Exact Solution of Modified Veselov–Novikov Equation and Some Applications in the Game Theory

The Gaussian curvatures at an isolated point are not defined, but in other points, K< 0 and |a| + |b| + |c|≠ 0 for all surfaces Σa,b,c in which potentials 􏽥 U are regular. Moreover, the inverted Enneper surface of higher order Sa,b,c(n≥ 3) is smoothly deformed into the saddle surface with surface motions Σa,b,c by a straight line given through parameters a, b, and c. Soliton deformation [2] of inverted Enneper surfaces will be found by parallel translation (depending on parameters a, b, and c) of these surfaces in the proof of'eorem 2 as the main result. Physical interpretation of exact solutions of the modified Veselov–Novikov equation (given below) corresponding to inverted Enneper surfaces analogically could be represented by solitons (depending on different parameters) for mixed coupled nonlinear Schrödinger equations [3]. Futhermore, soliton solutions were found by the generalized Darboux transformation [4]. The strategy of this work is to find the solution of the following modified Veselov–Novikov (mVN) equation [5]:


Introduction
Consider the following problem with the initial data (this is the minimal surface): where A � (z 3 /zz 3 ). e following theorems are principal results of this paper. Theorem 1. If ψ 0 is holomorphic function, then the general solution of (1)- (2) is given by the following formula: where Ap i− 1 � p i , p 0 � ψ 0 , i � 1, 2, . . .. This initial-value problem (1)- (2) is solved by the method of successive approximations [1]. Theorem 2. Let a, b, an d c be constants such that a ≫ 1, b ≫ 1, and c ≫ 1 and |α| 2 +|β| 2 � a 2 + b 2 + c 2 ⟺x � y � 0. (4) The Gaussian curvatures at an isolated point are not defined, but in other points, K < 0 and |a| + |b| + |c| ≠ 0 for all surfaces Σ a,b,c in which potentials U are regular.
Moreover, the inverted Enneper surface of higher order S a,b,c (n ≥ 3) is smoothly deformed into the saddle surface with surface motions Σ a,b,c by a straight line given through parameters a, b, and c.
Soliton deformation [2] of inverted Enneper surfaces will be found by parallel translation (depending on parameters a, b, and c) of these surfaces in the proof of eorem 2 as the main result. Physical interpretation of exact solutions of the modified Veselov-Novikov equation (given below) corresponding to inverted Enneper surfaces analogically could be represented by solitons (depending on different parameters) for mixed coupled nonlinear Schrödinger equations [3]. Futhermore, soliton solutions were found by the generalized Darboux transformation [4].
The strategy of this work is to find the solution of the following modified Veselov-Novikov (mVN) equation [5]: where z � x + iy and U(z, z, t) is a real-valued function by eorem 1 and Moutard transformations [2,6] to illustrate its applications in game theory.
Taimanov [7] found blowing-up solutions of mVN equations by Moutard transformations on the example of a first-order Enneper surface. eorem 1 generates all minimal surfaces (higher order Enneper surface, catenoid, and helicoid) and leads to the regular solutions of mVN equations formulated in eorem 2 by given transformations.
Theorems 1 and 2 are based on the geometric interpretations of the following Moutard transformations [7,8]. en, (1) Matrices K(Ψ 0 ) and M(Ψ 0 ) have the following form: where S − 1 is the inverse of the matrix S and (2) For every Ψsolution of the system (7), Ψ functions are also solutions for this system and have the form , Ψ) and the Dirac operator D with the potential U � ik 12 , and U with the potential V � k 2 11 − 2(m 21 k 21 + m 11 k 11 ) satisfies the following system of equations: where e previous matrix is chosen so that the system (9) is equivalent to Manakov's L, A, B triple [5,9], which is an operator representation of mVN equations.
(3) e real-valued function U and the function V satisfy the mVN equations.

Preliminaries
Every pair of holomorphic functions ψ 1 , ψ 2 : U ⟶ C corresponds to a minimal surface Σ: U ⟶ R 3 . Σ � (u 1 , u 2 , u 3 ) is given by the following Weierstrass representation [7,10]: and the initial point on the surface has some constant u 1 0 � − a, u 2 0 � b, and u 3 0 � c. We represent the surface in the following form [7]:

Solution of the mVN Equation
Let n ≥ 3. en, the matrix representations S t for all inverted minimal surfaces of Enneper (Σ t ) have to be presented in the following holomorphic functions: with potential U(z, z, t) having the following representation: e polynomial in (14) obtained by eorem 1 (proved in the next section) satisfies problem (1)

Main Results
In this section, we will prove eorems 1 and 2 Hence by using Taylor series of e tA , we obtain which is the formal solution of the problem (1)-(2). For example, initial data (2) could be given by one of the following minimal surfaces [11].

International Journal of Mathematics and Mathematical Sciences
where ψ 1 is given by (14) and k has the following form: For x � y � 0, |β| 2 � 4(a 2 + b 2 ) and |α| 2 � 4c 2 . If you give large parameters a ≫ 1, b ≫ 1, and c ≫ 1 for moving surfaces of Enneper of higher order Σ ⟶ Σ a,b,c by a predetermined straight line, then the soliton deformation [2] of the surfaces S a,b,c is defined globally; that is, all surfaces are deformed into the saddle surface. Potentials U in this case are regular by eorem 4.2. Also, the regularity of the surface follows the inverse minimal surfaces that are regular at the point x � 0, y � 0:  where n 1 � 2j and n 2 � 2j + 1, j � 0, 1, . . ., respectively, a s and b s are nonzero, and c s is some constant (s ∈ N).
Proof. From the potential formula of (3.6), we take partial derivatives with respect to two variables, briefly change to polar coordinates, and calculate limits at r ⟶ 0, φ � const, in the corresponding orders:    (d) What are the Nash equilibria?
Applying the backward algorithm and IDWDS procedure [12] to a given tree, we have the following solution.
(lost, notT, T): backward solution means all players have optimal payoff after comparing payoffs in terminal nodes if Camel chooses action lost with payoff − 1. Player 1 chooses action notT with payoff 1, and Player 2 chooses action T with payoff 100. e strategic form associated with this game has the following three cases: e meaning of the given proverb about the effective way for Player 2 (boy) is the Nash equilibria (lost, notT, T) with payoffs (− 1, 1, 100) where number − 1 is the most optimal payoff for Camel. But this strategy is unstable for the boy; if Player 1 (grandfather) is angry and bad, then he could tumble the boy.
Here, mVN equations are applied as the solution method as follows: Moutard or Darboux transformations [6] work for initial given solutions; furthermore, other solutions could be found. is result is significant for the Kazakh proverb with the number component as backward solutions; Nash equilibria were found in terms of trees as Figure 3. Likewise, other hundreds of examples with Kazakh proverbs could be represented as a rooted tree in game theory.
is example illustrates connection between mathematics and Kazakh proverbs. In addition, Kazakh proverbs have undoubted relation to the Muslim religion [13]. However, because of a huge number of historical facts [14], the readers can find alternative properties of mathematics and religion. e author means any other examples (given below) will adhere to the following paradoxical law: where in A 1 RA 2 , A 1 and A 2 do not have the relation. is law (27) illustrates the main result of this work as follows.
Let A 1 be solutions of mVN equations, A 2 be minimal surfaces with singular points, and A 3 be minimal surfaces with regular points. is assignment satisfies the law (27) for every order of surfaces n ≥ 3 because there is no relation between all solution and minimal surfaces with singular points and A 2 RA 3 shows relation between minimal surfaces with singular and regular points (see Corollary 1), implying A 1 RA 3 as solutions of mVN equations related to minimal surfaces with regular points (see eorem 2).
For n � 1, 2, the nonassociative law: e purpose of mVN equation application as the solution in the game theory is as follows.
Blowing-up or singular solutions of mVN equations led to regular solutions of mVN equations for every order of surfaces n ≥ 3, as initial nonlinear mVN equations led to the linear system of equations, and finally, the alternative between mathematics and religion led to the relation between mathematics and Kazakh proverbs.
In the future, this procedure will show a solution to some nonlinear disagreements or any other life circumstances.
In summary, scientists could find some set of relations in which all components satisfy the paradoxical law (27). Given about five components, e.g., A 1 , A 2 , A 3 , A 4 , an d A 5 , the readers may concentrate on one component and try to solve this problem (about 20 percent of all components) to improve the result of some goal and may gain about 80 percent efficiency according to the Pareto efficiency law. As this law contradicts the Pareto efficiency law, it is named "paradoxical."

Conclusions
In conclusion, to solve the mVN equation, eorem 1 was used not only with Enneper surfaces but also with other minimal surfaces as an example. e obtained results in this article ( eorem 2 and Corollary 1) were graphically represented and introduced to a paradoxical law of game theory. Further analysis of the potentials led to the study of Kazakh proverbs, which are rich in educational and meaningful words, presented in the form of a tree in game theory. Although the meaning of these two concepts as proverbs and potentials (not in an analytical form) is different, the readers can be sure about the fulfillment of paradoxical law in studies of the structures of these objects. e meaning of this law is that nonlinear objects can be reduced to the linear form, and blowing-up solutions can be reduced to the regular form in general.
erefore, the Darboux transformation and potentials are also applied in game theory.
Data Availability e author confirms that the data supporting the findings of this study are available in this published article.

Conflicts of Interest
e author declares that there are no conflicts of interest.