1. bookVolume 6 (2021): Issue 1 (January 2021)
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Optical soliton solutions to a (2+1) dimensional Schrödinger equation using a couple of integration architectures

Published Online: 26 Jun 2020
Page range: 381 - 396
Received: 20 May 2019
Accepted: 11 Feb 2020
Journal Details
License
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Abstract

In this work, we consider a (2+1) dimensional nonlinear Schrödinger system which appears in the theory of nonlinear optics and describe transmission of the optical pulses in optical fibers. We attain certain special type traveling wave solutions of the under investigated model by help of finite series expansion and auxiliary differential equations. In this manner, we exploit exp(−ϕ(ε)) and modified Kudryashov approaches as solution procedures. Moreover, we make tanh ansatz because of the being even order of the reduced ordinary differential equation. The obtained solutions are in the form of dark soliton, combined soliton, symmetrical Lucas sine, Lucas cosine functions, and periodic wave solutions. We present also some graphical simulations of the solutions corresponding to values of parameters which leads to a better understanding the phenomena.

Keywords

MSC 2010

Introduction

As well-known, many physical phenomena occurred in the nonlinear science are governed by the nonlinear evolution equations (NLEEs). We observe these phenomena in many models of applied sciences such as non-linear optics, nuclear physics, shallow water wave theory, plasma physics, biology, chemistry, etc. Therefore, solving these models via analytic mathematical methods is quite important for revealing the physical explanations of the considered model. In the last three decades, we have witnessed many powerful analytical methods such as the theory of Lie groups, Hirota's bilinear method, Painleve property, homogeneous balance method, inverse scattering method, Backlund transformation, Darboux transformation, Lax pairs, He's the exp-function method, Kudryashov's simplest equation method, sine-cosine method, etc. (see, [1],[2],[13] and references therein).

The one-dimensional nonlinear Schrödinger equation (NLSE) is the basic equation of physics for describing quantum mechanical behavior. It is also frequently called the Schrödinger wave equation, and is a partial differential equation (PDE), iqt+2|q|2q+qxx=0 {{iq}_t} + 2|q{|^2}q + {q_{xx}} = 0 for complex field q(x,t). The Eq.(1) models many nonlinearity effects in a fiber, including but not limited to self-phase modulation, four-wave mixing, second harmonic generation, stimulated Raman scattering, optical solitons, ultrashort pulses, etc. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides [15] and to Bose-Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime.

It has been noted [16] that NLEEs in higher order spatial dimensions correspond to certain holomorphic vector bundles over the twistor space, the connection being provided by the Penrose correspondence. The self-dual Yang Mills (SDYM) equation is the best known example admitting such an interpretation and interestingly demonstrated that the integrable many models have been found to be embedded within this equation [17]. By dimensionally reducing the SDYM equation, Strachan [18] introduced a (2+1) dimensional spatio-temporal non-local system [19], [20] kqt=12qxyqy[p.q]dx,kpt=12pxypy[p.q]dx. \matrix{&\hfill {{{kq}_t} = {1 \over 2}{q_{xy}} - q\int {{\partial _y}[p.q]{d_x}},} \cr &\hfill {- k{p_t} = {1 \over 2}{p_{xy}} - p\int {{\partial _y}[p.q]{d_x}}.}} which is a generalization of the NLS equation and showed it to be integrable from the point of view of geometrical consideration. Under potential function vx(p,q)=2y[p.q], {v_x}(p,q) = 2{\partial _y}[p.q], with the ansatz p = −q* = u and k = −i/2, Eq.(2) reduces to iut=uxy+uv,vx=2(|u|2)y. \matrix{{i{u_t} = {u_{xy}} + uv,} \cr {{v_x} = 2{{(|u{|^2})}_y}.} \cr} Eq.(4) converts to classical NLSE while x = y.

In [21], the authors constructed the possible explicit parametric representations of the bounded travelling wave solutions and all kinds of phase portraits in the parametric space of Eq.(4) by using the approach of dynamical systems and the theory of bifurcations. In [22] the authors obtained different nature exact solutions of Eq.(4) such as triangular-type, soliton-type, doubly periodic-like, single and combined non-degenerate Jacobi elliptic function like solutions by using the Fan sub-equation method.

The main aim of this study is to reveal the exact traveling solutions of the Eq.(2) via three distinct algorithms. In this regard, in the second section, we briefly present the model equation and converts it into the ordinary differential equation (ODE) system. In section 3, we present the exact solution schemas of exp(−ϕ(ε)), modified Kudryashov, and tanh function method. Section 4 is devoted to the application of the methods to our model separately. In the last section, we give numerical simulations and graphics of the obtained solutions for better understanding the physical phenomena. Moreover, in this section we give some concluding remarks on the presented methods and future works.

The derivation of reduced ODE of Eq.(4)

In order to reveal the exact solutions of Eq.(4), the following transformation u(x,y,t)=U(ε)eiψ,v(x,y,t)=V(ε) u(x,y,t) = U(\varepsilon){e^{i\psi}},v(x,y,t) = V(\varepsilon) is choosed in Eq.(4) where ψ=γ1x+γ2y+γ3tandε=η1x+η2y+η3t. \psi = {\gamma _1}x + {\gamma _2}y + {\gamma _3}t\,\, {\rm and}\,\,\varepsilon \, = {\eta _1}x + {\eta _2}y + {\eta _3}t.

In Eq.(6) γi and ηi (i = 1,2,3) are arbitrary constants which to be determined later [22]. By substitution of Eq.(5) into Eq.(4) and then decomposing the real and imaginary parts of it, we get respectively η1η2U(ε)+U(ε)V(ε)+(γ3γ1η2)U(ε)=0, {\eta _1}{\eta _2}U''(\varepsilon) + U(\varepsilon)V(\varepsilon) + ({\gamma _3} - {\gamma _1}{\eta _2})U(\varepsilon) = 0, (η1η2+γ1η2+η3)U(ε)=0, ({\eta _1}{\eta _2} + {\gamma _1}{\eta _2} + {\eta _3})U'(\varepsilon) = 0, and η1V(ε)=2η2(U2(ε)). {\eta _1}V'(\varepsilon) = 2{\eta _2}({U^2}(\varepsilon))'.

It is readily seen that the Eq.(8) should holds the following constraint: (η1η2+γ1η2+η3)=0. ({\eta _1}{\eta _2} + {\gamma _1}{\eta _2} + {\eta _3}) = 0.

After the integration of Eq.(9) with respect to ɛ, we yield V(ε)=2η2η1U2(ε). V(\varepsilon) = {{2{\eta _2}} \over {{\eta _1}}}{U^2}(\varepsilon).

Plugging the Eq.(11) into Eq.(7), we get second order nonlinear ordinary differential equation (NLODE) as follows [22]: U(ε)+2η12U3(ε)+(γ3γ1η2η1η2)U(ε)=0. U''(\varepsilon) + {2 \over {{\eta _{{1^2}}}}}{U^3}(\varepsilon) + \left({{{{\gamma _3} - {\gamma _1}{\eta _2}} \over {{\eta _1}{\eta _2}}}} \right)U(\varepsilon) = 0.

Methods
Exp(−ϕ(ε)) Method

The main properties of the method can be found in [23,24,25]. Let us consider (2+1) dimensional NLEEs (involving polynomial derivatives) F(U,Ux,Uy,Ut,Uxy,Uxt,Uyx,Uyt,...)=0 F(U,{U_x},{U_y},{U_t},{U_{xy}},{U_{xt}},{U_{yx}},{U_{yt}},...) = 0 where U is dependent variable and x,y and t are independent variables. Using the wave variable transformation u(x,y,t) = U(ε), where ε = η1x + η2y + η3t, Eq.(13) transforms to the following NLODE of integer order G(U,U,U,...)=0. G(U,U',U'',...) = 0.

We are looking for the following finite series expansion solution U(ε)=n=0Nanexp(Nϕ(ε))=a0+a1exp(ϕ(ε))+...+aNexp(Nϕ(ε)),aN0 U(\varepsilon) = \sum\limits_{n = 0}^N {{a_n}exp(- N\phi (\varepsilon)) = {a_0} + {a_1}exp(- \phi (\varepsilon)) +... + {a_N}exp(- N\phi (\varepsilon)),\,\,\,\,\,{a_N} \ne 0} for Eq.(14). In Eq.(15), the coefficients an n = 1,2,...,N will be determined later. The positif integer N is computed by the homogenous balance between highest order nonlinear terms and linear terms. The ϕ(ε) analytic function holds the following first order NLODE ϕ(ε)=exp(ϕ(ε))+μexp(ϕ(ε))+λ. \phi '(\varepsilon) = exp(- \phi (\varepsilon)) + \mu exp(\phi (\varepsilon)) + \lambda. We now present some solution sets of Eq.(16) with respect to coefficient classifications.

Case 1 If λ2 − 4μ > 0 and μ ≠ 0, then ϕ1(ε)=ln(λ24μtanh((λ24μ/2)(ε+C)λ2μ). {\phi _1}(\varepsilon) = ln\left({{{- \sqrt {{\lambda ^2} - 4\mu} \,\tanh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C) - \lambda} \over {2\mu}}} \right).

Case 2 If λ2 − 4μ > 0, μ = 0 and λ ≠ 0, then ϕ2(ε)=ln(λcosh(λ(ε+C))+sinh(λ(ε+C))1). {\phi _2}(\varepsilon) = - ln\left({{\lambda \over {\cosh(\lambda (\varepsilon + C)) + \sinh(\lambda (\varepsilon + C)) - 1}}} \right).

Case 3 If λ2 − 4μ < 0 and μ ≠ 0, then ϕ3(ε)=ln(4μλ2tan((4μλ2/2(ε+C))λ2μ). {\phi _3}(\varepsilon) = ln\left({{{\sqrt {4\mu - {\lambda ^2}} \tan ((\sqrt {4\mu - {\lambda ^2}} /2(\varepsilon + C)) - \lambda} \over {2\mu}}} \right).

By inserting Eq.(15) into Eq.(14) with the assistance of Maple and equating the coefficients of same terms of exp(−ϕ(ε)), we obtain an algebraic system which involves an's, η1, η2 and η3. Having solved this system we capture these coefficients easily. If necessary arrangements are made, Eq.(15) gives the traveling wave solutions of Eq.(14).

Modified Kudryashov Method

The preliminary steps of the modified Kudryashov method [25,26,27] are as previous method. With the same traveling wave transformation, the original NLEE can be converted to a NLODE. Supposed solution form is U(ε)=a0+a1Q(ε)+...+aNQN(ε),aN0, \matrix{\hfill {U(\varepsilon) = {a_0} + {a_1}Q(\varepsilon) +... + {a_N}{Q^N}(\varepsilon),} & \hfill {{a_N} \ne 0,} \cr} where the constants an, n = 0,1,2,..., N shall be determined later, N is a positive number which is assessed by the technique of homogeneous balance, and Q(ε)=1/(1+daε) Q(\varepsilon) = 1/(1 + {{da}^\varepsilon}) is an explicit function that satisfies the following NLODE Q(ε)=Q(ε)(Q(ε)1)lna. Q'(\varepsilon) = Q(\varepsilon)(Q(\varepsilon) - 1)\,\ln\,a.

By substituting Eq.(22) into Eq.(14) with the help of Maple and equating the coefficients of like terms of Q(ε), we will derive an algebraic system for obtaining an's, η1, η2 and η3. If necessary arrangements are made, Eq.(20) furnishes the exact traveling wave solutions of Eq.(14).

As stressed out in the works of Stakhov and Ruzin in [28] and Sayed and Alurrfi in [29] that the Eq.(21) could be described in terms of Lucas symmetric hyperbolic sine and cosine functions. From those works, we recognize that the Lucas symmetric hyperbolic sine and cosine functions are defined respectively as follows: sLs(ε)=aεaε, {sLs}(\varepsilon) = {a^\varepsilon} - {a^{- \varepsilon}}, cLs(ε)=aε+aε. {cLs}(\varepsilon) = {a^\varepsilon} + {a^{- \varepsilon}}.

Similarly, Lucas symmetric hyperbolic tangent and cotanget functions are read as tLs(ε)=aεaεaε+aε,ctLs(ε)=aε+aεaεaε \matrix{\hfill {tLs(\varepsilon) = {{{a^\varepsilon} - {a^{- \varepsilon}}} \over {{a^\varepsilon} + {a^{- \varepsilon}}}},} & \hfill {ctLs(\varepsilon) = {{{a^\varepsilon} + {a^{- \varepsilon}}} \over {{a^\varepsilon} - {a^{- \varepsilon}}}}} \cr} Moreover, it is demonstrated that there exist a identity ([28], [29]) between Lucas symmetric hyperbolic sine and cosine functions as the follows: [cLS(ε)]2[sLS(ε)]2=4 {[cLS(\varepsilon)]^2} - {[sLS(\varepsilon)]^2} = 4

If one chooses a = e in Eq. (22) then classical Kudryashov method is obtained [26].

Tanh Method

In this subsection, we give a detailed description of the tanh method [30,31,32]. The preliminary steps of the tanh method are again as exp(−ϕ(ε)) method. In order to integrate Eq.(13) and to deduce U(x,y,t) explicitly, one can pursue the following steps:

Step 1: Use the traveling wave transformation: U=U(ε),ε=η1x+η2y+η3t U = U(\varepsilon),\,\varepsilon = {\eta _1}x + {\eta _2}y + {\eta _3}t where, η1, η2 and η3 are constants to be fixed latter. Then, the NLEE Eq.(13) is reduced to a NLODE for U = U(ε) as follows: G(U,U,U,...)=0. G(U,U',U'',...) = 0.

Step 2: Suppose that the NLODE Eq.(26) has the following solution in the form of finite series expansion: U(ε)=n=0Nan(tanh(ε))n=a0+a1tanh(ε)+...+aN(tanh(ε))N,aN0, U(\varepsilon) = \sum\limits_{n = 0}^N {{a_n}{{(\tanh(\varepsilon))}^n} = {a_0} + {a_1}\,\tanh(\varepsilon) +... + {a_N}{{(\tanh(\varepsilon))}^N},\,\,\,\,\,{a_N} \ne 0,} where, an(n = 0,1,...,N) are constants to be fixed later and N is a positive integer to be determined in step 3.

Step 3: Determine the positive integer N by balancing the highest order derivatives of linear terms and nonlinear terms appearing in Eq.(26).

Step 4: Inserting Eq.(27) into Eq.(26) we get an algebraic equations involving an and ηi (i = 1,2,3). In this stage, we equate the expressions of different power of (tanh(ε))n to zero. Solving those equations sequently by Maple the coefficients an and the parameters η1, η2 and η3 are easily determined.

Step 5: Plugging an, η1, η2 and η3 into Eq.(27), we can yield the traveling wave solutions of Eq.(13).

Applications
Application of the exp(−ϕ(ε)) method to Eq.(4)

In this section, we are looking for the exact solutions of Eq.(4) using the exp(−ϕ(ε)) method. As noted above the Eq.(4) is transformed to the second order NLODE Eq.(12) by the wave variables Eq.(5). Firstly, balancing U″ with U3 in Eq.(12), one gets the finite series order as N = 1. Therefore our solution form is as follows U(ε)=a0+a1exp(ϕ(ε)). U(\varepsilon) = {a_0} + {a_1}exp(- \phi (\varepsilon)).

By substituting Eq.(28) into Eq.(12) and equating the coefficients of same terms of exp(−ϕ(ε)), we yield an algebraic system

exp(−3ϕ(ε)) Coeff.: 2a1η2(a12+η12)=0, 2{a_1}{\eta _2}(a_1^2 + \eta _1^2) = 0,

exp(−2ϕ(ε)) Coeff.: 3a1η2(λη12+2a0a1)=0, 3{a_1}{\eta _2}(\lambda \eta _1^2 + 2{a_0}{a_1}) = 0,

exp(−ϕ(ε)) Coeff.: 2a1(((12λ2+μ)η1212η1γ1+3a02)η2+12η1γ3)=0, 2{a_1}((({1 \over 2}{\lambda ^2} + \mu)\eta _1^2 - {1 \over 2}{\eta _1}{\gamma _1} + 3a_0^2){\eta _2} + {1 \over 2}{\eta _1}{\gamma _3}) = 0,

Const: (λμa1η12+2a03a0η1γ1)η2+a0η1γ3=0. (\lambda \mu {a_1}\eta _1^2 + 2a_0^3 - {a_0}{\eta _1}{\gamma _1}){\eta _2} + {a_0}{\eta _1}{\gamma _3} = 0.

which contain an's, η1, η2 and η3. Solving these equations by help of Maple the unknown aforementioned coefficients are fixed.

Four different coefficient sets are derived after solving the above system which are given as follows: Set1.a0=12iλη1,a1=iη1,η1=η1,η2=2γ3λ2η14μη1+2γ1,η3=η3. {\rm Set}\,1.\,{a_0} = {1 \over 2}i\lambda {\eta _1},{a_1} = i{\eta _1},{\eta _1} = {\eta _1},{\eta _2} = {{2{\gamma _3}} \over {{\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1}}},{\eta _3} = {\eta _3}. Set2.a0=12iλη1,a1=iη1,η1=η1,η2=2γ3λ2η14μη1+2γ1,η3=η3. {\rm Set}\,2.\,{a_0} = {1 \over 2}i\lambda {\eta _1},{a_1} = - i{\eta _1},{\eta _1} = {\eta _1},{\eta _2} = {{2{\gamma _3}} \over {{\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1}}},{\eta _3} = {\eta _3}. Set3.a0=12iλη1,a1=iη1,η1=η1,η2=2γ3λ2η14μη1+2γ1,η3=η3. {\rm Set}\,3.\,{a_0} = - {1 \over 2}i\lambda {\eta _1},{a_1} = i{\eta _1},{\eta _1} = {\eta _1},{\eta _2} = {{2{\gamma _3}} \over {{\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1}}},{\eta _3} = {\eta _3}. Set4.a0=12iλη1,a1=iη1,η1=η1,η2=2γ3λ2η14μη1+2γ1,η3=η3. {\rm Set}\,4.\,{a_0} = - {1 \over 2}i\lambda {\eta _1},{a_1} = - i{\eta _1},{\eta _1} = {\eta _1},{\eta _2} = {{2{\gamma _3}} \over {{\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1}}},{\eta _3} = {\eta _3}.

Therefore, the exact solution forms Eq.(5) of Eq.(4) (together with Eq.(11) and Eq.(28)) corresponding to above sets can be given as follows: u1(x,y,t)=(12iλη1+iη1eϕ(ε))eiψ,v1(x,y,t)=4γ3η1(λ2η14μη1+2γ1)(12iλη1+iη1eϕ(ε))2. \matrix{{{u_1}(x,y,t) = \left({{1 \over 2}i\lambda {\eta _1} + i{\eta _1}{e^{- \phi (\varepsilon)}}} \right){e^{i\psi}},} \hfill \cr {{v_1}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}}{{\left({{1 \over 2}i\lambda {\eta _1} + i{\eta _1}{e^{- \phi (\varepsilon)}}} \right)}^2}.} \hfill \cr} u2(x,y,t)=(12iλη1iη1eϕ(ε))eiψ,v2(x,y,t)=4γ3η1(λ2η14μη1+2γ1)(12iλη1iη1eϕ(ε))2. \matrix{{{u_2}(x,y,t) = \left({{1 \over 2}i\lambda {\eta _1} - i{\eta _1}{e^{- \phi (\varepsilon)}}} \right){e^{i\psi}},} \hfill \cr {{v_2}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}}{{\left({{1 \over 2}i\lambda {\eta _1} - i{\eta _1}{e^{- \phi (\varepsilon)}}} \right)}^2}.} \hfill \cr} u3(x,y,t)=(12iλη1+iη1eϕ(ε))eiψ,v3(x,y,t)=4γ3η1(λ2η14μη1+2γ1)(12iλη1+iη1eϕ(ε))2. \matrix{{{u_3}(x,y,t) = \left({- {1 \over 2}i\lambda {\eta _1} + i{\eta _1}{e^{- \phi (\varepsilon)}}} \right){e^{i\psi}},} \hfill \cr {{v_3}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}}{{\left({- {1 \over 2}i\lambda {\eta _1} + i{\eta _1}{e^{- \phi (\varepsilon)}}} \right)}^2}.} \hfill \cr} u4(x,y,t)=(12iλη1iη1eϕ(ε))eiψ,v4(x,y,t)=4γ3η1(λ2η14μη1+2γ1)(12iλη1iη1eϕ(ε))2. \matrix{{{u_4}(x,y,t) = \left({- {1 \over 2}i\lambda {\eta _1} - i{\eta _1}{e^{- \phi (\varepsilon)}}} \right){e^{i\psi}},} \hfill \cr {{v_4}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}}{{\left({- {1 \over 2}i\lambda {\eta _1} - i{\eta _1}{e^{- \phi (\varepsilon)}}} \right)}^2}.} \hfill \cr}

We now insert the ϕ(ε) which is classified in Eqs.(17)(19) into the Eqs.(29)(32).

Case 1: If λ2 − 4μ > 0 and μ ≠ 0, then the combined soliton solution as follows: u1(x,y,t)=(12iλη1+2μiη1cosh((λ24μ/2)(ε+C))λ24μsinh((λ24μ/2)(ε+C))λcosh((λ24μ/2)(ε+C)))eiψ,v1(x,y,t)=4γ3η1(λ2η14μη1+2γ1)×(12iλη1+2μiη1cosh((λ24μ/2)(ε+C))λ24μsinh((λ24μ/2)(ε+C))λcosh((λ24μ/2)(ε+C)))2 \matrix{{{u_1}(x,y,t) = \left({{1 \over 2}i\lambda {\eta _1} + {{2\mu i{\eta _1}\,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))} \over {- \sqrt {{\lambda ^2} - 4\mu \,} \sinh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C)) - \lambda \,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))}}} \right){e^{i\psi}},} \hfill \cr {{v_1}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}} \times {{\left({{1 \over 2}i\lambda {\eta _1} + {{2\mu i{\eta _1}\,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))} \over {- \sqrt {{\lambda ^2} - 4\mu \,} \sinh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C)) - \lambda \,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))}}} \right)}^2}} \hfill \cr} u2(x,y,t)=(12iλη12μiη1cosh((λ24μ/2)(ε+C))λ24μsinh((λ24μ/2)(ε+C))λcosh((λ24μ/2)(ε+C)))eiψ,v2(x,y,t)=4γ3η1(λ2η14μη1+2γ1)×(12iλη12μiη1cosh((λ24μ/2)(ε+C))λ24μsinh((λ24μ/2)(ε+C))λcosh((λ24μ/2)(ε+C)))2 \matrix{{{u_2}(x,y,t) = \left({{1 \over 2}i\lambda {\eta _1} - {{2\mu i{\eta _1}\,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))} \over {- \sqrt {{\lambda ^2} - 4\mu \,} \sinh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C)) - \lambda \,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))}}} \right){e^{i\psi}},} \hfill \cr {{v_2}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}} \times {{\left({{1 \over 2}i\lambda {\eta _1} - {{2\mu i{\eta _1}\,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))} \over {- \sqrt {{\lambda ^2} - 4\mu \,} \sinh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C)) - \lambda \,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))}}} \right)}^2}} \hfill \cr} u3(x,y,t)=(12iλη1+2μiη1cosh((λ24μ/2)(ε+C))λ24μsinh((λ24μ/2)(ε+C))λcosh((λ24μ/2)(ε+C)))eiψ,v3(x,y,t)=4γ3η1(λ2η14μη1+2γ1)×(12iλη1+2μiη1cosh((λ24μ/2)(ε+C))λ24μsinh((λ24μ/2)(ε+C))λcosh((λ24μ/2)(ε+C)))2. \matrix{{{u_3}(x,y,t) = \left({- {1 \over 2}i\lambda {\eta _1} + {{2\mu i{\eta _1}\,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))} \over {- \sqrt {{\lambda ^2} - 4\mu \,} \sinh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C)) - \lambda \,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))}}} \right){e^{i\psi}},} \hfill \cr {{v_3}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}} \times {{\left({- {1 \over 2}i\lambda {\eta _1} + {{2\mu i{\eta _1}\,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))} \over {- \sqrt {{\lambda ^2} - 4\mu \,} \sinh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C)) - \lambda \,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))}}} \right)}^2}.} \hfill \cr} u4(x,y,t)=(12iλη1+2μiη1cosh((λ24μ/2)(ε+C))λ24μsinh((λ24μ/2)(ε+C))+λcosh((λ24μ/2)(ε+C)))eiψ,v4(x,y,t)=4γ3η1(λ2η14μη1+2γ1)×(12iλη1+2μiη1cosh((λ24μ/2)(ε+C))λ24μsinh((λ24μ/2)(ε+C))+λcosh((λ24μ/2)(ε+C)))2. \matrix{{{u_4}(x,y,t) = \left({- {1 \over 2}i\lambda {\eta _1} + {{2\mu i{\eta _1}\,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))} \over {\sqrt {{\lambda ^2} - 4\mu \,} \sinh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C)) + \lambda \,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))}}} \right){e^{i\psi}},} \hfill \cr {{v_4}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}} \times {{\left({- {1 \over 2}i\lambda {\eta _1} + {{2\mu i{\eta _1}\,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))} \over {\sqrt {{\lambda ^2} - 4\mu \,} \sinh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C)) + \lambda \,\cosh((\sqrt {{\lambda ^2} - 4\mu} /2)(\varepsilon + C))}}} \right)}^2}.} \hfill \cr}

Case 2: If λ2 − 4μ > 0, μ = 0 and λ ≠ 0, then the combined soliton solution: u1(x,y,t)=(12iλη1+iη1λcosh(λ(ε+C))+sinh(λ(ε+C))1)eiψ,v1(x,y,t)=4γ3η1(λ2η14μη1+2γ1)(12iλη1+iη1λcosh(λ(ε+C))+sinh(λ(ε+C))1)2. \matrix{{{u_1}(x,y,t) = \left({{1 \over 2}i\lambda {\eta _1} + {{i{\eta _1}\lambda} \over {\cosh(\lambda (\varepsilon + C)) + \sinh(\lambda (\varepsilon + C)) - 1}}} \right){e^{i\psi}},} \hfill \cr {{v_1}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}}{{\left({{1 \over 2}i\lambda {\eta _1} + {{i{\eta _1}\lambda} \over {\cosh(\lambda (\varepsilon + C)) + \sinh(\lambda (\varepsilon + C)) - 1}}} \right)}^2}.} \hfill \cr} u2(x,y,t)=(12iλη1iη1λcosh(λ(ε+C))+sinh(λ(ε+C))1)eiψ,v2(x,y,t)=4γ3η1(λ2η14μη1+2γ1)(12iλη1iη1λcosh(λ(ε+C))+sinh(λ(ε+C))1)2. \matrix{{{u_2}(x,y,t) = \left({{1 \over 2}i\lambda {\eta _1} - {{i{\eta _1}\lambda} \over {\cosh(\lambda (\varepsilon + C)) + \sinh(\lambda (\varepsilon + C)) - 1}}} \right){e^{i\psi}},} \hfill \cr {{v_2}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}}{{\left({{1 \over 2}i\lambda {\eta _1} - {{i{\eta _1}\lambda} \over {\cosh(\lambda (\varepsilon + C)) + \sinh(\lambda (\varepsilon + C)) - 1}}} \right)}^2}.} \hfill \cr} u3(x,y,t)=(12iλη1+iη1λcosh(λ(ε+C))+sinh(λ(ε+C))1)eiψ,v3(x,y,t)=4γ3η1(λ2η14μη1+2γ1)(12iλη1+iη1λcosh(λ(ε+C))+sinh(λ(ε+C))1)2. \matrix{{{u_3}(x,y,t) = \left({- {1 \over 2}i\lambda {\eta _1} + {{i{\eta _1}\lambda} \over {\cosh(\lambda (\varepsilon + C)) + \sinh(\lambda (\varepsilon + C)) - 1}}} \right){e^{i\psi}},} \hfill \cr {{v_3}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}}{{\left({- {1 \over 2}i\lambda {\eta _1} + {{i{\eta _1}\lambda} \over {\cosh(\lambda (\varepsilon + C)) + \sinh(\lambda (\varepsilon + C)) - 1}}} \right)}^2}.} \hfill \cr} u4(x,y,t)=(12iλη1iη1λcosh(λ(ε+C))+sinh(λ(ε+C))1)eiψ,v4(x,y,t)=4γ3η1(λ2η14μη1+2γ1)(12iλη1iη1λcosh(λ(ε+C))+sinh(λ(ε+C))1)2. \matrix{{{u_4}(x,y,t) = \left({- {1 \over 2}i\lambda {\eta _1} - {{i{\eta _1}\lambda} \over {\cosh(\lambda (\varepsilon + C)) + \sinh(\lambda (\varepsilon + C)) - 1}}} \right){e^{i\psi}},} \hfill \cr {{v_4}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}}{{\left({- {1 \over 2}i\lambda {\eta _1} - {{i{\eta _1}\lambda} \over {\cosh(\lambda (\varepsilon + C)) + \sinh(\lambda (\varepsilon + C)) - 1}}} \right)}^2}.} \hfill \cr}

Case 3: If λ2 − 4μ < 0, μ ≠ 0, then the periodic wave solutions: u1(x,y,t)=(12iλη1+2μiη14μλ2tan((4μλ2/2)(ε+C))λ)eiψ,v1(x,y,t)=4γ3η1(λ2η14μη1+2γ1)(12iλη1+2μiη14μλ2tan((4μλ2/2)(ε+C))λ)2. \matrix{{{u_1}(x,y,t) = \left({{1 \over 2}i\lambda {\eta _1} + {{2\mu i{\eta _1}} \over {\sqrt {4\mu - {\lambda ^2}\,} \tan((\sqrt {4\mu - {\lambda ^2}} /2)(\varepsilon + C)) - \lambda}}} \right){e^{i\psi}},} \hfill \cr {{v_1}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}}{{\left({{1 \over 2}i\lambda {\eta _1} + {{2\mu i{\eta _1}} \over {\sqrt {4\mu - {\lambda ^2}\,} \\tan ((\sqrt {4\mu - {\lambda ^2}} /2)(\varepsilon + C)) - \lambda}}} \right)}^2}.} \hfill \cr} u2(x,y,t)=(12iλη12μiη14μλ2tan((4μλ2/2)(ε+C))λ)eiψ,v2(x,y,t)=4γ3η1(λ2η14μη1+2γ1)(12iλη12μiη14μλ2tan((4μλ2/2)(ε+C))λ)2. \matrix{{{u_2}(x,y,t) = \left({{1 \over 2}i\lambda {\eta _1} - {{2\mu i{\eta _1}} \over {\sqrt {4\mu - {\lambda ^2}\,} \tan((\sqrt {4\mu - {\lambda ^2}} /2)(\varepsilon + C)) - \lambda}}} \right){e^{i\psi}},} \hfill \cr {{v_2}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}}{{\left({{1 \over 2}i\lambda {\eta _1} - {{2\mu i{\eta _1}} \over {\sqrt {4\mu - {\lambda ^2}\,} \\tan ((\sqrt {4\mu - {\lambda ^2}} /2)(\varepsilon + C)) - \lambda}}} \right)}^2}.} \hfill \cr} u3(x,y,t)=(12iλη1+2μiη14μλ2tan((4μλ2/2)(ε+C))λ)eiψv3(x,y,t)=4γ3η1(λ2η14μη1+2γ1)(12iλη1+2μiη14μλ2tan((4μλ2/2)(ε+C))λλ)2. \matrix{{{u_3}(x,y,t) = \left({- {1 \over 2}i\lambda {\eta _1} + {{2\mu i{\eta _1}} \over {\sqrt {4\mu - {\lambda ^2}\,} \tan((\sqrt {4\mu - {\lambda ^2}} /2)(\varepsilon + C)) - \lambda}}} \right){e^{i\psi}}} \hfill \cr {{v_3}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}}{{\left({- {1 \over 2}i\lambda {\eta _1} + {{2\mu i{\eta _1}} \over {\sqrt {4\mu - {\lambda ^2}\,} \\tan ((\sqrt {4\mu - {\lambda ^2}} /2)(\varepsilon + C)) - \lambda \lambda}}} \right)}^2}.} \hfill \cr} u4(x,y,t)=(12iλη12μiη14μλ2tan((4μλ2/2)(ε+C))λ)eiψ,v4(x,y,t)=4γ3η1(λ2η14μη1+2γ1)(12iλη12μiη14μλ2tan((4μλ2/2)(ε+C))λ)2. \matrix{{{u_4}(x,y,t) = \left({- {1 \over 2}i\lambda {\eta _1} - {{2\mu i{\eta _1}} \over {\sqrt {4\mu - {\lambda ^2}\,} \tan((\sqrt {4\mu - {\lambda ^2}} /2)(\varepsilon + C)) - \lambda}}} \right){e^{i\psi}},} \hfill \cr {{v_4}(x,y,t) = {{4{\gamma _3}} \over {{\eta _1}({\lambda ^2}{\eta _1} - 4\mu {\eta _1} + 2{\gamma _1})}}{{\left({- {1 \over 2}i\lambda {\eta _1} - {{2\mu i{\eta _1}} \over {\sqrt {4\mu - {\lambda ^2}\,} \\tan ((\sqrt {4\mu - {\lambda ^2}} /2)(\varepsilon + C)) - \lambda}}} \right)}^2}.} \hfill \cr}

Application of the modified Kudryashov method to Eq.(4)

We note that again the order of series expansion is N = 1. Therefore, the solution form of Eq.(20) can be given as U(ε)=a0+a1Q(ε). U(\varepsilon) = {a_0} + {a_1}Q(\varepsilon).

By substituting Eq.(45) into Eq.(12) with the help of Maple and equating the coefficients of same terms of Q(ε), we reach to an algebraic system involving an's, η1, η2 and η3 as

Q3(ε) Coeff.: 2a1ln(a)2+2a13η12=0 2{a_1}\,\ln \,{(a)^2} + {{2a_1^3} \over {{\eta _{{1^2}}}}} = 0

Q2(ε) Coeff.: 3a1ln(a)2+6a0a12η12=0 - 3{a_1}\,\ln \,{(a)^2} + {{6{a_0}a_1^2} \over {{\eta _{{1^2}}}}} = 0

Q(ε) Coeff.: a1ln(a)2+6a02a1η12a1γ1η1+a1γ3η1η2=0 {a_1}\,\ln \,{(a)^2} + {{6a_0^2{a_1}} \over {{\eta _{{1^2}}}}} - {{{a_1}{\gamma _1}} \over {{\eta _1}}} + {{{a_1}{\gamma _3}} \over {{\eta _1}{\eta _2}}} = 0

Const: 2a03η12a0γ1η1+a0γ3η1η2=0 {{2a_0^3} \over {{\eta _{{1^2}}}}} - {{{a_0}{\gamma _1}} \over {{\eta _1}}} + {{{a_0}{\gamma _3}} \over {{\eta _1}{\eta _2}}} = 0

Four different results are obtained after solving the above system, Set1.a0=12ln(a)η1,a1=ln(a)η1,η1=η1,η2=2γ1ln(a)2η12γ3,η3=γ3, {\rm Set}\,1.\,{a_0} = - {1 \over 2}\ln (a){\eta _1},{a_1} = \ln (a){\eta _1},{\eta _1} = {\eta _1},{\eta _2} = {{2{\gamma _1} - \ln {{(a)}^2}{\eta _1}} \over {2{\gamma _3}}},{\eta _3} = {\gamma _3}, Set2.a0=12ln(a)η1,a1=ln(a)η1,η1=η1,η2=2γ1ln(a)2η12γ3,η3=γ3. {\rm Set}\,2.\,{a_0} = {1 \over 2}\ln (a){\eta _1},{a_1} = - \ln (a){\eta _1},{\eta _1} = {\eta _1},{\eta _2} = {{2{\gamma _1} - \ln {{(a)}^2}{\eta _1}} \over {2{\gamma _3}}},{\eta _3} = {\gamma _3}.

Thus, the exact traveling wave solutions of Eq.(5) can be given as follows: u1(x,y,t)=(12ln(a)η1+ln(a)η1Q(ε))eiψ,v1(x,y,t)=2γ1ln(a)2η1η1γ3(12ln(a)η1+ln(a)η1Q(ε))2, \matrix{{{u_1}(x,y,t) = \left({- {1 \over 2}\ln (a){\eta _1} + \ln (a){\eta _1}Q(\varepsilon)} \right){e^{i\psi}},} \hfill \cr {{v_1}(x,y,t) = {{2{\gamma _1} - \ln {{(a)}^2}{\eta _1}} \over {{\eta _1}{\gamma _3}}}{{\left({- {1 \over 2}\ln (a){\eta _1} + \ln (a){\eta _1}Q(\varepsilon)} \right)}^2},} \hfill \cr} u2(x,y,t)=(12ln(a)η1ln(a)η1Q(ε))eiψ,v2(x,y,t)=2γ1ln(a)2η1η1γ3(12ln(a)η1ln(a)η1Q(ε))2. \matrix{{{u_2}(x,y,t) = \left({{1 \over 2}\ln (a){\eta _1} - \ln (a){\eta _1}Q(\varepsilon)} \right){e^{i\psi}},} \hfill \cr {{v_2}(x,y,t) = {{2{\gamma _1} - \ln {{(a)}^2}{\eta _1}} \over {{\eta _1}{\gamma _3}}}{{\left({{1 \over 2}\ln (a){\eta _1} - \ln (a){\eta _1}Q(\varepsilon)} \right)}^2}.} \hfill \cr}

We now replace the term Q(ε)=11+deε Q(\varepsilon) = {1 \over {1 + {de}^\varepsilon}} in Eqs.(46)(47) u1(x,y,t)=(12ln(a)η1+ln(a)η11+daη1x+(2γ1ln(a)2η12γ3)y+γ3t)eiψ,v1(x,y,t)=2γ1ln(a)2η1η1γ3(12ln(a)η1+ln(a)η11+daη1x+(2γ1ln(a)2η12γ3)y+γ3t)2. \matrix{{{u_1}(x,y,t) = \left({- {1 \over 2}\ln (a){\eta _1} + {{\ln (a){\eta _1}} \over {1 + d{a^{{\eta _1}x + \left({{{2{\gamma _1} - \ln {{(a)}^2}{\eta _1}} \over {2{\gamma _3}}}} \right)y + {\gamma _3}t}}}}} \right){e^{i\psi}},} \hfill \cr {{v_1}(x,y,t) = {{2{\gamma _1} - \ln {{(a)}^2}{\eta _1}} \over {{\eta _1}{\gamma _3}}}{{\left({- {1 \over 2}\ln (a){\eta _1} + {{\ln (a){\eta _1}} \over {1 + d{a^{{\eta _1}x + \left({{{2{\gamma _1} - \ln {{(a)}^2}{\eta _1}} \over {2{\gamma _3}}}} \right)y + {\gamma _3}t}}}}} \right)}^2}.} \hfill \cr} u2(x,y,t)=(12ln(a)η1ln(a)η11+daη1x+(2γ1ln(a)2η12γ3)y+γ3t)eiψ,v2(x,y,t)=2γ1ln(a)2η1η1γ3(12ln(a)η1ln(a)η11+daη1x+(2γ1ln(a)2η12γ3)y+γ3t)2. \matrix{{{u_2}(x,y,t) = \left({{1 \over 2}\ln (a){\eta _1} - {{\ln (a){\eta _1}} \over {1 + d{a^{{\eta _1}x + \left({{{2{\gamma _1} - \ln {{(a)}^2}{\eta _1}} \over {2{\gamma _3}}}} \right)y + {\gamma _3}t}}}}} \right){e^{i\psi}},} \hfill \cr {{v_2}(x,y,t) = {{2{\gamma _1} - \ln {{(a)}^2}{\eta _1}} \over {{\eta _1}{\gamma _3}}}{{\left({{1 \over 2}\ln (a){\eta _1} - {{\ln (a){\eta _1}} \over {1 + d{a^{{\eta _1}x + \left({{{2{\gamma _1} - \ln {{(a)}^2}{\eta _1}} \over {2{\gamma _3}}}} \right)y + {\gamma _3}t}}}}} \right)}^2}.} \hfill \cr} where d and a are arbitrary constants. Eq. (48) can be described in terms of Lucas symmetric hyperbolic sine and cosine functions as u1(x,y,t)=(12ln(a)η1+2ln(a)η12+d(sLS(ε)+cLS(ε)))eiψ,v1(x,y,t)=2γ1ln(a)2η1η1γ3(12ln(a)η1+2ln(a)η1d(sLS(ε)+cLS(ε)))2. \matrix{{{u_1}(x,y,t) = \left({- {1 \over 2}\ln (a){\eta _1} + {{2\ln (a){\eta _1}} \over {2 + d({\rm{sLS}}(\varepsilon) + cLS(\varepsilon))}}} \right){e^{i\psi}},} \hfill \cr {{v_1}(x,y,t) = {{2{\gamma _1} - \ln {{(a)}^2}{\eta _1}} \over {{\eta _1}{\gamma _3}}}{{\left({- {1 \over 2}\ln (a){\eta _1} + {{2\ln (a){\eta _1}} \over {d({\rm{sLS}}(\varepsilon) + cLS(\varepsilon))}}} \right)}^2}.} \hfill \cr} where ε=η1x+(2γ1ln(a)2η12γ3)y+γ3t \varepsilon = {\,^{{\eta _1}x + \left({{{2{\gamma _1} - \ln {{(a)}^2}{\eta _1}} \over {2{\gamma _3}}}} \right)y + {\gamma _3}t}} .

Application of the tanh method to Eq.(4)

In this subsection, we are looking for the exact solutions of Eq.(4) by using the tanh method. Based on the order of finite series expansion N = 1, our solution form is as follows U(ε)=a0+a1tanh(ε). U(\varepsilon) = {a_0} + {a_1}\tanh(\varepsilon).

By substituting Eq.(50) into Eq.(12) with the help of Maple and equating the coefficients of same terms of tanh(ε), we yield an algebraic system containing an's, η1, η2 and η3. The coefficients of some various powers of tanh(ε) are listed as follows:

tanh3(ε) Coeff.: 2a12+2a13η12=0 2a_1^2 + {{2a_1^3} \over {{\eta _{{1^2}}}}} = 0

tanh2(ε) Coeff.: 6a0a12η12=0 {{6{a_0}a_1^2} \over {{\eta _{{1^2}}}}} = 0

tanh(ε) Coeff.: 2a12+6a02a1η12a1γ1η1+a1γ3η1η2=0 - 2a_1^2 + {{6a_0^2{a_1}} \over {{\eta _{{1^2}}}}} - {{{a_1}{\gamma _1}} \over {{\eta _1}}} + {{{a_1}{\gamma _3}} \over {{\eta _1}{\eta _2}}} = 0

Const: 2a03η12a0γ1η1+a0γ3η1η2=0 {{2a_0^3} \over {{\eta _{{1^2}}}}} - {{{a_0}{\gamma _1}} \over {{\eta _1}}} + {{{a_0}{\gamma _3}} \over {{\eta _1}{\eta _2}}} = 0

Having solved above algebraic system by the help of Maple, we attain the following coefficient sets: Set1.a0=0,a1=1,η1=i,η2=γ3+2γ1γ1,η3=η3. {\rm Set}\,1.\,{a_0} = 0,{a_1} = 1,{\eta _1} = i,{\eta _2} = - {{{\gamma _3} + 2{\gamma _1}} \over {{\gamma _1}}},{\eta _3} = {\eta _3}. Set2.a0=0,a1=1,η1=i,η2=γ32γ1γ1,η3=η3. {\rm Set}\,2.\,{a_0} = 0,{a_1} = 1,{\eta _1} = i,{\eta _2} = {{{\gamma _3} - 2{\gamma _1}} \over {{\gamma _1}}},{\eta _3} = {\eta _3}. Set3.a0=0,a1=1,η1=i,η2=γ3+2γ1γ1,η3=η3. {\rm Set}\,3.\,{a_0} = 0,{a_1} = 1,{\eta _1} = - i,{\eta _2} = - {{{\gamma _3} + 2{\gamma _1}} \over {{\gamma _1}}},{\eta _3} = {\eta _3}. Set4.a0=0,a1=1,η1=i,η2=γ32γ1γ1,η3=η3. {\rm Set}\,4.\,{a_0} = 0,{a_1} = 1,{\eta _1} = - i,{\eta _2} = {{{\gamma _3} - 2{\gamma _1}} \over {{\gamma _1}}},{\eta _3} = {\eta _3}.

Therefore, exact traveling wave solutions (dark soliton) of Eq.(4) can be given as follows: u1(x,y,t)=tanh(ix+(i2γ32γ1γ1)y+η3t)eiψ,v1(x,y,t)=2i2γ34γ1γ1itanh2(ix+(i2γ32γ1γ1)y+η3t). \matrix{{{u_1}(x,y,t) = \tanh\left({ix + \left({{{{i^2}{\gamma _3} - 2{\gamma _1}} \over {{\gamma _1}}}} \right)y + {\eta _3}t} \right){e^{i\psi}},} \hfill \cr {{v_1}(x,y,t) = {{2{i^2}{\gamma _3} - 4{\gamma _1}} \over {{\gamma _1}i}}{{\tanh}^2}\left({ix + \left({{{{i^2}{\gamma _3} - 2{\gamma _1}} \over {{\gamma _1}}}} \right)y + {\eta _3}t} \right).} \hfill \cr} u2(x,y,t)=tanh(ix+(i2γ3+2γ1γ1)y+η3t)eiψ,v2(x,y,t)=2i2γ3+4γ1γ1itanh2(ix+(i2γ3+2γ1γ1)y+η3t). \matrix{{{u_2}(x,y,t) = \tanh\left({ix + \left({- {{{i^2}{\gamma _3} + 2{\gamma _1}} \over {{\gamma _1}}}} \right)y + {\eta _3}t} \right){e^{i\psi}},} \hfill \cr {{v_2}(x,y,t) = - {{2{i^2}{\gamma _3} + 4{\gamma _1}} \over {{\gamma _1}i}}{{\tanh}^2}\left({ix + \left({- {{{i^2}{\gamma _3} + 2{\gamma _1}} \over {{\gamma _1}}}} \right)y + {\eta _3}t} \right).} \hfill \cr} u3(x,y,t)=tanh(ix+(i2γ32γ1γ1)y+η3t)eiψ,v3(x,y,t)=2i2γ3+4γ1γ1itanh2(ix+(i2γ32γ1γ1)y+η3t). \matrix{{{u_3}(x,y,t) = \tanh\left({- ix + \left({{{{i^2}{\gamma _3} - 2{\gamma _1}} \over {{\gamma _1}}}} \right)y + {\eta _3}t} \right){e^{i\psi}},} \hfill \cr {{v_3}(x,y,t) = {{- 2{i^2}{\gamma _3} + 4{\gamma _1}} \over {{\gamma _1}i}}{{\tanh}^2}\left({- ix + \left({{{{i^2}{\gamma _3} - 2{\gamma _1}} \over {{\gamma _1}}}} \right)y + {\eta _3}t} \right).} \hfill \cr} u4(x,y,t)=tanh(ix+(i2γ3+2γ1γ1)y+η3t)eiψ,v4(x,y,t)=2i2γ3+4γ1γ1itanh2(ix+(i2γ3+2γ1γ1)y+η3t). \matrix{{{u_4}(x,y,t) = \tanh\left({- ix + \left({- {{{i^2}{\gamma _3} + 2{\gamma _1}} \over {{\gamma _1}}}} \right)y + {\eta _3}t} \right){e^{i\psi}},} \hfill \cr {{v_4}(x,y,t) = {{2{i^2}{\gamma _3} + 4{\gamma _1}} \over {{\gamma _1}i}}{{\tanh}^2}\left({- ix + \left({- {{{i^2}{\gamma _3} + 2{\gamma _1}} \over {{\gamma _1}}}} \right)y + {\eta _3}t} \right).} \hfill \cr}

The solutions in (33) (see Figures 1 and 2), (48) (see Figures 3 and 4) and (51) (see Figures 5 and 6) of Eq.(4) are shown in Figs. 1–6 for y = 0.

Fig. 1

3D-plot for |v(x,y,t)|2 when y = 0 where μ = −1,λ = 1,γ1 = 1,γ3 = 1,η1 = 1,η3 = 1, C = 0. (combined soliton solution)

Fig. 2

3D-plot for |v(x,y,t)|2 when y = 0 where μ = −1,λ = 1,γ1 = 1,γ3 = 1,η1 = 1,η3 = 1, C = 0. (combined soliton solution)

Fig. 3

3D-plot for |v(x,y,t)|2 when y = 0 where a = 1.5 and all other parameter values are 1 (Lucas combined hyperbolic function)

Fig. 4

3D-plot for |v(x,y,t)|2 when y = 0 where a = 1.5 and all other parameter values are 1 (Lucas combined hyperbolic function)

Fig. 5

3D-plot for |v(x,y,t)|2 when y = 0 where all parameter values are 1 (dark soliton)

Conclusions and discussions

In this study, we have obtained various different structures of traveling wave solutions to a special (2+1) dimensional Schrödinger equation. These solutions are in the form of dark soliton, combined soliton, symmetrical Lucas sine, Lucas cosine functions, and periodic waves. Some of the solutions are the same (dark solitons and periodic waves) with those obtained by [22] while the rest of solutions seems to be new. With the comparison of the performed methods, we conclude that exp(−ϕ(ε)) give us many different solution structures yet modified Kudryashov and tanh ansatz approaches lead to merely one type solution prototype (see Eq.(21) and Eq.(27)). Thus, exp(−ϕ(ε)) method has an apparent advantage over the other two methods.

The obtained exact (traveling wave) solutions might be used as an initial value in the initial/boundary value problems. In addition, these solutions can be utilized in numerical schemas as a benchmark and stability theories. Though omitted in the present research, there are several topics which will be addressed in our future works:

The group invariant solutions can be sought which are different than the wave solutions by the theory of Lie groups.

The different types of soliton interactions can be investigated depend upon Hirota's bilinearization method.

The conservation laws of the model can be studied from the point of view of variational or Bell polynomials.

The Painleve analysis and interaction solutions can also be constructed.

Fig. 1

3D-plot for |v(x,y,t)|2 when y = 0 where μ = −1,λ = 1,γ1 = 1,γ3 = 1,η1 = 1,η3 = 1, C = 0. (combined soliton solution)
3D-plot for |v(x,y,t)|2 when y = 0 where μ = −1,λ = 1,γ1 = 1,γ3 = 1,η1 = 1,η3 = 1, C = 0. (combined soliton solution)

Fig. 2

3D-plot for |v(x,y,t)|2 when y = 0 where μ = −1,λ = 1,γ1 = 1,γ3 = 1,η1 = 1,η3 = 1, C = 0. (combined soliton solution)
3D-plot for |v(x,y,t)|2 when y = 0 where μ = −1,λ = 1,γ1 = 1,γ3 = 1,η1 = 1,η3 = 1, C = 0. (combined soliton solution)

Fig. 3

3D-plot for |v(x,y,t)|2 when y = 0 where a = 1.5 and all other parameter values are 1 (Lucas combined hyperbolic function)
3D-plot for |v(x,y,t)|2 when y = 0 where a = 1.5 and all other parameter values are 1 (Lucas combined hyperbolic function)

Fig. 4

3D-plot for |v(x,y,t)|2 when y = 0 where a = 1.5 and all other parameter values are 1 (Lucas combined hyperbolic function)
3D-plot for |v(x,y,t)|2 when y = 0 where a = 1.5 and all other parameter values are 1 (Lucas combined hyperbolic function)

Fig. 5

3D-plot for |v(x,y,t)|2 when y = 0 where all parameter values are 1 (dark soliton)
3D-plot for |v(x,y,t)|2 when y = 0 where all parameter values are 1 (dark soliton)

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