On the exact solutions to some system of complex nonlinear models

For the past two decades, the investigations of various travelling wave solutions to the nonlinear evolution equations have attracted the attentions of many scientist from all over the world. Nonlinear evolution equations (NLEEs) are used in describing many complex phenomena the arise on daily basis in the various fields of nonlinear sciences, such as; plasmas physics, quantum mechanics, biosciences, chemistry, water waves and so on. Various mathematical approaches have been formulated to tackle such type of problems, such as; the extended Conte's truncation method [1], the Hirota method [2], the local fractional Riccati differential equation method [3], the improved tan(ϕ/2)-expansion method [4], the generalized algebraic method [5], the simplified Hirota's method [6], the extended Jacobi elliptic function expansion method [7], the tanh function method [8], the generalized Kudryashov method [9], the sine-cosine method [10], the complex hyperbolic function method [11], the spectral-homotopy analysis method [12], the improved Bernoulli sub-equation function method [13], the modified exp (−φ(ξ))-expansion function method [14, 15, 16], sine-Gordon expansion method [17], the Adomian decomposition method [18], the Riccati equation method [19], the extended generalized Riccati equation mapping method [20] and many more other methods [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51].

However, in this study, we present the application of the extended sinh-Gordon equation expansion method (ShGEEM) [52] to the Davey-Stewartson equation [53] and the (2+1)-dimensional nonlinear complex coupled Maccari system [54, 55].

The Davey-Stewartson equation reads (1.1) iut+a(uxx+uyy)+b|u|2u−αuv=0,vxx+vyy−β(|u|2)xx=0. \matrix{{iu_t + a(u_{xx} + u_{yy}) + b|u|^2 u - \alpha uv} & = & {0,} \cr {v_{xx} + v_{yy} - \beta (|u|^2)_{xx}} & = & {0.} \cr}

Eq. (1.1) arises as a result of multiple-scale analysis of modulated nonlinear surface gravity waves propagating over a horizontal seabed [53]. Eq. (1.1) may also be used in modelling long-wave, short-wave resonances and other patterns of propagating waves [56, 57, 58]. Various studies have been conducted on Eq. (1.1) [59, 60, 61].

The (2+1)-dimensional nonlinear complex coupled Maccari equation reads (1.2) iut+uxx+uv=0,vt+vy+(|u|2)x=0, \matrix{{iu_t + u_{xx} + uv} & = & {0,} \cr {v_t + v_y + (|u|^2)_x} & = & {0,} \cr} where i=−1 i = \sqrt { - 1} .

Eq. (1.2) describes the motion of the isolated waves, localized in a small part of space, in many fields such as hydrodynamic, plasma physics, nonlinear optics etc. Eq. (1.2) was derived from the well known two-dimensional generalizations of the KdV equation [62, 63]. Several attempts by different scientists have been made to investigate Eq. (1.2) [64, 65, 66, 67, 68, 69, 70, 71].

In this sections, the general facts of the sinh-Gordon equation expansion method are presented.

To apply the ShGEEM, the following steps are followed:

Step-1: Consider the following nonlinear partial differential equation and the travelling wave transformation: (2.1) P(ux, u2uxx, ut, uxt, …)=0, P(u_x,\;u^2 u_{xx},\;u_t,\;u_{xt},\; \ldots) = 0, where P is a polynomial in u, the subscripts indicate the partial derivative of u with respect to x or t, and (2.2) u=Ψ(η), η=x−ct, u = \Psi (\eta),\;\;\;\eta = x - ct, respectively.

Substituting Eq. (2.2) into Eq. (2.1), we get the following nonlinear ordinary differential equation (NODE): (2.3) Q(Ψ, Ψ′, Ψ″, Ψ2Ψ′, …)=0, Q(\Psi,\;\Psi ^\prime,\;\Psi {''},\;\Psi ^2 \Psi ^\prime,\; \ldots) = 0, where Q is a polynomial in Ψ and the superscripts indicate the ordinary derivative of Ψ with respect to η.

Step-2: Eq. (2.3) is assumed to have solution of the form (2.4) Ψ(w)=∑j=1m[Bjsinh(w)+Ajcosh(w)]j+A0, \Psi (w) = \sum\limits_{j = 1}^m [B_j sinh(w) + A_j cosh(w)]^j + A_0, where A₀, A_j, B_j (j = 1, 2, ..., n) are constants to be determine later and w is a function of η that satisfies the following ordinary differential equations: (2.5) w′=sinh(w) w^\prime = sinh(w) and (2.6) w′=cosh(w) w^\prime = cosh(w)

To obtain the value of m, the homogeneous balance principle is used on the highest derivatives and highest power nonlinear term in Eq. (2.3).

Eqs. (2.5) and (2.6) have been extracted from the popularly known sinh-Gordon equation [52] given as (2.7) uxt=λsinh(u). u_{xt} = \lambda sinh(u).

Eq. (2.5) has the following solutions [52]: (2.8) sinh(w)=±csch(η) or sinh(w)=± i sech(η) sinh(w) = \pm csch(\eta)\;\;\;{\kern 1pt} {\rm or} {\kern 1pt} \;\;\;sinh(w) = \pm \;i\;sech(\eta) and (2.9) cosh(w)=±coth(η) or cosh(w)=± tanh(η), cosh(w) = \pm coth(\eta)\;\;\;{\kern 1pt} {\rm or}{\kern 1pt} \;\;\;cosh(w) = \pm \;\;tanh(\eta), where i=−1 i = \sqrt { - 1} .

Eq. (2.6) posses the following solutions [52]: (2.10) sinh(w)=tan(η) or sinh(w)=−cot(η) sinh(w) = tan(\eta)\;\;\;{\kern 1pt} {\rm or} {\kern 1pt} \;\;\;sinh(w) = - cot(\eta) and (2.11) cosh(w)=±sec(η) or cosh(w)=± csc(η). cosh(w) = \pm sec(\eta)\;\;\;{\kern 1pt} {\rm or} {\kern 1pt} \;\;\;cosh(w) = \pm \;\;csc(\eta).

Step-3: With fixed value of m, we substitute Eq. (2.4), its derivative along with Eq. (2.5) or (2.6) into Eq. (2.3) to obtain a polynomial equation in w^′ssinhⁱ(w)cosh^j(w) (s = 0,1 and i, j = 0, 1, 2, ...). We collect a set of over-determined nonlinear algebraic equations in A₀, A_j, B_j, c by setting the coefficients of w^′ssinhⁱ(w)cosh^j(w) to zero.

Step-4: The obtained set of over-determined nonlinear algebraic equations is then solved with aid of symbolic software to determine the values of the parameters A₀, A_j, B_j, c.

Step-5: Based on Eqs. (2.8), (2.9) and (2.10) and (2.11) solutions of Eq. (2.1) have the following forms: (2.12) Ψ(η)=∑j=1m[±iBj sech(η)±Ajtanh(η)]j+A0, \Psi (\eta) = \sum\limits_{j = 1}^m [ \pm iB_j \;\;sech(\eta) \pm A_j tanh(\eta)]^j + A_0, (2.13) Ψ(η)=∑j=1m[±Bj csch(η)±Ajcoth(η)]j+A0. \Psi (\eta) = \sum\limits_{j = 1}^m [ \pm B_j \;csch(\eta) \pm A_j coth(\eta)]^j + A_0. (2.14) Ψ(η)=∑j=1m[±Bjsec(η)+Ajtan(η)]j+A0 \Psi (\eta) = \sum\limits_{j = 1}^m [ \pm B_j sec(\eta) + A_j tan(\eta)]^j + A_0 and (2.15) Ψ(η)=∑j=1m[±Bjcsc(η)−Aj cot(η)]j+A0. \Psi (\eta) = \sum\limits_{j = 1}^m [ \pm B_j csc(\eta) - A_j \;cot(\eta)]^j + A_0.

In this section, the application of the extended ShGEEM to the Davey-Stewartson equation and the (2+1)-dimensional nonlinear complex coupled Maccari system is presented.

1. Consider the Davey-Stewartson equation [53] given in (1.1).

Substituting the complex travelling wave transformation (3.1) u(x,y,t)=eiθΨ(η), v(x,y,t)=V(η), η=x+y+ct, θ=σx+ny+rt u(x,y,t) = e^{i\theta} \Psi (\eta),\;\;v(x,y,t) = V(\eta),\;\;\eta = x + y + ct,\;\;\theta = \sigma x + ny + rt into (1.1), gives the following NODEs: (3.2) (−r−a(n2+σ2))Ψ+bΨ3−αΨV+2aΨ″=0, (- r - a(n^2 + \sigma ^2))\Psi + b\Psi ^3 - \alpha \Psi V + 2a\Psi {''} = 0, (3.3) β(Ψ′)2+βΨΨ″+V″=0, \beta (\Psi ^\prime)^2 + \beta \Psi \Psi {''} + V{''} = 0, from the real part, and the relation (3.4) c=−2a(n+σ). c = - 2a(n + \sigma).

Integrating Eq. (3.3) once, one can get (3.5) V=−β2Ψ2. V = - {\beta \over 2}\Psi ^2.

Substituting Eq. (3.5) into Eq. (3.2), we get (3.6) 2(−r−a(n2+σ2))Ψ+(2b+αβ)Ψ3+4aΨ″=0. 2(- r - a(n^2 + \sigma ^2))\Psi + (2b + \alpha \beta)\Psi ^3 + 4a\Psi {''} = 0.

Balancing Ψ³ and Ψ^″, we get m = 1.

With m = 1, Eqs. (2.4), (2.12), (2.13), (2.14) and (2.15) take the forms (3.7) Ψ(w)=B1sinh(w)+A1cosh(w)+A0, \Psi (w) = B_1 sinh(w) + A_1 cosh(w) + A_0, (3.8) Ψ(η)=±iB1 sech(η)±A1tanh(η)+A0, \Psi (\eta) = \pm iB_1 \;\;sech(\eta) \pm A_1 tanh(\eta) + A_0, (3.9) Ψ(η)=±B1 csch(η)±A1coth(η)+A0, \Psi (\eta) = \pm B_1 \;csch(\eta) \pm A_1 coth(\eta) + A_0, (3.10) Ψ(η)=±B1sec(η)+A1tan(η)+A0 \Psi (\eta) = \pm B_1 sec(\eta) + A_1 tan(\eta) + A_0 and (3.11) Ψ(η)=±B1csc(η)−A1 cot(η)+A0, \Psi (\eta) = \pm B_1 csc(\eta) - A_1 \;cot(\eta) + A_0, respectively.

Putting Eq. (3.7) and its second derivative along with Eq. (2.5) or (2.6) into Eq. (3.6), yields a polynomial in the power of hyperbolic functions. We collect a set of algebraic equations from the polynomial by equating each summations of the coefficients of the hyperbolic functions with the same power to zero. To obtain the values of the parameters involved, we simplify the set of the algebraic equations with aid of symbolic software. To get the new solutions to Eq. (1.1), we put the secured values of the parameters in each case into Eqs. (3.8), (3.9), (3.10) and (3.11), then into Eq. (3.1).

Case-1: When A0=0, A1=−2a−(2b+αβ), B1=A1, r=−a(1+n2+σ2), A_0 = 0,\;A_1 = - \sqrt {{{2a} \over {- (2b + \alpha \beta)}}},\;B_1 = A_1,\;r = - a(1 + n^2 + \sigma ^2), we get the following solutions: (3.12) u1(x,y,t)=2a−(2b+αβ)(±i sech[x+y−2a(n+σ)t]±tanh[x+y−2a(n+σ)t])ei(σx+ny−a(1+n2+σ2)t), \matrix{\hfill {u_1 (x,y,t) = \sqrt {{{2a} \over {- (2b + \alpha \beta)}}} (\pm i\;sech[x + y - 2a(n + \sigma)t]} \cr \hfill {\pm tanh[x + y - 2a(n + \sigma)t])e^{i(\sigma x + ny - a(1 + n^2 + \sigma ^2)t)},} \cr} (3.13) v1(x,y,t)=aβ2b+αβ(±i sech[x+y−2a(n+σ)t]±tanh[x+y−2a(n+σ)t])2 \matrix{{v_1 (x,y,t) = {{a\beta} \over {2b + \alpha \beta}}(\pm i\;sech[x + y - 2a(n + \sigma)t] \pm tanh[x + y - 2a(n + \sigma)t])^2} \hfill \cr} and (3.14) u2(x,y,t)=2a−(2b+αβ)tanh[12(x+y−2a(n+σ)t)]ei(σx+ny−a(1+n2+σ2)t), \matrix{{u_2 (x,y,t) = \sqrt {{{2a} \over {- (2b + \alpha \beta)}}} tanh[{1 \over 2}(x + y - 2a(n + \sigma)t)]e^{i(\sigma x + ny - a(1 + n^2 + \sigma ^2)t)},} \hfill \cr} (3.15) v2(x,y,t)=aβ2b+αβtanh2[12(x+y−2a(n+σ)t)]. \matrix{{v_2 (x,y,t) = {{a\beta} \over {2b + \alpha \beta}}tanh^2 [{1 \over 2}(x + y - 2a(n + \sigma)t)].} \hfill \cr}

Case-2: When A0=0, A1=22a−(2b+αβ), B1=0, r=−a(4+n2+σ2), A_0 = 0,\;A_1 = 2\sqrt {{{2a} \over {- (2b + \alpha \beta)}}},\;B_1 = 0,\;r = - a(4 + n^2 + \sigma ^2), we get the following solutions: (3.16) u3(x,y,t)=±22a−(2b+αβ)tanh[x+y−2a(n+σ)t]ei(σx+ny−a(4+n2+σ2)t), \matrix{{u_3 (x,y,t) = \pm 2\sqrt {{{2a} \over {- (2b + \alpha \beta)}}} tanh[x + y - 2a(n + \sigma)t]e^{i(\sigma x + ny - a(4 + n^2 + \sigma ^2)t)},} \hfill \cr} (3.17) v3(x,y,t)=4aβ2b+αβtanh2[x+y−2a(n+σ)t] \matrix{{v_3 (x,y,t) = {{4a\beta} \over {2b + \alpha \beta}}tanh^2 [x + y - 2a(n + \sigma)t]} \hfill \cr} and (3.18) u4(x,y,t)=±22a−(2b+αβ)coth[x+y−2a(n+σ)t]ei(σx+ny−a(4+n2+σ2)t), \matrix{{u_4 (x,y,t) = \pm 2\sqrt {{{2a} \over {- (2b + \alpha \beta)}}} coth[x + y - 2a(n + \sigma)t]e^{i(\sigma x + ny - a(4 + n^2 + \sigma ^2)t)},} \hfill \cr} (3.19) v4(x,y,t)=4aβ2b+αβcoth2[x+y−2a(n+σ)t]. \matrix{{v_4 (x,y,t) = {{4a\beta} \over {2b + \alpha \beta}}coth^2 [x + y - 2a(n + \sigma)t].} \hfill \cr}

Case-3: When A0=0, A1=0, B1=−22a−(2b+αβ), r=−a(n2+σ2−2), A_0 = 0,\;A_1 = 0,\;B_1 = - 2\sqrt {{{2a} \over {- (2b + \alpha \beta)}}},\;r = - a(n^2 + \sigma ^2 - 2), we get the following solutions: (3.20) u5(x,y,t)=±22a2b+αβsech[x+y−2a(n+σ)t]ei(σx+ny−a(n2+σ2−2)t), \matrix{{u_5 (x,y,t) = \pm 2\sqrt {{{2a} \over {2b + \alpha \beta}}} sech[x + y - 2a(n + \sigma)t]e^{i(\sigma x + ny - a(n^2 + \sigma ^2 - 2)t)},} \hfill \cr} (3.21) v5(x,y,t)=−4aβ2b+αβsech2[x+y−2a(n+σ)t] \matrix{{v_5 (x,y,t) = - {{4a\beta} \over {2b + \alpha \beta}}sech^2 [x + y - 2a(n + \sigma)t]} \hfill \cr} and (3.22) u6(x,y,t)=±22a−(2b+αβ)csch[x+y−2a(n+σ)t]ei(σx+ny−a(n2+σ2−2)t), \matrix{{u_6 (x,y,t) = \pm 2\sqrt {{{2a} \over {- (2b + \alpha \beta)}}} csch[x + y - 2a(n + \sigma)t]e^{i(\sigma x + ny - a(n^2 + \sigma ^2 - 2)t)},} \hfill \cr} (3.23) v6(x,y,t)=4aβ2b+αβcsch2[x+y−2a(n+σ)t]. \matrix{{v_6 (x,y,t) = {{4a\beta} \over {2b + \alpha \beta}}csch^2 [x + y - 2a(n + \sigma)t].} \hfill \cr}

Case-4: When A0=0, A1=−2a−(2b+αβ), B1=A1, r=−a(n2+σ2−1), A_0 = 0,\;A_1 = - \sqrt {{{2a} \over {- (2b + \alpha \beta)}}},\;B_1 = A_1,\;r = - a(n^2 + \sigma ^2 - 1), we get the following solutions: (3.24) u7(x,y,t)=2a−(2b+αβ)(±sec[x+y−2a(n+σ)t]±tan[x+y−2a(n+σ)t])ei(σx+ny−a(n2+σ2−1)t), \matrix{\hfill {u_7 (x,y,t) = \sqrt {{{2a} \over {- (2b + \alpha \beta)}}} (\pm sec[x + y - 2a(n + \sigma)t]} \cr \hfill {\pm tan[x + y - 2a(n + \sigma)t])e^{i(\sigma x + ny - a(n^2 + \sigma ^2 - 1)t)},} \cr} (3.25) v7(x,y,t)=aβ2b+αβ(±sec[x+y−2a(n+σ)t]±tan[x+y−2a(n+σ)t])2 \matrix{{v_7 (x,y,t) = {{a\beta} \over {2b + \alpha \beta}}(\pm sec[x + y - 2a(n + \sigma)t] \pm tan[x + y - 2a(n + \sigma)t])^2} \hfill \cr} and (3.26) u8(x,y,t)=2a−(2b+αβ)cot[12(x+y−2a(n+σ)t)]ei(σx+ny−a(n2+σ2−1)t), \matrix{{u_8 (x,y,t) = \sqrt {{{2a} \over {- (2b + \alpha \beta)}}} cot[{1 \over 2}(x + y - 2a(n + \sigma)t)]e^{i(\sigma x + ny - a(n^2 + \sigma ^2 - 1)t)},} \hfill \cr} (3.27) v8(x,y,t)=aβ2b+αβcot2[12(x+y−2a(n+σ)t)]. \matrix{{v_8 (x,y,t) = {{a\beta} \over {2b + \alpha \beta}}cot^2 [{1 \over 2}(x + y - 2a(n + \sigma)t)].} \hfill \cr}

2. Consider the (2+1)-dimensional nonlinear complex coupled Maccari equation [55] given in Eq. (1.2).

Substituting the complex wave transformation (3.28) u(x,y,t)=eiθΨ(η), v(x,y,t)=V(η), η=x+y+ct, θ=ax+by+rt u(x,y,t) = e^{i\theta} \Psi (\eta),\;v(x,y,t) = V(\eta),\;\eta = x + y + ct,\;\theta = ax + by + rt into Eq. (1.2), gives the following NODEs: (3.29) Ψ″+ΨV−(a2+r)Ψ=0, \Psi {''} + \Psi V - (a^2 + r)\Psi = 0, (3.30) 2ΨΨ′+(1+c)V′=0 2\Psi \Psi ^\prime + (1 + c)V^\prime = 0 from the real part and the relation (3.31) c=−2a c = - 2a from the imaginary part.

Integrating Eq. (3.30) once, we obtain (3.32) V=−11+cΨ. V = - {1 \over {1 + c}}\Psi.

Substituting Eq. (3.32) into Eq. (3.29), we have the following single NODE: (3.33) Ψ3+(1+c)(a2+r)Ψ−(1+c)Ψ″, \Psi ^3 + (1 + c)(a^2 + r)\Psi - (1 + c)\Psi {''},

Balancing the terms Ψ³ and Ψ^″ in Eq. (3.33), yields m = 1.

Proceedings as before, we obtained the following solutions for Eq. (1.2):

Case-1: When A0=0, A1=−1+−2−4r2, B1=A1, a=−−12−r, A_0 = 0,\;A_1 = - {{\sqrt {1 + \sqrt {- 2 - 4r}}} \over {\sqrt 2}},\;B_1 = A_1,\;a = - \sqrt {- {1 \over 2} - r}, we get the following solutions: (3.34) u1(x,y,t)=1+−2−4r2(±i sech[2−12−r t+x+y]±tanh[2−12−r t+x+y])ei(rt−−12−r x+by), \matrix{\hfill {u_1 (x,y,t) = {{\sqrt {1 + \sqrt {- 2 - 4r}}} \over {\sqrt 2}}(\pm i\;sech[2\sqrt {- {1 \over 2} - r} \;t + x + y]} \cr \hfill {\pm tanh[2\sqrt {- {1 \over 2} - r} \;t + x + y])e^{i(rt - \sqrt {- {1 \over 2} - r} \;x + by)},}} (3.35) v1(x,y,t)=−(1+−2−4r)2(1+2−12−r)(±i sech[2−12−r t+x+y]±tanh[2−12−r t+x+y])2 \matrix{\hfill{v_1 (x,y,t) = - {{(1 + \sqrt {- 2 - 4r})} \over {2(1 + 2\sqrt {- {1 \over 2} - r})}}(\pm i\;sech[2\sqrt {- {1 \over 2} - r} \;t + x + y]} \cr \hfill{\pm tanh[2\sqrt {- {1 \over 2} - r} \;t + x + y])^2}} and (3.36) u2(x,y,t)=±1+−2−4r2(coth[2−12−r t+x+y]csch[2−12−r t+x+y])ei(rt−−12−r x+by), \matrix{\hfill {u_2 (x,y,t) = \pm {{\sqrt {1 + \sqrt {- 2 - 4r}}} \over {\sqrt 2}}(coth[2\sqrt {- {1 \over 2} - r} \;t + x + y]} \cr \hfill{csch[2\sqrt {- {1 \over 2} - r} \;t + x + y])e^{i(rt - \sqrt {- {1 \over 2} - r} \;x + by)},}} (3.37) v2(x,y,t)=−(1+−2−4r)2(1+2−12−r)(±coth[2−12−r t+x+y]±csch[2−12−r t+x+y])2. \matrix{\hfill{v_2 (x,y,t) = - {{(1 + \sqrt {- 2 - 4r})} \over {2(1 + 2\sqrt {- {1 \over 2} - r})}}(\pm coth[2\sqrt {- {1 \over 2} - r} \;t + x + y]} \cr \hfill{\pm csch[2\sqrt {- {1 \over 2} - r} \;t + x + y])^2.}}

Case-2: When A0=0, A1=−2+4−2−r, B1=0, a=−−2−r, A_0 = 0,\;A_1 = - \sqrt {2 + 4\sqrt {- 2 - r}},\;B_1 = 0,\;a = - \sqrt {- 2 - r}, we get the following solutions: (3.38) u3(x,y,t)=±2+4−2−r tanh[2−2−r t+x+y]ei(rt−−2−r x+by), \matrix{{u_3 (x,y,t) = \pm \sqrt {2 + 4\sqrt {- 2 - r}} \;tanh[2\sqrt {- 2 - r} \;t + x + y]e^{i(rt - \sqrt {- 2 - r} \;x + by)},}} (3.39) v3(x,y,t)=−(2+4−2−r)1+2−2−rtanh2[2−2−r t+x+y] \matrix{{v_3 (x,y,t) = - {{(2 + 4\sqrt {- 2 - r})} \over {1 + 2\sqrt {- 2 - r}}}tanh^2 [2\sqrt {- 2 - r} \;t + x + y]}} and (3.40) u4(x,y,t)=±2+4−2−r coth[2−2−r t+x+y]ei(rt−−2−r x+by), \matrix{{u_4 (x,y,t) = \pm \sqrt {2 + 4\sqrt {- 2 - r}} \;coth[2\sqrt {- 2 - r} \;t + x + y]e^{i(rt - \sqrt {- 2 - r} \;x + by)},}} (3.41) v4(x,y,t)=−(2+4−2−r)1+2−2−rcoth2[2−2−r t+x+y]. \matrix{{v_4 (x,y,t) = - {{(2 + 4\sqrt {- 2 - r})} \over {1 + 2\sqrt {- 2 - r}}}coth^2 [2\sqrt {- 2 - r} \;t + x + y].}}

Case-3: When A0=0, A1=0, B1=−2+41−r, a=−1−r, A_0 = 0,\;A_1 = 0,\;B_1 = - \sqrt {2 + 4\sqrt {1 - r}},\;a = - \sqrt {1 - r}, we get the following solutions: (3.42) u5(x,y,t)=±2+41−ri sech[21−r t+x+y]ei(rt−1−r x+by), \matrix{{u_5 (x,y,t) = \pm \sqrt {2 + 4\sqrt {1 - r}} i\;sech[2\sqrt {1 - r} \;t + x + y]e^{i(rt - \sqrt {1 - r} \;x + by)},}} (3.43) v5(x,y,t)=2+41−r1+21−rsech2[21−r t+x+y] \matrix{{v_5 (x,y,t) = {{2 + 4\sqrt {1 - r}} \over {1 + 2\sqrt {1 - r}}}sech^2 [2\sqrt {1 - r} \;t + x + y]}} and (3.44) u6(x,y,t)=±2+41−rcsch[21−r t+x+y]ei(rt−1−r x+by), \matrix{{u_6 (x,y,t) = \pm \sqrt {2 + 4\sqrt {1 - r}} csch[2\sqrt {1 - r} \;t + x + y]e^{i(rt - \sqrt {1 - r} \;x + by)},}} (3.45) v6(x,y,t)=−(2+41−r)1+21−rcsch2[21−r t+x+y]. \matrix{{v_6 (x,y,t) = - {{(2 + 4\sqrt {1 - r})} \over {1 + 2\sqrt {1 - r}}}csch^2 [2\sqrt {1 - r} \;t + x + y].}}

Case-4: When A0=0, A1=−1+2−4r2, B1=0, a=−12−r, A_0 = 0,\;A_1 = - {{\sqrt {1 + \sqrt {2 - 4r}}} \over {\sqrt 2}},\;B_1 = 0,\;a = - \sqrt {{1 \over 2} - r}, we get the following solutions: (3.46) u7(x,y,t)=1+2−4r2(sec[212−r t+x+y]−tan[212−r t+x+y])ei(rt−12−r x+by), \matrix{\hfill{u_7 (x,y,t) = {{\sqrt {1 + \sqrt {2 - 4r}}} \over {\sqrt 2}}(sec[2\sqrt {{1 \over 2} - r} \;t + x + y]} \cr \hfill{- tan[2\sqrt {{1 \over 2} - r} \;t + x + y])e^{i(rt - \sqrt {{1 \over 2} - r} \;x + by)},}} (3.47) v7(x,y,t)=−(1+2−4r)2(1+212−r)(sec[212−r t+x+y]−tan[212−r t+x+y])2 \matrix{\hfill{v_7 (x,y,t) = - {{(1 + \sqrt {2 - 4r})} \over {2(1 + 2\sqrt {{1 \over 2} - r})}}(sec[2\sqrt {{1 \over 2} - r} \;t + x + y]} \cr \hfill{- tan[2\sqrt {{1 \over 2} - r} \;t + x + y])^2}} and (3.48) u8(x,y,t)=1+2−4r2(cot[212−r t+x+y]+csc[212−r t+x+y])ei(rt−12−r x+by), \matrix{\hfill{u_8 (x,y,t) = {{\sqrt {1 + \sqrt {2 - 4r}}} \over {\sqrt 2}}(cot[2\sqrt {{1 \over 2} - r} \;t + x + y]} \cr \hfill{+ csc[2\sqrt {{1 \over 2} - r} \;t + x + y])e^{i(rt - \sqrt {{1 \over 2} - r} \;x + by)},}} (3.49) v8(x,y,t)=−(1+2−4r)2(1+212−r)(cot[212−r t+x+y]+csc[212−r t+x+y])2. \matrix{\hfill{v_8 (x,y,t) = - {{(1 + \sqrt {2 - 4r})} \over {2(1 + 2\sqrt {{1 \over 2} - r})}}(cot[2\sqrt {{1 \over 2} - r} \;t + x + y]} \cr \hfill{+ csc[2\sqrt {{1 \over 2} - r} \;t + x + y])^2.}}

In this study, we successfully constructed some soliton, singular soliton and singular periodic wave solutions to the Davey-Stewartson equation and the (2+1)-dimensional nonlinear complex coupled Maccari system by using the extended sinh-Gordon equation expansion method. Under the choice of suitable parameters, the 2D, 3D and contour graphs to some of the obtained solutions are presented. The reported results in this study have some physical meanings, for instance; the hyperbolic tangent arises in the calculation of magnetic moment and rapidity of special relativity, the hyperbolic secant arises in the profile of a laminar jet, and hyperbolic cotangent arises in the Langevin function for magnetic polarization [72]. In order to have clear and good understanding of the physical properties of the reported topological, non-topological, singular solitons and singular periodic wave solutions, under the choice of the suitable values of parameters, the 3D, 2D and the contour graphs are plotted. The perspective view of the topological Eq. (3.16), non-topological Eq. (3.20) and mixed singular solitons Eq. (3.36) can be seen in the 3D graphs which appear in the (a) parts of figs. 1, 2 and 3, respectively. The propagation pattern of the wave along the x-axis for Eq. is illustrated in the 2D graphs which is located at the top right corner of the (a) parts of figs. 1, 2 and 3. The contour graphs is an alternative of the 3D plots. The the contour graph in the (b) part of Fig. 1 illustrates the unstable propagation of the exact toplogical soliton and contour graphs in the (b) parts of Fig. 2 illustrates the stable propagation of the exact fundamental non-toplogical soliton. The extended sinh-Gordon equation expansion method is powerful and efficient mathematical approach that can be used for investigating various nonlinear physical models. To the best of our knowledge the applications of the extended sinh-Gordon equation expansion method to the Davey-Stewartson equation and the (2+1)-dimensional nonlinear complex coupled Maccari system have not been submitted to the literature beforehand.

Figure 1

Figure 2

Figure 3