https://orcid.org/0000-0002-6291-4671
Abstract
The present study computes the Lie symmetries and exact solutions of some problems modeled by nonlinear partial differential equations. The (1 + 1)-dimensional integro-differential Ito, the first integro-differential KP hierarchy, the Calogero-Bogoyavlenskii-Schiff (CBS), the modified Calogero-Bogoyavlenskii-Schiff (CBS), and the modified KdV-CBS equations are some of the problems for which we want to find new exact solutions. We employ similarity variables to reduce the number of independent variables and inverse similarity transformations to obtain exact solutions to the equations under consideration. The sine-cosine method is then utilized to determine the exact solutions.
Citation: Amin A, Naeem I, Khan A (2023) Exact solutions of (1 + 1)-dimensional integro-differential Ito, KP hierarchy, CBS, MCBS and modified KdV-CBS equations. PLoS ONE 18(3): e0283569. https://doi.org/10.1371/journal.pone.0283569
Editor: Shou-Fu Tian, China University of Mining and Technology, CHINA
Received: September 12, 2022; Accepted: March 9, 2023; Published: March 30, 2023
Copyright: © 2023 Amin et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The article does not report data and the data availability policy is not applicable to my article.
Funding: The author(s) received no specific funding for this work. NO
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Nonlinear partial differential equations (NLPDEs) have been utilized to characterize various nonlinear occurrences in mathematical biology, physics, and several other areas of science and engineering. The most important problem in real-world phenomena is computing exact solutions of nonlinear PDEs. The homogeneous balance method [1], the Darboux transform method [2, 3], the first integral method [4, 5], the tanh function method [6], the modified simple equation method [7, 8], the method of the auxiliary equation [9, 10], the (G′/G)-expansion method [11, 12], the F-expansion method [13, 14], Jacobi elliptic function method [15, 16], and Lie symmetry method [17–19] are some of the important methods available to compute exact solutions of nonlinear PDEs. Although there is no universal strategy for solving nonlinear PDEs, Lie symmetry analysis is one of the most effective and reliable techniques for discovering new exact solutions to nonlinear PDEs arising in applied mathematics and physics. For the application of some well-known methods to compute numerical and exact solutions of differential equations, the interested reader is referred to see [20–25].
In this paper, the Lie point symmetry method is applied to nonlinear systems. The symmetry reductions related to the nonlocal symmetry can be analyzed in [26] where the truncated Painlevé analysis or the Möbious invariant form yields the nonlocal symmetry of the Gardner equation. Here, the extended system locates the nonlocal symmetry to the local Lie point symmetries. So the nonlocal symmetry is used to find possible reductions in symmetry using the localization technique. In [27] the nonlocal symmetries for the (2+1)-dimensional Konopelchenko–Dubrovsky equation are determined with the shortened Painlevé method and the Möbious (conformal) invariant form. Here the nonlocal symmetries are reduced to the Lie point symmetries by inserting auxiliary dependent variables. So, we can generate finite symmetry transformations by solving the initial value problem of the prolonged systems. Similarly using the truncated Painlevé approach and the Möbius (conformal) invariant form, we can drive the nonlocal symmetry for the Drinfel’d-Sokolov-Wilson equation in [28]. Meanwhile, for the use of symmetry reductions related to nonlocal symmetry, the nonlocal symmetry will be localized to the Lie point symmetry by introducing three dependent variables.
Recently, Li, Tian, Yang, and Fan have done some interesting work in deriving the solutions of the Wadati-Konno-Ichikawa equation and complex short pulse equation with the help of the Dbar-steepest descent method. They solved the long-time asymptotic behavior of the solutions of these equations and proved the soliton resolution conjecture and the asymptotic stability of solutions of these equations. (See: [29–31]).
The (1 + 1)- dimensional integro-differential Ito equation is a well known NLPDE, It can be governed by
The mathematical representation of first integro-differential KP hierarchy equation is
The Calogero—Bogoyavlenskii—schiff (CBS) equation is represented by
The modified Calogero-Bogoyavlenskii-schiff (CBS) equation can be expressed symbolically as
The standard form of modified KdV-CBS equation is
Developing techniques for the exact solutions of these models, which entail systems of nonlinear PDEs, has been important in the study of nonlinear PDEs.
In this article, we compute the exact solutions of five well-known NLPDEs, namely, the (1 + 1)-dimensional integro-differential Ito equation, the first integro-differential KP hierarchy equation, the Calogero-Bogoyavlenskii-Schiff (CBS) equation, the modified Calogero-Bogoyavlenskii-Schiff (CBS) equation, and the modified KdV-CBS equation. The classical Lie point symmetries are utilized to reduce the number of independent variables via similarity transformations, which ultimately give rise to the exact solutions of the prescribed equations. The sine-cosine method is also employed to derive general exact solutions. The exact solutions presented in this study are concise and straightforward and can be used to establish new solutions for other kinds of NLPDEs arising in different areas of mathematical physics.
The paper is organized in the following pattern: In Section 2, basic definitions, important relations, and the fundamental theorem of the sine-cosine method are presented. In Section 3, Lie symmetries and exact solutions of (1 + 1)-dimensional integro-differential Ito equations are constructed. In Section 4, Lie symmetries and exact solutions of the first integro-differential KP hierarchy equation are determined. In Section 5, Lie symmetries and exact solutions of the Calogero-Bogoyavlenskii-Schiff (CBS) equations are analyzed. In section 6, Lie symmetries and exact solutions of the modified Calogero-Bogoyavlenskii-Schiff (CBS) and in section 7, Lie symmetries and exact solutions of the modified Kdv-CBS equations are evaluated. In the last section, we summarize the concluding remarks.
2 Fundamental operators
Consider the following pth order system of differential equations
(2.1)
with m independent variables x = (x1, x2, x3, …, xm) and n dependent variables u = (u1, u2, u3…, un).
The differential function Eγ(x, u, u(1), …, u(p)), in (2.1), is a pth order differential invariant of a group G if
(2.2)
where X[p] is the pth prolongation of the Lie-Bäclund or generalized operator X defined by
(2.3)
where
can be determined from
The total derivative operator with respect to xi takes the form,
(2.4)
The invariants of the differential function Eγ in (2.1) can be obtained by solving characteristic equations derived from Eq (2.2).
The Euler operator is defined as
(2.5)
where Di is the total derivative operator.
2.1 Sine-Cosine method
There is no particular method that works for all types of nonlinear evolution equations. The sine-cosine method ([32–34]) is one such technique and it can be used to solve a wide variety of nonlinear evolution equations. The sine-cosine method works for those PDEs that admit translational symmetries. If a PDE possesses translational symmetries, we may convert it to an ODE by introducing a wave variable and assuming that the solution would take the form of a sine function or cosine function. The sine-cosine algorithm is described as follows:
Sine-Cosine algorithm:
- Integrate the ODE Q(u, us, uss, usss, …) = 0 as many times as possible and set the constants of integration to zero.
- Suppose the solution of the form
(2.6) or
(2.7) where we need to find the parameters α, β and k.
- Substitute (2.6) or (2.7) in Q(u, us, uss, usss, …) = 0 and balance the terms of sine functions when (2.6) is used or balance the terms of cosine functions when (2.7) is utilized.
- After defining the value of k, we separate the terms concerning powers of cosine or sine functions to obtain algebraic system of equations in terms of α and β.
After computing the values of α and β and inserting them into the main equation, we get at the solution.
3 Lie symmetries and exact solutions of (1 + 1)- dimensional Integro-differential Ito equation
The (1 + 1)- dimensional integro-differential Ito equation is governed by
(3.1)
Eq (3.1) can be expressed as
(3.2)
where
denotes the integral with respect to the subscripts.
Using Eq (2.2), the following overdetermined linear system of PDEs is obtained
The following Lie point symmetries can be generated by solving determining equations with one component equal to one and the remaining equal to zero
The translational symmetries corresponding to system (3.2) are
(3.3)
The following similarity transformations are obtained using combination of X1 and X2 i.e X = X1 + αX2
which further implies
(3.4)
Substituting (3.4) in system (3.2), we obtain
(3.5)
The system (3.5) gives rise to
Integrating twice yields
(3.6)
The particular solution of (3.6) is
(3.7)
The set of solutions (3.7) expressed in terms of original variables are
and
where
Now using the sine-cosine approach [32–34], we obtain exact solutions of Eq (3.6). Suppose (3.6) has a solution such as
(3.8)
Substituting the values of u from (3.8) in (3.6) yields
(3.9)
Eq (3.9) is satisfied if
Substituting k = −2 in (3.9), we obtain
Comparing the coefficients of powers of , we have
(3.10)
Solution of (3.10) gives
Eq (3.8) using the value of λ, ω and k results in
where r = x − αt. The solution of (3.6) in terms of original variables can be expressed as
which constitute the exact solutions of (1 + 1)- dimensional integro-differential Ito equation (3.2).
4 Lie symmetries and exact solutions of the first integro-differential KP hierarchy equation
The first integro-differential KP hierarchy equation is governed by
(4.1)
which can be rewritten as
(4.2)
where
denotes the integral with respect to the subscripts.
Using Eq (4.2), the following overdetermined linear set of PDEs are obtained
The following Lie point symmetries can be generated by solving determining equations with one component equal to one and the remaining equal to zero
Using the combination of translational symmetries
or
We obtain the similarity transformations
Substituting above values in Eq (4.2), we obtain
(4.3)
The system (4.3) admits the Lie point symmetries Using the combination of symmetries X1 and X2, we have
The similarity transformations are
(4.4)
Substituting (4.4) in system (4.3), we obtain
(4.5)
The system (4.5) gives rise to
Integration w.r.t g, yields
(4.6)
The particular solution of (4.6) is
The above solutions can be finally expressed in terms of original variables as
where
Now, Using the sine-cosine approach [32–34], we obtain the explicit solutions of Eq (4.6). Suppose (4.6) has a solution such as
(4.7)
Substituting the value of u from (4.7) in (4.6) yields
(4.8)
Eq (4.8) is satisfied if 2k = k − 2. Replacing k = −2 in (4.8) to obtain
Comparing the coefficients of powers of , we get
(4.9)
Solving (4.9) gives
Using the values of λ, ω and k in (4.7) result in
The solution of (4.6) in terms of original variables can be expressed as
and this constitute the exact solutions of the first integro-differential KP hierarchy equation (4.1).
5 Lie symmetries and exact solutions of the Calogero-Bogoyavlenskii-schiff (CBS) equation
The Calogero—Bogoyavlenskii—schiff (CBS) equation is governed by
(5.1)
Eq (5.1) can be expressed as system of following two equations
(5.2)
where
denotes the integral with respect to the subscripts.
Using Eq (5.2), the following overdetermined linear set of PDEs are obtained
The following Lie point symmetries can be generated by solving determining equations with one component equal to one and the remaining equal to zero
The combination of translational symmetries corresponding to system (5.2) are
or
The similarity transformations are obtained using combination of X = X1 + X2 + αX3
(5.3)
Substituting values from (5.3) in Eq (5.2), we obtain
(5.4)
The system (5.4) admits the translational symmetries We use the combination of X1 and X2, i.e
and compute the similarity transformations
(5.5)
Substituting (5.5) in Eq (5.4) gives rise to
(5.6)
where
From system (5.6), we conclude
Integrating w.r.t g and choosing the constant of integration to zero, we arrive at
(5.7)
which finally yields
Using v(g) in system (5.6), we obtain
We apply the inverse informations (5.3), the solution can be expressed in original variables as
where
.
which constitute an exact solution of Eq (5.1).
Now, Using the sine-cosine approach [32–34], we obtain the explicit solutions of Eq (5.7). Suppose (5.7) has a solution such as
(5.8)
Substituting v from (5.8) in (5.7), yields
(5.9)
Eq (5.9) is satisfied if 2k = k − 2. Substituting k = −2 in (5.9) to obtain
Comparing the coefficients of powers of , we obtain the following system
(5.10)
Eq (5.8) with the use of λ, ω and k results in
Using inverse transformations, the solution of (5.7) in terms of original variables can be expressed as
where
which constitute the exact solutions of Eq (5.1).
6 Lie symmetries and exact solutions of the modified Calogero-Bogoyavlenskii-schiff (MCBS) equation
The modified Calogero-Bogoyavlenskii-schiff (CBS) equation is regulated by
(6.1)
which can be expressed into system of two equations
(6.2)
where
denotes the integral with respect to the subscript.
Using Eq (6.2), the following overdetermined linear set of PDEs are obtained
The following Lie point symmetries can be generated by solving determining equations with one component equal to one and the remaining equal to zero
The combination of translational symmetries
or
The corresponding similarity transformations are
Substituting above values in Eq (6.2), we obtain
(6.3)
System (6.3) admits the translational symmetries Using combination of symmetries
we find similarity transformations
Substituting above values in system (6.3), results in
(6.4)
From (6.4), we conclude
Integrating w.r.t g and choosing constant of integration to zero gives rise to
(6.5)
Now using the sine-cosine approach [32–34], we obtain the explicit solutions of Eq (6.5). Suppose (6.5) has a solution such as
(6.6)
Substituting the values of v from (6.6) in (6.5) yields
(6.7)
Eq (6.7) is satisfied if 3k = k − 2. Replacing k = −1 in (6.7) to obtain
Comparing the coefficients of powers of , we have
(6.8)
Solving (6.8) gives
Using values of λ, ω and k in (6.6) results in
The solution of (6.5) in terms of original variables can be expressed as
(6.9)
Eq (6.9) constitute the exact solution of the modified Calogero-Bogoyavlenskii-schiff (CBS) Eq (6.2).
7 Lie symmetries and exact solutions of the modified KdV-CBS equation
The modified KdV-CBS equation is expressed as
(7.1)
Eq (7.1) can be re-written as
(7.2)
where
denotes the integral with respect to the subscripts.
Using Eq (7.2), we obtain the following set of overdetermined linear PDEs
(7.3)
We get the following Lie point symmetries
The combination of translational symmetries
rovide the similarity transformations
(7.4)
Using change of variables (7.4), system (7.2) transforms to
(7.5)
Eq (7.5) admits the Lie point symmetries . We use the combination of symmetries
to find the canonical variables
(7.6)
(7.5) with the help of similarity transformations and then integration w.r.t g, we find
(7.7)
The solution of above system gives the following equation:
Integration w.r.t g gives rise to
(7.8)
Now, using the sine-cosine approach [32–34], we obtain the explicit solutions of Eq (7.8). Suppose (7.8) has a solution such as
(7.9)
Substituting u from (7.9) in (7.8) yields
(7.10)
Eq (7.10) is satisfied if 2k = −2. Substituting k = −1 in (7.10), we obtain
Comparing the coefficients of powers of , we arrive at
(7.11)
Solving (7.11) gives
Using the values of λ, ω and k in (7.9) results in
Thus,
which constitute the exact solutions of the modified KdV-CBS equation (7.2).
8 Conclusion
The exact solutions of the (1 + 1)- dimensional integro-differential Ito, the first integro-differential KP hierarchy, the Calogero-Bogoyavlenskii-Schiff (CBS), the modified Calogero-Bogoyavlenskii-Schiff (CBS) and the modified KdV-CBS equations have been successfully established by utilizing the similarity transformations and the sine-cosine method. In the evaluation of exact solutions, the reduction in the number of independent variables via similarity variables and the use of inverse similarity transformations have been made. By substituting back, it has been checked that the acquired solutions satisfy the prescribed equations. Furthermore, we have done the Lie symmetry analysis to investigate these solutions. Thus, the proposed approach is more efficient, reliable, and concise by means of computational complexity. It can provide more exact solutions as compared to the other methods that exist in the literature. The obtained solutions are new and innovative to existing ones and therefore, more appropriate to understand. In fact, the proposed method is readily applicable to a large variety of nonlinear evolution equations which frequently appear in mathematical physics and nonlinear sciences.
Acknowledgments
The author expresses gratitude to the referees for their constructive criticism, which helped to strengthen the paper’s substance.
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