Optimal strategy analysis for adversarial differential games

Jiali Wang; Xin Jin; Yang Tang; Jiali Wang; Xin Jin; Yang Tang

doi:10.3934/era.2022189

Electronic Research Archive

2022, Volume 30, Issue 10: 3692-3710. doi: 10.3934/era.2022189

Previous Article Next Article

Research article Special Issues

Optimal strategy analysis for adversarial differential games

Key Laboratory of Smart Manufacturing in Energy Chemical Process, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China

Academic Editor: Wei Lin

Received: 25 May 2022 Revised: 04 July 2022 Accepted: 13 July 2022 Published: 05 August 2022

Optimal decision-making and winning-regions analysis in adversarial differential games are challenging theoretical problems because of the complex interactions between players. To solve these problems, we present an organized review for pursuit-evasion games, reach-avoid games and capture-the-flag games; we also outline recent developments in three types of games. First, we summarize recent results for pursuit-evasion games and classify them according to different numbers of players. As a special kind of pursuit-evasion games, target-attacker-defender games with an active target are analyzed from the perspectives of different speed ratios for players. Second, the related works for reach-avoid games and capture-the-flag games are compared in terms of analytical methods and geometric methods, respectively. These methods have different effects on the barriers and optimal strategy analysis between players. Future directions for the pursuit-evasion games, reach-avoid games, capture-the-flag games and their applications are discussed in the end.

Keywords:

Citation: Jiali Wang, Xin Jin, Yang Tang. Optimal strategy analysis for adversarial differential games[J]. Electronic Research Archive, 2022, 30(10): 3692-3710. doi: 10.3934/era.2022189

Related Papers:

[1]	Gonca Durmaz Güngör, Ishak Altun . Fixed point results for almost ( $\zeta -\theta _{\rho }$ )-contractions on quasi metric spaces and an application. AIMS Mathematics, 2024, 9(1): 763-774. doi: 10.3934/math.2024039
[2]	Hieu Doan . A new type of Kannan's fixed point theorem in strong $b$ - metric spaces. AIMS Mathematics, 2021, 6(7): 7895-7908. doi: 10.3934/math.2021458
[3]	Pragati Gautam, Vishnu Narayan Mishra, Rifaqat Ali, Swapnil Verma . Interpolative Chatterjea and cyclic Chatterjea contraction on quasi-partial $b$ -metric space. AIMS Mathematics, 2021, 6(2): 1727-1742. doi: 10.3934/math.2021103
[4]	Shaoyuan Xu, Yan Han, Suzana Aleksić, Stojan Radenović . Fixed point results for nonlinear contractions of Perov type in abstract metric spaces with applications. AIMS Mathematics, 2022, 7(8): 14895-14921. doi: 10.3934/math.2022817
[5]	Abdullah Shoaib, Poom Kumam, Shaif Saleh Alshoraify, Muhammad Arshad . Fixed point results in double controlled quasi metric type spaces. AIMS Mathematics, 2021, 6(2): 1851-1864. doi: 10.3934/math.2021112
[6]	Mi Zhou, Naeem Saleem, Xiao-lan Liu, Nihal Özgür . On two new contractions and discontinuity on fixed points. AIMS Mathematics, 2022, 7(2): 1628-1663. doi: 10.3934/math.2022095
[7]	Amjad Ali, Muhammad Arshad, Awais Asif, Ekrem Savas, Choonkil Park, Dong Yun Shin . On multivalued maps for $\varphi$ -contractions involving orbits with application. AIMS Mathematics, 2021, 6(7): 7532-7554. doi: 10.3934/math.2021440
[8]	Tahair Rasham, Abdullah Shoaib, Shaif Alshoraify, Choonkil Park, Jung Rye Lee . Study of multivalued fixed point problems for generalized contractions in double controlled dislocated quasi metric type spaces. AIMS Mathematics, 2022, 7(1): 1058-1073. doi: 10.3934/math.2022063
[9]	Xun Ge, Songlin Yang . Some fixed point results on generalized metric spaces. AIMS Mathematics, 2021, 6(2): 1769-1780. doi: 10.3934/math.2021106
[10]	Qing Yang, Chuanzhi Bai . Fixed point theorem for orthogonal contraction of Hardy-Rogers-type mapping on $O$ -complete metric spaces. AIMS Mathematics, 2020, 5(6): 5734-5742. doi: 10.3934/math.2020368

Abstract

1. Introduction

In various fields of science, nonlinear evolution equations practically model many natural, biological and engineering processes. For example, PDEs are very popular and are used in physics to study traveling wave solutions. They have played a crucial role in illustrating the nature of nonlinear problems. PDEs are collected to control the diffusion of chemical reactions. In biology, they play a fundamental role in describing various phenomena, such as population growth. In addition, natural phenomena such as fluid dynamics, plasma physics, optics and optical fibers, electromagnetism, quantum mechanics, ocean waves, and others are studied using PDEs. The qualitative and quantitative characteristics of these phenomena can be identified from the behaviors and shapes of their solutions. Therefore, finding the analytic solutions to such phenomena is a fundamental topic in mathematics. Scientists have developed sparse fundamental approaches to find analytic solutions for nonlinear PDEs. Among these techniques, I present integration methods from ^[1] and ^[2], the modified F-expansion and Generalized Algebraic methods, respectively. Bekir and Unsal ^[3] proposed the first integral method to find the analytical solution of nonlinear equations. Kumar, Seadawy and Joardar ^[4] used the improved Kudryashov technique to extract fractional differential equations. Adomian ^[5] proposed the Adomian decomposition technique to find the solution of frontier problems of physics. ^[6] uses an exploratory method to find explicit solutions of non-linear PDEs. Many different methods of solving equations arising from natural phenomena and some of their analytic solutions, such as dark and light solitons, non-local rogue waves, an occasional wave and mixed soliton solutions, are exhibited and can be found in ^{[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]}.

The Novikov-Veselov (NV) system ^[41,42] is given by

$\begin{equation} \begin{aligned} \Psi_t& +\alpha \Gamma_{xxy}+\beta \Phi_{xyy}+\gamma \Gamma_y \Phi+\gamma \left(\dfrac{\Psi^2}{2}\right)_y+\lambda\, \Gamma\, \Phi_x+\lambda\, \left(\dfrac{\Psi^2}{2}\right)_x = 0, \\ \Gamma_y& = \Psi_x, \\ \Psi_y& = \Phi_x, \\ \end{aligned} \end{equation}$

(1.1)

where $\alpha, \, \beta, \, \gamma$ and $\lambda$ are constants. Barman ^[42] declared that Eq (1.1) is involved to represent tidal and tsunami waves, electro-magnetic waves in communication cables and magneto-sound and ion waves in plasma. In ^[42], the generalized Kudryashov method was utilized to have traveling wave solutions for Eq (1.1). According to Croke ^[43], the Novikov-Veselov system is generalized from the KdV equations which were examined by Novikov and Veselov. Croke ^[43] used several approaches, (the extended mapping, the Hirota and the extended tanh-function approaches) in the proposed system to achieve numerous soliton solutions, such as breathers, and constrained analytic solutions. Boiti, Leon, and Manna ^[44] applied the inverse dispersion technique to solve (1.1) for a particular type of initial value. Numerical solutions and a study of the stability of solutions for the proposed equation were presented by Kazeykina and Klein ^[45]. The Nizhnik-Novikov-Veselov system for two dimensions was also solved using the Kansa technique to find the numerical results ^[46]. To the best of my knowledge, the stability and error analysis of the numerical scheme presented here has not yet been discussed for system (1.1). Therefore, this has motivated me enormously to do so. The primary purpose is to obtain multiple analytic solutions to system (1.1) by using both the modified F-expansion and Generalized Algebraic methods. In connection with the numerical solution, the method of finite differences is utilized to achieve numerical results for the studied system. I graphically and analytically compare the traveling wave solutions and numerical results. Undoubtedly, the presented results strongly contribute to describing physical problems in practice.

The outline of this article is provided in this paragraph. Section 2 summarizes the employed methods. All the analytic solutions are extracted in Section 3. The shooting and BVP results for the proposed system are presented in Section 4. In addition, I examine the numerical solution of the system (1.1) in Section 5. Sections 6 and 7 study the stability and error analysis of the numerical scheme, respectively. Section 8 presents the results and discussion.

2. Summary of proposed methods

Considering the development equation with physical fields $\Psi(x, y, t)$ , $\Phi(x, y, t)$ and $\Gamma(x, y, t)$ in the variables $x,$ $y$ and $t$ is given in the following form:

$\begin{equation} Q_1(\Psi, \Psi_t, \Psi_x, \Psi_y, \Gamma, \Gamma_y, \Gamma_{xxy}, \Phi, \Phi_{xyy}, \Phi_{x}, \dots) = 0. \end{equation}$

(2.1)

Step 1. We extract the traveling-wave solutions of System (1.1) that are formed as follows:

$\begin{equation} \begin{aligned} \Phi(x, y, t) = &\phi(\eta) , &&\qquad&\eta = x+y-wt, \\ \Psi(x, y, t) = &\psi(\eta), \\ \Gamma(x, y, t) = &\Theta(\eta), \end{aligned} \end{equation}$

(2.2)

where $w$ is the wave speed.

Step 2. The nonlinear evolution (2.1) is reduced to the following ODE:

$\begin{equation} Q_2(\psi, \psi_\eta, \Theta, \Theta_\eta, \Theta_{\eta\eta\eta}, \phi, \phi_{\eta\eta\eta}, \phi_{\eta}, \dots) = 0, \end{equation}$

(2.3)

where $Q_2$ is a polynomial in $\psi(\eta), \; \phi(\eta),$ $\Theta(\eta)$ and their total derivatives.

2.1. The modified F-expansion method

According to the modified F-expansion method, the solutions of (2.3) are given by the form

$\begin{equation} \psi(\eta) = \rho_0+\sum\limits_{k = 1}^{N} \left( \rho_{k} F(\eta)^{k}+\dfrac{ q_{k}}{F(\eta)^{k}} \right) , \end{equation}$

(2.4)

and $F(\eta)$ is a solution of the following differential equation:

$\begin{equation} F^{\prime}(\eta) = \mu_0+\mu_1 F(\eta)+\mu_2 F(\eta)^{2}, \end{equation}$

(2.5)

where $\mu_0$ , $\mu_1$ , $\mu_2$ , are given in ^[1], and $\rho_k$ , $q_k$ are to be determined later.

Table 1. The relations among

$\mu_0$ ,

$\mu_1$ ,

$\mu_2$ and the function

$F(\eta)$ .

$\mu_0$	$\mu_{1}$	$\mu_{2}$	$F(\eta)$
$\mu_0=0,$	$\mu_1=1,$	$\mu_2=-1,$	$F(\eta)=\dfrac{1}{2}+\dfrac{1}{2} \tanh \left(\dfrac{1}{2} \eta\right).$
$\mu_0=0,$	$\mu_1=-1,$	$\mu_2=1,$	$F(\eta)=\dfrac{1}{2}-\dfrac{1}{2} \operatorname{coth}\left(\dfrac{1}{2} \eta\right).$
$\mu_0=\dfrac{1}{2},$	$\mu_1=0,$	$\mu_2=\dfrac{-1}{2},$	$F(\eta)=\operatorname{coth}(\eta) \pm \operatorname{csch}(\eta),$ $\tanh (\eta) \pm \operatorname{sech}(\eta).$
$\mu_0=1,$	$\mu_1=0,$	$\mu_2=-1,$	$F(\eta)=\tanh (\eta),$ $\operatorname{coth}(\eta).$
$\mu_0=\dfrac{1}{2},$	$\mu_1=0,$	$\mu_2=\dfrac{1}{2},$	$F(\eta)=\sec (\eta)+\tan (\eta),$ $\csc (\eta)-\cot (\eta).$
$\mu_0=\dfrac{1}{2},$	$\mu_1=0,$	$\mu_2=\dfrac{1}{2},$	$F(\eta)=\sec (\eta)-\tan (\eta),$ $\csc (\eta)+\cot (\eta).$
$\mu_0=\pm1,$	$\mu_1=0,$	$\mu_2=\pm1,$	$F(\eta)=\tan (\eta),$ $\cot (\eta).$

| Show Table

DownLoad: CSV

2.2. The generalized algebraic method

According to the generalized direct algebraic method, the solutions of (2.3) are given by

$\begin{equation} \psi(\eta) = \nu_0+\sum\limits_{k = 1}^{N} \left( \nu_{k} G(\eta)^{k} +\dfrac{ r_{k}}{G(\eta) ^{k}} \right) , \end{equation}$

(2.6)

and $G(\eta)$ is a solution of the following differential equation:

$\begin{equation} G^{\prime}(\eta) = \varepsilon \sqrt{\sum\limits_{k = 0}^{4} \delta_{k} G^{k}(\eta)}, \end{equation}$

(2.7)

where $\nu_k,$ and $r_k$ are to be determined, and $N$ is an integer number obtained by the highest degree of the nonlinear terms and the highest order of the derivatives. $\varepsilon$ is user-specified, usually taken with $\varepsilon = \pm1$ , and $\delta_k$ , $k = 0, 1, 2, 3, 4,$ are given in Table 2 ^[2].

Table 2. The relations among

$\delta_k$ ,

$k = 0, 1, 2, 3, 4,$ and the function

$G(\eta)$ .

$\delta_{0}$	$\delta_{1}$	$\delta_{2}$	$\delta_{3}$	$\delta_{4}$	$G(\eta)$
$\delta_{0}=0$ ,	$\delta_{1}=0$ ,	$\delta_{2} > 0$ ,	$\delta_{3}=0$ ,	$\delta_{4} < 0$ ,	$G(\eta)= \varepsilon$ $\sqrt{-\dfrac{\delta_{2}}{\delta_{4}}} \operatorname{sech}\left(\sqrt{\delta_{2}} \eta\right)$ .
$\delta_{0}=\dfrac{\delta_{2}^{2}}{4 c 4}$ ,	$\delta_{1}=0$ ,	$\delta_{2} < 0$ ,	$\delta_{3}=0$ ,	$\delta_{4} > 0$ ,	$G(\eta)=\varepsilon \sqrt{-\dfrac{\delta_{2}}{2 \delta_{4}}} \tanh \left(\sqrt{-\dfrac{\delta_{2}}{2}} \eta\right)$ .
$\delta_{0}=0$ ,	$\delta_{1}=0$ ,	$\delta_{2} < 0$ ,	$\delta_{3}=0$ ,	$\delta_{4} > 0$ ,	$G(\eta)=\varepsilon$ $\sqrt{-\dfrac{\delta_{2}}{\delta_{4}}}$ $\sec$ $\left(\sqrt{-\delta_{2}} \eta\right)$ .
$\delta_{0}=\dfrac{\delta_{2}^{2}}{4 \delta_{4}},$	$\delta_{1}=0$ ,	$\delta_{2} > 0$ ,	$\delta_{3}=0,$	$\delta_{4} > 0,$	$G(\eta)=\varepsilon$ $\sqrt{\dfrac{\delta_{2}}{2 \delta_{4}}} \tan \left(\sqrt{\dfrac{\delta_{2}}{2}} \eta\right)$ .
$\delta_{0}=0,$	$\delta_{1}=0,$	$\delta_{2}=0,$	$\delta_{3}=0,$	$\delta_{4} > 0,$	$G(\eta)=-\dfrac{\varepsilon}{\sqrt{\delta_{4}} \eta}$ .
$\delta_{0}=0,$	$\delta_{1}=0,$	$\delta_{2} > 0,$	$\delta_{3}\neq 0,$	$\delta_{4}=0,$	$G(\eta)=-\dfrac{\delta_{2}}{\delta_{3}}. \operatorname{sech}^{2}\left(\dfrac{\sqrt{\delta_{2}}}{2} \eta\right)$ .

| Show Table

DownLoad: CSV

3. Methodology

Consider the Novikov-Veselov (NV) system

(3.1)

a system of PDEs in the unknown functions $\Psi = \Psi(x, y, t), \; \Phi = \Phi(x, y, t)$ , $\Gamma = \Gamma(x, y, t)$ and their partial derivatives. I plug the transformations

$\begin{equation} \begin{aligned} \Phi(x, y, t) = &\phi(\eta) , &&\qquad&\eta = x+y-wt, \\ \Psi(x, y, t) = &\psi(\eta), \\ \Gamma(x, y, t) = &\Theta(\eta), \end{aligned} \end{equation}$

(3.2)

into Eq (3.1) to reduce it to a system of ODEs given by

$\begin{equation} \begin{aligned} -w\; \psi_\eta& +\alpha \Theta_{\eta\eta\eta}+\beta \phi_{\eta\eta\eta}+\gamma \Theta_\eta \phi+\gamma \left(\dfrac{\psi^2}{2}\right)_\eta+\lambda\, \Theta\, \phi_\eta+\lambda\, \left(\dfrac{\psi^2}{2}\right)_\eta = 0, \\ \Theta_\eta& = \psi_\eta, \\ \psi_\eta& = \phi_\eta.\\ \end{aligned} \end{equation}$

(3.3)

Integrating $\Theta_\eta = \psi_\eta$ and $\phi_\eta = \psi_\eta$ yields

$\begin{equation} \Theta = \psi, \qquad \text{and } \qquad \phi = \psi. \end{equation}$

(3.4)

Substituting (3.4) into the first equation of (3.3) and integrating once with respect to $\eta$ yields

$\begin{equation} \begin{aligned} -w\; \psi& +\left( \alpha +\beta\right) \psi_{\eta\eta}+\left( \gamma+\lambda\right) \psi^2 = 0. \end{aligned} \end{equation}$

(3.5)

Balancing $\psi_{\eta\eta}$ with $\psi^2$ in (3.5) calculates the value of $N = 2.$

3.1. Application of modified F-expansion method

According to the modified F-expansion method with $N = 2$ , the solutions of (3.5) are

$\begin{equation} \psi(\eta) = \rho_0+ \rho_{1} F(\eta)+\dfrac{ q_{1}}{F(\eta)}+ \rho_{2} F(\eta)^{2}+\dfrac{ q_{2}}{F(\eta)^{2}} , \end{equation}$

(3.6)

and $F(\eta)$ is a solution of the following differential equation:

$\begin{equation} F^{\prime}(\eta) = \mu_0+\mu_1 F(\eta)+\mu_2 F(\eta)^{2}, \end{equation}$

(3.7)

where $\mu_0$ , $\mu_1$ , $\mu_2$ are given in Table 1. To explore the analytic solutions to (3.5), I ought to follow the subsequent steps.

$\textbf{Step 1.}$ Placing (3.6) along with (3.7) into Eq (3.5) and gathering the coefficients of $F(\eta)^j$ , $j$ = $-4, -3, -2, -1, 0, 1, 2, 3, 4,$ to zeros gives a system of equations for $\rho_0, \; \rho_k, \; q_k$ , $k = 1, 2.$

$\textbf{Step 2.}$ Solve the resulting system using mathematical software: for example, Mathematica or Maple.

$\textbf{Step 3.}$ Choosing the values of $\mu_0, \; \mu_1$ and $\mu_2$ and the function $F(\eta)$ from and substituting them along with $\rho_0, \; \rho_k, \; q_k$ , $k = 1, 2,$ in (3.6) produces a set of trigonometric function and rational solutions to (3.5).

Applying the above steps, I determine the values of $\rho_0, \; \rho_1, \; \rho_2, \; q_1, \; q_2$ and $w$ as follows:

$(1).$ When $\mu_0 = 0,$ $\mu_1 = 1$ and $\mu_2 = -1$ , I have two cases.

$\textbf{Case 1.}$

$\begin{equation} \begin{aligned} \rho_0 = &0, && \rho_1 = \dfrac{6 (\alpha +\beta)}{\gamma +\lambda}, &\rho_2 = & -\dfrac{6 (\alpha +\beta)}{\gamma + \lambda}, & q_1 = q_2 = 0, \quad \text{and}\quad w = \alpha+\beta. \end{aligned} \end{equation}$

(3.8)

The solution is given by

$\begin{equation} \begin{aligned} \Psi_1(x, y, t) = \dfrac{ 3(\alpha+\beta)}{2(\gamma+\lambda)} \operatorname{sech}^2\left(\dfrac{1}{2} \left( x+y-(\alpha +\beta) t+x0\right) \right). \end{aligned} \end{equation}$

(3.9)

$\textbf{Case 2.}$

$\rho_0 = -\dfrac{\alpha+\beta}{\gamma+\lambda}, \rho_1 = \dfrac{6 (\alpha +\beta)}{\gamma +\lambda}, \\ \rho_2 = -\dfrac{6 (\alpha +\beta)}{\gamma + \lambda}, q_1 = q_2 = 0, \quad \text{and}\quad w = -\left( \alpha+\beta\right) .$

(3.10)

The solution is given by

$\begin{equation} \begin{aligned} \Psi_2(x, y, t) = -\dfrac{ (\alpha+\beta)}{2(\gamma+\lambda)} \left( 3 \tanh ^2\left(\dfrac{1}{2} (x+y+t (\alpha+\beta))+x0\right)-1 \right) . \end{aligned} \end{equation}$

(3.11)

presents the time evolution of the analytic solutions $(a)$ $\Psi_1$ and $(b)$ $\Psi_2$ with $t = 0, 10, 20$ . The parameter values are $x0 = -20, \ \alpha = 0.50, \beta = 0.6, \ \gamma = -1.5,$ and $\lambda = 1.$ presents the wave behavior by changing a certain parameter value and fixing the values of the others. presents the behavior of $\Psi_1$ when I change the values of (a) $\alpha$ or $\beta$ and (b) $\gamma$ or $\lambda$ . In it can also be seen that the value of $\alpha$ or $\beta$ affects the direction and amplitude of the wave, such that a negative value always makes the wave negative, its amplitude decreases when $\alpha, \, \beta \to 0$ , and its amplitude increases when $\alpha, \, \beta \to \infty.$ In the value of $\gamma$ or $\lambda$ affects the direction and amplitude of the wave, such that a negative value always makes the wave negative, and its amplitude decreases when the value of $\gamma$ or $\lambda$ increases. In and present the wave behavior of $\Psi_2.$

Figure 1. Time evolution of the analytic solutions

$(a)$

$\Psi_1$ and

$(b)$

$\Psi_2$ with

$t = 0, 10, 20$ . The parameter are given by

$x0 = -20, \ \alpha = 0.50, \beta = 0.6, \ \gamma = -1.5,$ and

$\lambda = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. This figure present the wave behavior when changing a certain parameter value and fixing the values of the others.

$(a)$ presents the behavior when I change the value of

$\alpha$ or

$\beta,$ and

$(b)$ presents when I change the value of

$\gamma$ or

$\lambda$ for the solution

$\Psi_1.$

$(c)$ and

$(d)$ are for

$\Psi_2$ .

DownLoad: Full-Size Img PowerPoint

$(2).$ When $\mu_0 = 0,$ $\mu_1 = -1,$ and $\mu_2 = 1$ , I have two cases.

$\textbf{Case 3.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_3(x, y, t) = -\dfrac{ 3(\alpha+\beta)}{2(\gamma+\lambda)} \operatorname{csch}^2\left(\dfrac{1}{2} \left( x+y-(\alpha +\beta) t\right) \right). \end{aligned} \end{equation}$

(3.12)

$\textbf{Case 4.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_4(x, y, t) = -\dfrac{ (\alpha+\beta)}{2(\gamma+\lambda)} \left( 3 \coth ^2\left(\dfrac{1}{2} (x+y+t (\alpha+\beta))\right)-1 \right) . \end{aligned} \end{equation}$

(3.13)

$(3).$ When $\mu_0 = 1,$ $\mu_1 = 0,$ and $\mu_2 = -1$ , I have

$\textbf{Case 5.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_5(x, y, t) = -\dfrac{8 (\alpha+\beta) }{\gamma+\lambda}(\cosh (4 (16 t (\alpha+\beta)+x+y))+2) \operatorname{csch}^2(2 (16 t (\alpha+\beta)+x+y)). \end{aligned} \end{equation}$

(3.14)

$(4).$ When $\mu_0 = \pm 1,$ $\mu_1 = 0,$ and $\mu_2 = \pm 1$ , I have one case.

$\textbf{Case 6.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_6(x, y, t) = -\dfrac{24 (\alpha+\beta) }{\gamma+\lambda}\csc ^2(2 (16 t (\alpha+\beta)+x+y)) . \end{aligned} \end{equation}$

(3.15)

3.2. Application of the generalized algebraic method

According to the generalized algebraic method, the solutions of (3.5) are given by the form

$\begin{equation} \psi(\eta) = \nu_0+ \nu_{1} G(\eta)+ \nu_{2} G(\eta)^{2}+\dfrac{ r_{1}}{G(\eta)}+\dfrac{ r_{2}}{G(\eta)^{2}}, \end{equation}$

(3.16)

where $\nu_k$ , $r_k$ are to be determined later. $G(\eta)$ is a solution of the following differential equation:

$\begin{equation} G^{\prime}(\eta) = \varepsilon \sqrt{\sum\limits_{k = 0}^{4} \delta_{k} G^{k}(\eta)}, \end{equation}$

(3.17)

where $\delta_k$ , $k = 0, 1, 3, 4$ , are given in . In all the cases mentioned above and the subsequent solutions, I used the mathematical software Mathematica to find the values of the constants $\nu_0, \, \nu_1, \, \nu_2, \, r_1, \, r_2$ and $w$ . Thus, the analytic solutions to (3.5) using the generalized algebraic method will be presented here with different values of the constants $\delta_k$ , $k = 0, 1, 3, 4$ .

$(5).$ When $\delta_0 = \dfrac{\delta^2_2}{4 \delta_4},$ $\delta_1 = \delta_3 = 0,$ $\delta_2 < 0$ , $\delta_4 > 0$ , and $\varepsilon = \pm1,$

$\begin{equation} \begin{aligned} \nu_0 = &\frac{\pm\sqrt{4 \delta_2^2 \varepsilon^4 \left(\alpha+\beta\right)^2 \left(\gamma+\lambda\right)^2-3 \delta_2 \varepsilon^4 \left(\alpha+\beta\right)^2 \left(\gamma+\lambda\right)^2}-2 \delta_2 \varepsilon^2 \left(\alpha+\beta\right) \left(\gamma+\lambda\right)}{\left(\gamma+\lambda\right)^2}, \\ \nu_2 = &-\frac{6 \delta_4 \varepsilon^2 \left(\alpha+\beta\right)}{\left(\gamma+\lambda\right)}, \\ \nu_1 = &r_1 = r_2 = 0, \quad\varepsilon = \pm1.\\ w = &\pm\frac{2 \sqrt{\left(4 \delta_2^2-3 \delta_2\right) \varepsilon^4 \left(\alpha+\beta\right)^2 \left(\gamma+\lambda\right)^2}}{\left(\gamma+\lambda\right)}. \end{aligned} \end{equation}$

(3.18)

$\textbf{Case 7.}$ The solution is given by

$\Psi_7(x, y, t) = -\frac{1}{{(\gamma+\lambda)^2}}\\ \left( -3 \delta_2 \varepsilon^4 (\alpha+\beta) (\gamma+\lambda) \tanh ^2\left(\frac{\sqrt{-\delta_2} \left(\frac{2 t \sqrt{\delta_2 (4 \delta_2-3) \varepsilon^4 (\alpha+\beta)^2 (\gamma+\lambda)^2}}{\gamma+\lambda}+x+y\right)}{\sqrt{2}}\right)\right.\\ \left.+2 \delta_2 \varepsilon^2 (\alpha+\beta) (\gamma+\lambda)+\sqrt{\delta_2 (4 \delta_2-3) \varepsilon^4 (\alpha+\beta)^2 (\gamma+\lambda)^2}\right) .$

(3.19)

shows the time evolution of the analytic solutions. shows $\Psi_7$ with $t$ = $0:2:6$ . The parameter values are $\delta_2 = -1$ , $\delta_4 = 1,$ $\epsilon = -1,$ $\alpha = 0.50,$ $\beta = 0.6,$ $\gamma = -1.5,$ $\lambda = 1.8$ and $x0 = -10.$ shows $\Psi_8$ with $t = 0:2:8$ . The parameter values are $\delta_2 = 1$ , $\delta_4 = -1,$ $\epsilon = -1,$ $\alpha = 0.50,$ $\beta = 0.6,$ $\gamma = -1.5,$ $\lambda = 1.8$ and $x0 = -10.$ – present the $3D$ time evolution of the analytic solutions $\Psi_2$ (left) and the numerical solutions (right) obtained employing the scheme 5.1 with $t = 5, 15, 25$ , $M_x = 1600,$ $N_y = 100,$ $x = 0\to60$ and $y = 0\to1.$

Figure 3. Time evolution of the analytic solutions.

$(a)$

$\Psi_7$ with

$t = 0:2:6$ . The parameter values are

$\delta_2 = -1$ ,

$\delta_4 = 1,$

$\epsilon = -1,$

$\alpha = 0.50,$

$\beta = 0.6,$

$\gamma = -1.5,$

$\lambda = 1.8$ and

$x0 = -10.$

$(b)$

$\Psi_8$ with

$t = 0:2:8$ . The parameter values are

$\delta_2 = 1$ ,

$\delta_4 = -1,$

$\epsilon = -1,$

$\alpha = 0.50,$

$\beta = 0.6,$

$\gamma = -1.5,$

$\lambda = 1.8$ and

$x0 = -10$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. 3D graphs presenting the analytic (left) and the numerical (right) solutions of

$\Psi_2(x, y, t)$ at

$t = 5.$ The figures present the strength of agreement between analytic and numerical solutions.

DownLoad: Full-Size Img PowerPoint

Figure 5. 3D graphs presenting the analytic (left) and the numerical (right) solutions of

$\Psi_2(x, y, t)$ at

$t = 15.$ The figures present the strength of agreement between analytic and numerical solutions.

DownLoad: Full-Size Img PowerPoint

Figure 6. 3D graphs presenting the analytic (left) and the numerical (right) solutions of

$\Psi_2(x, y, t)$ at

$t = 25.$ The figures present the strength of agreement between analytic and numerical solutions.

DownLoad: Full-Size Img PowerPoint

$(6).$ When $\delta_0 = 0,$ $\delta_1 = \delta_3 = 0,$ $\delta_2 > 0$ , $\delta_4 < 0$ , and $\varepsilon = \pm1,$

$\begin{equation} \begin{aligned} \nu_2 = &-\frac{6 \delta_4 \varepsilon^2 (\alpha+\beta)}{\gamma+\lambda}, \quad \nu_1 = \nu_0 = r_1 = r_2 = 0, \quad w = &4 \delta_2 \varepsilon^2 (\alpha+\beta). \end{aligned} \end{equation}$

(3.20)

$\textbf{Case 8.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_8(x, y, t) = &\frac{6 \delta_2\epsilon^4 (\alpha+\beta) \operatorname{sech}^2\left(\sqrt{\delta_2} \left(-4 \delta_2\epsilon^2 t (\alpha+\beta)+x+y\right)\right)}{\gamma+\lambda} \end{aligned} \end{equation}$

(3.21)

$(7).$ When $\delta_0 = 0,$ $\delta_1 = \delta_4 = 0,$ $\delta_3\neq0,$ $\delta_2 > 0$ , $\varepsilon = \pm1$

$\begin{equation} \begin{aligned} \text{Set 1.}\, \nu_1 = & -\tfrac{3 \delta_3 \varepsilon^2 \left(\alpha+\beta\right)}{2 \left(\gamma+\lambda\right)}, \, \nu_0 = \nu_2 = r_1 = r_2 = 0, \, w = \delta_2 \varepsilon^2 \left(\alpha+\beta\right). \\ &\\ \text{Set 2.}\, \nu_0 = & -\tfrac{\delta_2 \varepsilon^2 \left(\alpha+\beta\right)}{\left(\gamma+\lambda\right)}, \, \nu_1 = -\tfrac{3 \delta_3 \varepsilon^2 \left(\alpha+\beta\right)}{2 \left(\gamma+\lambda\right)}, \quad\nu_2 = r_1 = r_2 = 0, \, w = -\delta_2 \varepsilon^2 \left(\alpha+\beta\right). \end{aligned} \end{equation}$

(3.22)

$\textbf{Case 9.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_9(x, y, t) = \frac{3 \delta_2 (\alpha+\beta) \operatorname{sech}^2\left(\frac{1}{2} \sqrt{\delta_2} (\delta_2 t (-(\alpha+\beta))+x+y)\right)}{2 (\gamma+\lambda)}. \end{aligned} \end{equation}$

(3.23)

$\textbf{Case 10.}$ The solution is given by

$\begin{equation} \begin{aligned} \Psi_{10}(x, y, t) = \frac{3 \delta_2 (\alpha+\beta) \operatorname{sech}^2\left(\frac{1}{2} \sqrt{\delta_2} (\delta_2 t (\alpha+\beta)+x+y)\right)}{2 (\gamma+\lambda)}-\frac{\delta_2 (\alpha+\beta)}{\gamma+\lambda}. \end{aligned} \end{equation}$

(3.24)

4. Numerical solution

In this section I extract numerical solutions to the resulting ODE system (3.5) using several numerical methods. The purpose of this procedure is to guarantee the accuracy of the analytic solutions. I picked one of the analytic solutions above to be a sample, (3.11). The nonlinear shooting and BVP methods, at $t = 0,$ are used by taking the value of $\psi$ at the right endpoint of the domain $\eta = 0$ with guessing the initial value for $\psi_\eta.$ The new target is to obtain the second boundary condition of $\psi$ at the left endpoint of a particular domain. Once the numerical result is obtained, I compare it with the analytic solution (3.11). The MATLAB solver ODE15s and FSOLVE ^[47] are used to get the numerical solution. The resulting ODE (3.5) is discretized as

$\begin{equation} f(\psi) = 0, \qquad f(\psi_i) = -w\; \psi_i +\dfrac{\alpha +\beta}{\Delta_\eta} \left( \psi_{i+1}-2\psi_i+\psi_{i-1}\right) +\dfrac{ \gamma+\lambda}{2\Delta_\eta} \left( \psi^2_{i+1}-\psi_{i-1}^2\right), \end{equation}$

(4.1)

for the BVP method and

$\begin{equation} \psi_{\eta\eta} = \dfrac{1}{\alpha+\beta}\left( w\psi-\left( \gamma+\lambda\right) \psi^2\right), \end{equation}$

(4.2)

for the shooting method. Figure 7 presents the comparison between the numerical solutions obtained using the above numerical methods and the analytic solution. Figure 7 shows that the solutions are identical to the analytic solution.

Figure 7. Comparing the numerical solutions resulting from the shooting and BVP methods with the analytic solution (3.11) at

$t = 0.$ The parameter values are taken as

$\alpha = 0.50$ ,

$\beta = 0.6$ ,

$\gamma = -1.5, \lambda = 1.8$ , with

$N = 600$ .

DownLoad: Full-Size Img PowerPoint

Thus, it is possible to verify the correctness of the analytic solution. I also accept the obtained numerical solution as an initial condition for the numerical scheme in the next section.

5. Numerical scheme on a fixed mesh

In this section, I use the finite-difference method to obtain the numerical results of system (1.1) over the domain $[a, \, b]\times [c, \, d].$ Here, $a, \; b, \; c$ and $d$ represent the endpoints of the rectangular domain in the $x$ and $y$ directions, respectively, and $T_f$ is a certain time. The domain $[a, \, b]\times [c, \, d]$ is split into $(M_x+1)\times (N_y+1)$ mesh points:

$\begin{equation*} \label{} \begin{aligned} x_m = &a+m\, \Delta_x, &&& m = 0, \, 1, \, 2, \, \dots, \, M_x, \\ y_n& = c+n\, \Delta_y, &&& n = 0, \, 1, \, 2, \, \dots, \, N_y, \end{aligned} \end{equation*}$

where $\Delta_x$ and $\Delta_y$ are the step-sizes of the $x$ and $y$ domains, respectively. The system (1.1) is converted to an ODE system by discretizing the space derivatives while keeping the time derivative continuous. Completing this yields

$\begin{equation} \begin{aligned} &\Psi_t|_{m, n}^k + \dfrac{\alpha}{2\Delta_y \Delta_x^2} \delta_x^2\left( \Gamma^{k+1}_{m, n+1}-\Gamma^{k+1}_{m, n-1}\right) +\dfrac{\beta}{2\Delta_x\Delta_y^2} \delta_y^2\left( \Phi^{k+1}_{m+1, n}-\Phi^{k+1}_{m-1, n}\right)\\ &\qquad \quad- \frac{ \gamma}{4\Delta_y} \left( \left( \Phi^{k+1}_{m, n+1}+\Phi^{k+1}_{m, n} \right) \Gamma^{k+1}_{m, n+1}- \left( \Phi^{k+1}_{m, n}+\Phi^{k+1}_{m, n-1} \right)\Gamma^{k+1}_{m, n-1}\right)\\ &\qquad \quad- \frac{ \lambda}{4\Delta_x} \left( \left( \Gamma^{k+1}_{m+1, n}+\Gamma^{k+1}_{m, n} \right) \Phi^{k+1}_{m+1, n}- \left( \Gamma^{k+1}_{m, n}+\Gamma^{k+1}_{m-1, n} \right)\Phi^{k+1}_{m-1, n}\right)\\ &\qquad \quad+\frac{ \gamma}{4\Delta_y} \left( \left( \Psi^{k+1}_{m, n+1}\right)^2- \left( \Psi^{k+1}_{m, n-1}\right)^2\right) +\frac{ \lambda}{4\Delta_x} \left( \left( \Psi^{k+1}_{m+1, n}\right)^2- \left( \Psi^{k+1}_{m-1, n}\right)^2\right) = 0, \\ &\frac{1}{2\Delta_y}(\Gamma_{m, n+1}^{k+1}-\Gamma_{m, n-1}^{k+1}) = \frac{1}{2\Delta_x}(\Psi_{m+1, n}^{k+1}-\Psi_{m-1, n}^{k+1}), \\ &\frac{1}{2\Delta_y}(\Psi_{m, n+1}^{k+1}-\Psi_{m, n-1}^{k+1}) = \frac{1}{2\Delta_x}(\Phi_{m+1, n}^{k+1}-\Phi_{m-1, n}^{k+1}), \end{aligned} \end{equation}$

(5.1)

where

$\begin{aligned} \delta_x^2\Gamma_{m, n}^{k+1} = &\left(\Gamma^{k+1}_{m+1, n}-2 \Gamma^{k+1}_{m, n}+ \Gamma^{k+1}_{m-1, n}\right), \\ \delta_y^2\Phi_{m, n}^{k+1} = &\left(\Phi^{k+1}_{m, n+1}-2 \Phi^{k+1}_{m, n}+ \Phi^{k+1}_{m, n-1}\right), \end{aligned}$

subject to the boundary conditions:

$\begin{equation} \begin{array}{l} \Psi_{x}(a, \, y, \, t) = \Psi_{x}(b, \, y, \, t) = 0, \, \, \, \, \, \, \, \, \forall y\in [c, \, d], \\ \\ \Psi_{y}(x, \, c, \, t) = \Psi_{y}(x, \, d, \, t) = 0, \, \, \, \, \, \forall x\in [a, \, b]. \end{array} \end{equation}$

(5.2)

Equation (5.2) permits us to use fictitious points in estimating the space derivatives at the domain's endpoints. The initial conditions are generated by

$\begin{equation} \begin{aligned} \Psi_2(x, y, 0) = -\dfrac{ (\alpha+\beta)}{2(\gamma+\lambda)} \left( 3 \tanh ^2\left(\dfrac{1}{2} (x+y+x0\right)-1 \right) , \end{aligned} \end{equation}$

(5.3)

where $\alpha, \, \beta, \, \gamma$ and $\lambda$ are user-defined parameters. In all the numerical results shown in this section, the parameter values are fixed as $\alpha = 0.50, \, \beta = 0.6, \, \gamma = -1.50, \, \lambda = 1.80, \, x0 = -45.0$ , $y = 0\to1,$ $x = 0\to60$ and $t = 0\to 25.$ The above system is solved by using an ODE solver in FORTRAN called the DDASPK solver ^[48]. This solver used a backward differentiation formula. Since I do not have the initial conditions for the space derivatives, I approximate the Jacobian matrix of the linearized system by using LU-Factorization. The obtained numerical results are acceptable. This can be observed from the Figures 8 and 9.

Figure 8. Time change for the numerical results while holding

$y = 0.5$ and

$M_x = 1600$ at

$t = 0:5:25.$ The wave at

$t = 25$ illustrates that the numerical and the analytic solutions are quite identical.

DownLoad: Full-Size Img PowerPoint

Figure 9. The convergence histories of the scheme with the fixation of both

$y = 0.5$ and

$M_x = 1600$ at

$t = 5$ .

DownLoad: Full-Size Img PowerPoint

6. Stability of the numerical scheme

The von Neumann analysis is used to examine the stability of the scheme (5.1). The von Neumann analysis is occasionally called Fourier analysis and is utilized exclusively when the scheme is linear. Hence, I suppose that the linear version is given by

$\begin{equation} \begin{aligned} \Psi_t& +\alpha \Gamma_{xxy}+\beta \Phi_{xyy}+s_0\, \Gamma_y+s_1\, \Psi_y+s_2\, \Phi_x+s_3\, \Psi_x = 0, \\ \Gamma_y& = \Psi_x, \\ \Psi_y& = \Phi_x, \\ \end{aligned} \end{equation}$

(6.1)

where $s_0 = \gamma \Phi$ , $s_1 = \gamma \Psi$ , $s_2 = \lambda \Gamma$ , $s_3 = \lambda \Psi$ are constants. Since $\Gamma_y = \Psi_x,$ and $\Psi_y = \Phi_x,$ the first equation of (6.1) is given by

$\begin{equation} \Psi_t +\alpha \Psi_{xxx}+\beta \Psi_{yyy}+ s_0\, \Psi_y +s_1\, \Psi_y+s_2\, \Psi_x+s_3\, \Psi_x = 0, \end{equation}$

(6.2)

where $\alpha, \, \beta, \, \gamma, \, \lambda, \, s_0, \, s_1, \, s_2, \, s_3, \, l_4$ are constants. I set directly

$\begin{equation} \begin{aligned} \Psi_{m, n}^k = &\mu^k\, \exp(\iota\, \pi\, \xi_0 \, n\, \Delta_x)\exp(\iota\, \pi\, \xi_1 \, m\, \Delta_y), \quad\\ \end{aligned} \end{equation}$

(6.3)

and also I can have

$\begin{aligned} \Psi_{m, n}^{k+1} = &\mu \, \Psi_{m, n}^k, \quad \Psi_{m+1, n}^{k} = \exp(\iota\, \pi\, \xi_0 \, \Delta_x) \, \Psi_{m, n}^k, \quad \Psi_{m, n+1}^{k} = \exp(\iota\, \pi\, \xi_1 \, \Delta_y) \, \Psi_{m, n}^k, \\ & \qquad\qquad\, \, \, \Psi_{m-1, n}^{k} = \exp(-\iota\, \pi\, \xi_0 \, \Delta_x) \, \Psi_{m, n}^k, \quad \Psi_{m, n-1}^{k} = \exp(-\iota\, \pi\, \xi_1 \, \Delta_y) \, \Psi_{m, n}^k , \\ m = &1, \, 2, \dots, \, N_x-1, \quad \quad n = 1, 2, \dots, \, N_y-1. \end{aligned}$

Substituting (6.3) into (6.2) and doing some operations, I have

$\begin{array}{l} 1 = \mu\left( 1-\iota\Delta_t\left( \frac{\sin(\xi_0\pi \Delta_x)\, }{\Delta_x} \left(\frac{4\, \alpha}{\Delta_x^2}\, \sin^2(\tfrac{\xi_0\pi\, \Delta_x}{2})-s_2-s_3\right)+\frac{ \sin(\xi_1\pi \Delta_y)\, }{\Delta_y} \left(\frac{4\, \beta}{\Delta_y^2}\, \sin^2(\tfrac{\xi_1\pi\, \Delta_y}{2})-s_0-s_1\right)\right)\right). \end{array}$

Hence,

$\begin{equation} \begin{array}{l} \mu = \dfrac{1}{ 1-a\iota}, \end{array} \end{equation}$

(6.4)

where

$a = \Delta_t\left( \frac{\sin(\xi_0\pi \Delta_x)\, }{\Delta_x} \left(\frac{4\, \alpha}{\Delta_x^2}\, \sin^2(\tfrac{\xi_0\pi\, \Delta_x}{2})-s_2-s_3\right)+\frac{ \sin(\xi_1\pi \Delta_y)\, }{\Delta_y} \left(\frac{4\, \beta}{\Delta_y^2}\, \sin^2(\tfrac{\xi_1\pi\, \Delta_y}{2})-s_0-s_1\right)\right).$

Thus,

$\begin{equation} \begin{array}{l} |\mu|^2 = \frac{1}{1+a^2}\leq 1.\\ \, \end{array} \end{equation}$

(6.5)

The stability condition of the von Neumann analysis is fulfilled. Consequently, from Eq (6.5), the scheme is unconditionally stable.

7. Error analysis

To examine the accuracy of the numerical scheme (5.1), I study the truncation error utilizing Taylor expansions. Suppose that the error is

$\begin{equation} \begin{array}{l} e^{k+1}_{m, n} = \Psi_{m, n}^{k+1}-\Psi(x_m, y_n, t_{k+1}), \end{array} \end{equation}$

(7.1)

where $\Psi(x_m, y_n, t_{k+1})$ and $\Psi_{m, n}^{k+1}$ are the analytic solution and an approximate solution, respectively. Substituting (7.1) into (5.1) gives

$\begin{equation*} \label{} \begin{aligned} \frac{e^{k+1}_{j, k}-e^{k}_{j, k}}{\Delta_t} = & T^{k+1}_{m, n}- \left( \alpha \frac{1}{2\Delta_x^3 } \delta_x^2\left( e^{k+1}_{m+1, n}-e^{k+1}_{m-1, n}\right) +\beta \frac{1}{2\Delta_y^3 } \delta_y^2\left( e^{k+1}_{m, n+1}-e^{k+1}_{m, n-1}\right)\right.\\ &\left.+ \frac{ s_2+s_3}{2\Delta_x} \left( e^{k+1}_{m+1, n}-e^{k+1}_{m-1, n}\right)+ \frac{ s_0+s_1}{2\Delta_y} \left( e^{k+1}_{m, n+1}-e^{k+1}_{m, n-1}\right)\right), \end{aligned} \end{equation*}$

where

$T^{k+1}_{m, n} = \frac{\alpha}{2\Delta_x^3 } \delta_x^2\left( \Psi(x_{m+1}, y_n, t_{k+1})-\Psi(x_{m-1}, y_n, t_{k+1})\right) \\ +\frac{\beta }{2\Delta_y^3 } \delta_y^2\left( \Psi(x_{m}, y_{n+1}, t_{k+1})-\Psi(x_{m}, y_{n-1}, t_{k+1})\right)\\ + \frac{ s_2+s_3}{2\Delta_x} \left( \Psi(x_{m+1}, y_n, t_{k+1})-\Psi(x_{m-1}, y_n, t_{k+1})\right)+\\ \frac{ s_0+s_1}{2\Delta_y} \left( \Psi(x_{m}, y_{n+1}, t_{k+1})-\Psi(x_{m}, y_{n-1}, t_{k+1})\right).$

Hence,

$\begin{equation*} \label{} \begin{aligned} T^{k+1}_{m, n}\leq& \frac{\Delta_t}{2}\frac{\partial^2\Psi(x_{m}, y_{n}, \xi_{k+1})}{\partial t^2} -\frac{\Delta_x^2}{2}\frac{\partial^5\Psi(\zeta_{m}, y_{n}, t_{k+1})}{\partial x^5} -\frac{\Delta_y^2}{2}\frac{\partial^5\Psi(x_{m}, \eta_{n}, t_{k+1})}{\partial x^5}\\ & -\frac{\Delta_y^2}{6}\frac{\partial^3\Psi(x_{m}, \eta_{n}, t_{k+1})}{\partial x^3}-\frac{\Delta_x^2}{6}\frac{\partial^3\Psi(\zeta_{m}, y_{n}, t_{k+1})}{\partial x^3}. \end{aligned} \end{equation*}$

Accordingly, the truncation error of the numerical scheme is

$\begin{equation*} \label{} \begin{array}{l} T^{k+1}_{m, n} = O(\Delta_t, \Delta_x^2, \Delta_y^2). \end{array} \end{equation*}$

8. Results and discussion

I have prosperously employed several analytical methods to extract the traveling wave solutions to the two-dimensional Novikov-Veselov system, confirming the solutions with numerical results obtained using the numerical scheme (5.1). The major highlights of the results are shown in Table 3 and –, which allow immediate comparison of the analytic solutions with the numerical results. Through these, I can notice that the solutions are identical to a large extent, and the error approaches zero whenever the value of $\Delta_x, \Delta_y\to 0$ . The numerical schemes are unconditionally stable for fixing the parameter values $\alpha = 0.50, \, \beta = 0.6, \, \gamma = -1.50, \, \lambda = 1.80, \, x0 = -45.0$ , $y = 0\to1,$ $x = 0\to60$ and $t = 0\to 25.$

Table 3. The relative error with

$L_2$ norm and CPU at

$t = 20.$ .

$\Delta_x$	The Relative Error	CPU
$0.6000 \qquad\qquad$	$5.600\times 10^{-3}\qquad$	$0.063\times10^{3}\, \text{m} \qquad$
$0.3000$	$2.100\times 10^{-3}$	$0.1524\times10^{3} \, s$
$0.1500$	$6.700\times 10^{-4}$	$0.3564\times10^{3}\, s$
$0.0750$	$2.100\times 10^{-4}$	$0.8892\times10^{3}\, s$
$0.0375$	$6.610\times 10^{-5}$	$1.7424\times10^{3}\, s$
$0.0187$	$2.310\times 10^{-5}$	$4.0230\times10^{3}\, s$

| Show Table

DownLoad: CSV

Figure 10. The convergence histories measured utilizing the relative error with

$l_2$ norm as a function of

$\Delta_x$ (see Table 3). Here, I picked a certain value of the variable

$y = 0.5$ at

$t = 20$ and

$x = 0\to60$ .

DownLoad: Full-Size Img PowerPoint

presents the time evolution of the analytic solutions $(a)$ $\Psi_1$ and $(b)$ $\Psi_2$ with $t = 0, 10, 20$ . The parameter values are $x0 = -20, \ \alpha = 0.50, \ \beta = 0.6, \ \gamma = -1.5,$ and $\lambda = 1.$ presents the wave behavior by changing a certain parameter value and fixing the values of the others. presents the behavior of $\Psi_1$ when I change the values of (a) $\alpha$ or $\beta$ and (b) $\gamma$ or $\lambda$ . In it can also be seen that the value of $\alpha$ or $\beta$ affects the direction and amplitude of the wave, such that a negative value always makes the wave negative, its amplitude decreases when $\alpha, \, \beta \to 0$ , and its amplitude increases when $\alpha, \, \beta \to \infty.$ In the value of $\gamma$ or $\lambda$ affects the direction and amplitude of the wave, such that a negative value always makes the wave negative, and its amplitude decreases when the value of $\gamma$ or $\lambda$ increases. In present the wave behavior of $\Psi_2.$ shows the time evolution of the analytic solutions. shows $\Psi_7$ with $t = 0:2:6$ . The parameter values are $\delta_2 = -1$ , $\delta_4 = 1,$ $\epsilon = -1,$ $\alpha = 0.50,$ $\beta = 0.6,$ $\gamma = -1.5,$ $\lambda = 1.8$ and $x0 = -10.$ shows $\Psi_8$ with $t = 0:2:8$ . The parameter values are $\delta_2 = 1$ , $\delta_4 = -1,$ $\epsilon = -1,$ $\alpha = 0.50,$ $\beta = 0.6,$ $\gamma = -1.5,$ $\lambda = 1.8$ and $x0 = -10.$ – present the $3D$ time evolution of the analytic solutions $\Psi_2$ (left) and the numerical solutions (right) obtained employing the scheme 5.1 with $t = 5, 15, 25$ , $M_x = 1600,$ $N_y = 100,$ $x = 0\to60$ and $y = 0\to1.$ These figures provide us with an adequate answer that the numerical and analytic solutions are quite identical. Barman et al. ^[42] accepted several traveling wave solutions for (1.1) as hyperbolic functions. The authors employed other parameters to develop new forms for the accepted solution. They proposed that Eq (1.1) describes tidal and tsunami waves, electromagnetic waves in transmission cables and magneto-sound and ion waves in plasma. In comparison, I have found numerous solutions also as hyperbolic functions. Furthermore, I obtained the numerical solutions to enhance the assurance that the solutions presented here are correct and accurate.

9. Conclusions

I have successfully utilized the generalized algebraic and modified F-expansion methods to acquire the soliton solutions for the two-dimensional Novikov-Veselov system, verifying these solutions with numerical results obtained by employing the numerical scheme (5.1). The major highlights of the results shown in Figures 8– and , which allow immediate comparison of the analytic solutions with the numerical results. Through these, I can notice that the solutions are identical to a large extent, and the error approaches zero whenever the value of $\Delta_x, \Delta_y\to 0$ . The numerical schemes are unconditionally stable for fixing the parameter values $\alpha = 0.50, \, \beta = 0.6, \, \gamma = -1.50, \, \lambda = 1.80,$ $x0 = -45.0$ , $y = 0\to1,$ $x = 0\to60$ and $t = 0\to 25.$ The Jacobi elliptic functions have effectively deteriorated to hyperbolic functions. The applied numerical schemes have provided reliable numerical solutions when using a small value of $\Delta_x, \Delta_y\to0$ .

Ultimately, I can deduce that the methods used are valuable and applicable to extract soliton solutions for other nonlinear evolutionary systems found in chemistry, engineering, physics and other sciences.

Conflict of interest

The author declares that he has no potential conflict of interest in this article.

References

[1]	R. Yan, Z. Shi, Y. Zhong, Task assignment for multiplayer reach–avoid games in convex domains via analytical barriers, IEEE Trans. Rob., 36 (2019), 107–124. https://doi.org/10.1109/TRO.2019.2935345 doi: 10.1109/TRO.2019.2935345
[2]	E. Garcia, I. Weintraub, D. W. Casbeer, M. Pachter, Optimal strategies for the game of protecting a plane in 3-d, preprint, arXiv: 2202.01826.
[3]	E. Garcia, D. W. Casbeer, M. Pachter, Optimal strategies of the differential game in a circular region, IEEE Control Syst. Lett., 4 (2019), 492–497. https://doi.org/10.1109/LCSYS.2019.2963173 doi: 10.1109/LCSYS.2019.2963173
[4]	J. Chen, W. Zha, Z. Peng, D. Gu, Multi-player pursuit–evasion games with one superior evader, Automatica, 71 (2016), 24–32. https://doi.org/10.1016/j.automatica.2016.04.012 doi: 10.1016/j.automatica.2016.04.012
[5]	K. Chen, W. He, Q. L. Han, M. Xue, Y. Tang, Leader selection in networks under switching topologies with antagonistic interactions, Automatica, 142 (2022), 110334. https://doi.org/10.1016/j.automatica.2022.110334 doi: 10.1016/j.automatica.2022.110334
[6]	Z. Li, X. Yu, J. Qiu, H. Gao, Cell division genetic algorithm for component allocation optimization in multifunctional placers, IEEE Trans. Ind. Inf., 18 (2021), 559–570. https://doi.org/10.1109/TⅡ.2021.3069459 doi: 10.1109/TⅡ.2021.3069459
[7]	Y. Tang, C. Zhao, J. Wang, C. Zhang, Q. Sun, W. Zheng, et al., An overview of perception and decision-making in autonomous systems in the era of learning, IEEE Trans. Neural Networks Learn. Syst., 2022. https://doi.org/10.1109/TNNLS.2022.3167688 doi: 10.1109/TNNLS.2022.3167688
[8]	E. Garcia, D. W. Casbeer, A. V. Moll, M. Pachter, Multiple pursuer multiple evader differential games, IEEE Trans. Autom. Control, 66 (2020), 2345–2350. https://doi.org/10.1109/TAC.2020.3003840 doi: 10.1109/TAC.2020.3003840
[9]	E. Garcia, D. W. Casbeer, M. Pachter, Optimal strategies for a class of multi-player reach-avoid differential games in 3d space, IEEE Rob. Autom. Lett., 5 (2020), 4257–4264, https://doi.org/10.1109/LRA.2020.2994023 doi: 10.1109/LRA.2020.2994023
[10]	H. Huang, J. Ding, W. Zhang, C. J. Tomlin, Automation-assisted capture-the-flag: A differential game approach, IEEE Trans. Control Syst. Technol., 23 (2014), 1014–1028. https://doi.org/10.1109/TCST.2014.2360502 doi: 10.1109/TCST.2014.2360502
[11]	Z. Zhou, J. Huang, J. Xu, Y. Tang, Two-phase jointly optimal strategies and winning regions of the capture-the-flag game, in IECON 2021 – 47th Annual Conference of the IEEE Industrial Electronics Society, (2021), 1–6. https://doi.org/10.1109/IECON48115.2021.9589624
[12]	E. Garcia, A. V. Moll, D. W. Casbeer, M. Pachter, Strategies for defending a coastline against multiple attackers, in 2019 IEEE 58th Conference on Decision and Control (CDC), (2019), 7319–7324. https://doi.org/10.1109/CDC40024.2019.9029340
[13]	I. E. Weintraub, M. Pachter, E. Garcia, An introduction to pursuit-evasion differential games, in 2020 American Control Conference (ACC), (2020), 1049–1066. https://doi.org/10.23919/ACC45564.2020.9147205
[14]	T. Başar, A tutorial on dynamic and differential games, Dyn. Games Appl. Econ., (1986), 1–25. https://doi.org/10.1007/978-3-642-61636-5_1 doi: 10.1007/978-3-642-61636-5_1
[15]	S. S. Kumkov, S. L. Ménec, V. S. Patsko, Zero-sum pursuit-evasion differential games with many objects: survey of publications, Dyn. Games Appl., 7 (2017), 609–633. https://doi.org/10.1007/s13235-016-0209-z doi: 10.1007/s13235-016-0209-z
[16]	R. Yan, Z. Shi, Y. Zhong, Defense game in a circular region, in 2017 IEEE 56th Annual Conference on Decision and Control (CDC), (2017), 5590–5595. https://doi.org/10.1109/CDC.2017.8264502
[17]	I. E. Weintraub, A. V. Moll, E. Garcia, D. Casbeer, Z. J. Demers, M. Pachter, Maximum observation of a faster non-maneuvering target by a slower observer, in 2020 American Control Conference (ACC), (2020), 100–105. https://doi.org/10.23919/ACC45564.2020.9147340
[18]	J. Wang, Y. Hong, J. Wang, J. Xu, Y. Tang, Q. L. Han, et al., Cooperative and competitive multi-agent systems:from optimization to games, IEEE/CAA J. Autom. Sin., 9 (2022), 763–783. https://doi.org/10.1109/JAS.2022.105506 doi: 10.1109/JAS.2022.105506
[19]	A. A. Al-Talabi, Multi-player pursuit-evasion differential game with equal speed, in 2017 International Automatic Control Conference (CACS), (2017), 1–6. https://doi.org/10.1109/CACS.2017.8284276
[20]	D. Shishika, J. Paulos, V. Kumar, Cooperative team strategies for multi-player perimeter-defense games, IEEE Rob. Autom. Lett., 5 (2020), 2738–2745. https://doi.org/10.1109/LRA.2020.2972818 doi: 10.1109/LRA.2020.2972818
[21]	E. Garcia, Z. E. Fuchs, D. Milutinovic, D. W. Casbeer, M. Pachter, A geometric approach for the cooperative two-pursuer one-evader differential game, IFAC-PapersOnLine, 50 (2017), 15209–15214. https://doi.org/10.1016/j.ifacol.2017.08.2366 doi: 10.1016/j.ifacol.2017.08.2366
[22]	A. V. Moll, D. Casbeer, E. Garcia, D. Milutinović, M. Pachter, The multi-pursuer single-evader game, J. Intell. Rob. Syst., 96 (2019), 193–207. https://doi.org/10.1007/s10846-018-0963-9 doi: 10.1007/s10846-018-0963-9
[23]	E. Garcia, S. D. Bopardikar, Cooperative containment of a high-speed evader, in 2021 American Control Conference (ACC), (2021), 4698–4703. https://doi.org/10.23919/ACC50511.2021.9483097
[24]	E. Garcia, D. W. Casbeer, D. Tran, M. Pachter, A differential game approach for beyond visual range tactics, in 2021 American Control Conference (ACC), (2021), 3210–3215. https://doi.org/10.23919/ACC50511.2021.9482650
[25]	Y. Xu, H. Yang, B. Jiang, M. M. Polycarpou, Multi-player pursuit-evasion differential games with malicious pursuers, IEEE Trans. Autom. Control, 2022. https://doi.org/10.1109/TAC.2022.3168430 doi: 10.1109/TAC.2022.3168430
[26]	W. Lin, Z. Qu, M. A. Simaan, Nash strategies for pursuit-evasion differential games involving limited observations, IEEE Trans. Aerosp. Electron. Syst., 51 (2015), 1347–1356. https://doi.org/10.1109/TAES.2014.130569 doi: 10.1109/TAES.2014.130569
[27]	M. Pachter, E. Garcia, D. W. Casbeer, Active target defense differential game, in 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton), (2014), 46–53. https://doi.org/10.1109/ALLERTON.2014.7028434
[28]	E. Garcia, D. W. Casbeer, M. Pachter, Active target defense using first order missile models, Automatica, 78 (2017), 139–143. https://doi.org/10.1016/j.automatica.2016.12.032 doi: 10.1016/j.automatica.2016.12.032
[29]	M. Coon, D. Panagou, Control strategies for multiplayer target-attacker-defender differential games with double integrator dynamics, in 2017 IEEE 56th Annual Conference on Decision and Control (CDC), (2017), 1496–1502. https://doi.org/10.1109/CDC.2017.8263864
[30]	I. E. Weintraub, E. Garcia, M. Pachter, A kinematic rejoin method for active defense of non-maneuverable aircraft, in 2018 Annual American Control Conference (ACC), (2018), 6533–6538. https://doi.org/10.23919/ACC.2018.8431129
[31]	E. Garcia, D. W. Casbeer, M. Pachter, Design and analysis of state-feedback optimal strategies for the differential game of active defense, IEEE Trans. Autom. Control, 64 (2018), 553–568. https://doi.org/10.1109/TAC.2018.2828088 doi: 10.1109/TAC.2018.2828088
[32]	E. Garcia, D. W. Casbeer, M. Pachter, Optimal target capture strategies in the target-attacker-defender differential game, in 2018 Annual American Control Conference (ACC), (2018), 68–73. https://doi.org/10.23919/ACC.2018.8431715
[33]	E. Garcia, D. W. Casbeer, M. Pachter, The complete differential game of active target defense, J. Optim. Theory Appl., 191 (2021), 675–699. https://doi.org/10.1007/s10957-021-01816-z doi: 10.1007/s10957-021-01816-z
[34]	E. Garcia, D. W. Casbeer, M. Pachter, Pursuit in the presence of a defender, Dyn. Games Appl., 9 (2019), 652–670. https://doi.org/10.1007/s13235-018-0271-9 doi: 10.1007/s13235-018-0271-9
[35]	M. Pachter, E. Garcia, D. W. Casbeer, Toward a solution of the active target defense differential game, Dyn. Games Appl., 9 (2019), 165–216. https://doi.org/10.1007/s13235-018-0250-1 doi: 10.1007/s13235-018-0250-1
[36]	E. Garcia, Cooperative target protection from a superior attacker, Automatica, 131 (2021), 109696. https://doi.org/10.1016/j.automatica.2021.109696 doi: 10.1016/j.automatica.2021.109696
[37]	M. Pachter, E. Garcia, R. Anderson, D. W. Casbeer, K. Pham, Maximizing the target's longevity in the active target defense differential game, in 2019 18th European Control Conference (ECC), (2019), 2036–2041. https://doi.org/10.23919/ECC.2019.8795650
[38]	E. Garcia, D. W. Casbeer, M. Pachter, Defense of a target against intelligent adversaries: A linear quadratic formulation, in 2020 IEEE Conference on Control Technology and Applications (CCTA), (2020), 619–624. https://doi.org/10.1109/CCTA41146.2020.9206368
[39]	E. Garcia, D. W. Casbeer, M. Pachter, Cooperative strategies for optimal aircraft defense from an attacking missile, J. Guid., Control, Dyn., 38 (2015), 1510–1520. https://doi.org/10.2514/1.G001083 doi: 10.2514/1.G001083
[40]	L. Liang, F. Deng, Z. Peng, X. Li, W. Zha, A differential game for cooperative target defense, Automatica, 102 (2019), 58–71. https://doi.org/10.1016/j.automatica.2018.12.034 doi: 10.1016/j.automatica.2018.12.034
[41]	Z. Zhou, J. Ding, H. Huang, R. Takei, C. Tomlin, Efficient path planning algorithms in reach-avoid problems, Automatica, 89 (2018), 28–36. https://doi.org/10.1016/j.automatica.2017.11.035 doi: 10.1016/j.automatica.2017.11.035
[42]	P. Shi, W. Sun, X. Yang, I. J. Rudas, H. Gao, Master-slave synchronous control of dual-drive gantry stage with cogging force compensation, IEEE Trans. Syst. Man Cybern.: Syst., https://doi.org/10.1109/TSMC.2022.3176952
[43]	J. Lorenzetti, M. Chen, B. Landry, M. Pavone, Reach-avoid games via mixed-integer second-order cone programming, in 2018 IEEE Conference on Decision and Control (CDC), (2018), 4409–4416. https://doi.org/10.1109/CDC.2018.8619382
[44]	R. Isaacs, Differential games: Their scope, nature, and future, J. Optim. Theory Appl., 3 (1969), 283–295. https://doi.org/10.1007/BF00931368 doi: 10.1007/BF00931368
[45]	R. Yan, Z. Shi, Y. Zhong, Guarding a subspace in high-dimensional space with two defenders and one attacker, IEEE Trans. Cybern., 2020. https://doi.org/10.1109/TCYB.2020.3015031 doi: 10.1109/TCYB.2020.3015031
[46]	R. Yan, Z. Shi, Y. Zhong, Construction of the barrier for reach-avoid differential games in three-dimensional space with four equal-speed players, in 2019 IEEE 58th Conference on Decision and Control (CDC), (2019), 4067–4072. https://doi.org/10.1109/CDC40024.2019.9029495
[47]	K. Margellos, J. Lygeros, Hamilton–jacobi formulation for reach–avoid differential games, IEEE Trans. Autom. Control, 56 (2011), 1849–1861. https://doi.org/10.1109/TAC.2011.2105730 doi: 10.1109/TAC.2011.2105730
[48]	J. F. Fisac, M. Chen, C. J. Tomlin, S. S. Sastry, Reach-avoid problems with time-varying dynamics, targets and constraints, in HSCC '15: Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control, (2015), 11–20. https://doi.org/10.1145/2728606.2728612
[49]	M. Chen, Z. Zhou, C. J. Tomlin, Multiplayer reach-avoid games via pairwise outcomes, IEEE Trans. Autom. Control, 62 (2016), 1451–1457. https://doi.org/10.1109/TAC.2016.2577619 doi: 10.1109/TAC.2016.2577619
[50]	V. Mnih, K. Kavukcuoglu, D. Silver, A. Graves, I. Antonoglou, D. Wierstra, et al., Playing atari with deep reinforcement learning, preprint, arXiv: 1312.5602.
[51]	S. Bansal, C. J. Tomlin, Deepreach: A deep learning approach to high-dimensional reachability, in 2021 IEEE International Conference on Robotics and Automation (ICRA), (2021), 1817–1824. https://doi.org/10.1109/ICRA48506.2021.9561949
[52]	J. Li, D. Lee, S. Sojoudi, C. J. Tomlin, Infinite-horizon reach-avoid zero-sum games via deep reinforcement learning, preprint, arXiv: 2203.10142.
[53]	K. C. Hsu, V. R. Royo, C. J. Tomlin, J. F. Fisac, Safety and liveness guarantees through reach-avoid reinforcement learning, preprint, arXiv: 2112.12288.
[54]	E. Garcia, D. W. Casbeer, A. V. Moll, M. Pachter, Cooperative two-pursuer one-evader blocking differential game, in 2019 American Control Conference (ACC), (2019), 2702–2709. https://doi.org/10.23919/ACC.2019.8814294
[55]	R. Yan, X. Duan, Z. Shi, Y. Zhong, F. Bullo, Matching-based capture strategies for 3d heterogeneous multiplayer reach-avoid differential games, Automatica, 140 (2022), 110207. https://doi.org/10.1016/j.automatica.2022.110207 doi: 10.1016/j.automatica.2022.110207
[56]	J. Selvakumar, E. Bakolas, Feedback strategies for a reach-avoid game with a single evader and multiple pursuers, IEEE Trans. Cybern., 51 (2019), 696–707. https://doi.org/10.1109/TCYB.2019.2914869 doi: 10.1109/TCYB.2019.2914869
[57]	E. Garcia, D. W. Casbeer, M. Pachter, J. W. Curtis, E. Doucette, A two-team linear quadratic differential game of defending a target, in 2020 American Control Conference (ACC), (2020), 1665–1670. https://doi.org/10.23919/ACC45564.2020.9147665
[58]	S. D. Bopardikar, F. Bullo, J. P. Hespanha, A cooperative homicidal chauffeur game, Automatica, 45 (2009), 1771–1777. https://doi.org/10.1016/j.automatica.2009.03.014 doi: 10.1016/j.automatica.2009.03.014
[59]	R. Lopez-Padilla, R. Murrieta-Cid, I. Becerra, G. Laguna, S. M. LaValle, Optimal navigation for a differential drive disc robot: A game against the polygonal environment, J. Intell. Rob. Syst., 89 (2018), 211–250. https://doi.org/10.1007/s10846-016-0433-1 doi: 10.1007/s10846-016-0433-1
[60]	A. Pierson, Z. Wang, M. Schwager, Intercepting rogue robots: An algorithm for capturing multiple evaders with multiple pursuers, IEEE Rob. Autom. Lett., 2 (2016), 530–537. https://doi.org/10.1109/LRA.2016.2645516 doi: 10.1109/LRA.2016.2645516
[61]	Z. Zhou, W. Zhang, J. Ding, H. Huang, D. M. Stipanović, C. J. Tomlin, Cooperative pursuit with voronoi partitions, Automatica, 72 (2016), 64–72. https://doi.org/10.1016/j.automatica.2016.05.007 doi: 10.1016/j.automatica.2016.05.007
[62]	E. Bakolas, P. Tsiotras, Relay pursuit of a maneuvering target using dynamic voronoi diagrams, Automatica, 48 (2012), 2213–2220. https://doi.org/10.1016/j.automatica.2012.06.003 doi: 10.1016/j.automatica.2012.06.003
[63]	R. Yan, Z. Shi, Y. Zhong, Reach-avoid games with two defenders and one attacker: An analytical approach, IEEE Trans. Cybern., 49 (2018), 1035–1046. https://doi.org/10.1109/TCYB.2018.2794769 doi: 10.1109/TCYB.2018.2794769
[64]	R. Yan, Z. Shi, Y. Zhong, Cooperative strategies for two-evader-one-pursuer reach-avoid differential games, Int. J. Syst. Sci., 52 (2021), 1894–1912. https://doi.org/10.1080/00207721.2021.1872116 doi: 10.1080/00207721.2021.1872116
[65]	J. Wang, J. Huang, Y. Tang, Swarm intelligence capture-the-flag game with imperfect information based on deep reinforcement learning, Sci. Sin. Technol., 2021. https://doi.org/10.1360/SST-2021-0382 doi: 10.1360/SST-2021-0382
[66]	I. M. Mitchell, A. M. Bayen, C. J. Tomlin, A time-dependent hamilton-jacobi formulation of reachable sets for continuous dynamic games, IEEE Trans. Autom. Control, 50 (2005), 947–957. https://doi.org/10.1109/TAC.2005.851439 doi: 10.1109/TAC.2005.851439
[67]	E. Garcia, D. W. Casbeer, M. Pachter, The capture-the-flag differential game, in 2018 IEEE Conference on Decision and Control (CDC), (2018), 4167–4172. https://doi.org/10.1109/CDC.2018.8619026
[68]	M. Pachter, D. W. Casbeer, E. Garcia, Capture-the-flag: A differential game, in 2020 IEEE Conference on Control Technology and Applications (CCTA), (2020), 606–610. https://doi.org/10.1109/CCTA41146.2020.9206333
[69]	Z. Liu, W. Lin, X. Yu, J. J. Rodríguez-Andina, H. Gao, Approximation-free robust synchronization control for dual-linear-motors-driven systems with uncertainties and disturbances, IEEE Trans. Ind. Electron., 69 (2021), 10500–10509. https://doi.org/10.1109/TIE.2021.3137619 doi: 10.1109/TIE.2021.3137619
[70]	Y. Tang, X. Jin, Y. Shi, W. Du, Event-triggered attitude synchronization of multiple rigid body systems with velocity-free measurements, Automatica, in press.
[71]	X. Jin, Y. Shi, Y. Tang, X. Wu, Event-triggered attitude consensus with absolute and relative attitude measurements, Automatica, 122 (2020), 109245. https://doi.org/10.1016/j.automatica.2020.109245 doi: 10.1016/j.automatica.2020.109245
[72]	R. R. Brooks, J. E. Pang, C. Griffin, Game and information theory analysis of electronic countermeasures in pursuit-evasion games, IEEE Trans. Syst. Man Cybern. Part A Syst. Humans, 38 (2008), 1281–1294. https://doi.org/10.1109/TSMCA.2008.2003970 doi: 10.1109/TSMCA.2008.2003970
[73]	J. Ni, S. X. Yang, Bioinspired neural network for real-time cooperative hunting by multirobots in unknown environments, IEEE Trans. Neural Networks, 22 (2011), 2062–2077. https://doi.org/10.1109/TNN.2011.2169808 doi: 10.1109/TNN.2011.2169808
[74]	J. Poropudas, K. Virtanen, Game-theoretic validation and analysis of air combat simulation models, IEEE Trans. Syst. Man Cybern. Part A Syst. Humans, 40 (2010), 1057–1070. https://doi.org/10.1109/TSMCA.2010.2044997 doi: 10.1109/TSMCA.2010.2044997
[75]	Z. E. Fuchs, P. P. Khargonekar, J. Evers, Cooperative defense within a single-pursuer, two-evader pursuit evasion differential game, in 49th IEEE Conference on Decision and Control (CDC), (2010), 3091–3097. https://doi.org/10.1109/CDC.2010.5717894
[76]	B. Goode, A. Kurdila, M. Roan, Pursuit-evasion with acoustic sensing using one step nash equilibria, in Proceedings of the 2010 American Control Conference, (2010), 1925–1930. https://doi.org/10.1109/ACC.2010.5531356
[77]	Y. Tang, D. Zhang, P. Shi, W. Zhang, F. Qian, Event-based formation control for nonlinear multiagent systems under DoS attacks, IEEE Trans. Autom. Control, 66 (2020), 452–459. https://doi.org/10.1109/TAC.2020.2979936 doi: 10.1109/TAC.2020.2979936
[78]	S. Wang, X. Jin, S. Mao, A. V. Vasilakos, Y. Tang, Model-free event-triggered optimal consensus control of multiple Euler-Lagrange systems via reinforcement learning, IEEE Trans. Network Sci. Eng., 8 (2020), 246–258. https://doi.org/10.1109/TNSE.2020.3036604 doi: 10.1109/TNSE.2020.3036604
[79]	H. Gao, Z. Li, X. Yu, J. Qiu, Hierarchical multiobjective heuristic for PCB assembly optimization in a beam-head surface mounter, IEEE Trans. Cybern., 2021. https://doi.org/10.1109/TCYB.2020.3040788 doi: 10.1109/TCYB.2020.3040788
[80]	Y. Tang, X. Wu, P. Shi, F. Qian, Input-to-state stability for nonlinear systems with stochastic impulses, Automatica, 113 (2020), 108766. https://doi.org/10.1016/j.automatica.2019.108766 doi: 10.1016/j.automatica.2019.108766

This article has been cited by:

1.	Erdal Karapinar, Andreea Fulga, Seher Sultan Yeşilkaya, Santosh Kumar, Fixed Points of Proinov Type Multivalued Mappings on Quasimetric Spaces, 2022, 2022, 2314-8888, 1, 10.1155/2022/7197541
2.	Mi Zhou, Xiaolan Liu, Naeem Saleem, Andreea Fulga, Nihal Özgür, A New Study on the Fixed Point Sets of Proinov-Type Contractions via Rational Forms, 2022, 14, 2073-8994, 93, 10.3390/sym14010093
3.	Salvador Romaguera, Basic Contractions of Suzuki-Type on Quasi-Metric Spaces and Fixed Point Results, 2022, 10, 2227-7390, 3931, 10.3390/math10213931
4.	Salvador Romaguera, On Protected Quasi-Metrics, 2024, 13, 2075-1680, 158, 10.3390/axioms13030158

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

Electronic Research Archive

1 1.3

Metrics

Article views(2749) PDF downloads(157) Cited by(4)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(6) / Tables(1)

Electronic Research Archive

Optimal strategy analysis for adversarial differential games

Related Papers:

Abstract

1. Introduction

2. Summary of proposed methods

2.1. The modified F-expansion method

2.2. The generalized algebraic method

3. Methodology

3.1. Application of modified F-expansion method

3.2. Application of the generalized algebraic method

4. Numerical solution

5. Numerical scheme on a fixed mesh

6. Stability of the numerical scheme

7. Error analysis

8. Results and discussion

9. Conclusions

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

Electronic Research Archive

Optimal strategy analysis for adversarial differential games

Related Papers:

Abstract

1. Introduction

2. Summary of proposed methods

2.1. The modified F-expansion method

2.2. The generalized algebraic method

3. Methodology

3.1. Application of modified F-expansion method

3.2. Application of the generalized algebraic method

4. Numerical solution

5. Numerical scheme on a fixed mesh

6. Stability of the numerical scheme

7. Error analysis

8. Results and discussion

9. Conclusions

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog