Elliptic and multiple-valued solutions of some higher order ordinary di ff erential equations

: In the present paper, by the complex method, the meromorphic solutions of the higher order ordinary di ff erential equation w (5) + aw ′′ + bw 2 − cw + d = 0 are investigated, where a , b , c , d are constant complex numbers, and b (cid:44) 0. Furthermore, by Theorem 1.1, we built elliptic and multiple-valued solutions for the higher order ordinary di ff erential equations u (6) − u (5) + u ′ 2 − 2 u ′ u + u 2 + 2 u ′ − 2 u + 1 = 0 and u (6) − u (5) + au ′′′ − au ′′ + bu ′ 2 − 2 bu ′ u + bu 2 − cu ′ + cu + d = 0. At the end, we give some new meromorphic solutions for two higher-order KdV-like equations.


Introduction
Non-linear differential equations are widely applied to represent complex phenomena in many natural sciences, and exact solutions contribute to well understanding of natural phenomena. Therefore, it is important to study the exact solutions of non-linear differential equations. There are many effective methods that are being used to find exact solutions of differential equations, such as the F-expansion method [1], the exponential function method [2,3], the tanh method [4], the inverse scattering transform method [5], the direct algebraic method [6], the sine-cosine method [7], the first integral method [8], the transformed rational function method [9], the Bäcklund transform method [10], the (G ′ /G)-expansion method [11] and the Lie group method [12].
We say w(z) is a meromorphic function, which means that w(z) is analytic in the complex plane C except for poles. In recent years, many researchers studied complex differential equations using the complex method [13] and Nevanlinna's theory, and build some new elliptic function solutions and simple periodic function solutions, for instance, see [14][15][16][17]. These results show that the complex method is an effective tool for constructing explicit meromorphic solutions for complex differential equations. In this paper, we consider the following partial differential equation u t + au xxx + 2buu x + u xxxxxx = 0, (1.1) where u(x, t) is a real-valued function, a, b( 0) are real constants. Equation (1.1) is a modified version of the Kuramoto-Sivashinsky equation in [18] which has aroused great interest in physical scientists in recent years. Define a meromorphic function f belongs to the class W (see [18]) if f is an elliptic function, a rational function of e αz (α ∈ C) or a rational function of z. The Kuramoto-Sivashinsky equation reads ϕ t + νϕ xxxx + bϕ xxx + µϕ xx + ϕϕ x = 0, ν, b, µ ∈ R, ν 0. By the traveling wave transformation ϕ(x, t) = c + w(z), z = x − ct, it reduces to the ordinary differential equation Eremenko applied the Nevanlinna theory and found that all meromorphic solutions of Eq (1.2) belong to the class W, and if for some values of parameters such solution w exists, then all other meromorphic solutions form a one-parametric family w(z − z 0 ). Further, elliptic solutions exist only if b 2 = 16 µν, non-constant rational solutions exist if and only if b = µ = A = 0, and all exponential solutions have the form of P(tan kz), where P is a polynomial [18]. In this direction, the motivation of this paper is, therefore, whether it is possible to study very high order differential equations, such as Eq (1.1), to study whether these equations have solutions in W, whether there are only solutions in W, and, further, find out the expressions for the solutions.
Take traveling wave transformation u(x, t) = w(z), z = x − ct into Eq (1.1) and get the fifth-order algebraic ordinary differential equation (ODE) where a, b( 0), c, d are constant complex numbers, and the superscript (k) denotes the kth derivative with respect to z. Conte and Ng used the subequations method to obtain meromorphic solutions for the generalized third-order differential equation (see [19], pp. 2, Eq (3)). Demina and Kudryashov used the Laurent series method to study some non-linear partial differential equations, such as the Kawahara equation [20]. However, higher-order ODEs are rarely touched. Starting from this point, our aim is to prove that all meromorphic solutions for Eq (1.3) belong to the class W, and use the complex method and direct method of substitution to construct non-trivial elliptic and multiple-valued solutions of Eqs (1.3), (1.4) and (1.7). Further, we will prove the following results.
where z 0 ∈ C, g 2 = 0, g 3 = 2a 2 226935 . Eq (1.3) is without rational and simply periodic function solution. 2) If a = 0, the only rational function solution is w r (z) = 15120 b 1 (z − z 0 ) 5 + Put z = x − ct into former solutions, and the traveling wave solutions for Eq (1.1) will be obtained immediately.
For some values of the parameters, Theorem 1.1 shows that Eq (1.3) has only elliptic function solutions and rational function solutions. We can use the results of Theorem 1.1 to evaluate the existence of solutions to more complex sixth-order differential equations through a kind of functional transformation, for instance, Eqs (1.4) and (1.7). The results show that we obtain a class of innovative multiple-valued solutions for some complex ordinary differential equations. By Theorem 1.1, we prove the following theorems.
a, b( 0), c, d are constant complex numbers. If 4bd = c 2 , Eq (1.7) has the following elliptic and multi-valued solution This paper is organized as follows. In Section 2, we will introduce some mathematical definitions, lemmas, and the complex method. In Section 3, we will prove the three theorems. In Section 4, we will give elliptic meromorphic solutions to the modified singularly perturbed generalized higher-order KdV equation and the special sixth-order KdV-like equation by virtue of Eq (1.3). In Section 5, we will give the conclusions and discussion and pose two unsolved conjectures for the readers.

Lemmas and the complex method
In this section, we introduce the related concepts, the lemmas, and the complex method [13].
Define differential polynomial where a r are constants, and I is a finite index set. The degree of P(w, w ′ , · · · , w (m) ) is defined by Consider an autonomous algebraic ODE where P is a polynomial in w(z) and its arguments with constant coefficients, b( 0) and c are complex constants. We investigate the solutions, which are in the form of the formal Laurent series If there are exactly p distinct formal Laurent series satisfy Eq (2.1), we say Eq (2.1) satisfies ⟨p, q⟩ condition. If we only determine p distinct principle parts w(z) = −1 k=−q c k z k (q > 0, c −q 0), we say Eq (2.1) satisfies weak ⟨p, q⟩ condition. If Eq (2.1) satisfies ⟨p, q⟩ condition, we say Eq (2.1) satisfies the finiteness property: has only finitely many formal Laurent series with finite principle part admitting the equation.

4)
where c −i j are given by Eq (2.2), (ii) Each rational function solution w := R(z) is of the form

5)
with l(≤ p) distinct poles of multiplicity q.
Apply the complex method [13], we will pose the following steps: Step 1 Substituting the transform T : u(x, y, t) → w(z), (x, y, t) → z into a given PDE gives a non-linear ordinary differential Eq (2.1).
Step 5 Substituting the inverse transform T −1 into the meromorphic solutions w(z − z 0 ), then get all exact solutions u(x, y, t) of the original given PDE.
There is no unified method to handle all types of differential equations and obtain all types of solutions. One of the fundamental reasons we apply the complex method in the current paper is that by applying this method, we can obtain new meromoprhic solutions on the complex domains, e.g., W-class solutions. Furthermore, the dominant term is E(z, w) = w ′′′′′ + bw 2 , therefore by Lemma 2.2, all meromorphic solutions w ∈ W. In the following, we are going to solve Eq (1.3). 1) a 0. By (2.4), we infer that the indeterminant of elliptic solution with pole z = 0 is Combining similar terms, we have Eliminating the coefficients for the above functional relation, we have 967680a 2 − 109800230400g 3 = 0, −12810026880g 2 2 = 0, 78109920ag 2 = 0, 51660a 2 g 2 + 6724db − 1681c 2 − 10065021120g 2 g 3 = 0, so g 2 = 0, g 3 = 2a 2 /226935, and 6724bd − 1681c 2 = 0. Therefore, we yield that Eq (1.3) is integrable provided that 4bd − c 2 = 0 (if 4bd − c 2 0, the constant terms in the expansion of Eq (1.3) can not be vanished), then Eq (1.3) has the following elliptic solution where z 0 ∈ C, g 2 = 0, g 3 = 2a 2 /226935. By additional formula and (2.4), we know that each elliptic function w can be written as w = R 1 (℘) + R 2 (℘)℘ ′ , where R 1 , R 2 are uniquely determined rational functions. By (2.5), we infer that the indeterminant of rational solution with pole z = 0 is It contradicts with a 0, therefore, Eq (1.3) doesn't have any rational function solution. By (2.6), we infer that the indeterminant of simply periodic solution with pole z = 0 is setting ξ = e αz , substituting (3.4) into Eq (1.3), we have then eliminating the coefficients, letting the leading terms equal to zero, we have α 5 e 5αz − 1 = 0, hence z = − 1 α log α, but it is contradict with z is arbitrary. Therefore Eq (1.3) doesn't have any simple periodic solution.
2) a = 0. By (3.2), letting g 3 = 0, it is obvious to see that is the unique rational solution when a = 0, where z 0 ∈ C. Thus, we complete the proof of Theorem 1.1.

The proof of Theorem 1.2
Proof. We take a transformation u(z) := ( w(z)e −z dz + β)e z into Eq (1.4), where β is an arbitrary constant and w(z) is a meromorphic function on the complex plane. Then, we reduce Eq (1.4) to 1) By Theorem 1.1, all the non-constant meromorphic solution with pole z = 0 of Eq (3.7) is Furthermore, the solutions with pole z = z 0 ∈ C of Eq (1.4) is where β, z 0 ∈ C arbitrary. Clearly, γ is a constant. According to the multiple-valued property of Logarithmic function log z, solution (1.5) demonstrates that Eq (1.4) has a class of multiple-valued solutions that do not belong to W.
The proof of Theorem 1.2 is completed.

The Proof of Theorem 1.3
Proof. By the transformation u(z) := ( w(z)e −z dz + β)e z , Eq (1.7) will be change into (1.3): where β is arbitrary, w(z) is meromorphic in the complex plane.

Applications
The complex method has been applied in the process of many higher order differential equations, such as the six-order thin-film equation [15], the seventh-order KdV equation [16] with the assistance of Painlevé analysis and Nevanlinna theory. Many meromorphic solutions are constructed. In this section, the following sixth-order KdV-like equations are considered again, and the exact solutions are derived with the aid of Eq (1.3).

The special sixth-order KdV-like equation
We second give an example, modified from Kaya ( [26], Example 2), consider a special sixth-order KdV equation as following Substituting the traveling wave transformation where z 0 ∈ C, g 2 = 0, g 3 = 2/226935. Then, substitute z = x − λt into (4.8), and the traveling wave solutions for Eq (4.5) will be built.

Conclusions and discussion
We prove that all meromorphic solutions for Eq (1.3) belong to the class W, and construct them by the complex method. Using a functional transformation u(z) = ( w(z)e −z dz + β)e z , we obtain the elliptic and multiple-valued solutions for the high order nonlinear Eqs (1.4) and (1.7). At last, we give two applications on the KdV-like equations for Theorem 1.1. In conclusion, the complex method is an effective method for constructing explicit traveling wave solutions for some high-order nonlinear differential equations, such as elliptic solutions, simple periodic solutions and rational solutions. Most recently, the non-traveling wave rational solutions of a KdV-like equation [27] and a KP-like equation [28], the non-traveling wave soliton solutions of two types of nonlocal integrable nonlinear Schrödinger equation were investigated [29,30]. It is of great interest to investigate the traveling wave reduced KdVlike equation and the KP-like equation, and the nonlocal integrable nonlinear Schrödinger equations using the complex method to construct rational solutions and meromorphic solutions.
Furthermore, we would like to raise the unsolved conjectures for readers:

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.