In this manuscript, we focus on the application of recently developed analytical scheme, namely, the rational sine-Gordon expansion method (SGEM). Some new exact solutions of Riemann wave system and the nonlinear Gross−Pitaevskii equation (GPE) by using this method are extracted. This method is based on the general properties of the SGEM which uses the fundamental properties of trigonometric functions. Many novel analytical solutions such as dark, bright, mixed dark–bright, hyperbolic, and periodic wave solutions are successfully extracted. Physical meanings of solutions are simulated by the various figures in 2D and 3D along with the contour graphs. Strain conditions of the existence are also reported in detail. Finally, modulation instability analysis of the nonlinear GPE is studied in detail.

1. Introduction

In modern century, the real-world problems arising in various fields of daily life, particularly, engineering such as computer viruses, cyber security, artificial intelligence, magnetism, physics, oceanography, and so on have been symbolized via mathematical norms. Scientists have investigated their connections among multidisciplinary properties. For the last several decades, newly developed mathematical properties have been used to explain many physical problems [1–5]. In [6], scientists have applied the Hopf Bifurcation theorem on the models arising in the group competitive martial arts. In [7], the modified double Laplace transform method is handled to extract some results of the pseudo-hyperbolic telegraph model. The relativistic wave equation associated with the Schrödinger equation was studied in terms of traveling wave distribution in [8]. Wazwaz [9, 10] introduced the several optical solitons for (2 + 1)-dimensional Schrödinger (NLS) equation. In [11], modified exponential function method have been applied to the nonlinear Gerdjikov–Ivanov equation with the M-fractional. Hu et al. [12] focused on the rational and semirational properties of B-type Kadomtsev Petviashvili Boussinesq model. Kudryashov [13] presented the Lax pair and the first integrals for some mathematical properties. Ghanbari et al. [14] observed the rational function solutions for the extended Zakharov Kuzetsov equation. M-type dark soliton facts in optical fibers have been investigated by Yao et al. [15]. Hybrid method has been applied into Rosenau–Hyman equation [16]. Real quadratic fields have been focused on the Handy method in [17]. Generalized (3 + 1) Shallow Water-Like (SWL) equation was observed by Dusunceli [18]. The perturbed Schrödinger and the Heisenberg ferromagnetic spin chain have been investigated in [19, 20]. Park et al. [21] have observed the fifth-order Korteweg-de Vries equations. The multiple exp-function method has been applied on some mathematical models [22]. Chebyshev series have been used to investigate numerically the integro-differential equations in [23]. Duffing and diffusion reaction models have been investigated in the M-derivative operator in [24]. W-shape properties of the (3 + 1)-dimensional nonlinear evolution equations have been reported in [25]. The general and wide-spectral explicit properties of the complex coupled Maccari system have been introduced in [26]. The superconductivity models have been studied in [27]. The Benjamin–Bona–Mahony–Peregrine equation with power law nonlinearity has been studied in [28]. The finite element method have been used to observe a surface plasmon resonance (SPR) sensor in [29]. A current decoupling control strategy was considered on combining a fast terminal sliding mode control and an adaptive extended state observer [30]. A Robust Hammerstein–Wiener Model identification method was applied in [31]. A robust model predictive current control scheme was proposed in [32]. Liu and Liu [32] have studied on the photoacoustic microfluidic pumps [33]. A linear AC unit commitment formulation have been presented in [34]. Moreover, seismic-wave attenuation, shear-wave attenuations, and waves with a wide range of frequencies in digital core have been introduced [35–38]. Some control problems and optimization application have been presented [39–45]. In [46], bipolar cubic fuzzy graphs have been studied. The shallow ocean-waves and rogue waves with translational coordination have been observed in [47, 48]. The influence of Woods–Saxon potential was presented in [49]. Soliton, numerical, and closed solutions of nonlinear models have been investigated by using various schemes [49–56]. To represent the wave’s amplitude, simplest equation algorithms have been used by Inc et al. [57]. Az-Zo’bi et al. [58] studied and used to observe the pulses propagation with power nonlinearity via conformable. Some novel liquid crystals models have been investigated by using some analytical schemes in [59]. Ur Rahman et al. [60] extracted the deeper properties of the fractional regularized long-wave Burgers problem by using two different fractional operators, Beta and M-truncated. In [61], the fractional Huxley equation with Beta and M-truncated derivatives have been considered for reporting nonlinear coherent structures arising in a variety of environments, like spectrum energy, applied mathematics, mechanics, control theory, biology, seismology, and many more.

One of such models belongs to the Riemann wave systems (RWS) defined by the following:

(1)

being α, β, and γ are constants. RWS studied in this paper are used to explain the tidal and tsunami waves in ocean [62]. The nonlinear terms of first equation in Equation (1) give the stationary wave propagation properties in the frame of physical sense. Kundu et al. [63] investigated the parametric analysis via sine-Gordon expansion method (SGEM). Gurevich et al. [64] has used the slow modulation method. They have applied the generalized hodograph method to find some analytical solutions.

Another important model is considered as follows:

(2)

in which λ ≪ 1, N is a constant value and a_s is the axial and transverse harmonic oscillator frequencies. This equation is used to describe the rogue wave in the evolution of the macroscopic wave function of Bose–Einstein condensates (BECs) and also to investigate the deeper properties at the mean-field level [65], via

(3)

in which equation

, the nonlinear Gross−Pitaevskii equation (GPE) can be rewritten as follows [66, 67]:

(4)

in which

is conformable derivative order in 0 < θ < 1. Equation (4) arises in the magnetic field theory observed by magnetically tuning the interatomic interaction.

The paper is organized as follows. In Section 2, the rational sine-Gordon expansion method (RSGEM) is given in detail. In Section 3, RSGEM is applied to extract the stationary optical solitons, mixed dark, and bright solitons to the Equations (1) and (4). Additionally, stability properties of the complex mixed dark–bright soliton solutions are also reported. Finally, the paper is completed by presenting the important novelties of this paper in Section 4.

2. General Properties of RSGEM

This part introduces the general properties of RSGEM, First, let us consider the following sine-Gordon equation [68–70].

(5)

Here, m is a real constant with nonzero. Taking the traveling wave transformation as u(x, t) = U(ξ), ξ = μ(x − ct) into Equation (5) produces

(6)

where U = U(ξ),

and c, μ are also real constants with nonzero. With some calculations, it may be obtained as follows:

(7)

where k is an integral constant. For simplicity, by considering as

and

Equation (7) reads as follows:

(8)

in which w = w(ξ). Gaining a = 1 in Equation (8), we obtain the following two interesting and important relationship

(9)

(10)

Let us take into account the general mathematical model as follows:

(11)

Taking into account as Ψ = Ψ(x, t) = U(ξ), ξ = μ(x − ct), it is converted as follows:

(12)

Here

. In Equation (12), the trial equation for solution function may be considered as follows:

(13)

Equation (13) may be rewritten with the help of Equations (9) and (10) as following:

(14)

in which the values of n, m will be determined later via balance principle. After putting the necessary derivations of Equation (14) into Equation (12), we obtain an equation of sinⁱ(ω)cos^j(ω). Taking all these terms to zero yields a system of equations. Solving this system by using some computational programs, gives the values of A_i, B_i, c_i, d_i, μ, and c. Via these values of parameters A_i, B_i, c_i, d_i, μ, and c in Equation (13), we obtain the new traveling wave solutions to Equation (11).

3. Applications of Projected Scheme

In this part of the paper, the stationary soliton and new analytical solutions to the Riemann wave system and conformable GPE is studied by using RSGEM.

3.1. RSGEM to the Riemann Wave System of Equations

First of all, we apply RSGEM to the Riemann wave system of equations for reporting new stationary soliton solutions. Applying the traveling wave transformation defined by

(15)

into Equation (1) results in the following NODE:

(16)

where

(17)

After taking n = m especially in this paper, via the balance rule in Equation (16) gives n = m = 2. Getting n = m = 2 in Equation (14), one can write the trial equation of solution function given as follows:

(18)

Substituting Equation (18) and its second derivative into Equation (16), we obtain an equation of sinⁱ(w)cosⁱ(w). Getting all coefficients of these terms to zero, we gain a system of equations. Solving this system produces the values of parameters A_i, B_i, c₁, d₁, μ, ω, and δ which results in many entirely new traveling wave solutions to the Equation (1).

Case-1: when

into Equation (13) with n = 2, it yields the following entirely complex mixed dark–bright soliton solution to the RWE as follows

(19)

(20)

(21)

where

. Selecting several parameters values from the physical meanings of the RWE, we can observe the simulations of Equation (19) via Figure 1 in 3D, Figure 2 contour sense, and Figure 3 in 2D according to the specific time scala. From these simulations, one can see that this solution has singular properties.

Details are in the caption following the image — **Figure 1**
Open in figure viewer PowerPoint

The 3D simulations of V₁.

Case-2: choosing as

; for Equation (13) with n = 2, it produces entirely new complex mixed dark–bright soliton solution given by

(22)

(23)

(24)

where

. More strict simulations of Equations (22)–(26) are also seen by Figure 4 in three-dimensional, Figure 5 contour sense, and Figure 6 in two-dimensional in a specific time. From the Figures 4 to –6, it may be observed that this solution has dark–bright properties.

Case-3: once it is selected as

for Equation (13) with n = 2, and inserting these values into Equation (13) along with n = 2, we obtain dark–bright soliton solution as following:

(25)

(26)

(27)

where κ = αδμ², f(x, y, t) = δy + μx + κt. Moreover, −1 < B₁ < 1 for strain condition. The intersection points of Equations (25)–(27) are also observed by Figure 7 in three- and two-dimensional in a specific time and Figure 8 in the low region. From the Figures 7 and 8, it may be also seen that Equations (25)–(27) have singular property.

Case-4: if it is chosen as

, mixed-dark soliton solution to the Equation (1) is obtained as following

(28)

(29)

(30)

where κ = αδμ², f(x, y, t) = δy + μx + κt, and also δ, μ, α, B₁ are real constants and nonzero. The periodic wave behaviors of Equations (28)–(30) may be also presented by Figure 9. Figure 9 has stable wave propagation.

Case-5: when

results in the another entirely new exact solution as follows:

(31)

(32)

(33)

where

, and also δ, μ, A₂, d₁ are real constants and nonzero. Smooth wave distribution of Equations(31)–(33) are also plotted by Figure 10 in three-dimensional, Figure 11 contour sense, and Figure 12 in two-dimensional in a specific time. From the Figures 10–12, these solution have the dark–bright properties.

Case-6: using

produces combined dark–bright stationary soliton solution as follows:

(34)

(35)

(36)

where

and δ, μ, A₂ are real constants and nonzero. Intercrossing wave distribution of Equations (34)–(36) are also introduced by Figures 13 and 14. Such sketches is estimated that soliton wavelength is closely related to the gravitational potential. Thus, it may be said that the gravitational potential affect the wavelength and its power. From the Figures 13 and 14, it is estimated that they may be used to explain the gravitational potential energy properties.

3.2. RSGEM for the Nonlinear GPE in Conformable Operator

This subsection of the manuscript extracts the stationary soliton, mixed dark–bright and complex traveling wave solutions to the conformable GPE by using RSGEM. Applying wave transform given as follows:

(37)

in which α, β, κ are nonzero and also 0 < θ < 1 into Equation (4) converts

(38)

Here Υ is a real constant. Considering balance properties in Equation (38), we find n = 1 for Equations (13) and (14). Using n = 1 Equation (14) reaches the following trial solution function given as:

(39)

where B₁, d₁ are each nonzero constants. Putting Equation (39) and its second derivation in Equation (38) gives various terms containing sin^s⁡(w)cos^s⁡(w). This produces a system of algebraic equations. Solving this system, we find the desired analytical solutions as follows to the conformable GPE.

Case-1: using

produces mixed dark–bright stationary soliton solution as follows:

(40)

where α, A₀, A₁, p, d₁, 0 < θ < 1 are real constants and nonzero. Selecting suitable to the physical properties of conformable GPE, we plot various wave distributions Figure 15. From the Figure 15, it is estimated that it has mixed dark–bright stationary.

3.3. Stability Properties of q₁(X, T)

In this subsection of this paper, Hamiltonian system is considered and applied on the mixed dark–bright stationary soliton solution to investigate its stability on a general range. This system is introduced in detail [71, 72] as following

(41)

in which Π symbolizes the momentum function and also w is used to express the wave speed and Ψ(ζ) is the projected analytical solution. Sandstede et al. [72] observed some important models by using various spaces properties such as Banach and so on. The sufficient condition for the stability is as follows:

(42)

If we take into account Equations (41), and (42) on Equation (40), we obtain

(43)

Thus, the Equation (40) solution is unconditionally stable.

Case-2: when

reaches the mixed dark–bright stationary soliton solution as follows:

(44)

where p, α, A₀, A₁, d₁, κ, 0 < θ < 1 are real constants and nonzero. With the suitable values related to the physical properties for governing model, various wave distributions may be observed Figure 16. From the Figure 16, it is estimated that it has some important wave propagations.

3.4. Stability Properties of q₂(X, T)

Here, if we reconsider the momentum function as follows:

(45)

and the sufficient condition for the stability condition by

(46)

on Equation (44), we obtain

(47)

which is the mixed dark–bright stationary soliton solution given as Equation (44) is unconditionally stable.

4. Modulation Instability Analysis of the Nonlinear GPE

In this section of the article, we discuss the modulation instability (MI) analysis for the stationary solutions of Equation (4) by assuming the following stationary solutions

(48)

where Ω represents the incident power. We investigate the evolution of the perturbation Ξ(X, T) using the concept of linear stability analysis. Substituting Equation (48) into Equation (4) and linearizing the result in Ξ(X, T), we acquire

(49)

supposing solutions of Equation (49) are in the following

(50)

where β is the wave number, and α is the frequency. Putting Equation (50) in Equation (49) gives a set of two homogenous equations as follows:

(51)

(52)

From the Equations (51) and (52), one can easily obtain the following coefficients matrix of ζ and ξ as follows:

(53)

The coefficient matrix in Equation (53) has a nontrivial solution if the determinant equal to zero. By expanding the determinant, we obtain the following

(54)

Equation (54) has the following solutions

(55)

and

(56)

The stability of the steady state is determined by Equations (55) and (56). If the wave number has an imaginary part, the steady-state solution is unstable since the perturbation grows exponentially. But if the wave number is real, the steady state is stable against small perturbation. Thus, the condition is necessary for the existence of MI to occur from Equations (55) and (56) is when either

(57)

(58)

Now for investigating instability modulation gain spectrum it should be noticed that

(59)

We have the following cases

Case 1: when

(60)

We have the following subcases

Case 1.1: for these values

, of constants in Equation (59) we have

(61)

Case 1.2: when it is considered as

, of constants in Equation (59) we have

(62)

Case 1.3: taking as

, of constants in Equation (59) we have

(63)

Case 1.4: once they are considered as

of constants in Equation (59) we have

(64)

These subcases can be expressed in Figure 17 between −10 < α < 10.

5. Conclusions

In this work, we introduced a recently developed scheme being RSGEM. RSGEM has been handled to the Riemann wave system and nonlinear GPE in conformable frame. Main novelty of this paper is the solutions such as complex, mixed dark–bright, and hyperbolic in conformable operator were extracted. Via graphical illustrations, the dynamical behaviors of solutions found have been reported. The stability properties of the obtained solutions have been introduced, as well. Moreover, the strain conditions of solutions for valid have been also given.

It is estimated that these dark solitons may be closely related to the gravitational dynamical potential [73]. MI analysis of the nonlinear Gross–Pitaevskii equation has been studied, as well. Its various wave simulations have been also plotted in Figure 17. From these results, it may be suggested that the RSGEM is a power tool to solve such nonlinear partial models arising in applied and engineering sciences. Therefore, it may be also applied to investigate deeper properties of some real-world problems [74–77].

As the future direction of this topic, it is envisaged and suggested that some important analytical schemes will be developed by the experts studying in these areas of nonlinear sciences to solve more complex PDEs. This comes from the properties of real-world problems.

Conflicts of Interest

The authors declare that they have no known conflicts of interest.

Authors’ Contributions

W. Gao majorly contributed and H.M. Baskonus contributed on typing collection and formal analyzing.

Open Research

Data Availability

The data that support the findings of this study are available within the article.

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The Modulation Instability Analysis and Analytical Solutions of the Nonlinear Gross−Pitaevskii Model with Conformable Operator and Riemann Wave Equations via Recently Developed Scheme

Abstract

1. Introduction

2. General Properties of RSGEM

3. Applications of Projected Scheme

3.1. RSGEM to the Riemann Wave System of Equations

3.2. RSGEM for the Nonlinear GPE in Conformable Operator

3.3. Stability Properties of q₁(X, T)

3.4. Stability Properties of q₂(X, T)

4. Modulation Instability Analysis of the Nonlinear GPE

5. Conclusions

Conflicts of Interest

Authors’ Contributions

Open Research

Data Availability

References

Citing Literature

Figures

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The Modulation Instability Analysis and Analytical Solutions of the Nonlinear Gross−Pitaevskii Model with Conformable Operator and Riemann Wave Equations via Recently Developed Scheme

Abstract

1. Introduction

2. General Properties of RSGEM

3. Applications of Projected Scheme

3.1. RSGEM to the Riemann Wave System of Equations

3.2. RSGEM for the Nonlinear GPE in Conformable Operator

3.3. Stability Properties of q1(X, T)

3.4. Stability Properties of q2(X, T)

4. Modulation Instability Analysis of the Nonlinear GPE

5. Conclusions

Conflicts of Interest

Authors’ Contributions

Open Research

Data Availability

References

Citing Literature

Figures

References

Related

Information

3.3. Stability Properties of q₁(X, T)

3.4. Stability Properties of q₂(X, T)