Optimal system and dynamics of optical soliton solutions for the Schamel KdV equation

In this research, we investigate the integrability properties of the Schamel–Korteweg–de Vries (S-KdV) equation, which is important for understanding the effect of electron trapping in the nonlinear interaction of ion-acoustic waves. Using the optimal system, we come over reduced ordinary differential equations (ODEs). To deal with reduced ODEs for this problem, Lie symmetry analysis is combined with the modified auxiliary equation (MAE) procedure and the generalized Jacobi elliptic function expansion (JEF) method. The analytical solutions reported here are novel and have a wide range of applications in mathematical physics.

In this research, we investigate the integrability properties of the Schamel-Korteweg-de Vries (S-KdV) equation, which is important for understanding the effect of electron trapping in the nonlinear interaction of ion-acoustic waves.Using the optimal system, we come over reduced ordinary differential equations (ODEs).To deal with reduced ODEs for this problem, Lie symmetry analysis is combined with the modified auxiliary equation (MAE) procedure and the generalized Jacobi elliptic function expansion (JEF) method.The analytical solutions reported here are novel and have a wide range of applications in mathematical physics.
Many dynamical systems are commonly described using nonlinear evolution equations as models in a variety of scientific disciplines, notably fluid mechanics, plasma physics, solid-state physics, nonlinear optics, chemical kinematics, astrophysics, optical fiber, geochemistry, and chemical chemistry.To better understand the physical phenomena besides further applications in practical life, it is important to seek as many exact solutions as we can.Numerous important techniques, like the variational iteration method 1 , the sine-cosine technique 2 , the homotopy perturbation technique 3 , the first integral method 4 , the Bäcklund transformations 5 , the Jacobi elliptic function expansion procedure 6 , the (G ′ /G)-expansion method 7 , the exponential function technique 8 , the Weierstrass elliptic function method 9 , the tanh function technique and its extensions 10 , the simplest equation technique 11 and the Lie group analysis method [12][13][14][15] have been developed during the past few decades to provide exact solutions as well as understand their features.In this study, the S-KdV equation is taken into account 16 where α 1 , α 2 and α 3 are constants.Equation (1) is relevant to the study of ion-acoustic solitons in plasma physics when electron entrapment occurs.It also controls the electrostatic potential for a specific electron distribution in velocity space.Tagare and Chakraborti demonstrated in 17 using the direct integral approach that Eq. (1) has a single wave solution.Lee and Sakthivel provided some precise traveling wave solutions for Eq.(1) in 18 using the exp-function approach.A generalized KdV equation is an instance of Eq. (1) that has been examined in several situations arising in mathematical physics.Equation (1) turns into the Schamel equation for α 2 = 0, Whereas Eq. (1) becomes a well-known KdV equation when α 1 = 0 that has been thoroughly examined by several authors.By using the transformation U(x, t) = w 2 (x, t) .Then Eq. (1) can be written as Here we study this equation from the point of view of Lie symmetry analysis.Our main concern is to discuss the optimal system for the S-KdV Eq. ( 4) and then to utilize it for symmetry reductions to obtain analytical solutions.Our findings will include periodic solutions, double periodic, shock waves, bell-shaped, and solitary waves.Lie symmetry method is a powerful method and is popular among recent techniques.This technique effectively (1) (2)

Lie symmetries
Let us consider a one-parameter (local) Lie group of transformations ( ϑ parameter) given by The infinitesimal generator associated with the above transformations is The coefficient functions ϕ 1 , ϕ 2 and η are to be found, and using the Lie invariance condition where [3] is the third extension of and By solving the determining equations originating from Eq. ( 7), we can derive the symmetry generators of Eq. ( 4), Case-1 (α 1 , α 2 = 0): Case-2 (α 2 = 0): One-dimensional optimal systems.The idea of an optimal system of subalgebras for a certain Lie algebra to obtain fundamentally distinct invariant solutions was perhaps originally introduced by Ovsyannikov 27 .
Then it was followed by Ibragimov 28 and continued by Olver 29 .To determine the optimal system, we identify the sets of equivalent classes of one-dimensional subalgebras by observing their behavior under the influence of the adjoint representation.The representation of the adjoint action can be expressed as, where ϑ represents a real number and [ m , n ] indicates the Lie product defined by, Optimal system for Case-1.In this case, the algebra is two-dimensional and satisfies the commutation relation, We take into account following general element of symmetry algebra L 2 given by, (5)

Case-I
When taking into account that the commutator relations are equal to zero, it becomes clear that the vector form cannot be further simplified.
Case-I. 2 = 0 , so that We get Case-II. 1 = 0, 2 � = 0 , so that We have Case-III. 1 = 0, 2 = 0 , so that We get Therefore, the set of symmetry subalgebras representing the optimal system can be described as follows Optimal system for Case-2.In this case, the algebra is three-dimensional with non-zero commutators, Table 1 displays the adjoint table, which assists us in calculating the optimal system of one-dimensional subalgebras.

We get
Case-III. 1 = 0, 3 = 0, 2 = 0 , so that We get Case-IV. 1 = 0, 2 = 0, 3 = 0 , so that We get Therefore, the set of symmetry subalgebras representing the optimal system can be described as follows Optimal system for Case-3.In this case, the algebra is four dimensional with nonzero commutators, Table 2 displays the adjoint table, which assists us in calculating the optimal system of one-dimensional subalgebras.

Invariant solutions and symmetry reductions
Symmetry reductions for Case-1.Case-I.

The associated Lagrange equation is
We obtain the similarity variables w = τ (θ), θ = x .By using this transformation, it becomes possible to express the simplified version of Eq. ( 4) in the following manner We recommend obtaining a numerical solution for the aforementioned ODE.

Case-II. Consider
The associated Lagrange equation is We obtain the similarity variables w = τ (θ), θ = t .By using this transformation, it becomes possible to express the simplified version of Eq. ( 4) in the following manner Since, τ = 0, this imples τ ′ = 0 which gives τ (θ) = c 1 .Hence, exact solution of (4) invariant under 2 is

The associated Lagrange equation is
We obtain the similarity variables w = τ (θ), θ = t − x c .By using this transformation, it becomes possible to express the simplified version of Eq. ( 4) in the following manner

The associated Lagrange equation is
We obtain the similarity variables w = τ (θ), θ = t .By using this transformation, it becomes possible to express the simplified version of Eq. ( 4) in the following manner Since, τ = 0, this imples τ ′ = 0 which gives τ (θ) = c 1 .Hence, exact solution of Eq. ( 4) invariant under 2 is Case-II.Consider Q 3 = 1 .The associated Lagrange equation is (36) www.nature.com/scientificreports/ We obtain the similarity variables w = τ (θ), θ = x .By using this transformation, it becomes possible to express the simplified version of Eq. ( 4) in the following manner We recommend obtaining a numerical solution for the aforementioned ODE.

Case-III. Consider
The associated Lagrange equation is We obtain the similarity variables w = τ (θ), θ = t − x .By using this transformation, it becomes possible to express the simplified version of Eq. ( 4) in the following manner This particular ODE cannot be solved using conventional analytical methods.Therefore, we recommend approaching it through numerical methods. Case-IV.
The associated Lagrange equation is We obtain the similarity variables w = τ (θ), θ = t + x .By using this transformation, it becomes possible to express the simplified version of Eq. ( 4) in the following manner We propose solving this ODE using numerical methods.
The associated Lagrange equation is We obtain the similarity variables w = τ (θ), θ = t .By using this transformation, it becomes possible to express the simplified version of Eq. ( 4) in the following manner Since, τ = 0, this imples τ ′ = 0 which gives τ (θ) = c 1 .Hence, exact solution of (4) invariant under 2 is

Case-II. Consider
t , θ = t .By using this transformation, it becomes possible to express the simplified version of Eq. ( 4) in the following manner This gives, Hence, exact solution of (4) invariant under 3 is

The associated Lagrange equation is
We obtain the similarity variables w = τ (θ), θ = x .By using this transformation, it becomes possible to express the simplified version of Eq. ( 4) in the following manner We obtain the similarity variables w = √ τ (θ) + 2t, θ = t 2 − x α 2 .By using this transformation, it becomes possible to express the simplified version of Eq. ( 4) in the following manner So, the solution of (4) in main variables becomes Hence, the solution of (4) in main variables becomes

The associated Lagrange equation is
We obtain the similarity variables w = √ τ (θ) − 2t, θ = t 2 + x α 2 .By using this transformation, it becomes possible to express the simplified version of Eq. ( 4) in the following manner In the next sections, we introduce some methods to deal with reduced ODEs arising in the reductions.

Generalized JEF method
In this section, we invoke the main steps for the generalized JEF method.This method is very efficient in generating periodic, double periodic, solitary wave, rational, and exponential function solutions.This method has a wide range of applications in engineering, biological sciences, chemical sciences, and mathematical physics.We briefly mention here the algorithm for the generalized JEF method.
(Step 1): Consider a nonlinear PDE in the explicit form of the dependent variable Step 2): We look for the transformation based on scaling and translation given by In the above equation, the parameter V indicates the speed of the wave, and for the stationary solutions V = 0 .Note that Eq. ( 56) is invariant under the transformation (57) this usually works as an existence criterion for the waveform solutions.The transformation (57) reduces the number of independent variables in the nonlinear PDE (56) and turns it into an ODE of the form This ODE (58) helps us to classify the waveform solutions of the nonlinear PDE (56).(Step 3): At this stage, we consider the general assumed solution for the ODE (58) as follows In the equation provided, the constants b q (where q = 1, 2, . . ., N ) represent coefficient values, and the func- tion S(θ) must fulfill the subsequent Jacobi elliptic equation the coefficients a 1 , a 2 and a 3 are nonzero real parameters.The values of these parameters are listed in Table 3.
(Step 4): Subsequently, the balance principle is applied to determine the appropriate balancing number N. (Step 5): By solving the system described as S q = 0 for all values of q, we obtain a collection of parameters.The subsequent relations are valid for the elliptic functions; (51) (56) X(w, w x , w t , w xx , w xt , . ..) = 0.

Periodic solutions of the S-KdV Eq. (4).
By employing the balance principle to the ODE (40), we determine the balancing number to be N = 1 .Subsequently, the following assumed form arises from Eq. (58) Continuing with step 5 of the algorithm mentioned earlier, we obtain Therefore, Eq. (62) transforms into Equation (64) results in a diverse range of solutions for the S-KdV Eq. ( 4), achieved by altering the parameters as outlined in Table 3.
For this case, we take S(θ) = sn(θ, ζ ) to obtain periodic wave solutions of the Eq.(40); When ζ is set to 1, Eq. ( 65) transforms into the solution corresponding to a shock wave By applying the transformation (57), the solution for the S-KdV Eq. ( 4) can be expressed as For this case, we take S(θ) = ds(θ, ζ ) to obtain periodic wave solutions of the Eq.(40);     Table 3. Types of solutions of (60).

Equation (71) degenerates to ζ → 1 in this case
Employing the transformation (57), the solution for the S-KdV Eq. ( 4) is obtained as We follow in a similar way for ζ → 1 For this case, we take S(θ) = cn(θ, ζ ) to obtain periodic wave solutions of the Eq.(40);
For this case, we take S(θ) = De θ to obtain exponential functions-based solutions of the Eq. (40); Employing the transformation (57), the solution for the S-KdV Eq. ( 4) is obtained as where, in the aforementioned subcases, F is a constant.

A description of the MAE procedure
The algorithm for the MAE procedure 22 is described here to be used for the soliton solutions for the reduced ODEs.We follow the first two steps of the algorithm for the MAE procedure as we did for the generalized JEF method in the previous section.(Step 3): At this stage, we proceed with the subsequently assumed solution for the ODE (58), (92) where , a φ , b φ , and a 0 are arbitrary constants, and τ (θ) satisfies the auxiliary equation In the above equation 1 , 2 , and 3 are arbitrary constants and � > 0, � = 1 .The subsequent solution of the auxiliary Eq. ( 104) is presented, taking into consideration different cases.56) for the solutions and subsequently, we replace the transformation defined in step 2. Now, we apply the method described above from the MAE procedure to the S-KdV Eq. ( 4).
Exact solutions for the S-KdV Eq. ( 4).By applying the balance principle to the ordinary differential equation (51), we deduce the balancing number to be N = 1 .Subsequently, the following assumed form emerges from Eq. ( 103), By substituting the aforementioned assumption into Eq.( 51) and utilizing Eq. (104), we can then solve the system � τ = 0 to obtain the subsequent set of parameters

Physical interpretations of the solutions
In this section, we demonstrate the physical interpretations of the solutions obtained in the above sections.Our findings include rational, trigonometric, hyperbolic and exponential functions-based analytical solutions.The elliptic sine amplitude function solutions lead to the shock wave of the S-KdV Eq. ( 4