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Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations
Pavel N. Ryabov, Dmitry I. Sinelshchikov, Mark B. Kochanov
TL;DR
The paper addresses exact solitary-wave solutions for high-order nonlinear evolution equations. It applies the Kudryashov method to fifth-, sixth-, and seventh-order equation families and reports that the method yields solitary-wave solutions, including eight cases for one fifth-order equation. The manuscript omits two fifth-order cases because their parameter and solution forms are cumbersome.
Problem
The paper seeks an effective way to find exact solitary-wave solutions for high-order nonlinear evolution equations.
Method
The Kudryashov method is applied to families of fifth-, sixth-, and seventh-order nonlinear evolution equations.
Results
The study obtains solitary-wave solutions for three equation families and finds that one fifth-order equation has solutions in eight cases.
Takeaways & Limitations
The paper demonstrates the Kudryashov method’s efficiency for finding exact solutions of high-order nonlinear evolution equations.
Takeaways & Limitations
Two fifth-order cases are omitted because their parameter values and solution presentations are very cumbersome.
Abstract
from arXiv · showhide
The application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is considered. Some classes of solitary wave solutions for the families of nonlinear evolution equations of fifth, sixth and seventh order are obtained. The efficiency of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is demonstrated.
1 Introduction
The paper presents the Kudryashov method as an effective approach for constructing exact solitary-wave solutions of high-order nonlinear evolution equations. It applies the method to equation families of fifth, sixth, and seventh order.
- The Kudryashov method uses a special singularity-manifold form within the truncation method.
- The method reduces exact-solution construction to solving an overdetermined algebraic system.
- The authors describe the method as more effective than other exact-solution methods for high-order nonlinear evolution equations.
- The study targets solitary-wave solutions for families of fifth-, sixth-, and seventh-order nonlinear evolution equations.
- The paper is organized around the method algorithm, constructions for the three equation families, and a final discussion of results.
2 Method applied
The Kudryashov algorithm transforms a nonlinear evolution equation into an algebraic problem by using a structured Q(z)-based solution ansatz. Dominant-balance analysis determines the pole order, after which derivatives and polynomial coefficients are matched to solve for unknown parameters.
- The first step. Determination of the dominant terms.: The first step determines dominant terms and the pole order by comparing the degrees of equation terms.
- The second step. The solution structure.: The solution is assumed as a finite polynomial in Q(z) with unknown coefficients.
- The third step. Derivatives calculation.: Q(z) satisfies a first-order ordinary differential equation used to calculate derivatives of the ansatz.
- Computer algebra systems such as Maple or Mathematica can calculate derivatives and support the method’s implementation.
- The fourth step. Defining the values of unknown parameters.: After substitution, equal powers of Q(z) are collected and set to zero, producing an algebraic system for the unknown parameters.
- The method is presented as effective for high-order equations and related to tanh, coth, (G’/G), and Exp-function approaches.
- The Exp-function method is described as inapplicable to high-order equations, marking a scope boundary for that comparison.
3 Exact solitary wave solutions of the fifth order evolution equation
The fifth-order evolution equation combines fifth-order Korteweg–de Vries dynamics with additional dispersive and dissipative terms, and the Kudryashov method yields exact solitary-wave solutions in several parameter cases.
- Equation (11) combines the fifth-order Korteweg–de Vries equation with additional dispersive and dissipative terms.The equation may arise in wave processes in active dispersive-dissipative media.
- The traveling-wave reduction transforms the fifth-order equation into an equation for y(z), whose pole order is N = 2.The solution is therefore sought using a quadratic Q-block ansatz with unknown constants a0, a1, and a2.
- Equating coefficients of equal powers of Q(z) produces an algebraic system whose solution exists in eight cases.
- The parameter and solution presentations for cases k = k(5,6) and k = k(7,8) are omitted because they are very cumbersome.Solutions (19) and (20) are plotted in Figure 1, while solutions (21) and (22) are not pictured because they have the same structure.
- For cases k = k(1,2) and k = k(3,4), the paper gives parameter values and corresponding exact solutions.
4 Exact solitary waves of the sixth order evolution equation
The sixth-order equation is analyzed through a traveling-wave reduction and a fifth-degree Q-block ansatz, producing several real parameter families and exact solution forms.
- The equation considered has applications in modeling longitudinal seismic waves, soft turbulence, and chemical reactions in reaction-diffusion systems.
- After applying the traveling-wave transformation and integrating, the sixth-order equation is reduced to an ordinary differential equation.
- Dominant terms give pole order N = 5, motivating the ansatz y(z) = a0 + a1Q + a2Q2 + a3Q3 + a4Q4 + a5Q5.The coefficients a0 through a5 are constants to be determined.
- Solving the resulting algebraic system yields four real families of unknown parameters, but one cumbersome family is omitted.
- The paper identifies conditions for real solutions and presents exact solution forms corresponding to three parameter relations.Solutions (31) and (32) are shown in Figure 2; the third solution also has kink form.
5 Exact solitary waves of the seventh order evolution equation
For the seventh-order evolution equation, the Kudryashov method yields exact solutions for n = 1, 2, and 3, including solitary-wave, kink, and traveling-wave forms.
- Relation to prior work: The work generalizes results previously obtained for this equation in the case n = 2.The authors explicitly present the seventh-order results as a generalization of earlier work.
- Reduction to an ordinary differential equation: The seventh-order evolution equation is reduced using a traveling-wave ansatz and integration with respect to z.The ansatz is u(x, t) = y(z), z = kx − wt.
- Case n = 1: For n = 1, the solution uses a degree-six polynomial in Q because its pole order is N = 6.The resulting solutions and corresponding parameters are reported after substitution.
- Case n = 2: The solution (44) is identified as a kink and illustrated in Figure 4 at γ = −1 and k = 1.The figure caption specifies the parameter values used for the illustration.
- Case n = 3: For n = 3, substitution of a degree-two ansatz produces an exact solution whose form is a traveling wave, illustrated in Figure 5.The construction equates coefficients of equal powers of Q(z) to obtain an algebraic system for the unknown parameters.
6 Conclusion
The paper demonstrates the Kudryashov method's efficiency for high-order nonlinear evolution equations and obtains solitary-wave solutions for fifth-, sixth-, and seventh-order families.
- The Kudryashov method is presented as efficient for finding exact solutions of high-order nonlinear evolution equations.
- Solitary-wave solutions are obtained for three families of nonlinear evolution equations of fifth, sixth, and seventh orders.Graphical representations of the exact solutions are also presented.