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Exact solutions of Hirota–Maccari system forced by multiplicative noise in the Itô sense:

Exact solutions of Hirota–Maccari system forced by multiplicative noise in the Itô sense: In this article, the stochastic Hirota–Maccari system forced in the Ito ˆ sense by multiplicative noise is considered. We just use the He’s semi-inverse method, sine–cosine method, and Riccati–Bernoulli sub-ODE method to get new stochastic solutions which are hyperbolic, trigonometric, and rational. The major benefit of these three approaches is that they can be used to solve similar models. Furthermore, we plot 3D surfaces of analytical solutions obtained in this article by using MATLAB to illustrate the effect of multiplicative noise on the exact solution of the Hirota–Maccari method. Keywords Stochastic Hirota–Maccari system, multiplicative noise, sine–cosine method, He’s semi-inverse method Introduction Nonlinear partial differential equations (NLPDEs) are used extensively in the modeling of physiology, fluid me- chanics, physiology, chemistry, physical phenomena, etc. The analytical solutions of nonlinear differential equations which describe mathematical models are not easy to obtain. For NLPDEs, there are many appropriate numerical and analytical methods in the literature, developed by authors such as the exp(f(ς))-expansion method, the Jacobi 2 3,4 5,6 elliptic function method, the variational iteration method, the ðG =GÞ-expansion method, the homotopy per- 7 8 9 turbation method, the modified homotopy perturbation method, the Riccati–Bernoulli sub-ODE method, per- 10,11 12,13 14–16 17 turbation method, He’s frequency formulation, the sine–cosine method, the Hirota’s method, the tanh– 18,19 sech method, etc. While there are some methods to find the analytical solutions of fractional partial differential 20–23 24,25 equations such as the fractal variational principle, the Lie group analysis method, and the perturbation 26–28 method. Deterministic models of differential equations were widely used before the 1950s to explain the system’s dynamics in applications. Nevertheless, it is clear that the events that exist in the world today are generally not deterministic in nature. Therefore, random influences have been important when modeling various physical phenomena occurring in environmental sciences, meteorology, biology, engineering, oceanography, physics, and so on. Equations that include noise or fluctuations are called stochastic differential equations. Obtaining exact solutions for a stochastic NLPDEs is one of the most essential parts of nonlinear science. For obtaining exact stochastic solutions, there are many papers such as Refs. 29–34, and the references therein. Department of Mathematics, Faculty of Science, University of Ha’il, Saudi Arabia Department of Mathematics, Faculty of Science, Mansoura University, Egypt Section of Mathematics, International Telematic University Uninettuno, Italy Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj, Saudi Arabia Corresponding author: Wael W. Mohammed, Department of Mathematics, Faculty of Science, University of Ha’il, Saudi Arabia. Email: wael.mohammed@mans.edu.eg Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). Mohammed et al. 75 In this article, the following stochastic Hirota–Maccari system (HMs) with multiplicative noise is considered in the Itos ˆ ense iφ þ φ þ iφ þ φψ  ijφj φ þ σφW ¼ 0 (1) t xy xxx x 3ψ þ jφj ¼ 0 (2) where φ = φ(t, x, y) and ψ = ψ(t, x, y) are complex and real functions, respectively. σ is a noise strength, and W ¼ dW =dt is the time derivative of standard Wiener process W(t). We say the stochastic integral φðsÞdWðsÞ is Ito, ˆ if we calculate the stochastic integral at the left-end. We restrict ourselves in this article to the case of noise is a constant in space. The HMs (1–2), which was recently obtained by Maccari, has been shown to be integrable in the sense of Lax. Moreover, it was suggested in Ref. 35 that this system would pass the Painleve test. The authors described the modulation stability of the HMs, which implies that this system is stable under small perturbation. In the deterministic case, that is, without noise (σ = 0), many authors have studied a number of approaches to get the 36 37 38 exact solutions of the H–M system (1–2) such as Hirota bilinear method, Painleve test, Painleve approach, Weierstrass 39 40 elliptic function expansion method, complex hyperbolic function method, the sech–csch method, sec–tan ansatz method, the rational sinh–cosh ansatz method and the tanh–coth method, the improved tanðφðρÞ=2Þ-expansion method and general projective Riccati equation method, the generalized Kudryashov method, and the extended trial equation method. While the stochastic H–M system is not yet studied. The main objective of this article is to get the exact solutions of stochastic Hirota–Maccari system (1–2) forced by multiplicative noise in the Ito ˆ sense by using three different methods as the He’s semi-inverse method, sine–cosine method, and Riccati–Bernoulli sub-ODE method. In addition, we investigate the influence of multiplicative noise on the exact solution of stochastic HMs (1–2) by introducing some graphical representations using the MATLAB package. This article is, to the best of our knowledge, the first to obtain the exact solution for stochastic HMs (1–2) forced by multiplicative noise in the Itosense. The rest of this article is described as follows. In the next section which is Traveling wave eq. for stochastic HMs, the traveling wave equation for the stochastic HMs is obtained (1–2). While in The exact solutions of stochastic HMs section, the exact stochastic solutions of the stochastic HMs (1–2) are obtained by using three different methods as He’s semi- inverse method, the sine–cosine method, and Riccati–Bernoulli sub-ODE method. In The impact of noise on the solutions of HMs section, the influence of multiplicative noise on the exact solution of HMs (1–2) is studied. Finally, the conclusions of this article are given. Traveling wave eq. for stochastic HMs To obtain the traveling wave equation of the HMs (1–2), let us use the next wave transformation iðθþσWðtÞÞ φðt, x, yÞ¼ uðςÞe , ψðt, x, yÞ¼ vðςÞ (3) with ς ¼ðb x þ b y þ b tÞ, θ ¼ a x þ a y þ a t 1 2 3 1 2 3 where a , b are nonzero constants for k = 1,2,3,and σ is the noise strength. Substituting equation (3) into (1) and (2) and k k using dφ i 0 2 iðθþσWðtÞÞ i ¼ ib u  a u  σuW  σ u e , 3 3 t dt 2 dφ 0 2 3 iðθþσWðtÞÞ ijφj ¼  ib u u þ a u e , 1 1 dx d φ 3 000 2 00 2 0 2 0 3 iðθþσWðtÞÞ (4) ¼ b u þ 3ia b u  2a b u  a b u  ia u e 1 1 1 3 1 1 1 1 1 dx d φ 00 0 0 iðθþσWðtÞÞ ¼ðb b u þ ib a u þ ib a u  a a uÞe , 1 2 1 2 2 1 1 2 dxdy dψ 0 2 ¼ b v , jφj ¼ b u 1 2 dx 76 Journal of Low Frequency Noise, Vibration and Active Control 41(1) to get the following system 2 00 3 3 b b  3a b u þ a u  a þ a a þ a u þ uv ¼ 0 (5) 1 2 1 1 3 1 2 1 1 0 2 3b v þ b u ¼ 0 (6) 1 2 Integrating equation (6) once with respect to ς to obtain 3b v þ b u ¼ c (7) 1 2 where c is an integration constant. For simplicity here, we put c = 0, hence equation (7) becomes v ¼ u (8) 3b Substituting equation (8) into (5), we have the following traveling wave equation 00 3 u  Θ u  Θ u ¼ 0 (9) 1 2 where 1 a þ a a þ a 3 1 2 Θ ¼ and Θ ¼ (10) 1 2 2 2 3b b b  3a b 1 1 2 1 1 We note that when Θ is negative, then equation (9) adopts a periodic solution (see for instance Refs. 44,45). The exact solutions of stochastic HMs In this section, we implement three different methods as He’s semi-inverse method, sin–cosine method, and Riccati– Bernoulli sub-ODE method, respectively, to obtain the solutions of equation (9). After then, we have the stochastic exact solution of the HMs (1–2). He’s semi-inverse method The first method we are using to seek the exact solution of the HMs (1–2) is He’s semi-inverse method. The following variational formulation can be constructed from equation (9)by using He’s semi-inverse method mentioned in Refs. 46–48 as 1 1 1 0 4 2 JðuÞ¼ ðu Þ þ Θ u þ Θ u dς (11) 1 2 2 4 2 We suppose the solitary wave solution of equation (9) according to Ref. 49 takes the form uðςÞ¼ K sechðςÞ (12) where K is an unknown constant. Substituting equation (12) into (11), we get ∞ 2 4 2 K K K 2 2 4 2 J ¼ sech ðςÞtanh ðςÞ Θ sech ðςÞþ Θ sech ðςÞ dς 1 2 2 4 2 2 4 2 K K K ¼  Θ þ Θ 1 2 6 6 2 Differentiating J with respect to K and putting ð∂J =∂KÞ¼ 0 ∂J 1 2Θ ¼ ð1 þ 3Θ ÞK  K ¼ 0 (13) ∂K 3 3 Mohammed et al. 77 We get by solving equation (13) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ 3Θ Þ 3 K ¼ ¼ b ð1 þ 3Θ Þ 1 2 2Θ 2 Hence, the solution of equation (9)is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðςÞ¼ b ð1 þ 3Θ ÞsechðςÞ 1 2 Now, the exact solution of the HMs (1–2) is iðθþσWðtÞÞ φ ðt, x, yÞ¼ uðςÞe 1;1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (14) iðθþσWðtÞÞ ¼ e b ð1 þ Θ ÞsechðςÞ 1 2 and ψ ðt, x, yÞ¼ vðςÞ¼ u 1;1 3b (15) ¼ b b ð1 þ Θ Þsech ðςÞ 1 2 2 Analogously, we can take the solution in the following form uðςÞ¼ K sech ðςÞ, uðςÞ¼ K cschðςÞ, uðςÞ¼ K tanhðςÞ and uðςÞ¼ K cothðςÞ to get different forms of exact solutions of the HMs (1–2). Sine–Cosine method In this subsection, we use the sine–cosine method to get the solitary wave solutions of equation (9) and consequently the exact solution of the HMs (1–2). According to Refs. 14–16, let the solution u of equation (9) take the form uðςÞ¼ αY (16) where Y ¼ sinðλζ Þ or Y ¼ cosðλζÞ (17) Substituting equation (16) into (9), we have 2 2 m m2 3 3m m αλ m Y þ mðm  1ÞY Θ α Y  Θ αY ¼ 0 1 2 Rewriting the above equation 2 2 m 2 m2 3 3m Θ α  αλ m Y þ mðm  1Þαλ Y þ Θ α Y ¼ 0 (18) 2 1 Balancing the term of Y in equation (18), we get m  2 ¼ 3m 0 m ¼1 (19) Substituting equation (19) into (18) 2 1 3 2 3 Θ α  αλ Y þ Θ α þ 2αλ Y ¼ 0 (20) 2 1 1 3 Equating each coefficient of Y and Y to zero, we obtain Θ α  αλ ¼ 0 (21) 2 78 Journal of Low Frequency Noise, Vibration and Active Control 41(1) and Θ α þ 2αλ ¼ 0 (22) By solving these equations, we get rffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 2Θ λ ¼ ± Θ and α ¼ ± (23) Using equation (10), we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi λ ¼ ± Θ and α ¼ ± 6b Θ (24) 2 2 There are two cases depend on Θ : First case: If Θ > 0, then the solitary wave solution takes the form qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 2 uðςÞ¼ ± i 6b Θ sec Θ ς or uðςÞ¼ ± i 6b Θ csc Θ ς 2 2 2 2 1 1 In this case, the exact solution of the stochastic HMs (1–2) takes the form qffiffiffiffiffiffiffiffiffiffiffiffi h i pffiffiffiffiffiffi iðθþσWðtÞÞ φ ðt, x, yÞ¼ ± i 6b Θ sec Θ ς e (25) 2 2 1;1 1 h i pffiffiffiffiffiffi ψ ðt, x, yÞ¼ 2b b Θ sec Θ ς (26) 1 2 2 2 1;1 or qffiffiffiffiffiffiffiffiffiffiffiffi h i pffiffiffiffiffiffi iðθþσWðtÞÞ φ ðt, x, yÞ¼ ± i 6b Θ csc Θ ς e (27) 2 2 1;2 1 h i pffiffiffiffiffiffi ψ ðt, x, yÞ¼ 2b b Θ csc Θ ς (28) 1 2 2 2 1;2 Second case: If Θ < 0, then the solitary wave solution takes the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 2 2 uðςÞ¼ 6b Θ sech Θ ς or uðςÞ¼ 6b Θ csch Θ ς 2 2 2 2 1 1 In this case, the exact solution of the stochastic HMs (1–2) is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i pffiffiffiffiffiffiffiffiffi iðθþσWðtÞÞ φ ðt, x, yÞ¼ 6b Θ sech Θ ς e (29) 2 2 2;1 1 h i pffiffiffiffiffiffiffiffiffi ψ ðt, x, yÞ¼ 2b b Θ sech Θ ς (30) 1 2 2 2 2;1 or qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i pffiffiffiffiffiffiffiffiffi iðθþσWðtÞÞ φ ðt, x, yÞ¼ 6b Θ csch Θ ς e (31) 2 2 2;2 1 h i pffiffiffiffiffiffiffiffiffi 2 2 ψ ðt, x, yÞ¼ 2b b Θ csch Θ ς (32) 1 2 2 2;2 2 Remark 1. If we put σ = 0 in equations (25)–(32), we obtain the same solutions (see, equations (106), (108), (111), and (113)) stated in Ref. 41. Riccati–Bernoulli sub-ODE method Here, we take the following Riccati–Bernoulli equation (9) Mohammed et al. 79 0 2 u ¼ ρ u þ ρ u þ ρ (33) 1 2 3 where ρ , ρ , ρ are constants. 1 2 3 Differentiating equation (33) once with respect to ς, we obtain 00 0 0 u ¼ 2ρ uu þ ρ u 1 2 utilizing equation (33) gives 00 2 3 2 2 u ¼ 2ρ u þ 3ρ ρ u þ 2ρ ρ þ ρ u þ ρ ρ (34) 1 1 2 1 3 2 2 3 Substituting (34) into (9), we have 2 3 2 2 2ρ  Θ u þ 3ρ ρ u þ 2ρ ρ þ ρ  Θ u þ ρ ρ ¼ 0 1 2 1 1 2 1 3 2 2 3 Putting each coefficient of u (i = 0, 1, 2, 3) equal zero yields system of algebraic equations. We obtain by solving these equations sffiffiffiffiffiffiffi rffiffiffiffiffiffi Θ 1 ρ ¼ ± ¼ ± (35) 2 6b ρ ¼ 0 (36) and pffiffiffi Θ 6b Θ 2 1 2 ρ ¼ ¼ ± (37) 2ρ 2 where we used equation (35). To find the solutions of equation (33), we consider many cases depending on ρ Θ ρ Θ where Θ and Θ are defined in equation (10). 1 2 First case: If ðρ =ρ Þ > 0, then the solution of equation (33)is 3 1 rffiffiffiffiffi rffiffiffiffiffi ρ ρ 3 3 uðζÞ¼ tan ðρ ζ þ CÞ ρ ρ 1 1 or rffiffiffiffiffi rffiffiffiffiffi ρ ρ 3 3 uðζÞ¼ cot ðρ ζ þ CÞ ρ ρ 1 1 In this case, the exact solution of the stochastic HMs (1–2) is rffiffiffiffiffiffi rffiffiffiffiffiffi Θ Θ 2 2 iðθþσWðtÞÞ φ ðt, x, yÞ¼ tan ðρ ζ þ CÞ e (38) 3;1 1 Θ 2 rffiffiffiffiffiffi ψ ðt, x, yÞ¼b b Θ tan ðρ ζ þ CÞ (39) 1 2 2 3;1 1 or rffiffiffiffiffiffi rffiffiffiffiffiffi Θ Θ 2 2 iðθþσWðtÞÞ φ ðt, x, yÞ¼ cot ðρ ζ þ CÞ e (40) 3;2 1 Θ 2 rffiffiffiffiffiffi ψ ðt, x, yÞ¼b b Θ cot ðρ ζ þ CÞ (41) 1 2 2 3;2 1 2 80 Journal of Low Frequency Noise, Vibration and Active Control 41(1) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Second case: If ð ρ =ρ Þ < 0 and juj < ð ρ =ρ Þ, then the solution of equation (33)is 3 1 3 1 rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi ρ ρ 3 3 uðξÞ¼ tanh ð ρ ξ þ CÞ ρ ρ 1 1 In this case, the exact solution of the stochastic HMs (1–2) is rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi Θ Θ 2 2 iðθþσWðtÞÞ φ ðt, x, yÞ¼ tanh ð ρ ξ þ CÞ e (42) 3;3 1 Θ 2 rffiffiffiffiffiffiffiffi ψ ðt, x, yÞ¼ b b Θ tanh ðρ ξ þ CÞ (43) 3;3 1 2 2 1 Third case: If ðρ =ρ Þ < 0 and u > ðρ =ρ Þ, then the solution of equation (33)is 3 1 3 1 rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi ρ ρ 3 3 uðξÞ¼ coth ðρ ξ þ CÞ ρ ρ 1 1 In this case, the exact solution of the stochastic HMs (1–2) is rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi Θ Θ 2 2 iðθþσWðtÞÞ φ ðt, x, yÞ¼ coth ð ρ ξ þ CÞ e (44) 3;4 1 Θ 2 rffiffiffiffiffiffiffiffi ψ ðt, x, yÞ¼ b b Θ coth ðρ ξ þ CÞ (45) 1 2 2 3;4 1 where Θ and Θ are defined in equation (10). 1 2 Remark 2. If we put σ = 0 in equations (38)–(45), we obtain the same solutions (see, equations (93), (95), (100), and (102)) stated in Ref. 41. Remark 3. We can apply different methods such as the Hirota bilinear method, the Painleve test, the Painleve approach, Weierstrass elliptic function expansion method, the complex hyperbolic function method, the extended trial equation method, the extended tanh method, the improved tanðfðρÞ=2Þ-expansion method, the exp(φ)-expansion method, etc. to get some various solutions. Thus, we can generalize the results found in Refs. 36–43. The impact of noise on the solutions of HMs In this section, we show the effect of multiplicative noise on the exact solution of the HMs (1–2). In the following section, we provide some graphical representations to illustrate the behavior of these solutions. We use the MATLAB package to simulate the solution (42) for various σ (noise strength) and for fixed parameters a = 0.9, a =0, a = 1.4, b = 1.7, b =0 1 2 3 1 2 and b = 3.06 as follows: In Figure 1, we see that the solution of the HMs (1–2) fluctuates and has a pattern if the noise intensity σ =0: In Figures 2–4, we see that the pattern begins to destroy if the noise intensity σ increases. We see that when σ = 0, then the solution u takes the value between 400 and 400 as depicted in Figure 1. Moreover, we see that the values of the solution u begin to decrease and go to zero when the noise increases as seen in Figures 1–4. Also, we note that the blue and yellow color in the pattern indicates the maximum and minimum amplitude of the solution of the given system (1–2). Figure 5 illustrates the 2D graph corresponding to the solution (42) for σ =0,1, 3,5. Mohammed et al. 81 Figure 1. The solution in equation (42) with σ =0. Figure 2. The solution in equation (42) with σ =1. Figure 3. The solution in equation (42) with σ =3. 82 Journal of Low Frequency Noise, Vibration and Active Control 41(1) Figure 4. The solution in equation (42) with σ =5. Figure 5. The picture profile of solution in equation (42). Mohammed et al. 83 Conclusions In this article, we obtained various solutions of stochastic exact solution for the stochastic Hirota–Maccari system (1–2) forced in the Ito sense by multiplicative noise. We used three different methods as He’s semi-inverse method, sin–cosine method, and Riccati–Bernoulli sub-ODE method to obtain exact solutions of the stochastic Hirota–Maccari system (1–2). By applying these methods, we extended and improved some results such as the results stated in Ref. 41. 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Exact solutions of Hirota–Maccari system forced by multiplicative noise in the Itô sense:

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Abstract

In this article, the stochastic Hirota–Maccari system forced in the Ito ˆ sense by multiplicative noise is considered. We just use the He’s semi-inverse method, sine–cosine method, and Riccati–Bernoulli sub-ODE method to get new stochastic solutions which are hyperbolic, trigonometric, and rational. The major benefit of these three approaches is that they can be used to solve similar models. Furthermore, we plot 3D surfaces of analytical solutions obtained in this article by using MATLAB to illustrate the effect of multiplicative noise on the exact solution of the Hirota–Maccari method. Keywords Stochastic Hirota–Maccari system, multiplicative noise, sine–cosine method, He’s semi-inverse method Introduction Nonlinear partial differential equations (NLPDEs) are used extensively in the modeling of physiology, fluid me- chanics, physiology, chemistry, physical phenomena, etc. The analytical solutions of nonlinear differential equations which describe mathematical models are not easy to obtain. For NLPDEs, there are many appropriate numerical and analytical methods in the literature, developed by authors such as the exp(f(ς))-expansion method, the Jacobi 2 3,4 5,6 elliptic function method, the variational iteration method, the ðG =GÞ-expansion method, the homotopy per- 7 8 9 turbation method, the modified homotopy perturbation method, the Riccati–Bernoulli sub-ODE method, per- 10,11 12,13 14–16 17 turbation method, He’s frequency formulation, the sine–cosine method, the Hirota’s method, the tanh– 18,19 sech method, etc. While there are some methods to find the analytical solutions of fractional partial differential 20–23 24,25 equations such as the fractal variational principle, the Lie group analysis method, and the perturbation 26–28 method. Deterministic models of differential equations were widely used before the 1950s to explain the system’s dynamics in applications. Nevertheless, it is clear that the events that exist in the world today are generally not deterministic in nature. Therefore, random influences have been important when modeling various physical phenomena occurring in environmental sciences, meteorology, biology, engineering, oceanography, physics, and so on. Equations that include noise or fluctuations are called stochastic differential equations. Obtaining exact solutions for a stochastic NLPDEs is one of the most essential parts of nonlinear science. For obtaining exact stochastic solutions, there are many papers such as Refs. 29–34, and the references therein. Department of Mathematics, Faculty of Science, University of Ha’il, Saudi Arabia Department of Mathematics, Faculty of Science, Mansoura University, Egypt Section of Mathematics, International Telematic University Uninettuno, Italy Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj, Saudi Arabia Corresponding author: Wael W. Mohammed, Department of Mathematics, Faculty of Science, University of Ha’il, Saudi Arabia. Email: wael.mohammed@mans.edu.eg Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). Mohammed et al. 75 In this article, the following stochastic Hirota–Maccari system (HMs) with multiplicative noise is considered in the Itos ˆ ense iφ þ φ þ iφ þ φψ  ijφj φ þ σφW ¼ 0 (1) t xy xxx x 3ψ þ jφj ¼ 0 (2) where φ = φ(t, x, y) and ψ = ψ(t, x, y) are complex and real functions, respectively. σ is a noise strength, and W ¼ dW =dt is the time derivative of standard Wiener process W(t). We say the stochastic integral φðsÞdWðsÞ is Ito, ˆ if we calculate the stochastic integral at the left-end. We restrict ourselves in this article to the case of noise is a constant in space. The HMs (1–2), which was recently obtained by Maccari, has been shown to be integrable in the sense of Lax. Moreover, it was suggested in Ref. 35 that this system would pass the Painleve test. The authors described the modulation stability of the HMs, which implies that this system is stable under small perturbation. In the deterministic case, that is, without noise (σ = 0), many authors have studied a number of approaches to get the 36 37 38 exact solutions of the H–M system (1–2) such as Hirota bilinear method, Painleve test, Painleve approach, Weierstrass 39 40 elliptic function expansion method, complex hyperbolic function method, the sech–csch method, sec–tan ansatz method, the rational sinh–cosh ansatz method and the tanh–coth method, the improved tanðφðρÞ=2Þ-expansion method and general projective Riccati equation method, the generalized Kudryashov method, and the extended trial equation method. While the stochastic H–M system is not yet studied. The main objective of this article is to get the exact solutions of stochastic Hirota–Maccari system (1–2) forced by multiplicative noise in the Ito ˆ sense by using three different methods as the He’s semi-inverse method, sine–cosine method, and Riccati–Bernoulli sub-ODE method. In addition, we investigate the influence of multiplicative noise on the exact solution of stochastic HMs (1–2) by introducing some graphical representations using the MATLAB package. This article is, to the best of our knowledge, the first to obtain the exact solution for stochastic HMs (1–2) forced by multiplicative noise in the Itosense. The rest of this article is described as follows. In the next section which is Traveling wave eq. for stochastic HMs, the traveling wave equation for the stochastic HMs is obtained (1–2). While in The exact solutions of stochastic HMs section, the exact stochastic solutions of the stochastic HMs (1–2) are obtained by using three different methods as He’s semi- inverse method, the sine–cosine method, and Riccati–Bernoulli sub-ODE method. In The impact of noise on the solutions of HMs section, the influence of multiplicative noise on the exact solution of HMs (1–2) is studied. Finally, the conclusions of this article are given. Traveling wave eq. for stochastic HMs To obtain the traveling wave equation of the HMs (1–2), let us use the next wave transformation iðθþσWðtÞÞ φðt, x, yÞ¼ uðςÞe , ψðt, x, yÞ¼ vðςÞ (3) with ς ¼ðb x þ b y þ b tÞ, θ ¼ a x þ a y þ a t 1 2 3 1 2 3 where a , b are nonzero constants for k = 1,2,3,and σ is the noise strength. Substituting equation (3) into (1) and (2) and k k using dφ i 0 2 iðθþσWðtÞÞ i ¼ ib u  a u  σuW  σ u e , 3 3 t dt 2 dφ 0 2 3 iðθþσWðtÞÞ ijφj ¼  ib u u þ a u e , 1 1 dx d φ 3 000 2 00 2 0 2 0 3 iðθþσWðtÞÞ (4) ¼ b u þ 3ia b u  2a b u  a b u  ia u e 1 1 1 3 1 1 1 1 1 dx d φ 00 0 0 iðθþσWðtÞÞ ¼ðb b u þ ib a u þ ib a u  a a uÞe , 1 2 1 2 2 1 1 2 dxdy dψ 0 2 ¼ b v , jφj ¼ b u 1 2 dx 76 Journal of Low Frequency Noise, Vibration and Active Control 41(1) to get the following system 2 00 3 3 b b  3a b u þ a u  a þ a a þ a u þ uv ¼ 0 (5) 1 2 1 1 3 1 2 1 1 0 2 3b v þ b u ¼ 0 (6) 1 2 Integrating equation (6) once with respect to ς to obtain 3b v þ b u ¼ c (7) 1 2 where c is an integration constant. For simplicity here, we put c = 0, hence equation (7) becomes v ¼ u (8) 3b Substituting equation (8) into (5), we have the following traveling wave equation 00 3 u  Θ u  Θ u ¼ 0 (9) 1 2 where 1 a þ a a þ a 3 1 2 Θ ¼ and Θ ¼ (10) 1 2 2 2 3b b b  3a b 1 1 2 1 1 We note that when Θ is negative, then equation (9) adopts a periodic solution (see for instance Refs. 44,45). The exact solutions of stochastic HMs In this section, we implement three different methods as He’s semi-inverse method, sin–cosine method, and Riccati– Bernoulli sub-ODE method, respectively, to obtain the solutions of equation (9). After then, we have the stochastic exact solution of the HMs (1–2). He’s semi-inverse method The first method we are using to seek the exact solution of the HMs (1–2) is He’s semi-inverse method. The following variational formulation can be constructed from equation (9)by using He’s semi-inverse method mentioned in Refs. 46–48 as 1 1 1 0 4 2 JðuÞ¼ ðu Þ þ Θ u þ Θ u dς (11) 1 2 2 4 2 We suppose the solitary wave solution of equation (9) according to Ref. 49 takes the form uðςÞ¼ K sechðςÞ (12) where K is an unknown constant. Substituting equation (12) into (11), we get ∞ 2 4 2 K K K 2 2 4 2 J ¼ sech ðςÞtanh ðςÞ Θ sech ðςÞþ Θ sech ðςÞ dς 1 2 2 4 2 2 4 2 K K K ¼  Θ þ Θ 1 2 6 6 2 Differentiating J with respect to K and putting ð∂J =∂KÞ¼ 0 ∂J 1 2Θ ¼ ð1 þ 3Θ ÞK  K ¼ 0 (13) ∂K 3 3 Mohammed et al. 77 We get by solving equation (13) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ 3Θ Þ 3 K ¼ ¼ b ð1 þ 3Θ Þ 1 2 2Θ 2 Hence, the solution of equation (9)is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðςÞ¼ b ð1 þ 3Θ ÞsechðςÞ 1 2 Now, the exact solution of the HMs (1–2) is iðθþσWðtÞÞ φ ðt, x, yÞ¼ uðςÞe 1;1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (14) iðθþσWðtÞÞ ¼ e b ð1 þ Θ ÞsechðςÞ 1 2 and ψ ðt, x, yÞ¼ vðςÞ¼ u 1;1 3b (15) ¼ b b ð1 þ Θ Þsech ðςÞ 1 2 2 Analogously, we can take the solution in the following form uðςÞ¼ K sech ðςÞ, uðςÞ¼ K cschðςÞ, uðςÞ¼ K tanhðςÞ and uðςÞ¼ K cothðςÞ to get different forms of exact solutions of the HMs (1–2). Sine–Cosine method In this subsection, we use the sine–cosine method to get the solitary wave solutions of equation (9) and consequently the exact solution of the HMs (1–2). According to Refs. 14–16, let the solution u of equation (9) take the form uðςÞ¼ αY (16) where Y ¼ sinðλζ Þ or Y ¼ cosðλζÞ (17) Substituting equation (16) into (9), we have 2 2 m m2 3 3m m αλ m Y þ mðm  1ÞY Θ α Y  Θ αY ¼ 0 1 2 Rewriting the above equation 2 2 m 2 m2 3 3m Θ α  αλ m Y þ mðm  1Þαλ Y þ Θ α Y ¼ 0 (18) 2 1 Balancing the term of Y in equation (18), we get m  2 ¼ 3m 0 m ¼1 (19) Substituting equation (19) into (18) 2 1 3 2 3 Θ α  αλ Y þ Θ α þ 2αλ Y ¼ 0 (20) 2 1 1 3 Equating each coefficient of Y and Y to zero, we obtain Θ α  αλ ¼ 0 (21) 2 78 Journal of Low Frequency Noise, Vibration and Active Control 41(1) and Θ α þ 2αλ ¼ 0 (22) By solving these equations, we get rffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 2Θ λ ¼ ± Θ and α ¼ ± (23) Using equation (10), we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi λ ¼ ± Θ and α ¼ ± 6b Θ (24) 2 2 There are two cases depend on Θ : First case: If Θ > 0, then the solitary wave solution takes the form qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 2 uðςÞ¼ ± i 6b Θ sec Θ ς or uðςÞ¼ ± i 6b Θ csc Θ ς 2 2 2 2 1 1 In this case, the exact solution of the stochastic HMs (1–2) takes the form qffiffiffiffiffiffiffiffiffiffiffiffi h i pffiffiffiffiffiffi iðθþσWðtÞÞ φ ðt, x, yÞ¼ ± i 6b Θ sec Θ ς e (25) 2 2 1;1 1 h i pffiffiffiffiffiffi ψ ðt, x, yÞ¼ 2b b Θ sec Θ ς (26) 1 2 2 2 1;1 or qffiffiffiffiffiffiffiffiffiffiffiffi h i pffiffiffiffiffiffi iðθþσWðtÞÞ φ ðt, x, yÞ¼ ± i 6b Θ csc Θ ς e (27) 2 2 1;2 1 h i pffiffiffiffiffiffi ψ ðt, x, yÞ¼ 2b b Θ csc Θ ς (28) 1 2 2 2 1;2 Second case: If Θ < 0, then the solitary wave solution takes the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 2 2 uðςÞ¼ 6b Θ sech Θ ς or uðςÞ¼ 6b Θ csch Θ ς 2 2 2 2 1 1 In this case, the exact solution of the stochastic HMs (1–2) is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i pffiffiffiffiffiffiffiffiffi iðθþσWðtÞÞ φ ðt, x, yÞ¼ 6b Θ sech Θ ς e (29) 2 2 2;1 1 h i pffiffiffiffiffiffiffiffiffi ψ ðt, x, yÞ¼ 2b b Θ sech Θ ς (30) 1 2 2 2 2;1 or qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i pffiffiffiffiffiffiffiffiffi iðθþσWðtÞÞ φ ðt, x, yÞ¼ 6b Θ csch Θ ς e (31) 2 2 2;2 1 h i pffiffiffiffiffiffiffiffiffi 2 2 ψ ðt, x, yÞ¼ 2b b Θ csch Θ ς (32) 1 2 2 2;2 2 Remark 1. If we put σ = 0 in equations (25)–(32), we obtain the same solutions (see, equations (106), (108), (111), and (113)) stated in Ref. 41. Riccati–Bernoulli sub-ODE method Here, we take the following Riccati–Bernoulli equation (9) Mohammed et al. 79 0 2 u ¼ ρ u þ ρ u þ ρ (33) 1 2 3 where ρ , ρ , ρ are constants. 1 2 3 Differentiating equation (33) once with respect to ς, we obtain 00 0 0 u ¼ 2ρ uu þ ρ u 1 2 utilizing equation (33) gives 00 2 3 2 2 u ¼ 2ρ u þ 3ρ ρ u þ 2ρ ρ þ ρ u þ ρ ρ (34) 1 1 2 1 3 2 2 3 Substituting (34) into (9), we have 2 3 2 2 2ρ  Θ u þ 3ρ ρ u þ 2ρ ρ þ ρ  Θ u þ ρ ρ ¼ 0 1 2 1 1 2 1 3 2 2 3 Putting each coefficient of u (i = 0, 1, 2, 3) equal zero yields system of algebraic equations. We obtain by solving these equations sffiffiffiffiffiffiffi rffiffiffiffiffiffi Θ 1 ρ ¼ ± ¼ ± (35) 2 6b ρ ¼ 0 (36) and pffiffiffi Θ 6b Θ 2 1 2 ρ ¼ ¼ ± (37) 2ρ 2 where we used equation (35). To find the solutions of equation (33), we consider many cases depending on ρ Θ ρ Θ where Θ and Θ are defined in equation (10). 1 2 First case: If ðρ =ρ Þ > 0, then the solution of equation (33)is 3 1 rffiffiffiffiffi rffiffiffiffiffi ρ ρ 3 3 uðζÞ¼ tan ðρ ζ þ CÞ ρ ρ 1 1 or rffiffiffiffiffi rffiffiffiffiffi ρ ρ 3 3 uðζÞ¼ cot ðρ ζ þ CÞ ρ ρ 1 1 In this case, the exact solution of the stochastic HMs (1–2) is rffiffiffiffiffiffi rffiffiffiffiffiffi Θ Θ 2 2 iðθþσWðtÞÞ φ ðt, x, yÞ¼ tan ðρ ζ þ CÞ e (38) 3;1 1 Θ 2 rffiffiffiffiffiffi ψ ðt, x, yÞ¼b b Θ tan ðρ ζ þ CÞ (39) 1 2 2 3;1 1 or rffiffiffiffiffiffi rffiffiffiffiffiffi Θ Θ 2 2 iðθþσWðtÞÞ φ ðt, x, yÞ¼ cot ðρ ζ þ CÞ e (40) 3;2 1 Θ 2 rffiffiffiffiffiffi ψ ðt, x, yÞ¼b b Θ cot ðρ ζ þ CÞ (41) 1 2 2 3;2 1 2 80 Journal of Low Frequency Noise, Vibration and Active Control 41(1) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Second case: If ð ρ =ρ Þ < 0 and juj < ð ρ =ρ Þ, then the solution of equation (33)is 3 1 3 1 rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi ρ ρ 3 3 uðξÞ¼ tanh ð ρ ξ þ CÞ ρ ρ 1 1 In this case, the exact solution of the stochastic HMs (1–2) is rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi Θ Θ 2 2 iðθþσWðtÞÞ φ ðt, x, yÞ¼ tanh ð ρ ξ þ CÞ e (42) 3;3 1 Θ 2 rffiffiffiffiffiffiffiffi ψ ðt, x, yÞ¼ b b Θ tanh ðρ ξ þ CÞ (43) 3;3 1 2 2 1 Third case: If ðρ =ρ Þ < 0 and u > ðρ =ρ Þ, then the solution of equation (33)is 3 1 3 1 rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi ρ ρ 3 3 uðξÞ¼ coth ðρ ξ þ CÞ ρ ρ 1 1 In this case, the exact solution of the stochastic HMs (1–2) is rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi Θ Θ 2 2 iðθþσWðtÞÞ φ ðt, x, yÞ¼ coth ð ρ ξ þ CÞ e (44) 3;4 1 Θ 2 rffiffiffiffiffiffiffiffi ψ ðt, x, yÞ¼ b b Θ coth ðρ ξ þ CÞ (45) 1 2 2 3;4 1 where Θ and Θ are defined in equation (10). 1 2 Remark 2. If we put σ = 0 in equations (38)–(45), we obtain the same solutions (see, equations (93), (95), (100), and (102)) stated in Ref. 41. Remark 3. We can apply different methods such as the Hirota bilinear method, the Painleve test, the Painleve approach, Weierstrass elliptic function expansion method, the complex hyperbolic function method, the extended trial equation method, the extended tanh method, the improved tanðfðρÞ=2Þ-expansion method, the exp(φ)-expansion method, etc. to get some various solutions. Thus, we can generalize the results found in Refs. 36–43. The impact of noise on the solutions of HMs In this section, we show the effect of multiplicative noise on the exact solution of the HMs (1–2). In the following section, we provide some graphical representations to illustrate the behavior of these solutions. We use the MATLAB package to simulate the solution (42) for various σ (noise strength) and for fixed parameters a = 0.9, a =0, a = 1.4, b = 1.7, b =0 1 2 3 1 2 and b = 3.06 as follows: In Figure 1, we see that the solution of the HMs (1–2) fluctuates and has a pattern if the noise intensity σ =0: In Figures 2–4, we see that the pattern begins to destroy if the noise intensity σ increases. We see that when σ = 0, then the solution u takes the value between 400 and 400 as depicted in Figure 1. Moreover, we see that the values of the solution u begin to decrease and go to zero when the noise increases as seen in Figures 1–4. Also, we note that the blue and yellow color in the pattern indicates the maximum and minimum amplitude of the solution of the given system (1–2). Figure 5 illustrates the 2D graph corresponding to the solution (42) for σ =0,1, 3,5. Mohammed et al. 81 Figure 1. The solution in equation (42) with σ =0. Figure 2. The solution in equation (42) with σ =1. Figure 3. The solution in equation (42) with σ =3. 82 Journal of Low Frequency Noise, Vibration and Active Control 41(1) Figure 4. The solution in equation (42) with σ =5. Figure 5. The picture profile of solution in equation (42). Mohammed et al. 83 Conclusions In this article, we obtained various solutions of stochastic exact solution for the stochastic Hirota–Maccari system (1–2) forced in the Ito sense by multiplicative noise. We used three different methods as He’s semi-inverse method, sin–cosine method, and Riccati–Bernoulli sub-ODE method to obtain exact solutions of the stochastic Hirota–Maccari system (1–2). By applying these methods, we extended and improved some results such as the results stated in Ref. 41. 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Journal

"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE

Published: Jun 24, 2021

Keywords: Stochastic Hirota–Maccari system; multiplicative noise; sine–cosine method; He’s semi-inverse method

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