]} \over {{{\left[ {{a_2} + {a_1}\sinh (\mu \eta )} \right]}^2}}}, (20) \ (21) Θ″(η)=a0μ2cosh(μη)[2a12−a1sinh(μη)a2+a22][a2+a1sinh(μη)]3. \Theta ''(\eta ) = {{{a_0}{\mu ^2}\cosh (\mu \eta )\left[ {2a_1^2 - {a_1}\sinh (\mu \eta ){a_2} + a_2^2} \right]} \over {{{\left[ {{a_2} + {a_1}\sinh (\mu \eta )} \right]}^3}}}. in the second form.

We substitute Eqs. (16) or (19) into the reduced form of the governing equation obtained above in Eq. (5). Collecting the coefficients of the cosh^m(μη) and/or sinh^m(μη) of the same power, and equating each summation to zero, we get a set of algebraic equations. We solve the system of equations to get the solutions of the equation into consideration.

Here, we apply the methods the Klein-Gordon-Zakharov equations.

Consider the wave transformation (22) χ=Θ(η)eiθ, ψ=Φ(η), η=x−kt, θ=px+rt. \chi = \Theta (\eta ){e^{i\theta }},\;\;\psi = \Phi (\eta ),\;\;\eta = x - kt,\;\;\theta = px + rt. Placing Eq. (22) in Eq. (1), provides: (23) (k2−1)(1+p2−r2)Θ+λσΘ3+μ2(k2−1)2Θ″=0 ({k^2} - 1)(1 + {p^2} - {r^2})\Theta + \lambda \sigma {\Theta ^3} + {\mu ^2}{({k^2} - 1)^2}{\Theta ^{''}} = 0 from the real part, and k=−pr k = - {p \over r} from the imaginary part.

Suppose that Eq. (23) have the solutions of the form (24) Θ(η)=a0sin(μη)a2+a1cos(μη) \Theta (\eta ) = {{{a_0}\sin (\mu \eta )} \over {{a_2} + {a_1}\cos (\mu \eta )}} Placing Eq. (24) and its derivatives into Eq. (23) gives a polynomial in power of trigonometric functions. Collecting the coefficients of the same power of cos(μη), and equating each summation to zero, provides a system of equations. Solving this system of equations, provides

Class-1: When a0=a1p4−2p2r2+p2+r4−r2λrσ, a2=0, μ=−−p2r2+r4−r22p2−r2, {a_0} = {{{a_1}\sqrt {{p^4} - 2{p^2}{r^2} + {p^2} + {r^4} - {r^2}} } \over {\sqrt \lambda r\sqrt \sigma }},\;{a_2} = 0,\;\mu = - {{\sqrt { - {p^2}{r^2} + {r^4} - {r^2}} } \over {\sqrt 2 \sqrt {{p^2} - {r^2}} }}, we get (25) χ1a(x,t)=−p4−2p2r2+p2+r4−r2ei(px+rt)tan(|−p2r2+r4−r2|(ptr+x)2|p2−r2|)λrσ, {\chi _{1a}}(x,t) = - {{\sqrt {{p^4} - 2{p^2}{r^2} + {p^2} + {r^4} - {r^2}} {e^{i(px + rt)}}\tan \left( {{{\sqrt {| - {p^2}{r^2} + {r^4} - {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt 2 \sqrt {|{p^2} - {r^2}|} }}} \right)} \over {\sqrt \lambda r\sqrt \sigma }}, (26) ψ1a(x,t)=(p4−2p2r2+p2+r4−r2)tan2(|−p2r2+r4−r2|(ptr+x)2|p2−r2|)λr2(p2r2−1). {\psi _{1a}}(x,t) = {{\left( {{p^4} - 2{p^2}{r^2} + {p^2} + {r^4} - {r^2}} \right)\mathop {\tan }\nolimits^2 \left( {{{\sqrt {| - {p^2}{r^2} + {r^4} - {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt 2 \sqrt {|{p^2} - {r^2}|} }}} \right)} \over {\lambda {r^2}\left( {{{{p^2}} \over {{r^2}}} - 1} \right)}}.

Class-2: When a0=a1p4−2p2r2+p2+r4−r2λrσ, a2=a1, μ=−2−p2r2+r4−r2p2−r2, {a_0} = {{{a_1}\sqrt {{p^4} - 2{p^2}{r^2} + {p^2} + {r^4} - {r^2}} } \over {\sqrt \lambda r\sqrt \sigma }},\;{a_2} = {a_1},\;\mu = - {{\sqrt 2 \sqrt { - {p^2}{r^2} + {r^4} - {r^2}} } \over {\sqrt {{p^2} - {r^2}} }}, we get (27) χ2a(x,t)=−p4−2p2r2+p2+r4−r2ei(px+rt)sin(2|−p2r2+r4−r2|(ptr+x)|p2−r2|)λrσ(cos(2|−p2r2+r4−r2|(ptr+x)|p2−r2|)+1), {\chi _{2a}}(x,t) = - {{\sqrt {{p^4} - 2{p^2}{r^2} + {p^2} + {r^4} - {r^2}} {e^{i(px + rt)}}\sin \left( {{{\sqrt 2 \sqrt {| - {p^2}{r^2} + {r^4} - {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right)} \over {\sqrt \lambda r\sqrt \sigma \left( {\cos \left( {{{\sqrt 2 \sqrt {| - {p^2}{r^2} + {r^4} - {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right) + 1} \right)}}, (28) ψ2a(x,t)=(p4−2p2r2+p2+r4−r2)sin2(2|−p2r2+r4−r2|(ptr+x)|p2−r2|)λr2(p2r2−1)(cos(2|−p2r2+r4−r2|(ptr+x)|p2−r2|)+1)2. {\psi _{2a}}(x,t) = {{\left( {{p^4} - 2{p^2}{r^2} + {p^2} + {r^4} - {r^2}} \right)\mathop {\sin }\nolimits^2 \left( {{{\sqrt 2 \sqrt {| - {p^2}{r^2} + {r^4} - {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right)} \over {\lambda {r^2}\left( {{{{p^2}} \over {{r^2}}} - 1} \right){{\left( {\cos \left( {{{\sqrt 2 \sqrt {| - {p^2}{r^2} + {r^4} - {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right) + 1} \right)}^2}}}.

Suppose that Eq. (23) secures the solution of the form (29) Θ(η)=a0cos(μη)a2+a1sin(μη). \Theta (\eta ) = {{{a_0}\cos (\mu \eta )} \over {{a_2} + {a_1}\sin (\mu \eta )}}.

Placing Eq. (29) and its derivatives into Eq. (23) gives a polynomial in power of trigonometric functions. Collecting the coefficients of the same power of sin(μη), and equating each summation to zero, provides a system of equations. Solving this system of equations, provides

Class-1: When a0=a1p4−2p2r2+p2+r4−r2λrσ, a2=0, μ=−p2r2+r4−r22p2−r2, {a_0} = {{{a_1}\sqrt {{p^4} - 2{p^2}{r^2} + {p^2} + {r^4} - {r^2}} } \over {\sqrt \lambda r\sqrt \sigma }},\;{a_2} = 0,\;\mu = {{\sqrt { - {p^2}{r^2} + {r^4} - {r^2}} } \over {\sqrt 2 \sqrt {{p^2} - {r^2}} }}, we get (30) χ1b(x,t)=p4−2p2r2+p2+r4−r2ei(px+rt)cot(|−p2r2+r4−r2|(ptr+x)2|p2−r2|)λrσ, {\chi _{1b}}(x,t) = {{\sqrt {{p^4} - 2{p^2}{r^2} + {p^2} + {r^4} - {r^2}} {e^{i(px + rt)}}\cot \left( {{{\sqrt {| - {p^2}{r^2} + {r^4} - {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt 2 \sqrt {|{p^2} - {r^2}|} }}} \right)} \over {\sqrt \lambda r\sqrt \sigma }}, (31) ψ1b(x,t)=(p4−2p2r2+p2+r4−r2)cot2(|−p2r2+r4−r2|(ptr+x)2|p2−r2|)λr2(p2r2−1). {\psi _{1b}}(x,t) = {{\left( {{p^4} - 2{p^2}{r^2} + {p^2} + {r^4} - {r^2}} \right)\mathop {\cot }\nolimits^2 \left( {{{\sqrt {| - {p^2}{r^2} + {r^4} - {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt 2 \sqrt {|{p^2} - {r^2}|} }}} \right)} \over {\lambda {r^2}\left( {{{{p^2}} \over {{r^2}}} - 1} \right)}}.

Class-2: When a0=a1p4−2p2r2+p2+r4−r2λrσ, a2=−a1, μ=2−p2r2+r4−r2p2−r2, {a_0} = {{{a_1}\sqrt {{p^4} - 2{p^2}{r^2} + {p^2} + {r^4} - {r^2}} } \over {\sqrt \lambda r\sqrt \sigma }},\;{a_2} = - {a_1},\;\mu = {{\sqrt 2 \sqrt { - {p^2}{r^2} + {r^4} - {r^2}} } \over {\sqrt {{p^2} - {r^2}} }}, we get (32) χ2b(x,t)=p4−2p2r2+p2+r4−r2ei(px+rt)cos(2|−p2r2+r4−r2|(ptr+x)|p2−r2|)λrσ(sin(2|−p2r2+r4−r2|(ptr+x)|p2−r2|)−1), {\chi _{2b}}(x,t) = {{\sqrt {{p^4} - 2{p^2}{r^2} + {p^2} + {r^4} - {r^2}} {e^{i(px + rt)}}\cos \left( {{{\sqrt 2 \sqrt {| - {p^2}{r^2} + {r^4} - {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right)} \over {\sqrt \lambda r\sqrt \sigma \left( {\sin \left( {{{\sqrt 2 \sqrt {| - {p^2}{r^2} + {r^4} - {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right) - 1} \right)}}, (33) ψ2b(x,t)=(p4−2p2r2+p2+r4−r2)cos2(2|−p2r2+r4−r2|(ptr+x)|p2−r2|)λr2(p2r2−1)(sin(2|−p2r2+r4−r2|(ptr+x)|p2−r2|)−1)2. {\psi _{2b}}(x,t) = {{\left( {{p^4} - 2{p^2}{r^2} + {p^2} + {r^4} - {r^2}} \right)\mathop {\cos }\nolimits^2 \left( {{{\sqrt 2 \sqrt {| - {p^2}{r^2} + {r^4} - {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right)} \over {\lambda {r^2}\left( {{{{p^2}} \over {{r^2}}} - 1} \right){{\left( {\sin \left( {{{\sqrt 2 \sqrt {| - {p^2}{r^2} + {r^4} - {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right) - 1} \right)}^2}}}.

Assuming that Eq. (23) secures the solution of the form (34) Θ(η)=a0sinh(μη)a2+a1cosh(μη). \Theta (\eta ) = {{{a_0}\sinh (\mu \eta )} \over {{a_2} + {a_1}\cosh (\mu \eta )}}.

Placing Eq. (34) and its derivatives into Eq. (23) gives a polynomial in power of hyperbolic functions. Collecting the coefficients of the same power of cosh(μη), and equating each summation to zero, provides a system of equations. Solving this system of equations, provides

Class-1: When a0=a1−p4+2p2r2−p2−r4+r2λrσ, a2=0, μ=−rp2−r2+12p2−2r2, {a_0} = {{{a_1}\sqrt { - {p^4} + 2{p^2}{r^2} - {p^2} - {r^4} + {r^2}} } \over {\sqrt \lambda r\sqrt \sigma }},\;{a_2} = 0,\;\mu = - {{r\sqrt {{p^2} - {r^2} + 1} } \over {\sqrt {2{p^2} - 2{r^2}} }}, we get (35) χ1c(x,t)=−−p4+2p2r2−p2−r4+r2ei(px+rt)tanh(r|p2−r2+1|(ptr+x)|2p2−2r2|)λrσ, {\chi _{1c}}(x,t) = - {{\sqrt { - {p^4} + 2{p^2}{r^2} - {p^2} - {r^4} + {r^2}} {e^{i(px + rt)}}\tanh \left( {{{r\sqrt {|{p^2} - {r^2} + 1|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|2{p^2} - 2{r^2}|} }}} \right)} \over {\sqrt \lambda r\sqrt \sigma }}, (36) ψ1c(x,t)=(−p4+2p2r2−p2−r4+r2)tanh2(r|p2−r2+1|(ptr+x)|2p2−2r2|)λr2(p2r2−1). {\psi _{1c}}(x,t) = {{\left( { - {p^4} + 2{p^2}{r^2} - {p^2} - {r^4} + {r^2}} \right)\mathop {\tanh }\nolimits^2 \left( {{{r\sqrt {|{p^2} - {r^2} + 1|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|2{p^2} - 2{r^2}|} }}} \right)} \over {\lambda {r^2}\left( {{{{p^2}} \over {{r^2}}} - 1} \right)}}.

Class-2: When a0=a1−p4+2p2r2−p2−r4+r2λrσ, a2=−a1, μ=2p2r2−r4+r2p2−r2, {a_0} = {{{a_1}\sqrt { - {p^4} + 2{p^2}{r^2} - {p^2} - {r^4} + {r^2}} } \over {\sqrt \lambda r\sqrt \sigma }},\;{a_2} = - {a_1},\;\mu = {{\sqrt 2 \sqrt {{p^2}{r^2} - {r^4} + {r^2}} } \over {\sqrt {{p^2} - {r^2}} }}, (37) χ2c(x,t)=−p4+2p2r2−p2−r4+r2ei(px+rt)sinh(2|p2r2−r4+r2|(ptr+x)|p2−r2|)λrσ(cosh(2|p2r2−r4+r2|(ptr+x)|p2−r2|)−1), {\chi _{2c}}(x,t) = {{\sqrt { - {p^4} + 2{p^2}{r^2} - {p^2} - {r^4} + {r^2}} {e^{i(px + rt)}}\sinh \left( {{{\sqrt 2 \sqrt {|{p^2}{r^2} - {r^4} + {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right)} \over {\sqrt \lambda r\sqrt \sigma \left( {\cosh \left( {{{\sqrt 2 \sqrt {|{p^2}{r^2} - {r^4} + {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right) - 1} \right)}}, (38) ψ2c(x,t)=(−p4+2p2r2−p2−r4+r2)sinh2(2|p2r2−r4+r2|(ptr+x)|p2−r2|)λr2(p2r2−1)(cosh(2|p2r2−r4+r2|(ptr+x)|p2−r2|)−1)2. {\psi _{2c}}(x,t) = {{\left( { - {p^4} + 2{p^2}{r^2} - {p^2} - {r^4} + {r^2}} \right)\mathop {\sinh }\nolimits^2 \left( {{{\sqrt 2 \sqrt {|{p^2}{r^2} - {r^4} + {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right)} \over {\lambda {r^2}\left( {{{{p^2}} \over {{r^2}}} - 1} \right){{\left( {\cosh \left( {{{\sqrt 2 \sqrt {|{p^2}{r^2} - {r^4} + {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right) - 1} \right)}^2}}}.

Assuming that Eq. (23) secures the solution of the form (39) Θ(η)=a0cosh(μη)a2+a1sinh(μη). \Theta (\eta ) = {{{a_0}\cosh (\mu \eta )} \over {{a_2} + {a_1}\sinh (\mu \eta )}}.

Placing Eq. (39) and its derivatives into Eq. (23) gives a polynomial in power of hyperbolic functions. Collecting the coefficients of the same power of sinh(μη), and equating each summation to zero, provides a system of equations. Solving this system of equations, provides

Class-1: When a0=a1−p4+2p2r2−p2−r4+r2λrσ, a2=0, μ=−rp2−r2+12p2−2r2, {a_0} = {{{a_1}\sqrt { - {p^4} + 2{p^2}{r^2} - {p^2} - {r^4} + {r^2}} } \over {\sqrt \lambda r\sqrt \sigma }},\;{a_2} = 0,\;\mu = - {{r\sqrt {{p^2} - {r^2} + 1} } \over {\sqrt {2{p^2} - 2{r^2}} }}, we get (40) χ1d(x,t)=−−p4+2p2r2−p2−r4+r2ei(px+rt)coth(r|p2−r2+1|(ptr+x)|2p2−2r2|)λrσ, {\chi _{1d}}(x,t) = - {{\sqrt { - {p^4} + 2{p^2}{r^2} - {p^2} - {r^4} + {r^2}} {e^{i(px + rt)}}\coth \left( {{{r\sqrt {|{p^2} - {r^2} + 1|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|2{p^2} - 2{r^2}|} }}} \right)} \over {\sqrt \lambda r\sqrt \sigma }}, (41) ψ1d(x,t)=(−p4+2p2r2−p2−r4+r2)coth2(r|p2−r2+1|(ptr+x)|2p2−2r2|)λr2(p2r2−1). {\psi _{1d}}(x,t) = {{\left( { - {p^4} + 2{p^2}{r^2} - {p^2} - {r^4} + {r^2}} \right)\mathop {\coth }\nolimits^2 \left( {{{r\sqrt {|{p^2} - {r^2} + 1|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|2{p^2} - 2{r^2}|} }}} \right)} \over {\lambda {r^2}\left( {{{{p^2}} \over {{r^2}}} - 1} \right)}}.

Class-2: When a0=a1−p4+2p2r2−p2−r4+r2λrσ, a2=ia1, μ=−2p2r2−r4+r2p2−r2, {a_0} = {{{a_1}\sqrt { - {p^4} + 2{p^2}{r^2} - {p^2} - {r^4} + {r^2}} } \over {\sqrt \lambda r\sqrt \sigma }},\;{a_2} = i{a_1},\;\mu = - {{\sqrt 2 \sqrt {{p^2}{r^2} - {r^4} + {r^2}} } \over {\sqrt {{p^2} - {r^2}} }}, we get (42) χ2d(x,t)=−p4+2p2r2−p2−r4+r2ei(px+rt)cosh(2|p2r2−r4+r2|(ptr+x)|p2−r2|)λrσ(−sinh(2|p2r2−r4+r2|(ptr+x)|p2−r2|)+i), {\chi _{2d}}(x,t) = {{\sqrt { - {p^4} + 2{p^2}{r^2} - {p^2} - {r^4} + {r^2}} {e^{i(px + rt)}}\cosh \left( {{{\sqrt 2 \sqrt {|{p^2}{r^2} - {r^4} + {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right)} \over {\sqrt \lambda r\sqrt \sigma \left( { - \sinh \left( {{{\sqrt 2 \sqrt {|{p^2}{r^2} - {r^4} + {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right) + i} \right)}}, (43) ψ2d(x,t)=(−p4+2p2r2−p2−r4+r2)cosh2(2|p2r2−r4+r2|(ptr+x)|p2−r2|)λr2(p2r2−1)(−sinh(2|p2r2−r4+r2|(ptr+x)|p2−r2|)+i)2. {\psi _{2d}}(x,t) = {{\left( { - {p^4} + 2{p^2}{r^2} - {p^2} - {r^4} + {r^2}} \right)\mathop {\cosh }\nolimits^2 \left( {{{\sqrt 2 \sqrt {|{p^2}{r^2} - {r^4} + {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right)} \over {\lambda {r^2}\left( {{{{p^2}} \over {{r^2}}} - 1} \right){{\left( { - \sinh \left( {{{\sqrt 2 \sqrt {|{p^2}{r^2} - {r^4} + {r^2}|} \left( {{{pt} \over r} + x} \right)} \over {\sqrt {|{p^2} - {r^2}|} }}} \right) + i} \right)}^2}}}.

Here, we present the application of the extended rational sine/cosine and sinh/cosh methods to the coupled Maccari system.

Consider the wave transformation (44) χ(x,y,t)=eiθΘ(η), ψ(x,y,t)=Φ(η), η=x+y+kt, θ=ax+by+rt. \chi (x,y,t) = {e^{i\theta }}\Theta (\eta ),\;\psi (x,y,t) = \Phi (\eta ),\;\eta = x + y + kt,\;\theta = ax + by + rt. Placing Eq. (44) into Eq. (2), provides the following NODE: (45) Θ3+(1+c)(a2+r)Θ−(1+c)Θ″=0 {\Theta ^3} + (1 + c)({a^2} + r)\Theta - (1 + c){\Theta ^{''}} = 0 from the real part, and the relation k = −2a from the imaginary part.

Assuming that Eq. (45) secures (24) as its trial solution

Proceedings as before, we secure set of solutions as:

Class-1: When a0=a1(−(1−2a)(a2+r)), a2=0, μ=a2+r2, {a_0} = {a_1}\left( { - \sqrt {(1 - 2a)\left( {{a^2} + r} \right)} } \right),\;{a_2} = 0,\;\mu = {{\sqrt {{a^2} + r} } \over {\sqrt 2 }}, we get (46) χ1a(x,y,t)=−a2+(1−2a)rei(ax+by+rt)tan(|a2+r|(−2at+x+y)2), {\chi _{1a}}(x,y,t) = - \sqrt {{a^2} + (1 - 2a)r} {e^{i(ax + by + rt)}}\tan \left( {{{\sqrt {|{a^2} + r|} ( - 2at + x + y)} \over {\sqrt 2 }}} \right), (47) ψ1a(x,y,t)=−(a2+r)tan2(|a2+r|(−2at+x+y)2). {\psi _{1a}}(x,y,t) = - \left( {{a^2} + r} \right)\mathop {\tan }\nolimits^2 \left( {{{\sqrt {|{a^2} + r|} ( - 2at + x + y)} \over {\sqrt 2 }}} \right).

Class-2: a0=a1(1−2a)(a2+r), a2=−a1, μ=−2a2+r, {a_0} = {a_1}\sqrt {(1 - 2a)\left( {{a^2} + r} \right)} ,\;{a_2} = - {a_1},\;\mu = - \sqrt 2 \sqrt {{a^2} + r} , we get (48) χ2a(x,y,t)=−(1−2a)(a2+r)ei(ax+by+rt)sin(2|a2+r|(−2at+x+y))cos(2|a2+r|(−2at+x+y))−1, {\chi _{2a}}(x,y,t) = - {{\sqrt {(1 - 2a)\left( {{a^2} + r} \right)} {e^{i(ax + by + rt)}}\sin \left( {\sqrt 2 \sqrt {|{a^2} + r|} ( - 2at + x + y)} \right)} \over {\cos \left( {\sqrt 2 \sqrt {|{a^2} + r|} ( - 2at + x + y)} \right) - 1}}, (49) ψ2a(x,y,t)=−(a2+r)sin2(2|a2+r|(−2at+x+y))(cos(2|a2+r|(−2at+x+y))−1)2. {\psi _{2a}}(x,y,t) = - {{\left( {{a^2} + r} \right)\mathop {\sin }\nolimits^2 \left( {\sqrt 2 \sqrt {|{a^2} + r|} ( - 2at + x + y)} \right)} \over {{{\left( {\cos \left( {\sqrt 2 \sqrt {|{a^2} + r|} ( - 2at + x + y)} \right) - 1} \right)}^2}}}.

Assuming that Eq. (45) secures (29) as its trial solution

Proceedings as before, we get the following set of solutions:

Class-1: When a0=a1(−(1−2a)(a2+r)), a2=0, μ=a2+r2, {a_0} = {a_1}\left( { - \sqrt {(1 - 2a)\left( {{a^2} + r} \right)} } \right),\;{a_2} = 0,\;\mu = {{\sqrt {{a^2} + r} } \over {\sqrt 2 }}, we get (50) χ1b(x,y,t)=(1−2a)(a2+r)(−ei(ax+by+rt))cot(|a2+r|(−2at+x+y)2), {\chi _{1b}}(x,y,t) = \sqrt {(1 - 2a)\left( {{a^2} + r} \right)} \left( { - {e^{i(ax + by + rt)}}} \right)\cot \left( {{{\sqrt {|{a^2} + r|} ( - 2at + x + y)} \over {\sqrt 2 }}} \right), (51) ψ1b(x,y,t)=−(a2+r)cot2(|a2+r|(−2at+x+y)2). {\psi _{1b}}(x,y,t) = - \left( {{a^2} + r} \right)\mathop {\cot }\nolimits^2 \left( {{{\sqrt {|{a^2} + r|} ( - 2at + x + y)} \over {\sqrt 2 }}} \right).

Class-2: When a0=a1(−(1−2a)(a2+r)), a2=−a1, μ=2a2+r, {a_0} = {a_1}\left( { - \sqrt {(1 - 2a)\left( {{a^2} + r} \right)} } \right),\;{a_2} = - {a_1},\;\mu = \sqrt 2 \sqrt {{a^2} + r} , we get (52) χ2b(x,y,t)=−(1−2a)(a2+r)ei(ax+by+rt)cos(2|a2+r|(−2at+x+y))sin(2|a2+r|(−2at+x+y))−1, {\chi _{2b}}(x,y,t) = - {{\sqrt {(1 - 2a)\left( {{a^2} + r} \right)} {e^{i(ax + by + rt)}}\cos \left( {\sqrt 2 \sqrt {|{a^2} + r|} ( - 2at + x + y)} \right)} \over {\sin \left( {\sqrt 2 \sqrt {|{a^2} + r|} ( - 2at + x + y)} \right) - 1}}, (53) ψ2b(x,y,t)=−(a2+r)cos2(2|a2+r|(−2at+x+y))(sin(2|a2+r|(−2at+x+y))−1)2. {\psi _{2b}}(x,y,t) = - {{\left( {{a^2} + r} \right)\mathop {\cos }\nolimits^2 \left( {\sqrt 2 \sqrt {|{a^2} + r|} ( - 2at + x + y)} \right)} \over {{{\left( {\sin \left( {\sqrt 2 \sqrt {|{a^2} + r|} ( - 2at + x + y)} \right) - 1} \right)}^2}}}.

Assuming that Eq. (45) secures (34) as its trial solution

Proceedings as before, we get the following set of solutions:

Class-1: When a0=a1(2a−1)(a2+r), a2=0, μ=−−a2−r2, {a_0} = {a_1}\sqrt {(2a - 1)\left( {{a^2} + r} \right)} ,\;{a_2} = 0,\;\mu = - {{\sqrt { - {a^2} - r} } \over {\sqrt 2 }}, we get (54) χ1c(x,y,t)=(2a−1)(a2+r)(−ei(ax+by+rt))tanh(|−a2−r|(−2at+x+y)2), {\chi _{1c}}(x,y,t) = \sqrt {(2a - 1)\left( {{a^2} + r} \right)} \left( { - {e^{i(ax + by + rt)}}} \right)\tanh \left( {{{\sqrt {| - {a^2} - r|} ( - 2at + x + y)} \over {\sqrt 2 }}} \right), (55) ψ1c(x,y,t)=−(2a−1)(a2+r)tanh2(|−a2−r|(−2at+x+y)2)1−2a. {\psi _{1c}}(x,y,t) = - {{(2a - 1)\left( {{a^2} + r} \right)\mathop {\tanh }\nolimits^2 \left( {{{\sqrt {| - {a^2} - r|} ( - 2at + x + y)} \over {\sqrt 2 }}} \right)} \over {1 - 2a}}.

Class-2: When a0=a1(−(2a−1)(a2+r)), a2=−a1, μ=2−a2−r, {a_0} = {a_1}\left( { - \sqrt {(2a - 1)\left( {{a^2} + r} \right)} } \right),\;{a_2} = - {a_1},\;\mu = \sqrt 2 \sqrt { - {a^2} - r} , we get (56) χ2c(x,y,t)=−(2a−1)(a2+r)ei(ax+by+rt)sinh(2|−a2−r|(−2at+x+y))cosh(2|−a2−r|(−2at+x+y))−1, {\chi _{2c}}(x,y,t) = - {{\sqrt {(2a - 1)\left( {{a^2} + r} \right)} {e^{i(ax + by + rt)}}\sinh \left( {\sqrt 2 \sqrt {| - {a^2} - r|} ( - 2at + x + y)} \right)} \over {\cosh \left( {\sqrt 2 \sqrt {| - {a^2} - r|} ( - 2at + x + y)} \right) - 1}}, (57) ψ2c(x,y,t)=−(2a−1)(a2+r)sinh2(2|−a2−r|(−2at+x+y))(1−2a)(cosh(2|−a2−r|(−2at+x+y))−1)2. {\psi _{2c}}(x,y,t) = - {{(2a - 1)\left( {{a^2} + r} \right)\mathop {\sinh }\nolimits^2 \left( {\sqrt 2 \sqrt {| - {a^2} - r|} ( - 2at + x + y)} \right)} \over {(1 - 2a){{\left( {\cosh \left( {\sqrt 2 \sqrt {| - {a^2} - r|} ( - 2at + x + y)} \right) - 1} \right)}^2}}}.

Assuming that Eq. (45) secures (39) as its trial solution

Proceedings as before, we secure the following set of solutions:

Class-1: When a0=a1(2a−1)(a2+r), a2=0, μ=−a2−r2, {a_0} = {a_1}\sqrt {(2a - 1)\left( {{a^2} + r} \right)} ,\;{a_2} = 0,\;\mu = {{\sqrt { - {a^2} - r} } \over {\sqrt 2 }}, we get (58) χ1d(x,y,t)=(2a−1)a2+rei(ax+by+rt)coth(|−a2−r|(−2at+x+y)2), {\chi _{1d}}(x,y,t) = \sqrt {(2a - 1){a^2} + r} {e^{i(ax + by + rt)}}\coth \left( {{{\sqrt {| - {a^2} - r|} ( - 2at + x + y)} \over {\sqrt 2 }}} \right), (59) ψ1d(x,y,t)=−(2a−1)(a2+r)coth2(|−a2−r|(−2at+x+y)2)1−2a. {\psi _{1d}}(x,y,t) = - {{(2a - 1)\left( {{a^2} + r} \right)\mathop {\coth }\nolimits^2 \left( {{{\sqrt {| - {a^2} - r|} ( - 2at + x + y)} \over {\sqrt 2 }}} \right)} \over {1 - 2a}}.

Class-2: When a0=a1(−(2a−1)(a2+r)), a2=−ia1, μ=−2−a2−r, {a_0} = {a_1}\left( { - \sqrt {(2a - 1)\left( {{a^2} + r} \right)} } \right),\;{a_2} = - i{a_1},\;\mu = - \sqrt 2 \sqrt { - {a^2} - r} , we get (60) χ2d(x,y,t)=−(2a−1)(a2+r)ei(ax+by+rt)cosh(2|−a2−r|(−2at+x+y))−sinh(2|−a2−r|(−2at+x+y))−i, {\chi _{2d}}(x,y,t) = - {{\sqrt {(2a - 1)\left( {{a^2} + r} \right)} {e^{i(ax + by + rt)}}\cosh \left( {\sqrt 2 \sqrt {| - {a^2} - r|} ( - 2at + x + y)} \right)} \over { - \sinh \left( {\sqrt 2 \sqrt {| - {a^2} - r|} ( - 2at + x + y)} \right) - i}}, (61) ψ2d(x,y,t)=−(2a−1)(a2+r)cosh2(2|−a2−r|(−2at+x+y))(1−2a)(−sinh(2|−a2−r|(−2at+x+y))−i)2. {\psi _{2d}}(x,y,t) = - {{(2a - 1)\left( {{a^2} + r} \right)\mathop {\cosh }\nolimits^2 \left( {\sqrt 2 \sqrt {| - {a^2} - r|} ( - 2at + x + y)} \right)} \over {(1 - 2a){{\left( { - \sinh \left( {\sqrt 2 \sqrt {| - {a^2} - r|} ( - 2at + x + y)} \right) - i} \right)}^2}}}.