Abstract
In this paper, we present a modified version of the potential-KdV equation by adding a new stochastic term. The new stochastic potential-KdV describes the propagation of nonlinear optical solitons and photons and appears in the applications of electric-circuits and multi-component plasmas. By using the Cole-Hopf transformation and Hirota bilinear method, we derive novel multi-solitons, lumps, and breather wave-solutions to the proposed model. Also, we provide some graphical analysis to study the impact of the model’s coefficients on the propagation of the recovery solutions. Finally, all the reported solutions in this work are checked by direct substitution in the governing equation.
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1 Introduction
Finding explicit solutions to nonlinear equations is one of the main pillars of understanding the dynamics of many physical applications and complex processes in chemistry, biology, geophysics, fluids, and nonlinear optical fibers. Over the past years, many effective schemes were used to solve nonlinear equations such as the tanh-method, sine-cosine function method (Wazwaz 2007; Alquran and Al-Khaled 2011; Alquran 2012), exp-function method (He and Wu 2006; Jaradat and Alquran 2022; Inan et al. 2022), mapping method El-Wakil and Abdou (2006), direct method Ma and Chen (2009), polynomial method (Huang 2006; Alquran et al. 2021, 2022), Kudryashov-expansion (Jaradat et al. 2021; Jaradat and Alquran 2020; Alquran 2021), Riccati-expansion Conte and Musette (1992), modified rational sine-cosine and sinh-cosh method (Ali et al. 2022; Alquran and Alhami 2022; Alquran 2021; Alquran and Alqawaqneh 2022), and many others (Akinyemi et al. 2022, 2022; Arnous et al. 2022; Sulaiman 2020; Baskonus et al. 2017; Alquran 2022).
Recently, W.X. Ma and other scholars implemented the Hirota bilinear form and Cole-Hopf transformation of the form \(u=\alpha (\ln {\phi })_x\) and \(u=\beta (\ln {\phi })_{xx}\) to extract new types of singular solutions known as lumps and breather waves (Ma 2015, 2013; Ma et al. 2016; Ma 2013; Ma et al. 2009). For example, if the test function \(\phi \) is a quadratic polynomial, the resulting solution is of type lump-soliton. While as, a linear combination of quadratic polynomial with the sine or cosine function will produce lump-periodic. Moreover, linear combinations of the sine or cosine with exponential function, trigonometric with hyperbolic functions, and combined exponential-trigonometric-hyperbolic functions, will produce different types of breather waves (Sulaiman et al. 2021, 2021; Alquran and Alhami 2022a, b; Feng and Bilige 2021; Sulaiman et al. 2021; Kumar et al. 2022).
In this work, we present for the first time a modified version of the potential-KdV under the name stochastic potential-KdV (spKdV) which takes the following form:
where \(\alpha \) is the stochastic parameter, \(\beta \) is the nonlinearity coefficient, and \(\gamma \) refers to the dispersion coefficient. In the absence of the stochastic term, \(\alpha =0\), the above equation is known as the potential-KdV and generally seen during the study of water waves. The spKdV serves as an approximate model for the description of week dispersive effects on the propagation of nonlinear optical-soliton and photons, and appears in the applications of electric-circuits and multi-component plasmas.
The contribution of the current paper is threefold. First, we derive the Hirota bilinear form of the new proposed model. Then, we construct multi-waves, lump-type. and breather-type solutions upon different selections of the Cole-Hopf test-function. Finally, we study the impact of the model’s coefficients acting on the propagation of the obtained solutions.
2 Cole-Hopf transformation
To find a suitable Cole-Hopf transformation to (1.1), we apply the bilinear method which requires that the function
satisfies the linear terms of (1.1). As a result, we get the dispersion relation:
Then, we assume the solution of (1.1) in the following form
Substitute (2.3) in (1.1) to obtain
Finally, we assign the following Cole-Hopf transformation
where f(x, t) is known as the test function.
3 One-soliton and two-soliton solutions
To extract one-soliton and two-soliton solutions to (1.1), we consider the following steps:
-
Generalize the dispersion relation (2.2) into
$$\begin{aligned} b_i= a_i \alpha + a_i^3 \gamma . \end{aligned}$$(3.1) -
Choose the function f(x, t) as
$$\begin{aligned} f(x,t)=1+e^{a_1 x-\left( a_1 \alpha + a_1^3 \gamma \right) t}. \end{aligned}$$(3.2) -
Substitute (2.5) and (3.2) in (1.1) to get the one-soliton solution labeled as \(\psi _1\)
$$\begin{aligned} \psi _1(x,t)=\frac{6\gamma a_1 e^{a_1 x}}{\beta (e^{a_1\left( \alpha + a_1^2 \gamma \right) t}+e^{a_1 x})}. \end{aligned}$$(3.3) -
Consider the following two dispersion relations:
$$\begin{aligned} b_1= & {} a_1 \alpha +c a_1^3 \gamma , \nonumber \\ b_2= & {} a_2 \alpha +c a_2^3 \gamma . \end{aligned}$$(3.4) -
Consider the new form of f(x, t),
$$\begin{aligned} f(x,t)=1+e^{a_1 x-b_1 t}+e^{a_2 x-b_2 t}+\frac{(a_1-a_2)^2}{(a_1+a_2)^2}\ e^{(a_1 x-b_1 t)+(a_2 x-b_2 t)}. \end{aligned}$$(3.5) -
Substitute (2.5) and (3.5) in (1.1) to get the two-soliton solution labeled as \(\psi _2\)
$$\begin{aligned} \psi _2(x,t)=\frac{6\gamma \left( a_1 e^{a_1 \delta _1}+a_2 e^{a_2 \delta _2}+\frac{(a_1-a_2)^2}{a_1+a_2}\ e^{(a_1+a_2)\theta }\right) }{\beta \left( 1+e^{a_1 \delta _1}+e^{a_2 \delta _2}+\frac{(a_1-a_2)^2}{(a_1+a_2)^2}\ e^{(a_1+a_2)\theta }\right) }, \end{aligned}$$(3.6)where \(\delta _i=-\alpha t+x-\gamma a_i^2 t\) and \(\theta =\delta _1+\gamma a_1 a_2 t-\gamma a_2^2 t\).
4 Hirota bilinear equation
In this section, we construct the Hirota bilinear form to the proposed spKdV (1.1). First, we recall the definition of the Hirota operator D
where \(p,\ q \in {\textbf {C}}^{\infty }({\mathbb {R}}^{2})\). Next, we consider the following assumption
Then, the function v(x, t) is to be assumed as
Substitution of (4.4) in (4.3), produces the Hirota’s form to spKdV as
which is equivalent to
5 Lump solutions
The aim of this section is to derive two types of lump solutions to the spKdV.
5.1 Lump-soliton
To obtain lump-soliton, we choose the test-function \(\phi (x,t)\) to be a quadratic polynomial as:
where
The parameters \(a_{i,j},\ (i,j=1,2,3)\) and \(\sigma \) are real constants to be determined. Substitute (5.1) in (4.5), equate the coefficients of different polynomials of x, t to zero, then solve the resulting system to get two solution’s sets:
Set I:
Thus,
Accordingly, the first lump-soliton solution labeled as \(\psi _{L_1}\) is
Set II:
Therefore,
and the second lump-soliton solution labeled as \(\psi _{L_2}\) is
5.2 Lump-periodic
Linear combination of quadratic and trigonometric functions produces lump-periodic solution. Therefore, the test function is to be chosen as:
where \(\quad Y=\begin{bmatrix} 1 \\ t \\ x \\ \end{bmatrix}\). Now, insert (5.7) in (4.5) leads to:
Set I:
Let \(\triangle =p_1(x-\alpha t )+\gamma p_1^3 t+p_3\), then
and the first lump-periodic solution labeled as \(\psi _{L_3}\) is
Set II:
Accordingly, the test function \(\phi _2\) is of the form
Thus, the second lump-periodic solution labeled as \(\psi _{L_4}\) is
6 Breather wave solutions
Different types of breather wave solutions can be constructed based on the selection of the test function \(\phi (x,t)\). Here, we derive three types of breather solutions.
6.1 Type-1
This type is constructed by a linear combination of cosine and exponential functions defined as:
Substitute (6.1) in (4.5) to get
As a result, the breather type-1 solution is
where \(T=k_1(x-(\alpha +\gamma k_1^2-3\gamma k_2^2)t)\) and \(P=k_2(x-(\alpha +3\gamma k_1^2-\gamma k_2^2)t)\).
6.2 Type-2
This type is a linear combination of trigonometric and hyperbolic functions, i.e.,
Substitution of (6.4) in (4.5) produces the following outputs:
Thus, the breather type-2 solution is
where \(Y= k_1 x+k_3-(\alpha k_1+\gamma k_1^3-3\gamma k_1 l_1^2)t\) and \(Z= l_1 x+l_3-(\alpha l_1+3\gamma k_1^2 l_1-\gamma l_1^3)t\).
6.3 Type-3
This type is a linear combination of three functions; exponential, trigonometric and hyperbolic, i.e.,
Substitution of (6.7) in (4.5) gives three cases.
Case I: \(\kappa _3=0\) and \(\varrho =\alpha _2=-(\alpha +4\gamma )\). Then,
Case II: \(\kappa _2=-\frac{\kappa _4^2}{4\kappa _1},\kappa _3=0,\alpha _2=-2\alpha -8\gamma -\varrho \). Then,
Case III: \(\kappa _2=-\frac{\kappa _3^2}{4\kappa _1},\kappa _4=0,\varrho =-(\alpha -2\gamma )\) and \(\alpha _1=-(\alpha +2\gamma )\). Then,
7 Graphical analysis
The aim of this section is twofold. First, we plot the obtained solutions to recognize the physical structures of (1.1). Second, we study the impact of the stochastic, nonlinearity, and dispersion parameters, \(\alpha \), \(\beta \), \(\gamma \), being acting on the propagation of the stochastic potential-KdV. Figure 1, shows the one- and the two-soliton solutions. Figure 2, shows the lump-soliton and the lump-periodic. Figure 3, shows three different types of breather waves.
To investigate the impact of the aforementioned parameters on the propagation of (1.1), we study the solution-function \(\psi _{B_5}(x,t)\), see Fig. 4, where we observe the following physical properties:
-
The propagation is transitive when \(\alpha \) changes its sign.
-
The propagation has a reflexive relation when \(\beta \) changes its sign.
-
The propagation is symmetric due to the sign of \(\gamma \).
8 Conclusion
This work included the study of a new extension of the potential KdV model by adding a new stochastic term \(\alpha \psi _x(x,t)\). The new model describes the propagation of nonlinear optical solitons and photons and appears in the applications of electric-circuits and multi-component plasmas. Several solutions of types multi-waves, lumps, and breathers are generated to the proposed model by means of Cole-Hopf transformation and Hirota bilinear method. Also, we performed a graphical analysis to study the impacts of the model’s parameters acting on the propagations of the recovery solutions.
For future work, we aim to extract lumps, breathers, and rogue-wave solutions to nonlinear models involving complex-valued field-functions. Moreover, we may study systems of nonlinear equations and investigate the possibility of having such types of solutions similar to those reported in this work.
References
Akinyemi, L., Inc, M., Khater, M.M.A., Rezazadeh, H.: Dynamical behaviour of Chiral nonlinear Schrodinger equation. Opt. Quant. Electron. 54, 191 (2022)
Akinyemi, L., Senol, M., Osman, M.S.: Analytical and approximate solutions of nonlinear Schrodinger equation with higher dimension in the anomalous dispersion regime. J. Ocean Eng. Sci. 7(2), 143–154 (2022)
Ali, M., Alquran, M., Salman, O. Bani.: A variety of new periodic solutions to the damped \((2+1)\)-dimensional Schrodinger equation via the novel modified rational sine-cosine functions and the extended tanh-coth expansion methods. Results. Phys. 37, 105462 (2022)
Alquran, M.: Solitons and periodic solutions to nonlinear partial differential equations by the sine-cosine method. Appl. Math. Inf. Sci. 6(1), 85–88 (2012)
Alquran, M.: Optical bidirectional wave-solutions to new two-mode extension of the coupled KdV-Schrodinger equations. Opt. Quant. Electron. 53, 588 (2021)
Alquran, M.: Physical properties for bidirectional wave solutions to a generalized fifth-order equation with third-order time-dispersion term. Results Phys. 28, 104577 (2021)
Alquran, M.: New symmetric bidirectional progressive surface-wave solutions to a generalized fourth-order nonlinear partial differential equation involving second-order time-derivative. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.06.021
Alquran, M., Al-Khaled, K.: The tanh and sine-cosine methods for higher order equations of Korteweg-de Vries type. Phys. Scr. 84, 025010 (2011)
Alquran, M., Alhami, R.: Convex-periodic, kink-periodic, peakon-soliton and kink bidirectional wave-solutions to new established two-mode generalization of Cahn-Allen equation. Results Phys. 34, 105257 (2022)
Alquran, M., Alhami, R.: Dynamics and bidirectional lumps of the generalized Boussinesq equation with time-space dispersion term: Application of surface gravity waves. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.05.010
Alquran, M., Alhami, R.: Analysis of lumps, single-stripe, breather-wave, and two-wave solutions to the generalized perturbed-KdV equation by means of Hirota’s bilinear method. Nonlinear Dyn. (2022). https://doi.org/10.1007/s11071-022-07509-0
Alquran, M., Ali, M., Jadallah, H.: New topological and non-topological unidirectional-wave solutions for the modified-mixed KdV equation and bidirectional-waves solutions for the Benjamin Ono equation using recent techniques. J. Ocean Eng. Sci. 7(2), 163–169 (2022)
Alquran, M., Alqawaqneh, A.: New bidirectional wave solutions with different physical structures to the complex coupled Higgs model via recent ansatze methods: applications in plasma physics and nonlinear optics. Opt. Quant. Electron. 54, 301 (2022)
Alquran, M., Sulaiman, T.A., Yusuf, A.: Kink-soliton, singular-kink-soliton and singular-periodic solutions for a new two-mode version of the Burger-Huxley model: applications in nerve fibers and liquid crystals. Opt. Quant. Electron. 53, 227 (2021)
Arnous, A.H., Mirzazadeh, M., Akinyemi, L., Akbulut, A.: New solitary waves and exact solutions for the fifth-order nonlinear wave equation using two integration techniques. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.02.012
Baskonus, H.M., Bulut, H., Sulaiman, T.A.: Investigation of various travelling wave solutions to the extended \((2+1)\)-dimensional quantum ZK equation. Eur. Phys. J. Plus 132, 482 (2017)
Conte, R., Musette, M.: Link between solitary waves and projective Riccati equations. J. Phys. A: Math. Gen. 25(21), 5609 (1992)
El-Wakil, S.A., Abdou, M.A.: The extended mapping method and its applications for nonlinear evolution equations. Phys. Lett. A 358(4), 275–282 (2006)
Feng, Y., Bilige, S.: Resonant multi-soliton, M-breather, M-lump and hybrid solutions of a combined pKP-BKP equation. J. Geom. Phys. 169, 104322 (2021)
He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos, Solitons Fractals 30(3), 700–708 (2006)
Huang, W.: A polynomial expansion method and its application in the coupled Zakharov-Kuznetsov equations. Chaos, Solitons & Fractals 29(2), 365–371 (2006)
Inan, I.E., Inc, M., Rezazadeh, H., Akinyemi, L.: Optical solitons of \((3+1)\)-dimensional and coupled nonlinear Schrodinger equations. Opt. Quant. Electron. 54, 246 (2022)
Jaradat, I., Alquran, M.: Construction of solitary two-wave solutions for a new two-mode version of the Zakharov-Kuznetsov equation. Mathematics 8(7), 1127 (2020)
Jaradat, I., Alquran, M.: Geometric perspectives of the two-mode upgrade of a generalized Fisher-Burgers equation that governs the propagation of two simultaneously moving waves. J. Comput. Appl. Math. 404, 113908 (2022)
Jaradat, I., Alquran, M., Ali, M., Sulaiman, T.A., Yusuf, A., Katatbeh, Q.: New mathematical model governing the propagation of two-wave modes moving in the same direction: classical and fractional potential KdV equation. Rom. Rep. Phys. 73(3), 118 (2021)
Kumar, S., Malik, S., Rezazadeh, H., Akinyemi, L.: The integrable Boussinesq equation and it’s breather, lump and soliton solutions. Nonlinear Dyn. 107, 2703–2716 (2022)
Ma, W.X.: Bilinear equations, Bell polynomials and linear superposition principle. J. Phys: Conf. Ser. 411(1), 012021 (2013)
Ma, W.X.: Bilinear equations and resonant solutions characterized by Bell polynomials. Rep. Math. Phys. 72(1), 41–56 (2013)
Ma, W.X.: Lump solutions to the Kadomtsev-Petviashvili equation. Phys. Lett. A 379(36), 1975–1978 (2015)
Ma, W.X., Chen, M.: Direct search for exact solutions to the nonlinear Schrodinger equation. Appl. Math. Comput. 215(8), 2835–2842 (2009)
Ma, W.X., Qin, Z.Y., Lu, X.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn. 84, 923–931 (2016)
Ma, W.X., Zhou, R., Gao, L.: Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in \((2+1)\) dimensions. Mod. Phys. Lett. A 24(21), 1677–1688 (2009)
Sulaiman, T.A.: Three-component coupled nonlinear Schrodinger equation: optical soliton and modulation instability analysis. Phys. Scr. 95(6), 065201 (2020)
Sulaiman, T.A., Yusuf, A., Alquran, M.: Dynamics of lump solutions to the variable coefficients \((2+1)\)-dimensional Burger’s and Chaffee-infante equations. J. Geom. Phys. 168, 104315 (2021)
Sulaiman, T.A., Yusuf, A., Alrazi, A., Alquran, M.: Dynamics of lump collision phenomena to the \((3+1)\)-dimensional nonlinear evolution equation. J. Geom. Phys. 169, 104347 (2021)
Sulaiman, T.A., Yusuf, A., Alrazi, A., Alquran, M.: Breather waves, analytical solutions and conservation laws using Lie-Backlund symmetries to the \((2+1)\)-dimensional Chaffee-Infante equation. J. Ocean Eng. Sci. (2021). https://doi.org/10.1016/j.joes.2021.12.008
Wazwaz, A.M.: The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations. Appl. Math. Comput. 184(2), 1002–1014 (2007)
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Alhami, R., Alquran, M. Extracted different types of optical lumps and breathers to the new generalized stochastic potential-KdV equation via using the Cole-Hopf transformation and Hirota bilinear method. Opt Quant Electron 54, 553 (2022). https://doi.org/10.1007/s11082-022-03984-2
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DOI: https://doi.org/10.1007/s11082-022-03984-2