Darboux polynomials and rational first integrals of the nonstretching Rolie–Poly model

doi:10.1016/j.aml.2020.106329

Applied Mathematics Letters

Volume 105, July 2020, 106329

https://doi.org/10.1016/j.aml.2020.106329 Get rights and content

Abstract

In this paper, we classify irreducible Darboux polynomials and rational first integrals of the nonstretching Rolie–Poly model by a new method based on the classical characteristic curves method. The classical method has been successfully applied to many systems, but it is not suitable for the nonstretching Rolie–Poly model. We take it as an auxiliary tool to give a preliminary understanding of the structure of the Darboux polynomials, and obtain partial integrability results of the system.

Section snippets

Introduction and statement of the main results

In this work, we consider the nonstretching Rolie–Poly model [1] $\dot{C_{11}} = 2 κ C_{12} - \frac{3}{2} κ C_{12} ((1 + β) C_{11} - β) - ε (C_{11} - 1),$ $\dot{C_{12}} = κ C_{22} - \frac{3}{2} (1 + β) κ C_{12}^{2} - ε C_{12},$ $\dot{C_{22}} = - \frac{3}{2} κ C_{12} ((1 + β) C_{22} - β) - ε (C_{22} - 1)$ which describes entangled linear polymer melts, where $C = (\begin{matrix} C_{11} & C_{12} & 0 \\ C_{12} & C_{22} & 0 \\ 0 & 0 & C_{22} \end{matrix})$ denotes a conformation tensor and the dot denotes the ordinary time derivative. Parameters $κ$ , $β$ and $ε$ denote the shear rate, the convective constraint release coefficient and the ratio of the Rouse time $τ_{R}$ and the reptation time $τ_{d}$ , respectively. With the

Proof of Theorem 1

Let $f (x, y, z) = \sum_{i = 0}^{n} f_{i} (x, y, z)$ be a Darboux polynomial of system (2), where each $f_{i}$ is a homogeneous polynomial of degree $i$ for $i = 0, 1, \dots, n$ . The cofactor $k (x, y, z)$ is of the form $k = k_{1} x + k_{2} y + k_{3} z + k_{0}$ .

By substituting $f$ and $k$ into Eq. (3) we have $(a b x y + c x + \frac{4}{3} a y) \sum_{i = 0}^{n} \frac{\partial f_{i}}{\partial x} + (a b y^{2} + c y + a z + a) \sum_{i = 0}^{n} \frac{\partial f_{i}}{\partial y} + (a b y z + c z - \frac{2}{3} a y) \sum_{i = 0}^{n} \frac{\partial f_{i}}{\partial z} = (k_{1} x + k_{2} y + k_{3} z + k_{0}) \sum_{i = 0}^{n} f_{n} .$

Identifying the homogeneous items of degree $n + 1$ , we get $a b x y \frac{\partial f_{n}}{\partial x} + a b y^{2} \frac{\partial f_{n}}{\partial y} + a b y z \frac{\partial f_{n}}{\partial z} = (k_{1} x + k_{2} y + k_{3} z) f_{n} .$

Here we use the method of characteristic curves to solve it.

Conclusion

In this paper we have proved that the nonstretching Rolie–Poly system has no polynomial first integral when $a b \neq 0$ . And there always exists an invariant algebraic surface $x + 2 z = 0$ which gives this system the prefix “nonstretching”. We also find out many Darboux polynomials and show the cases when this system is completely integrable. But it should be noted that we cannot exclude the existence of irreducible Darboux polynomials of degree more than two. Computation for these Darboux polynomials is

CRediT authorship contribution statement

Jiankun Wu: Methodology, Formal analysis, Investigation, Writing - original draft. Feng Xie: Conceptualization, Methodology, Supervision, Writing - review & editing.

Acknowledgments

The authors are grateful to the referees for valuable comments and suggestions.

The authors were supported by the Natural Science Foundation of Shanghai, China (No. 19ZR1400500).

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Darboux polynomials and rational first integrals of the nonstretching Rolie–Poly model

Abstract

Section snippets

Introduction and statement of the main results

Proof of Theorem 1

Conclusion

CRediT authorship contribution statement

Acknowledgments

J. Non-Newton. Fluid Mech.

J. Non-Newton. Fluid Mech.

J. Geom. Phys.

Physica D

Transient shear banding in entangled polymers: A study using the Rolie–Poly model

J. Rheol.

Nonmonotonic models are not necessary to obtain shear banding phenomena in entangled polymer solutions

Phys. Rev. Lett.

Invariant algebraic surfaces of the Rikitake system

J. Phys. A: Math. Gen.

On the integrability of a Muthuswamy–Chua system

J. Nonlinear Math. Phys.