Optical soliton solutions to a (2+1) dimensional Schrödinger equation using a couple of integration architectures

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), (1) ${iq}_{t} + 2 {| q |}^{2} q + q_{xx} = 0$ {{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] (2) $\begin{array}{r} {kq}_{t} = \frac{1}{2} q_{xy} - q \int \partial_{y} [p . q] d_{x}, \\ - {kp}_{t} = \frac{1}{2} p_{xy} - p \int \partial_{y} [p . q] d_{x} . \end{array}$ \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 (3) $v_{x} (p, q) = 2 \partial_{y} [p . q],$ {v_x}(p,q) = 2{\partial _y}[p.q], with the ansatz p = −q^* = u and k = −i/2, Eq.(2) reduces to (4) $\begin{array}{c} {iu}_{t} = u_{xy} + uv, \\ v_{x} = 2 {(| u |^{2})}_{y} . \end{array}$ \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 (5) $u (x, y, t) = U (ε) e^{i ψ}, v (x, y, t) = V (ε)$ u(x,y,t) = U(\varepsilon){e^{i\psi}},v(x,y,t) = V(\varepsilon) is choosed in Eq.(4) where (6) $ψ = γ_{1} x + γ_{2} y + γ_{3} t and ε = η_{1} x + η_{2} y + η_{3} t .$ \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) γ_i and η_i (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 (7) $η_{1} η_{2} U^{″} (ε) + U (ε) V (ε) + (γ_{3} - γ_{1} η_{2}) U (ε) = 0,$ {\eta _1}{\eta _2}U''(\varepsilon) + U(\varepsilon)V(\varepsilon) + ({\gamma _3} - {\gamma _1}{\eta _2})U(\varepsilon) = 0, (8) $(η_{1} η_{2} + γ_{1} η_{2} + η_{3}) U^{'} (ε) = 0,$ ({\eta _1}{\eta _2} + {\gamma _1}{\eta _2} + {\eta _3})U'(\varepsilon) = 0, and (9) $η_{1} V^{'} (ε) = 2 η_{2} {(U^{2} (ε))}^{'} .$ {\eta _1}V'(\varepsilon) = 2{\eta _2}({U^2}(\varepsilon))'.

It is readily seen that the Eq.(8) should holds the following constraint: (10) $(η_{1} η_{2} + γ_{1} η_{2} + η_{3}) = 0 .$ ({\eta _1}{\eta _2} + {\gamma _1}{\eta _2} + {\eta _3}) = 0.

After the integration of Eq.(9) with respect to ɛ, we yield (11) $V (ε) = \frac{2 η_{2}}{η_{1}} U^{2} (ε) .$ 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]: (12) $U^{″} (ε) + \frac{2}{η_{1^{2}}} U^{3} (ε) + (\frac{γ_{3} - γ_{1} η_{2}}{η_{1} η_{2}}) U (ε) = 0 .$ 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

3.1

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) (13) $F (U, U_{x}, U_{y}, U_{t}, U_{xy}, U_{xt}, U_{yx}, U_{yt}, ...) = 0$ 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 (14) $G (U, U^{'}, U^{″}, ...) = 0 .$ G(U,U',U'',...) = 0.

We are looking for the following finite series expansion solution (15) $U (ε) = \sum_{n = 0}^{N} a_{n} \exp (- N ϕ (ε)) = a_{0} + a_{1} \exp (- ϕ (ε)) + ... + a_{N} \exp (- N ϕ (ε)), a_{N} \neq 0$ 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 a_n 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 (16) $ϕ^{'} (ε) = \exp (- ϕ (ε)) + μ \exp (ϕ (ε)) + λ .$ \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 (17) $ϕ_{1} (ε) = \ln (\frac{- \sqrt{λ^{2} - 4 μ} tanh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C) - λ}{2 μ}) .$ {\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 (18) $ϕ_{2} (ε) = - \ln (\frac{λ}{cosh (λ (ε + C)) + sinh (λ (ε + C)) - 1}) .$ {\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 (19) $ϕ_{3} (ε) = \ln (\frac{\sqrt{4 μ - λ^{2}} tan ((\sqrt{4 μ - λ^{2}} / 2 (ε + C)) - λ}{2 μ}) .$ {\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 a_n'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).

3.2

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 (20) $\begin{array}{r} U (ε) = a_{0} + a_{1} Q (ε) + ... + a_{N} Q^{N} (ε), & a_{N} \neq 0, \end{array}$ \matrix{\hfill {U(\varepsilon) = {a_0} + {a_1}Q(\varepsilon) +... + {a_N}{Q^N}(\varepsilon),} & \hfill {{a_N} \ne 0,} \cr} where the constants a_n, n = 0,1,2,..., N shall be determined later, N is a positive number which is assessed by the technique of homogeneous balance, and (21) $Q (ε) = 1 / (1 + {da}^{ε})$ Q(\varepsilon) = 1/(1 + {{da}^\varepsilon}) is an explicit function that satisfies the following NLODE (22) $Q^{'} (ε) = Q (ε) (Q (ε) - 1) ln a .$ 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 a_n'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: (23) $sLs (ε) = a^{ε} - a^{- ε},$ {sLs}(\varepsilon) = {a^\varepsilon} - {a^{- \varepsilon}}, (24) $cLs (ε) = a^{ε} + a^{- ε} .$ {cLs}(\varepsilon) = {a^\varepsilon} + {a^{- \varepsilon}}.

Similarly, Lucas symmetric hyperbolic tangent and cotanget functions are read as $\begin{array}{r} tLs (ε) = \frac{a^{ε} - a^{- ε}}{a^{ε} + a^{- ε}}, & ctLs (ε) = \frac{a^{ε} + a^{- ε}}{a^{ε} - a^{- ε}} \end{array}$ \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].

3.3

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: (25) $U = U (ε), ε = η_{1} x + η_{2} y + η_{3} t$ 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: (26) $G (U, U^{'}, U^{″}, ...) = 0 .$ G(U,U',U'',...) = 0.

Step 2: Suppose that the NLODE Eq.(26) has the following solution in the form of finite series expansion: (27) $U (ε) = \sum_{n = 0}^{N} a_{n} {(tanh (ε))}^{n} = a_{0} + a_{1} tanh (ε) + ... + a_{N} {(tanh (ε))}^{N}, a_{N} \neq 0,$ 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, a_n(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 a_n and η_i (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 a_n and the parameters η₁, η₂ and η₃ are easily determined.

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

Applications

4.1

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 (28) $U (ε) = a_{0} + a_{1} \exp (- ϕ (ε)) .$ 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.: $2 a_{1} η_{2} (a_{1}^{2} + η_{1}^{2}) = 0,$ 2{a_1}{\eta _2}(a_1^2 + \eta _1^2) = 0,

exp(−2ϕ(ε)) Coeff.: $3 a_{1} η_{2} (λ η_{1}^{2} + 2 a_{0} a_{1}) = 0,$ 3{a_1}{\eta _2}(\lambda \eta _1^2 + 2{a_0}{a_1}) = 0,

exp(−ϕ(ε)) Coeff.: $2 a_{1} (((\frac{1}{2} λ^{2} + μ) η_{1}^{2} - \frac{1}{2} η_{1} γ_{1} + 3 a_{0}^{2}) η_{2} + \frac{1}{2} η_{1} γ_{3}) = 0,$ 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: $(λ μ a_{1} η_{1}^{2} + 2 a_{0}^{3} - a_{0} η_{1} γ_{1}) η_{2} + a_{0} η_{1} γ_{3} = 0 .$ (\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 a_n'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: $Set 1 . a_{0} = \frac{1}{2} i λ η_{1}, a_{1} = i η_{1}, η_{1} = η_{1}, η_{2} = \frac{2 γ_{3}}{λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1}}, η_{3} = η_{3} .$ {\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}. $Set 2 . a_{0} = \frac{1}{2} i λ η_{1}, a_{1} = - i η_{1}, η_{1} = η_{1}, η_{2} = \frac{2 γ_{3}}{λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1}}, η_{3} = η_{3} .$ {\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}. $Set 3 . a_{0} = - \frac{1}{2} i λ η_{1}, a_{1} = i η_{1}, η_{1} = η_{1}, η_{2} = \frac{2 γ_{3}}{λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1}}, η_{3} = η_{3} .$ {\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}. $Set 4 . a_{0} = - \frac{1}{2} i λ η_{1}, a_{1} = - i η_{1}, η_{1} = η_{1}, η_{2} = \frac{2 γ_{3}}{λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1}}, η_{3} = η_{3} .$ {\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: (29) $\begin{array}{l} u_{1} (x, y, t) = (\frac{1}{2} i λ η_{1} + i η_{1} e^{- ϕ (ε)}) e^{i ψ}, \\ v_{1} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} {(\frac{1}{2} i λ η_{1} + i η_{1} e^{- ϕ (ε)})}^{2} . \end{array}$ \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} (30) $\begin{array}{l} u_{2} (x, y, t) = (\frac{1}{2} i λ η_{1} - i η_{1} e^{- ϕ (ε)}) e^{i ψ}, \\ v_{2} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} {(\frac{1}{2} i λ η_{1} - i η_{1} e^{- ϕ (ε)})}^{2} . \end{array}$ \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} (31) $\begin{array}{l} u_{3} (x, y, t) = (- \frac{1}{2} i λ η_{1} + i η_{1} e^{- ϕ (ε)}) e^{i ψ}, \\ v_{3} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} {(- \frac{1}{2} i λ η_{1} + i η_{1} e^{- ϕ (ε)})}^{2} . \end{array}$ \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} (32) $\begin{array}{l} u_{4} (x, y, t) = (- \frac{1}{2} i λ η_{1} - i η_{1} e^{- ϕ (ε)}) e^{i ψ}, \\ v_{4} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} {(- \frac{1}{2} i λ η_{1} - i η_{1} e^{- ϕ (ε)})}^{2} . \end{array}$ \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: (33) $\begin{array}{l} u_{1} (x, y, t) = (\frac{1}{2} i λ η_{1} + \frac{2 μ i η_{1} cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))}{- \sqrt{λ^{2} - 4 μ} sinh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C)) - λ cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))}) e^{i ψ}, \\ v_{1} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} \times {(\frac{1}{2} i λ η_{1} + \frac{2 μ i η_{1} cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))}{- \sqrt{λ^{2} - 4 μ} sinh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C)) - λ cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))})}^{2} \end{array}$ \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} (34) $\begin{array}{l} u_{2} (x, y, t) = (\frac{1}{2} i λ η_{1} - \frac{2 μ i η_{1} cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))}{- \sqrt{λ^{2} - 4 μ} sinh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C)) - λ cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))}) e^{i ψ}, \\ v_{2} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} \times {(\frac{1}{2} i λ η_{1} - \frac{2 μ i η_{1} cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))}{- \sqrt{λ^{2} - 4 μ} sinh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C)) - λ cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))})}^{2} \end{array}$ \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} (35) $\begin{array}{l} u_{3} (x, y, t) = (- \frac{1}{2} i λ η_{1} + \frac{2 μ i η_{1} cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))}{- \sqrt{λ^{2} - 4 μ} sinh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C)) - λ cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))}) e^{i ψ}, \\ v_{3} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} \times {(- \frac{1}{2} i λ η_{1} + \frac{2 μ i η_{1} cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))}{- \sqrt{λ^{2} - 4 μ} sinh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C)) - λ cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))})}^{2} . \end{array}$ \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} (36) $\begin{array}{l} u_{4} (x, y, t) = (- \frac{1}{2} i λ η_{1} + \frac{2 μ i η_{1} cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))}{\sqrt{λ^{2} - 4 μ} sinh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C)) + λ cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))}) e^{i ψ}, \\ v_{4} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} \times {(- \frac{1}{2} i λ η_{1} + \frac{2 μ i η_{1} cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))}{\sqrt{λ^{2} - 4 μ} sinh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C)) + λ cosh ((\sqrt{λ^{2} - 4 μ} / 2) (ε + C))})}^{2} . \end{array}$ \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: (37) $\begin{array}{l} u_{1} (x, y, t) = (\frac{1}{2} i λ η_{1} + \frac{i η_{1} λ}{cosh (λ (ε + C)) + sinh (λ (ε + C)) - 1}) e^{i ψ}, \\ v_{1} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} {(\frac{1}{2} i λ η_{1} + \frac{i η_{1} λ}{cosh (λ (ε + C)) + sinh (λ (ε + C)) - 1})}^{2} . \end{array}$ \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} (38) $\begin{array}{l} u_{2} (x, y, t) = (\frac{1}{2} i λ η_{1} - \frac{i η_{1} λ}{cosh (λ (ε + C)) + sinh (λ (ε + C)) - 1}) e^{i ψ}, \\ v_{2} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} {(\frac{1}{2} i λ η_{1} - \frac{i η_{1} λ}{cosh (λ (ε + C)) + sinh (λ (ε + C)) - 1})}^{2} . \end{array}$ \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} (39) $\begin{array}{l} u_{3} (x, y, t) = (- \frac{1}{2} i λ η_{1} + \frac{i η_{1} λ}{cosh (λ (ε + C)) + sinh (λ (ε + C)) - 1}) e^{i ψ}, \\ v_{3} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} {(- \frac{1}{2} i λ η_{1} + \frac{i η_{1} λ}{cosh (λ (ε + C)) + sinh (λ (ε + C)) - 1})}^{2} . \end{array}$ \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} (40) $\begin{array}{l} u_{4} (x, y, t) = (- \frac{1}{2} i λ η_{1} - \frac{i η_{1} λ}{cosh (λ (ε + C)) + sinh (λ (ε + C)) - 1}) e^{i ψ}, \\ v_{4} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} {(- \frac{1}{2} i λ η_{1} - \frac{i η_{1} λ}{cosh (λ (ε + C)) + sinh (λ (ε + C)) - 1})}^{2} . \end{array}$ \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: (41) $\begin{array}{l} u_{1} (x, y, t) = (\frac{1}{2} i λ η_{1} + \frac{2 μ i η_{1}}{\sqrt{4 μ - λ^{2}} tan ((\sqrt{4 μ - λ^{2}} / 2) (ε + C)) - λ}) e^{i ψ}, \\ v_{1} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} {(\frac{1}{2} i λ η_{1} + \frac{2 μ i η_{1}}{\sqrt{4 μ - λ^{2}} tan ((\sqrt{4 μ - λ^{2}} / 2) (ε + C)) - λ})}^{2} . \end{array}$ \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} (42) $\begin{array}{l} u_{2} (x, y, t) = (\frac{1}{2} i λ η_{1} - \frac{2 μ i η_{1}}{\sqrt{4 μ - λ^{2}} tan ((\sqrt{4 μ - λ^{2}} / 2) (ε + C)) - λ}) e^{i ψ}, \\ v_{2} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} {(\frac{1}{2} i λ η_{1} - \frac{2 μ i η_{1}}{\sqrt{4 μ - λ^{2}} tan ((\sqrt{4 μ - λ^{2}} / 2) (ε + C)) - λ})}^{2} . \end{array}$ \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} (43) $\begin{array}{l} u_{3} (x, y, t) = (- \frac{1}{2} i λ η_{1} + \frac{2 μ i η_{1}}{\sqrt{4 μ - λ^{2}} tan ((\sqrt{4 μ - λ^{2}} / 2) (ε + C)) - λ}) e^{i ψ} \\ v_{3} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} {(- \frac{1}{2} i λ η_{1} + \frac{2 μ i η_{1}}{\sqrt{4 μ - λ^{2}} tan ((\sqrt{4 μ - λ^{2}} / 2) (ε + C)) - λ λ})}^{2} . \end{array}$ \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} (44) $\begin{array}{l} u_{4} (x, y, t) = (- \frac{1}{2} i λ η_{1} - \frac{2 μ i η_{1}}{\sqrt{4 μ - λ^{2}} tan ((\sqrt{4 μ - λ^{2}} / 2) (ε + C)) - λ}) e^{i ψ}, \\ v_{4} (x, y, t) = \frac{4 γ_{3}}{η_{1} (λ^{2} η_{1} - 4 μ η_{1} + 2 γ_{1})} {(- \frac{1}{2} i λ η_{1} - \frac{2 μ i η_{1}}{\sqrt{4 μ - λ^{2}} tan ((\sqrt{4 μ - λ^{2}} / 2) (ε + C)) - λ})}^{2} . \end{array}$ \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}

4.2

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 (45) $U (ε) = a_{0} + a_{1} Q (ε) .$ 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 a_n's, η₁, η₂ and η₃ as

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

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

Q(ε) Coeff.: $a_{1} ln {(a)}^{2} + \frac{6 a_{0}^{2} a_{1}}{η_{1^{2}}} - \frac{a_{1} γ_{1}}{η_{1}} + \frac{a_{1} γ_{3}}{η_{1} η_{2}} = 0$ {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: $\frac{2 a_{0}^{3}}{η_{1^{2}}} - \frac{a_{0} γ_{1}}{η_{1}} + \frac{a_{0} γ_{3}}{η_{1} η_{2}} = 0$ {{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, $Set 1 . a_{0} = - \frac{1}{2} ln (a) η_{1}, a_{1} = ln (a) η_{1}, η_{1} = η_{1}, η_{2} = \frac{2 γ_{1} - ln {(a)}^{2} η_{1}}{2 γ_{3}}, η_{3} = γ_{3},$ {\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}, $Set 2 . a_{0} = \frac{1}{2} ln (a) η_{1}, a_{1} = - ln (a) η_{1}, η_{1} = η_{1}, η_{2} = \frac{2 γ_{1} - ln {(a)}^{2} η_{1}}{2 γ_{3}}, η_{3} = γ_{3} .$ {\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: (46) $\begin{array}{l} u_{1} (x, y, t) = (- \frac{1}{2} ln (a) η_{1} + ln (a) η_{1} Q (ε)) e^{i ψ}, \\ v_{1} (x, y, t) = \frac{2 γ_{1} - ln {(a)}^{2} η_{1}}{η_{1} γ_{3}} {(- \frac{1}{2} ln (a) η_{1} + ln (a) η_{1} Q (ε))}^{2}, \end{array}$ \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} (47) $\begin{array}{l} u_{2} (x, y, t) = (\frac{1}{2} ln (a) η_{1} - ln (a) η_{1} Q (ε)) e^{i ψ}, \\ v_{2} (x, y, t) = \frac{2 γ_{1} - ln {(a)}^{2} η_{1}}{η_{1} γ_{3}} {(\frac{1}{2} ln (a) η_{1} - ln (a) η_{1} Q (ε))}^{2} . \end{array}$ \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 $Q (ε) = \frac{1}{1 + {de}^{ε}}$ Q(\varepsilon) = {1 \over {1 + {de}^\varepsilon}} in Eqs.(46)–(47) (48) $\begin{array}{l} u_{1} (x, y, t) = (- \frac{1}{2} ln (a) η_{1} + \frac{ln (a) η_{1}}{1 + d a^{η_{1} x + (\frac{2 γ_{1} - ln {(a)}^{2} η_{1}}{2 γ_{3}}) y + γ_{3} t}}) e^{i ψ}, \\ v_{1} (x, y, t) = \frac{2 γ_{1} - ln {(a)}^{2} η_{1}}{η_{1} γ_{3}} {(- \frac{1}{2} ln (a) η_{1} + \frac{ln (a) η_{1}}{1 + d a^{η_{1} x + (\frac{2 γ_{1} - ln {(a)}^{2} η_{1}}{2 γ_{3}}) y + γ_{3} t}})}^{2} . \end{array}$ \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} (49) $\begin{array}{l} u_{2} (x, y, t) = (\frac{1}{2} ln (a) η_{1} - \frac{ln (a) η_{1}}{1 + d a^{η_{1} x + (\frac{2 γ_{1} - ln {(a)}^{2} η_{1}}{2 γ_{3}}) y + γ_{3} t}}) e^{i ψ}, \\ v_{2} (x, y, t) = \frac{2 γ_{1} - ln {(a)}^{2} η_{1}}{η_{1} γ_{3}} {(\frac{1}{2} ln (a) η_{1} - \frac{ln (a) η_{1}}{1 + d a^{η_{1} x + (\frac{2 γ_{1} - ln {(a)}^{2} η_{1}}{2 γ_{3}}) y + γ_{3} t}})}^{2} . \end{array}$ \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 $\begin{array}{l} u_{1} (x, y, t) = (- \frac{1}{2} ln (a) η_{1} + \frac{2 ln (a) η_{1}}{2 + d (sLS (ε) + c L S (ε))}) e^{i ψ}, \\ v_{1} (x, y, t) = \frac{2 γ_{1} - ln {(a)}^{2} η_{1}}{η_{1} γ_{3}} {(- \frac{1}{2} ln (a) η_{1} + \frac{2 ln (a) η_{1}}{d (sLS (ε) + c L S (ε))})}^{2} . \end{array}$ \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 $ε =^{η_{1} x + (\frac{2 γ_{1} - ln {(a)}^{2} η_{1}}{2 γ_{3}}) y + γ_{3} t}$ \varepsilon = {\,^{{\eta _1}x + \left({{{2{\gamma _1} - \ln {{(a)}^2}{\eta _1}} \over {2{\gamma _3}}}} \right)y + {\gamma _3}t}} .

4.3

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 (50) $U (ε) = a_{0} + a_{1} tanh (ε) .$ 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 a_n's, η₁, η₂ and η₃. The coefficients of some various powers of tanh(ε) are listed as follows:

tanh³(ε) Coeff.: $2 a_{1}^{2} + \frac{2 a_{1}^{3}}{η_{1^{2}}} = 0$ 2a_1^2 + {{2a_1^3} \over {{\eta _{{1^2}}}}} = 0

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

tanh(ε) Coeff.: $- 2 a_{1}^{2} + \frac{6 a_{0}^{2} a_{1}}{η_{1^{2}}} - \frac{a_{1} γ_{1}}{η_{1}} + \frac{a_{1} γ_{3}}{η_{1} η_{2}} = 0$ - 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

Having solved above algebraic system by the help of Maple, we attain the following coefficient sets: $Set 1 . a_{0} = 0, a_{1} = 1, η_{1} = i, η_{2} = - \frac{γ_{3} + 2 γ_{1}}{γ_{1}}, η_{3} = η_{3} .$ {\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}. $Set 2 . a_{0} = 0, a_{1} = 1, η_{1} = i, η_{2} = \frac{γ_{3} - 2 γ_{1}}{γ_{1}}, η_{3} = η_{3} .$ {\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}. $Set 3 . a_{0} = 0, a_{1} = 1, η_{1} = - i, η_{2} = - \frac{γ_{3} + 2 γ_{1}}{γ_{1}}, η_{3} = η_{3} .$ {\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}. $Set 4 . a_{0} = 0, a_{1} = 1, η_{1} = - i, η_{2} = \frac{γ_{3} - 2 γ_{1}}{γ_{1}}, η_{3} = η_{3} .$ {\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: (51) $\begin{array}{l} u_{1} (x, y, t) = tanh (i x + (\frac{i^{2} γ_{3} - 2 γ_{1}}{γ_{1}}) y + η_{3} t) e^{i ψ}, \\ v_{1} (x, y, t) = \frac{2 i^{2} γ_{3} - 4 γ_{1}}{γ_{1} i} {tanh}^{2} (i x + (\frac{i^{2} γ_{3} - 2 γ_{1}}{γ_{1}}) y + η_{3} t) . \end{array}$ \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} (52) $\begin{array}{l} u_{2} (x, y, t) = tanh (i x + (- \frac{i^{2} γ_{3} + 2 γ_{1}}{γ_{1}}) y + η_{3} t) e^{i ψ}, \\ v_{2} (x, y, t) = - \frac{2 i^{2} γ_{3} + 4 γ_{1}}{γ_{1} i} {tanh}^{2} (i x + (- \frac{i^{2} γ_{3} + 2 γ_{1}}{γ_{1}}) y + η_{3} t) . \end{array}$ \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} (53) $\begin{array}{l} u_{3} (x, y, t) = tanh (- i x + (\frac{i^{2} γ_{3} - 2 γ_{1}}{γ_{1}}) y + η_{3} t) e^{i ψ}, \\ v_{3} (x, y, t) = \frac{- 2 i^{2} γ_{3} + 4 γ_{1}}{γ_{1} i} {tanh}^{2} (- i x + (\frac{i^{2} γ_{3} - 2 γ_{1}}{γ_{1}}) y + η_{3} t) . \end{array}$ \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} (54) $\begin{array}{l} u_{4} (x, y, t) = tanh (- i x + (- \frac{i^{2} γ_{3} + 2 γ_{1}}{γ_{1}}) y + η_{3} t) e^{i ψ}, \\ v_{4} (x, y, t) = \frac{2 i^{2} γ_{3} + 4 γ_{1}}{γ_{1} i} {tanh}^{2} (- i x + (- \frac{i^{2} γ_{3} + 2 γ_{1}}{γ_{1}}) y + η_{3} t) . \end{array}$ \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.

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 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 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: 1)

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.

eISSN:: 2444-8656
Idioma:: Inglés

Calendario de la edición:: Volume Open
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Optical soliton solutions to a (2+1) dimensional Schrödinger equation using a couple of integration architectures

Publicado en línea: 26 jun 2020

Páginas: 381 - 396

Recibido: 20 may 2019

Aceptado: 11 feb 2020

DOI: https://doi.org/10.2478/amns.2020.2.00010

Palabras clave
(2+1) dimensional nonlinear Schrodinger equation, traveling wave solutions, soliton solutions

© 2020 Pelin Doğan Çankal et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Optical soliton solutions to a (2+1) dimensional Schrödinger equation using a couple of integration architectures

Publicado en línea: 26 jun 2020

Páginas: 381 - 396

Recibido: 20 may 2019

Aceptado: 11 feb 2020

DOI: https://doi.org/10.2478/amns.2020.2.00010

Palabras clave(2+1) dimensional nonlinear Schrodinger equation, traveling wave solutions, soliton solutions

© 2020 Pelin Doğan Çankal et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Palabras clave
(2+1) dimensional nonlinear Schrodinger equation, traveling wave solutions, soliton solutions