The Formation and Propagation of Soliton Wave Profiles for the Shynaray-IIa Equation

The Shynaray-IIA is a coupled partial differential equation, a significant nonlinear partial differential equation (PDE), arises in numerous branches of physical and mathematical sciences, like as fluid mechanics, quantum physics and plasma physics. Its complex nonlinear nature presents a substantial challenge in finding exact analytical solutions, leading researchers to explore innovative and efficient methods for resolution such as tanh method [1], extended auxiliary equation method [2,3,4], variational method [5], modified and extended simple equation method [6,7,8], direct algebraic method [9], generalized exponential rational function technique [10], extended F-expansion scheme [11,12], GG′ {}^{G}\!\!\diagup\!\!{}_{{{G}^{'}}}\; – expansion algorithm [13], sine-Gordon expansion method [14], modified sub-equation method [15], darboux method [16], homogeneous balance [17], and so on [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. Among the abundance of mathematical tools available, the Jacobi elliptic function approach has emerged as a promising scheme for solving the non-linear partial differential equations (PDEs). This technique is particularly valuable in handling nonlinear equations with high nonlinearity, as it enables researchers to obtain exact solutions by transforming the original equation into a more manageable elliptic equation. In this research article, we focus on investigating the application of the Jacobi elliptic function approach to handle the Shynaray-IIA equation. The considered model is given as, (1) iqt+qxt−i(vq)x=0,irt−rxt−i(vr)x=0,vx−n2α(rq)t=0. \matrix{ {{iq_t} + {q_{xt}} - i{{(vq)}_x} = 0,} \hfill \cr {{ir_t} - {r_{xt}} - i{{(vr)}_x} = 0,} \hfill \cr {{v_x} - {{{n^2}} \over \alpha }{{(rq)}_t} = 0.} \hfill \cr }

We aimed to construct exact analytical solutions that shed light on the intricate dynamics described by the equation. The obtained solutions not only contribute to a deeper understanding of underlying physical processes but also offer a valuable standard for validating numerical and approximate method in solving this challenging PDE. The Jacobi elliptic function expansion method serves as a powerful mathematical tool to solve the Shynaray-IIA (S-IIA) equation, allowing us to gain deeper insight into the behavior of complex physical systems. The exact analytical solutions obtained through this research contribute to the existing body of knowledge, paving the way for further advancement in the study of nonlinear Partial differential equations (PDEs) and their implications in diverse scientific disciplines. Sachin et al. [34,35,36,37,38] have examined the Konopelchenko–Dubrovsky (KD) equation, generalized complex coupled Schrödinger–Boussinesq equations, Sakovich equation, Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZK-BBM) equation and Vakhnenko-Parkes equation to develop the solitary wave solution and visualized their propagation by utilizing the distinct analytical techniques. Rani et al. [39] constructed exact analytical solutions for complex modified Kortewegde-Vries. Nonlaopon et al. [40] performed Painlevé analysis to form the exact soliton solutions.

The remainder of this article is presented in the following structure: Section 1, provides a brief overview of the Shynaray-IIA equation and its relevance in various scientific fields. Section II outlines the theoretical basis of considered method. In Section III, we present the step-by-step implementation of the method to obtain exact solutions for the Shynaray-IIA equation. In section IV, provide the analysis of graphs for direct study. Section V, discusses the conclusion and applicability of the proposed approach.

An overview of the Jacobi elliptic function methodology is given in this section. We will use nonlinear partial differential equations, which typically have the following mathematical conclusion, (2) Nu,∂u∂t,∂u∂x,∂2u∂t2,∂2u∂x2,…=0. N\left( {u,{{\partial u} \over {\partial t}},{{\partial u} \over {\partial x}},{{{\partial ^2}u} \over {\partial {t^2}}},{{{\partial ^2}u} \over {\partial {x^2}}}, \ldots } \right) = 0.

Utilizing the following wave transformation to convert Eq. (1) into an ordinary differential equation, (3) u=uξ, ξ=k(x−ct), u = u\left( \xi \right),\;\xi = k(x - ct), where the symbols for frequency and wave number, respectively, are c and k. Equation (1) has been successfully transformed into an ordinary differential equation (ODE) by the procedure described in Equation (2). (4) ∂.∂t=−ckd.dξ,∂.∂x=kd.dξ, {{\partial \left( . \right)} \over {\partial t}} = - ck{{d\left( . \right)} \over {d\xi }},{{\partial \left( . \right)} \over {\partial x}} = k{{d\left( . \right)} \over {d\xi }}, (5) Nu′,u″,u‴,….=0. N\left( {u',u'',u''', \ldots .} \right) = 0.

In conjunction with utilizing this advanced directed technique, the underlying principle entails augmenting the likelihood of resolving an auxiliary ODE, namely first-order Jacobian problem with the three parameters. This method aims to produce a multitude of Jacobian elliptic solutions for the given issue. Visualizing the auxiliary equation is a feasible step in understanding this process. (6) (F′)2ξ=PF4ξ+QF2ξ+R. {(F')^2}\left( \xi \right) = P{F^4}\left( \xi \right) + Q{F^2}\left( \xi \right) + R.

Let F′=dFdξ F' = {{dF} \over {d\xi }} , where ξ = ξ(x, t), and the constants P, Q and R are involved. The solution for equation (5) is provided in Tab. 1. It is important to note that i² = −1. Additionally, the Jacobi elliptic functions are denoted as snξ = sn(ξ, m), cnξ = cn(ξ, m), and dnξ = dn(ξ, m), where m lies in the range 0 < m < 1 and represents the modulus.

The elliptic functions exhibit a distinctive double periodic, pro-cessing distinct properties as outline below: sn2ξ+cn2ξ=1,dn2ξ+m2sn2ξ=1,ddξsnξ=cnξdnξ,ddξcnξ=−snξdnξ,ddξdnξ=−m2snξcnξ. \matrix{ {{sn^2}\xi + c{n^2}\xi = 1,} \hfill \cr {{dn^2}\xi + {m^2}s{n^2}\xi = 1,} \hfill \cr {{d \over {d\xi }}sn\xi = cn\xi dn\xi ,} \hfill \cr {{d \over {d\xi }}cn\xi = - sn\xi dn\xi ,} \hfill \cr {{d \over {d\xi }}dn\xi = - {m^2}sn\xi cn\xi .} \hfill \cr }

With reference to Tab. 2, this reduction makes it possible to derive the solutions for the given problem using the trigonometric function and solitons. The Jacobi elliptic function expansion method can be used to describe the function as a finite series of Jacobi elliptic functions. (7) uξ=∑i=1n aiFiξ. u\left( \xi \right) = \mathop \sum \nolimits_{i = 1}^n \;{a_i}{F^i}\left( \xi \right).

Here the function F(ξ) represents solution to the non-linear ordinary equation denoted as Eq. 5. The constants n and a_i (where i = 0, 1, 2, ... , n) are parameters that have to be found. The determination of the integer n in Eq. 6 involves an analysis of the highest order linear term. (8) Odpudξp=n+p, p=0,1,2,3, … …, O\left( {{{{d^p}u} \over {{d\xi ^p}}}} \right) = n + p,\;\;\;p = 0,1,2,3,\; \ldots \; \ldots , thus, the most significant nonlinear terms at the highest order are (9) Ouqdpudξp=q+1n+p, p=0,1,2,3, … …,q=1,2,3, …, \matrix{ {O\left( {{u^q}{{{d^p}u} \over {{d\xi ^p}}}} \right) = \left( {q + 1} \right)n + p,\;\;\;p = 0,1,2,3,\; \ldots \; \ldots ,} \hfill \cr {q = 1,2,3,\; \ldots ,} \hfill \cr } in Eq. 4.

Utilizing Eq. 6 and setting all coefficients of powers F to zero, we derive a set of nonlinear algebraic equations for the variables a_i, (where i = 0,1,2,3, ….). Employing Mathematica, we proceed to solve this system of algebraic equations and put all the values for P, Q, and R as per Eq. 5 in Tab. 1. This approach, integrating the information from Eq. 6 with the selected auxiliary equation, allows for the determination of exact solutions for Eq. 1.

The precise solutions to Shynaray-IIA Eq. 1 using the Jacobi elliptic function expansion approach are shown in this section, iqt+qxt−i(vq)x=0,irt−rxt−i(vr)x=0,vx−n2α(rq)t=0. \matrix{ {{iq_t} + {q_{xt}} - i{{(vq)}_x} = 0,} \hfill \cr {{ir_t} - {r_{xt}} - i{{(vr)}_x} = 0,} \hfill \cr {{v_x} - {{{n^2}} \over \alpha }{{(rq)}_t} = 0.} \hfill \cr }

In case when = εq¯ε=±1 \varepsilon \bar q\left( {\varepsilon = \pm 1} \right) , the S-IIAE takes the following form: (10) iqt+qxt−i(vq)x=0,vx−n2εαq2t=0. \matrix{ {{iq_t} + {q_{xt}} - i{{(vq)}_x} = 0,} \hfill \cr {{v_x} - {{{n^2}\varepsilon } \over \alpha }{{\left( {{{\left| q \right|}^2}} \right)}_t} = 0.} \hfill \cr }

In the above equation m, n and ɛ are constants. By using the traveling wave transformation Eq. 11 is reduced into the following ODE: (11) qx,t=Uηeiξ(x,t),vx,t=G(η),ξx,t=−δx+ωt+θ,η=x−ct, \matrix{ {q\left( {x,t} \right) = U\left( \eta \right){e^{i\xi (x,t)}},v\left( {x,t} \right) = G(\eta ),} \hfill \cr {\xi \left( {x,t} \right) = - \delta x + \omega t + \theta ,\eta = x - ct,} \hfill \cr } where ν, θ, ω, δ characterize the frequency, the phase constant, the wave number and the velocity, respectively. The Eq. 27 is plugging into the first part of Eq. 26 and getting the real and imaginary parts, (12) cU″η+ω1−δUη+δGηUη+i(ω −c1−δ)U′η−GηU′η−G′ηUη=0,G′η+2cεn2αUηU′η=0. \matrix{ {cU''\left( \eta \right) + \omega \left( {1 - \delta } \right)U\left( \eta \right) + \delta G\left( \eta \right)U\left( \eta \right) + i(\omega \; - } \hfill \cr {c\left( {1 - \delta } \right))U'\left( \eta \right) - G\left( \eta \right)U'\left( \eta \right) - G'\left( \eta \right)U\left( \eta \right) = 0,} \hfill \cr {G'\left( \eta \right) + {{2c\varepsilon {n^2}} \over \alpha }U\left( \eta \right)U'\left( \eta \right) = 0.} \hfill \cr }

The second Eq. 28 is integrated, and we get (13) Gη=−cεn2αU2η. G\left( \eta \right) = - {{c\varepsilon {n^2}} \over \alpha }{U^2}\left( \eta \right).

Substitute the Eq. 13 into the first part of 12 and separating the real and imaginary parts as (14) cU″η+ω1−δUη−δcεn2αU3η=0. cU''\left( \eta \right) + \omega \left( {1 - \delta } \right)U\left( \eta \right) - {{\delta c\varepsilon {n^2}} \over \alpha }{U^3}\left( \eta \right) = 0.

And we have the imaginary part as, (15) ω−c1−δU′η+3cεn2αU″ηU′η=0. \left( {\omega - c\left( {1 - \delta } \right)} \right)U'\left( \eta \right) + {{3c\varepsilon {n^2}} \over \alpha }U''\left( \eta \right)U'\left( \eta \right) = 0.

By using the homogeneous balancing procedure, we obtained n = 1, the determine value of n is substituted in Eg. 7 we obtained the simple form of the solution as: (16) Uη=a0+a1F(η), U\left( \eta \right) = {a_0} + {a_1}F(\eta ), (17) U3η=a03+a13F3η+3a0a12F2η+3a02a1F {U^3}\left( \eta \right) = a_0^3 + a_1^3{F^3}\left( \eta \right) + 3{a_0}a_1^2{F^2}\left( \eta \right) + 3a_0^2{a_1}F and (18) U″η=a12PF3η+QFη. U''\left( \eta \right) = {a_1}\left( {2P{F^3}\left( \eta \right) + QF\left( \eta \right)} \right).

Substitute Eq. 16–18 into Eq. 15, we get, (19) ca12PF3η+QFη+ω1−δa0+a1Fη−δcεn2αa03+a13F3η+3a0a12F2η+3a02a1Fη=0. \matrix{ {{ca_1}\left( {2{PF^3}\left( \eta \right) + QF\left( \eta \right)} \right) + \omega \left( {1 - \delta } \right)\left( {{a_0} + {a_1}F\left( \eta \right)} \right) - } \hfill \cr {{{\delta c\varepsilon {n^2}} \over \alpha }\left( {a_0^3 + a_1^3{F^3}\left( \eta \right) + 3{a_0}a_1^2{F^2}\left( \eta \right) + 3a_0^2{a_1}F\left( \eta \right)} \right) = 0.} \hfill \cr }

By collecting the various coefficients of Fⁱ(η), we get the system of equations, (20) U0:ω1−δ−δcεn2αa02a0=0, {U^0}:\left( {\omega \left( {1 - \delta } \right) - {{\delta c\varepsilon {n^2}} \over \alpha }a_0^2} \right){a_0} = 0, (21) U1:cQ+ω1−δ−3a02δcεn2αa1=0, {U^1}:\left( {cQ + \omega \left( {1 - \delta } \right) - 3a_0^2{{\delta c\varepsilon {n^2}} \over \alpha }} \right){a_1} = 0, (22) U2:−3a0a13δcεn2α=0, {U^2}: - 3{a_0}a_1^3{{\delta c\varepsilon {n^2}} \over \alpha } = 0, (23) U3=2Pc−a12δcεn2αa1=0. {U^3} = \left( {2Pc - a_1^2{{\delta c\varepsilon {n^2}} \over \alpha }} \right){a_1} = 0.

Upon solving the aforementioned system by using Maple software, we obtain the coefficients pertaining to the series 16, (24) a0=0, a1=±2αPδεn. {a_0} = 0,\;\;\;{a_1} = \pm {{\sqrt {{{2\alpha P} \over {\delta \varepsilon }}} } \over n}.

The obtained solution is of the form, (25) U=±2αPδεnFη. U = \pm {{\sqrt {{{2\alpha P} \over {\delta \varepsilon }}} } \over n}F\left( \eta \right).

When the values P = m², Q = −(1 + m²), and R = 1 are chosen, table 1 provides the corresponding values of F = sn. Therefore, the periodic solution of Equation 1 can be represented as, (26) q1,1=±2αm2δεnsn(x−ct), {q_{1,1}} = \pm {{\sqrt {{{2\alpha {m^2}} \over {\delta \varepsilon }}} } \over n}sn(x - ct), (27) v1,1=−2cm2δsn2x−ct. {v_{1,1}} = - {{2c{m^2}} \over \delta }s{n^2}\left( {x - ct} \right).

Supposing m → 1, hence, by referring to table 2, one may derive the solitary wave solution of Eq. 1. (28) q1,2=±2αδεntanh(x−ct), {q_{1,2}} = \pm {{\sqrt {{{2\alpha } \over {\delta \varepsilon }}} } \over n}tanh(x - ct), (29) v1,2=−2cδtanh2x−ct. {v_{1,2}} = - {{2c} \over \delta }tan{h^2}\left( {x - ct} \right).

Choosing P = −m², Q = 2m² − 1, R = 1 − m², based on the data supplied in Table 1, it can be inferred that the variable F can be mathematically expressed as F = cn. Consequently, the periodic solution of Equation (1) can be derived as follows: (30) q1,3=±−2αm2δεncn(x−ct), {q_{1,3}} = \pm {{\sqrt {{{ - 2\alpha {m^2}} \over {\delta \varepsilon }}} } \over n}cn(x - ct), (31) v1,3=2cm2δcn2x−ct. {v_{1,3}} = {{2c{m^2}} \over \delta }c{n^2}\left( {x - ct} \right).

Considering m → 1 the solitary wave solution of Eq. 1 can be expressed as per the information provided in Tab. 2. (32) q1,4=±−2αδεnsech(x−ct), {q_{1,4}} = \pm {{\sqrt {{{ - 2\alpha } \over {\delta \varepsilon }}} } \over n}sech(x - ct), (33) v1,4=2cδsech2x−ct. {v_{1,4}} = {{2c} \over \delta }sec{h^2}\left( {x - ct} \right).

Setting P = −1, Q = 2 − m², R = m² − 1, based on the data shown in Tab. 1, it can be inferred that the periodic solution of Eq. 1 can be mathematically represented as follows: (34) q1,5=±−2αδεndn(x−ct), {q_{1,5}} = \pm {{\sqrt {{{ - 2\alpha } \over {\delta \varepsilon }}} } \over n}dn(x - ct), (35) v1,5=2cδdn2x−ct. {v_{1,5}} = {{2c} \over \delta }d{n^2}\left( {x - ct} \right).

In the context of m → 1 from Tab. 2, the similarity between the solution shown and the solution derived in Eq. 27 is clearly demonstrated.

While P = 1, Q = −(1 + m²), R = m², F = ns, according to the data presented in Tab. 1, the answer to Eq. 1 can be represented as follows: (36) q1,6=±2αδεnns(x−ct), {q_{1,6}} = \pm {{\sqrt {{{2\alpha } \over {\delta \varepsilon }}} } \over n}ns(x - ct), (37) v1,6=−2cδns2x−ct. {v_{1,6}} = - {{2c} \over \delta }n{s^2}\left( {x - ct} \right).

Additionally, when m → 1 the solitary wave solution of Eq. 1 is presented in Tab. 2. (38) q1,7=±2αδεncoth(x−ct), {q_{1,7}} = \pm {{\sqrt {{{2\alpha } \over {\delta \varepsilon }}} } \over n}coth(x - ct), (39) v1,7=−2cδcoth2x−ct. {v_{1,7}} = - {{2c} \over \delta }cot{h^2}\left( {x - ct} \right).

Using Tab. 2, the periodic solution of Eq. 1 can be stated as follows if m → 0: (40) q1,8=±2αδεncsc(x−ct), {q_{1,8}} = \pm {{\sqrt {{{2\alpha } \over {\delta \varepsilon }}} } \over n}csc(x - ct), (41) v1,8=−2cδcsc2x−ct. {v_{1,8}} = - {{2c} \over \delta }cs{c^2}\left( {x - ct} \right).

Supposing P = 1, Q = −(1 + m²), R = m².

Thus, F = dc, (42) q1,8=±2αδεndc(x−ct), {q_{1,8}} = \pm {{\sqrt {{{2\alpha } \over {\delta \varepsilon }}} } \over n}dc(x - ct), (43) v1,8=−2cδdc2x−ct. {v_{1,8}} = - {{2c} \over \delta }d{c^2}\left( {x - ct} \right).

Using Tab. 2, the periodic solution of Eq. 1 can be stated as follows if m → 0: (44) q1,10=±2αδεnsec(x−ct), {q_{1,10}} = \pm {{\sqrt {{{2\alpha } \over {\delta \varepsilon }}} } \over n}sec(x - ct), (45) v1110=−2cδscc2x−ct. {v_{{1_1}10}} = - {{2c} \over \delta }sc{c^2}\left( {x - ct} \right).

When P = 1 − m², Q = 2m² − 1, R = −m². Thus, F = nc and the solution of periodic nature of Eq. 1 as: (46) q1,11=±2α1−m2δεnnc(x−ct), {q_{1,11}} = \pm {{\sqrt {{{2\alpha \left( {1 - {m^2}} \right)} \over {\delta \varepsilon }}} } \over n}nc(x - ct), (47) v1,11=−2c(1−m)2δnc2x−ct. {v_{1,11}} = - {{2c{{(1 - m)}^2}} \over \delta }n{c^2}\left( {x - ct} \right).

As m → 0 from Tab. 2, it is shown that the solution found as that of 33.

Also regarding P = 1 − m², Q = 2 − m², R = 1.

Thus, F = sc: (48) q1,12=±2α1−m2δεnsc(x−ct), {q_{1,12}} = \pm {{\sqrt {{{2\alpha \left( {1 - {m^2}} \right)} \over {\delta \varepsilon }}} } \over n}sc(x - ct), (49) v1,12=−2c(1−m)2δsc2x−ct. {v_{1,12}} = - {{2c{{(1 - m)}^2}} \over \delta }s{c^2}\left( {x - ct} \right).

Furthermore, we find the periodic solution of Eq. 1 as follows for m → 0, as shown in Tab. 2: (50) q1,13=±2αδεntan(x−ct), {q_{1,13}} = \pm {{\sqrt {{{2\alpha } \over {\delta \varepsilon }}} } \over n}tan(x - ct), (51) v1,13=−2cδtan2x−ct. {v_{1,13}} = - {{2c} \over \delta }ta{n^2}\left( {x - ct} \right).

Considering P = 1, Q = 2 − m², R = 1 − m² and F = cs, thus: (52) q1,14=±2αδεncs(x−ct), {q_{1,14}} = \pm {{\sqrt {{{2\alpha } \over {\delta \varepsilon }}} } \over n}cs(x - ct), (53) v1,14=−2cδcs2x−ct. {v_{1,14}} = - {{2c} \over \delta }c{s^2}\left( {x - ct} \right).

The solitary wave solution Eq. (1) is given as follows as m→1, per Tab. 2: (54) q1,15=±2αδεncsch(x−ct), {q_{1,15}} = \pm {{\sqrt {{{2\alpha } \over {\delta \varepsilon }}} } \over n}csch(x - ct), (55) v1,15=−2cδcsch2x−ct. {v_{1,15}} = - {{2c} \over \delta }csc{h^2}\left( {x - ct} \right).

The solitary wave solution Eq. 1 is given as follows as m→0, per Tab. 2, (56) q1,16=±2αδεncot(x−ct), {q_{1,16}} = \pm {{\sqrt {{{2\alpha } \over {\delta \varepsilon }}} } \over n}cot(x - ct), (57) v1,16=−2cδcot2x−ct. {v_{1,16}} = - {{2c} \over \delta }co{t^2}\left( {x - ct} \right).

Also assigning P = 1, Q = 2m² − 1, R = m⁴ − m² and F = ds. Thus, (58) q1,17=±2αδεnds(x−ct), {q_{1,17}} = \pm {{\sqrt {{{2\alpha } \over {\delta \varepsilon }}} } \over n}ds(x - ct), (59) v1,17=−2cδdc2x−ct. {v_{1,17}} = - {{2c} \over \delta }d{c^2}\left( {x - ct} \right).

In this family, the soliton solution is the similar to Eq. 30. If the limit of m → 0, the solution can be articulated as per Eq. 38 with reference to Tab. 2.

Assuming P, Q, R as P=−14 P = {{ - 1} \over 4} , Q=m2+12 Q = {{{m^2} + 1} \over 2} , R=−1−m24 R = {{ - \left( {1 - {m^2}} \right)} \over 4} , according to Tab. 1, F formulated as F = mcn ∓ dn, the solution is determined as, (60) q1,18=±−α2δεnmcnx−ct∓dn(x−ct), {q_{1,18}} = \pm {{\sqrt {{{ - \alpha } \over {2\delta \varepsilon }}} } \over n}mcn\left( {x - ct} \right) \mp dn(x - ct), (61) v1,18=c2δmcnx−ct∓dnx−ct2. {v_{1,18}} = {c \over {2\delta }}{\left( {mcn\left( {x - ct} \right) \mp dn\left( {x - ct} \right)} \right)^2}.

Additionally, when m → 1, the obtained solution is similar the solution found in Eq. (25).

If we select P, Q, R as P=14 P = {1 \over 4} , Q=−2m2+12 Q = {{ - 2{m^2} + 1} \over 2} , R=14 R = {1 \over 4} , and evaluate F from table 1 where = ns ∓ cs, thus solution of Eq. (1) can be indicated as, (62) q1,19=±α2δεnnsx−ct∓csx−ct, {q_{1,19}} = \pm {{\sqrt {{\alpha \over {2\delta \varepsilon }}} } \over n}\left( {ns\left( {x - ct} \right) \mp cs\left( {x - ct} \right)} \right), (63) v1,19=−c2δnsx−ct∓csx−ct2. {v_{1,19}} = - {c \over {2\delta }}{\left( {ns\left( {x - ct} \right) \mp cs\left( {x - ct} \right)} \right)^2}.

The solitary wave solution for m → 1 in Eq. 1 is identified as, (64) q1,20=±α2δεncothx−ct∓csch(x−ct), {q_{1,20}} = \pm {{\sqrt {{\alpha \over {2\delta \varepsilon }}} } \over n}coth\left( {x - ct} \right) \mp csch(x - ct), (65) v1,20=−c2δcothx−ct∓cschx−ct2. {v_{1,20}} = - {c \over {2\delta }}{\left( {{\rm{coth\;}}\left( {x - ct} \right) \mp {\rm{csch}}\left( {x - ct} \right)} \right)^2}.

Additionally, in the case where m → 0, based on Tab. 2, obtaining a periodic solution is evident. (66) q1,21=±α2δεncscx−ct∓cot(x−ct), {q_{1,21}} = \pm {{\sqrt {{\alpha \over {2\delta \varepsilon }}} } \over n}csc\left( {x - ct} \right) \mp cot(x - ct), (67) v1,21=−c2δcscx−ct∓cotx−ct2. {v_{1,21}} = - {c \over {2\delta }}{\left( {{\rm{csc\;}}\left( {x - ct} \right) \mp {\rm{\;cot\;}}\left( {x - ct} \right)} \right)^2}.

If P=1−m24 P = {{1 - {m^2}} \over 4} , Q=m2+12 Q = {{{m^2} + 1} \over 2} , R=1−m24 R = {{1 - {m^2}} \over 4} and F = nc ∓ sc, the solution of Eq. (1) can be found as, (68) q1,22=±α1−m22δεnncx−ct∓scx−ct, {q_{1,22}} = \pm {{\sqrt {{{\alpha \left( {1 - {m^2}} \right)} \over {2\delta \varepsilon }}} } \over n}\left( {nc\left( {x - ct} \right) \mp sc\left( {x - ct} \right)} \right), (69) v1,22=−c1+m22δncx−ct∓scx−ct2. {v_{1,22}} = {{ - c\left( {1 + {m^2}} \right)} \over {2\delta }}{\left( {nc\left( {x - ct} \right) \mp sc\left( {x - ct} \right)} \right)^2}.

The solitary wave solution for m → 0 in Eq. 1 is identified as, (70) q1,23=±α2δεnsecx−ct∓tanx−ct, {q_{1,23}} = \pm {{\sqrt {{\alpha \over {2\delta \varepsilon }}} } \over n}\left( {sec\left( {x - ct} \right) \mp tan\left( {x - ct} \right)} \right), (71) v1,23=−c2δsecx−ct∓tanx−ct2. {v_{1,23}} = - {c \over {2\delta }}{\left( {{\rm{sec\;}}\left( {{\rm x} - {\rm ct}} \right) \mp {\rm{\;tan\;}}\left( {\rm {x - ct}} \right)} \right)^2}.

Setting P=m24 P = {{{m^2}} \over 4} , Q=m2−22 Q = {{{m^2} - 2} \over 2} , R=m24 R = {{{m^2}} \over 4} , as per Table 1, F = sn ∓ icn, due to this setting the solution of Eq. 1 can be found as: (72) q1,24=±αm22δεnsnx−ct∓icnx−ct, {q_{1,24}} = \pm {{\sqrt {{{\alpha {m^2}} \over {2\delta \varepsilon }}} } \over n}\left( {sn\left( {x - ct} \right) \mp icn\left( {x - ct} \right)} \right), (73) v1,24=−cm22δsnx−ct∓icnx−ct2. {v_{1,24}} = - {{{cm^2}} \over {2\delta }}{\left( {sn\left( {x - ct} \right) \mp icn\left( {x - ct} \right)} \right)^2}.

The solitary wave solution for m → 1 in Eq. (1) is identified as, (74) q1,25=±α2δεntanhx−ct∓isechx−ct, {q_{1,25}} = \pm {{\sqrt {{\alpha \over {2\delta \varepsilon }}} } \over n}\left( {tanh\left( {x - ct} \right) \mp isech\left( {x - ct} \right)} \right), (75) v1,25=−c2δtanhx−ct∓isechx−ct2. {v_{1,25}} = {{ - c} \over {2\delta }}{\left( {{\rm{tanh\;}}\left( {x - ct} \right) \mp isech\left( {x - ct} \right)} \right)^2}.

Regarding P=14 P = {1 \over 4} , Q=−2m2+12 Q = {{ - 2{m^2} + 1} \over 2} , R=14 R = {1 \over 4} , and F = msn ∓ idn from the Table 1, thus, the solution of Eq. 1 can be expressed as, (76) q1,26=±α2δεnmsnx−ct∓idnx−ct, {q_{1,26}} = \pm {{\sqrt {{\alpha \over {2\delta \varepsilon }}} } \over n}\left( {msn\left( {x - ct} \right) \mp idn\left( {x - ct} \right)} \right), (77) v1,26=−c2δmsnx−ct∓idnx−ct2. {v_{1,26}} = {{ - c} \over {2\delta }}{\left( {msn\left( {x - ct} \right) \mp idn\left( {x - ct} \right)} \right)^2}.

For m → 1, the solution obtained as that of 48.

Considering P=14,Q=1−2m22,R=14, P = {1 \over 4},Q = {{1 - 2{m^2}} \over 2},R = {1 \over 4}, and F=sn1∓cn, F = {{sn} \over {1 \mp cn}}, from Tab. 1, thus, the solution of Eq. 1 can be found as, (78) q1,27=±α2δεnsnx−ct1∓cnx−ct, {q_{1,27}} = \pm {{\sqrt {{\alpha \over {2\delta \varepsilon }}} } \over n}{{sn\left( {x - ct} \right)} \over {1 \mp cn\left( {x - ct} \right)}}, (79) v1,27=−c2δsnx−ct1∓cnx−ct2. {v_{1,27}} = {{ - c} \over {2\delta }}{\left( {{{sn\left( {x - ct} \right)} \over {1 \mp cn\left( {x - ct} \right)}}} \right)^2}.

If we take a look at Tab. 2, we can determine the solitary wave solution of Eq. 1 for m → 1, (80) q1,28=±α2δεntanhx−ct1∓sechx−ct, {q_{1,28}} = \pm {{\sqrt {{\alpha \over {2\delta \varepsilon }}} } \over n}{{tanh \left( {x - ct} \right)} \over {1 \mp {\mathop{sech}\nolimits} \left( {x - ct} \right)}}, (81) v1,28=−c2δtanhx−ct1∓sechx−ct2. {v_{1,28}} = {{ - c} \over {2\delta }}{\left( {{{{\rm{\;tanh\;}}\left( {x - ct} \right)} \over {1 \mp {\rm{sech}}\left( {x - ct} \right)}}} \right)^2}.

If we take a look at Tab. 2, we can determine the solitary wave solution of Eq. 1 for m → 0, (82) q1,29=±α2δεnsinx−ct1∓cosx−ct. {q_{1,29}} = \pm {{\sqrt {{\alpha \over {2\delta \varepsilon }}} } \over n}{{sin\left( {x - ct} \right)} \over {1 \mp cos\left( {x - ct} \right)}}. (83) v1,29=−c2δsinx−ct1∓cosx−ct2. {v_{1,29}} = {{ - c} \over {2\delta }}{\left( {{{{\rm{\;sin\;}}\left( {x - ct} \right)} \over {1 \mp {\rm{\;cos\;}}\left( {x - ct} \right)}}} \right)^2}.

Supposing P=m24 P = {{{m^2}} \over 4} , Q=m2−22 Q = {{{m^2} - 2} \over 2} , R=14 R = {1 \over 4} it can be concluded from Tab. 1 F=sn1∓dn F = {{sn} \over {1 \mp dn}} , so the solution of Eq. 1 can be found as, (84) q1,30=±αm22δεnsnx−ct1∓dnx−ct, {q_{1,30}} = \pm {{\sqrt {{{\alpha {m^2}} \over {2\delta \varepsilon }}} } \over n}{{sn\left( {x - ct} \right)} \over {1 \mp dn\left( {x - ct} \right)}}, (85) v1,30=−cm22δsnx−ct1∓dnx−ct2. {v_{1,30}} = - {{{cm^2}} \over {2\delta }}{\left( {{{sn\left( {x - ct} \right)} \over {1 \mp dn\left( {x - ct} \right)}}} \right)^2}.

When m → 1, the solution is determined by the solution in equation 52.

From Tab. 1, allocating P=1−m24 P = {{1 - {m^2}} \over 4} , Q=m2+12 Q = {{{m^2} + 1} \over 2} , R=1−m24 R = {{1 - {m^2}} \over 4} and F=cn1∓sn, F = {{cn} \over {1 \mp sn}}, thus, (86) q1,31=±α1−m22δεncnx−ct1∓snx−ct, {q_{1,31}} = \pm {{\sqrt {{{\alpha \left( {1 - {m^2}} \right)} \over {2\delta \varepsilon }}} } \over n}{{cn\left( {x - ct} \right)} \over {1 \mp sn\left( {x - ct} \right)}}, (87) v1,31=−c1+m22δcnx−ct1∓snx−ct2. {v_{1,31}} = {{ - c\left( {1 + {m^2}} \right)} \over {2\delta }}{\left( {{{cn\left( {x - ct} \right)} \over {1 \mp sn\left( {x - ct} \right)}}} \right)^2}.

If we take a look at Tab. 2, we can determine the solitary wave solution of Eq. 1 for m → 0, (88) q1,32=±α2δεncosx−ct1∓sinx−ct, {q_{1,32}} = \pm {{\sqrt {{\alpha \over {2\delta \varepsilon }}} } \over n}{{cos\left( {x - ct} \right)} \over {1 \mp sin\left( {x - ct} \right)}}, (89) v1,32=−c2δcosx−ct1∓sinx−ct2. {v_{1,32}} = {{ - c} \over {2\delta }}{\left( {{{{\rm{\;cos\;}}\left( {x - ct} \right)} \over {1 \mp {\rm{\;sin\;}}\left( {x - ct} \right)}}} \right)^2}.

Choosing P=(1−m2)24 P = {{{{(1 - {m^2})}^2}} \over 4} , Q=m2+12 Q = {{{m^2} + 1} \over 2} , R=14 R = {1 \over 4} and F=sndn∓cn F = {{sn} \over {dn \mp cn}} , so that the solution of Eq. 1 can be obtained as, (90) q1,33=±α(1−m2)22δεnsnx−ctdnx−ct∓cnx−ct, {q_{1,33}} = \pm {{\sqrt {{{\alpha {{(1 - {m^2})}^2}} \over {2\delta \varepsilon }}} } \over n}{{sn\left( {x - ct} \right)} \over {dn\left( {x - ct} \right) \mp cn\left( {x - ct} \right)}}, (91) v1,33=−c(1+m2)22δsnx−ctdnx−ct∓cnx−ct2. {v_{1,33}} = {{ - c{{(1 + {m^2})}^2}} \over {2\delta }}{\left( {{{sn\left( {x - ct} \right)} \over {dn\left( {x - ct} \right) \mp cn\left( {x - ct} \right)}}} \right)^2}.

For m → 0, the solution is obtained as that of 51.

This section offers physical explanation of Figure [1–12] and selection of wave solutions that have been obtained by applying the Jacobi elliptic function expansion method to the S-IIAE equation. In order to create visual representations of different soliton wave patterns, we have carefully selected and used certain parameter values. These patterns are illustrated in the accompanying figures. For every scenario, we have produced surface and contour visualization plots in two and three dimensions. These visual aids are important because they can verify that the theoretical conclusions, we came to earlier are accurate. It's important to keep in mind that these graphs and figures were produced using Mathematica. Consequently, one can notice that, the above-mentioned graphics are presenting the dark-bright, periodic, composite and bright soliton behavior respectively, under the influence of variation of wave number. On the other hand, the influence of wave is also discussed and noticed that, researchers and physicists can acquire their required results by controlling the propagation of soliton with wave number.

Fig. 1.

3-D, contour visualization and 2-D propagation of q_1,1 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, m = 0.9, c = 0.1

Fig. 2.

3-D, contour visualization and 2-D propagation of q_1,1 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, m = 0.9, c = 01

Fig. 3.

3-D, contour visualization and 2-D propagation of q_1,1 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, m = 0.9, c = 2.5

Fig. 4.

3-D, contour visualization and 2-D propagation of ν_1,1 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, n = 1.5, c = −1.5, m = 0.9, c = 0.1.

Fig. 5.

3-D, contour visualization and 2-D propagation of ν_1,1 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, n = 1.5, c = −1.5, m = 0.9, c = 01

Fig. 6.

3-D, contour visualization and 2-D propagation of ν_1,1 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, n = 1.5, c = −1.5, m = 0.9, c = 2.5

Fig. 7.

3-D, contour visualization and 2-D propagation of q_1,2 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, m = 0.5, c = 0.1

Fig. 8.

3-D, contour visualization and 2-D propagation of q_1,2 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, m = 0.5, c = 01

Fig. 9.

3-D, contour visualization and 2-D propagation of q_1,2 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, m = 0.5, c = 2.5

Fig. 10.

3-D, contour visualization and 2-D propagation of ν_1,2 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, n = 1.5, c = −1.5, m = 0.5, c = 0.1

Fig. 11.

3-D, contour visualization and 2-D propagation of ν_1,2 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, n = 1.5, c = −1.5, m = 0.5, c = 01

Fig. 12.

3-D, contour visualization and 2-D propagation of ν_1,2 for specific values of the parameters are ɛ = 1.2, α = 1.3, δ = 0.5, n = 1.5, c = −1.5, m = 0.5, c = 2.5

In conclusion, this research article explored the application of the Jacobi elliptic function expansion method for the Shynaray-IIA Equation (S-IIAE). The partial differential model is transformed into ordinary differential equation by employing the next travelling wave transformation according to considered analytical technique. Numerous properties of a particular class of solutions, called the Jacobi elliptic functions, make them useful for the analytical solution of a wide range of nonlinear problems. Using this powerful method, we derive a set of exact solutions for the Shynaray-IIA (S-IIA) equation, shedding light on its complex dynamics and behavior. The proposed method is shown to be highly effective in obtaining exact solutions in terms of Jacobi elliptic functions, such as dark, bright, periodic, dark-bright, dark-periodic, bright periodic, singular, and other various types of solitons. Additionally, a thorough examination of the accuracy and convergence of the obtained solutions is carried out. Overall, this research enriches the theoretical framework for the S-IIAE and presents a valuable tool for researchers and practitioners working in the field of nonlinear differential equations and mathematical physics.