Lagrangian formulation of a generalised coupled hyperbolic system

Ben Muatjetjeja; Chaudry Masood Khalique

doi:10.1016/j.jksus.2020.07.013

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Original article

32 (

); 2886-2891

doi:

10.1016/j.jksus.2020.07.013

Lagrangian formulation of a generalised coupled hyperbolic system

Ben Muatjetjeja^{^a,^b}, Chaudry Masood Khalique^{^b,^c,^d,⁎}

aDepartment of Mathematics, Faculty of Science, University of Botswana, Private Bag 0022, Gaborone, Botswana

bInternational Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa

cCollege of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong, 266590, China

dDepartment of Mathematics and Informatics, Azerbaijan University, Jeyhun Hajibeyli str., 71, AZ1007 Baku, Azerbaijan

⁎Corresponding author at: International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa. Masood.Khalique@nwu.ac.za (Chaudry Masood Khalique)

Received: 2019-9-30, Accepted: 2020-7-19, Published: 2020-09-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Abstract

We perform Noether classification of the generalised system of coupled (2 + 1)-dimensional hyperbolic equations, namely $u_{tt} - u_{xx} - u_{yy} + P (v) = 0, v_{tt} - v_{xx} - v_{yy} + Q (u) = 0$ . Besides this we compute conservation laws corresponding to cases that have Noether symmetries for the underlying coupled hyperbolic system.

Keywords

Noether operators

Lagrangian

Conservation laws

Generalised system of coupled (2 + 1)-D hyperbolic equations

Potential function

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1 Introduction

In Escobedo and Herrero (1991), system of equations

(1.11)

u_{t} - ▵ u = v^{q},

(1.12)

v_{t} - ▵ v = u^{p},

with

x \in R^{N}, N ⩾ 1, t, p, q > 0

was studied in which boundedness properties and blow-up of its solutions were analysed. After a few years same authors studied global existence and uniqueness of solutions for system (1.11), (1.12) in Escobedo and Herrero (1993). Similar problems of parabolic nature arise in numerous areas of applied mathematics and model many physical phenomena, for instance, population dynamics, chemical reactions or heat transfer.

Later in Gao and Gao (2013), blow-up and existence of solutions to initial & boundary-value problems to non-linear hyperbolic and parabolic systems including variable exponents were investigated. Recently, the system of coupled (2 + 1)-D hyperbolic equations

(1.13)

u_{tt} - u_{xx} - u_{yy} + α v^{q} = 0,

(1.14)

v_{tt} - v_{xx} - v_{yy} + β u^{p} = 0,

with

q, p

constants and

α, β \neq 0

constants, has been studied in Muatjetjeja and Khalique (2015) from the symmetry stand point. Noether and Lie symmetry classification of (1.13)–(1.14) were performed.

In this work, we examine a generalisation of system (1.13)–(1.14), which we obtain by substituting arbitrary functions $P (v)$ and $Q (u)$ for $v^{q}$ and $u^{p}$ , respectively in (1.13)–(1.14). Thus, we analyze generalised system of two coupled (2 + 1)-D hyperbolic equations

(1.15)

u_{tt} - u_{xx} - u_{yy} + P (v) = 0,

(1.16)

v_{tt} - v_{xx} - v_{yy} + Q (u) = 0,

with

P (v), Q (u)

being arbitrary elements. The purpose of our study is to classify Noether symmetries for system (1.15)–(1.16) and construct conservation laws corresponding to Noether operators admitted by system (1.15)–(1.16).

Lie symmetry analysis, originally developed by Sophus Lie (1842–1899) in the latter half of the nineteenth century, is one of the most systematic methods for studying differential equations. Recently it has attracted a lot of attention from the scientists and has been applied to different areas of research. See for example Ovsiannikov (1982), Ibragimov (1994-1996), Butt and Ahmad (2020), Wang et al. (2016), Yildirim and Mohyud-Din (2010),

The paper is planned in following manner. In Section 2 we give a few salient features regarding Noether point symmetries. Section 3 establishes Noether operators and corresponding conservation laws for system (1.15)–(1.16) are obtained. In Section 4 we present Concluding remarks.

2 Preliminaries and notations

We give some notations and results concerning the Noether operators, which will be used later. See for example Noether (1918) and Muatjetjeja and Khalique (2013) for details.

Let

(2.7)

Y = τ \frac{\partial}{\partial t} + ξ^{1} \frac{\partial}{\partial x} + ξ^{2} \frac{\partial}{\partial y} + η^{1} \frac{\partial}{\partial u} + η^{2} \frac{\partial}{\partial v}

be a vector field, where

τ, ξ^{1}, ξ^{2}, η^{1}

and

η^{2}

are functions of

(t, x, y, u, v)

. The first prolongation is defined as

\begin{matrix} Y^{[1]} = & τ \frac{\partial}{\partial t} + ξ^{1} \frac{\partial}{\partial x} + ξ^{2} \frac{\partial}{\partial y} + η^{1} \frac{\partial}{\partial u} + η^{2} \frac{\partial}{\partial v} + ζ_{t}^{1} \frac{\partial}{\partial u_{t}} + ζ_{t}^{2} \frac{\partial}{\partial v_{t}} + ζ_{x}^{1} \frac{\partial}{\partial u_{x}} \\ + ζ_{x}^{2} \frac{\partial}{\partial v_{x}} + ζ_{y}^{1} \frac{\partial}{\partial u_{y}} + ζ_{y}^{2} \frac{\partial}{\partial v_{y}}, \end{matrix}

where

\begin{matrix} ζ_{t}^{1} = & D_{t} (η^{1}) - u_{t} D_{t} (τ) - u_{x} D_{t} (ξ^{1}) - u_{y} D_{t} (ξ^{2}), \\ ζ_{x}^{1} = & D_{x} (η^{1}) - u_{t} D_{x} (τ) - u_{x} D_{x} (ξ^{1}) - u_{y} D_{x} (ξ^{2}), \\ ζ_{y}^{1} = & D_{y} (η^{1}) - u_{t} D_{y} (τ) - u_{x} D_{y} (ξ^{1}) - u_{y} D_{y} (ξ^{2}), \\ ζ_{t}^{2} = & D_{t} (η^{2}) - v_{t} D_{t} (τ) - v_{x} D_{t} (ξ^{1}) - v_{y} D_{t} (ξ^{2}), \\ ζ_{x}^{2} = & D_{x} (η^{2}) - v_{t} D_{x} (τ) - v_{x} D_{x} (ξ^{1}) - v_{y} D_{x} (ξ^{2}), \\ ζ_{y}^{2} = & D_{y} (η^{2}) - v_{t} D_{y} (τ) - v_{x} D_{y} (ξ^{1}) - v_{y} D_{y} (ξ^{2}) \end{matrix}

and

\begin{matrix} D_{t} = & \frac{\partial}{\partial t} + u_{t} \frac{\partial}{\partial u} + v_{t} \frac{\partial}{\partial v} + u_{tt} \frac{\partial}{\partial u_{x}} + v_{tt} \frac{\partial}{\partial v_{x}} + u_{tx} \frac{\partial}{\partial u_{x}} + v_{tx} \frac{\partial}{\partial v_{x}} \\ + u_{ty} \frac{\partial}{\partial u_{y}} + v_{ty} \frac{\partial}{\partial v_{y}} + \dots, \\ D_{x} = & \frac{\partial}{\partial x} + u_{x} \frac{\partial}{\partial u} + v_{x} \frac{\partial}{\partial v} + u_{xx} \frac{\partial}{\partial u_{x}} + v_{xx} \frac{\partial}{\partial v_{x}} + u_{tx} \frac{\partial}{\partial u_{t}} + v_{tx} \frac{\partial}{\partial v_{t}} \\ + u_{xy} \frac{\partial}{\partial u_{y}} + v_{xy} \frac{\partial}{\partial v_{y}} + \dots, \\ D_{y} = & \frac{\partial}{\partial y} + u_{y} \frac{\partial}{\partial u} + v_{y} \frac{\partial}{\partial v} + u_{yy} \frac{\partial}{\partial u_{y}} + v_{yy} \frac{\partial}{\partial v_{y}} + u_{ty} \frac{\partial}{\partial u_{t}} + v_{ty} \frac{\partial}{\partial v_{t}} \\ + u_{xy} \frac{\partial}{\partial u_{x}} + v_{xy} \frac{\partial}{\partial v_{x}} + \dots . \end{matrix}

Recall, from calculus of variations, Euler–Lagrange operators $\begin{matrix} \frac{δ}{δ u} = & \frac{\partial}{\partial u} - D_{t} \frac{\partial}{\partial u_{t}} - D_{x} \frac{\partial}{\partial u_{x}} - D_{y} \frac{\partial}{\partial u_{y}} + D_{t}^{2} \frac{\partial}{\partial u_{tt}} + D_{x}^{2} \frac{\partial}{\partial u_{xx}} + D_{y}^{2} \frac{\partial}{\partial u_{yy}} + \dots, \\ \frac{δ}{δ v} = & \frac{\partial}{\partial v} - D_{t} \frac{\partial}{\partial v_{t}} - D_{x} \frac{\partial}{\partial v_{x}} - D_{y} \frac{\partial}{\partial v_{y}} + D_{t}^{2} \frac{\partial}{\partial v_{tt}} + D_{x}^{2} \frac{\partial}{\partial v_{xx}} + D_{y}^{2} \frac{\partial}{\partial v_{yy}} + \dots . \end{matrix}$

Definition 1

A function $L (t, x, y, u, v, u_{t}, v_{t}, u_{x}, v_{x}, u_{y}, v_{y})$ is a first order Lagrangian of second order system of PDEs

(2.8)

E_{1} = 0,

(2.9)

E_{2} = 0,

if the system (2.8)–(2.9) is identical to Euler–Lagrange equations

(2.10)

\frac{δ L}{δ u} = 0, \frac{δ L}{δ v} = 0 .

Definition 2

The operator Y of (2.7), is a Noether operator corresponding to first order Lagrangian $L$ of system (2.8)–(2.9) if there exists potential functions $B^{1}, B^{2}$ and $B^{3}$ that depend on $(t, x, y, u, v)$ such that

(2.11)

Y^{[1]} (L) + L {D_{x} (ξ^{1}) + D_{y} (ξ^{2}) + D_{t} (τ)} = D_{t} (B^{1}) + D_{x} (B^{2}) + D_{y} (B^{3}) .

We state the acclaimed Noether theorem.

Theorem 1

(Noether Noether, 1918) If operator Y given by (2.7) is Noether corresponding to first order Lagrangian $L$ of system (2.8)–(2.9), then $T = (T^{1}, T^{2}, T^{3})$ , where

(2.12)

\begin{matrix} T^{1} = & τ L + (η^{1} - u_{t} τ - u_{x} ξ^{1} - u_{y} ξ^{2}) \frac{\partial L}{\partial u_{t}} \\ + (η^{2} - v_{t} τ - v_{x} ξ^{1} - v_{y} ξ^{2}) \frac{\partial L}{\partial v_{t}} - B^{1}, \end{matrix}

(2.13)

\begin{matrix} T^{2} = & ξ^{1} L + (η^{1} - u_{t} τ - u_{x} ξ^{1} - u_{y} ξ^{2}) \frac{\partial L}{\partial u_{x}} \\ + (η^{2} - v_{t} τ - v_{x} ξ^{1} - v_{y} ξ^{2}) \frac{\partial L}{\partial v_{x}} - B^{2}, \end{matrix}

(2.14)

\begin{matrix} T^{3} = & ξ^{2} L + (η^{1} - u_{t} τ - u_{x} ξ^{1} - u_{y} ξ^{2}) \frac{\partial L}{\partial u_{y}} \\ + (η^{2} - v_{t} τ - v_{x} ξ^{1} - v_{y} ξ^{2}) \frac{\partial L}{\partial v_{y}} - B^{3}, \end{matrix}

represents a conserved vector of system (2.8)–(2.9) corresponding to the operator Y.

3 Noether symmetries and conservation laws of the system (1.15)–(1.16)

It can ready be confirmed that the system (1.15)–(1.16) possess a first-order Lagrangian given by

(3.15)

L = u_{y} v_{y} + u_{x} v_{x} - u_{t} v_{t} + \int P (v) dv + \int Q (u) du .

Substituting the above value of $L$ in the determining Eq. (2.11) and splitting on different derivatives with respect to $u$ and $v$ , one obtains a linear homogeneous overdetermined system of PDEs $\begin{matrix} τ_{v} = 0, τ_{u} = 0, ξ_{u}^{1} = 0, ξ_{v}^{1} = 0, ξ_{v}^{2} = 0, ξ_{u}^{2} = 0, ξ_{t}^{2} - τ_{y} = 0, ξ_{t}^{1} - τ_{x} = 0, \\ ξ_{y}^{1} + ξ_{x}^{2} = 0, η_{u}^{1} + η_{v}^{2} - τ_{t} + ξ_{x}^{1} + ξ_{y}^{2} = 0, η_{u}^{1} + η_{v}^{2} + τ_{t} - ξ_{x}^{1} + ξ_{y}^{2} = 0, \\ η_{t}^{1} = - B_{v}^{1}, η_{t}^{2} = - B_{u}^{1}, η_{x}^{1} = B_{v}^{2}, η_{x}^{2} = B_{u}^{2}, η_{y}^{1} = B_{v}^{3}, η_{y}^{2} = B_{u}^{3}, \\ η^{1} Q (u) + η^{2} P (v) + τ_{t} (\int P (v) dv + \int Q (u) du) + ξ_{x}^{1} (\int P (v) dv + \int Q (u) du) + \\ ξ_{y}^{2} (\int P (v) dv + \int Q (u) du) = B_{t}^{1} + B_{x}^{2} + B_{y}^{3} . \end{matrix}$

The solution of above system is $\begin{matrix} ξ^{1} = & a (t, x, y), \\ ξ^{2} = & b (t, x, y), \\ τ = & c (t, x, y), \\ η^{1} = & - b_{y} u - j (t, x, y) u + h (t, x, y), \\ η^{2} = & jv + k (t, x, y), \\ B^{1} = & - j_{t} uv - h_{t} v - k_{t} u + n (t, x, y), \\ B^{2} = & j_{x} uv + k_{x} u + h_{x} v + r (t, x, y), \\ B^{3} = & j_{y} uv + k_{y} u + h_{y} v + s (t, x, y), \end{matrix}$

(3.16)

\begin{matrix} (- b_{y} u - ju + h) Q (u) + (jv + k) P (v) + c_{t} (\int P (v) dv + \int Q (u) du) \\ + a_{x} (\int P (v) dv + \int Q (u) du) + b_{y} (\int P (v) dv + \int Q (u) du) \\ = (j_{xx} + j_{yy} - j_{tt}) uv + (k_{xx} + k_{yy} - k_{tt}) u + (h_{xx} + h_{yy} - h_{tt}) v \\ + n_{t} + r_{x} + s_{y} . \end{matrix}

The study of Eq. (3.16) results in 5 cases:

Case 1

$P, Q$ arbitrary, but not as contained in cases 2–5 below.

We have following six Noether point symmetries

(3.17)

X_{1} = \frac{\partial}{\partial t}, B^{1} = n, B^{2} = r, B^{3} = s, n_{t} + r_{x} + s_{y} = 0,

(3.18)

X_{2} = \frac{\partial}{\partial x}, B^{1} = n, B^{2} = r, B^{3} = s, n_{t} + r_{x} + s_{y} = 0,

(3.19)

X_{3} = \frac{\partial}{\partial y}, B^{1} = n, B^{2} = r, B^{3} = s, n_{t} + r_{x} + s_{y} = 0,

(3.20)

X_{4} = y \frac{\partial}{\partial t} + t \frac{\partial}{\partial y}, B^{1} = n, B^{2} = r, B^{3} = s, n_{t} + r_{x} + s_{y} = 0,

(3.21)

X_{5} = x \frac{\partial}{\partial t} + t \frac{\partial}{\partial x}, B^{1} = n, B^{2} = r, B^{3} = s, n_{t} + r_{x} + s_{y} = 0,

(3.22)

X_{6} = y \frac{\partial}{\partial x} - x \frac{\partial}{\partial y}, B^{1} = n, B^{2} = r, B^{3} = s, n_{t} + r_{x} + s_{y} = 0 .

The invocation of the celebrated Noether theorem gives the six conserved vectors corresponding to these six Noether symmetries: $\begin{matrix} T_{1}^{1} = & u_{y} v_{y} + u_{x} v_{x} + u_{t} v_{t} + \int P (v) dv + \int Q (u) du - n, \\ T_{1}^{2} = & - u_{t} v_{x} - u_{x} v_{t} - r, \\ T_{1}^{3} = & - u_{t} v_{y} - u_{y} v_{t} - s; \end{matrix}$ $\begin{matrix} T_{2}^{1} = & u_{x} v_{t} + u_{t} v_{x} - n, \\ T_{2}^{2} = & u_{y} v_{y} - u_{x} v_{x} - u_{t} v_{t} + \int P (v) dv + \int Q (u) du - r, \\ T_{2}^{3} = & - u_{x} v_{y} - u_{y} v_{x} - s; \end{matrix}$ $\begin{matrix} T_{3}^{1} = & u_{y} v_{t} + u_{t} v_{y} - n, \\ T_{3}^{2} = & - u_{y} v_{x} - u_{x} v_{y} - r, \\ T_{3}^{3} = & u_{x} v_{x} - u_{y} v_{y} - u_{t} v_{t} + \int P (v) dv + \int Q (u) du - s; \end{matrix}$ $\begin{matrix} T_{4}^{1} = & {yu}_{x} v_{x} + {yu}_{y} v_{y} + {tu}_{t} v_{y} + {yu}_{t} v_{t} + {tu}_{y} v_{t} + y \int P (v) dv + y \int Q (u) du - n, \\ T_{4}^{2} = & - {yu}_{t} v_{x} - {tu}_{y} v_{x} - {yu}_{x} v_{t} - {tu}_{x} v_{y} - r, \\ T_{4}^{3} = & {tu}_{x} v_{x} - {tu}_{t} v_{t} - {yu}_{t} v_{y} - {tu}_{y} v_{y} - {yu}_{y} v_{t} + t \int P (v) dv + t \int Q (u) du - s; \end{matrix}$ $\begin{matrix} T_{5}^{1} = & {xu}_{x} v_{x} + {xu}_{y} v_{y} + {xu}_{t} v_{t} + {tu}_{x} v_{t} + {tu}_{t} v_{x} + x \int P (v) dv + x \int Q (u) du - n, \\ T_{5}^{2} = & {tu}_{y} v_{y} - {tu}_{t} v_{t} - {xu}_{x} v_{t} - {tu}_{x} v_{x} - {xu}_{t} v_{x} + t \int P (v) dv + t \int Q (u) du - r, \\ T_{5}^{3} = & - {xu}_{t} v_{y} - {tu}_{x} v_{y} - {xu}_{y} v_{t} - {tv}_{x} u_{y} - s; \end{matrix}$ $\begin{matrix} T_{6}^{1} = & {yu}_{x} v_{t} - {xu}_{y} v_{t} + {yu}_{t} v_{x} - {xv}_{y} u_{t} - n, \\ T_{6}^{2} = & {yu}_{y} v_{y} - {yu}_{t} v_{t} + {xu}_{x} v_{y} - {yu}_{x} v_{x} + {xu}_{y} v_{x} + y \int P (v) dv + y \int Q (u) du - r, \\ T_{6}^{3} = & {xu}_{t} v_{t} - {xu}_{x} v_{x} - {yu}_{y} v_{x} - {yu}_{x} v_{y} + {xu}_{y} v_{y} - x \int P (v) dv - x \int Q (u) du - s . \end{matrix}$

Case 2

$P = α v + β, Q = γ u + λ$ , $α, β, γ, λ$ constants, $α, γ \neq 0$ .

Here we have two sub-cases, viz.,

2.1. $β, λ \neq 0$ . This subcase yields six Noether symmetries, $X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}$ given by operators (3.17)–(3.22) and (2.12)–(2.14) yields six conserved vectors corresponding to six Noether symmetries given by $\begin{matrix} T_{1}^{1} = & u_{y} v_{y} + u_{x} v_{x} + u_{t} v_{t} + \frac{α}{2} v^{2} + β v + \frac{γ}{2} u^{2} + λ u - n, \\ T_{1}^{2} = & - u_{t} v_{x} - u_{x} v_{t} - r, \\ T_{1}^{3} = & - u_{t} v_{y} - u_{y} v_{t} - s; \end{matrix}$ $\begin{matrix} T_{2}^{1} = & u_{x} v_{t} + u_{t} v_{x} - n, \\ T_{2}^{2} = & u_{y} v_{y} - u_{x} v_{x} - u_{t} v_{t} + \frac{α}{2} v^{2} + β v + \frac{γ}{2} u^{2} + λ u - r, \\ T_{2}^{3} = & - u_{x} v_{y} - u_{y} v_{x} - s; \end{matrix}$ $\begin{matrix} T_{3}^{1} = & u_{y} v_{t} + u_{t} v_{y} - n, \\ T_{3}^{2} = & - u_{y} v_{x} - u_{x} v_{y} - r, \\ T_{3}^{3} = & u_{x} v_{x} - u_{y} v_{y} - u_{t} v_{t} + \frac{α}{2} v^{2} + β v + \frac{γ}{2} u^{2} + λ u - s; \end{matrix}$ $\begin{matrix} T_{4}^{1} = & {yu}_{x} v_{x} + {yu}_{y} v_{y} + {tu}_{t} v_{y} + {yu}_{t} v_{t} + {tu}_{y} v_{t} + \frac{α}{2} {yv}^{2} + β yv + \frac{γ}{2} {yu}^{2} + λ yu - n, \\ T_{4}^{2} = & - {yu}_{t} v_{x} - {tu}_{y} v_{x} - {yu}_{x} v_{t} - {tu}_{x} v_{y} - r, \\ T_{4}^{3} = & {tu}_{x} v_{x} - {tu}_{t} v_{t} - {yu}_{t} v_{y} - {tu}_{y} v_{y} - {yu}_{y} v_{t} + \frac{α}{2} {tv}^{2} + β tv + \frac{γ}{2} {tu}^{2} + λ tu - s; \end{matrix}$ $\begin{matrix} T_{5}^{1} = & {xu}_{x} v_{x} + {xu}_{y} v_{y} + {xu}_{t} v_{t} + {tu}_{x} v_{t} + {tu}_{t} v_{x} + x \frac{α}{2} v^{2} + β xv + x \frac{γ}{2} u^{2} + λ xu - n, \\ T_{5}^{2} = & {tu}_{y} v_{y} - {tu}_{t} v_{t} - {xu}_{x} v_{t} - {tu}_{x} v_{x} - {xu}_{t} v_{x} + \frac{α}{2} {tv}^{2} + β tv + \frac{γ}{2} {tu}^{2} + λ tu - r, \\ T_{5}^{3} = & - {xu}_{t} v_{y} - {tu}_{x} v_{y} - {xu}_{y} v_{t} - {tv}_{x} u_{y} - s; \end{matrix}$ $\begin{matrix} T_{6}^{1} = & {yu}_{x} v_{t} - {xu}_{y} v_{t} + {yu}_{t} v_{x} - {xv}_{y} u_{t} - n, \\ T_{6}^{2} = & {yu}_{y} v_{y} - {yu}_{t} v_{t} + {xu}_{x} v_{y} - {yu}_{x} v_{x} + {xu}_{y} v_{x} + \frac{α}{2} {yv}^{2} + β yv + \frac{γ}{2} {yu}^{2} + λ yu - r, \\ T_{6}^{3} = & {xu}_{t} v_{t} - {xu}_{x} v_{x} - {yu}_{y} v_{x} - {yu}_{x} v_{y} + {xu}_{y} v_{y} - \frac{α}{2} {xv}^{2} + β xv - \frac{γ}{2} {xu}^{2} + λ xu - s . \end{matrix}$

2.2. $β, λ = 0$ . Here we have seven Noether symmetry operators, namely, $X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}$ given by operators (3.17)-(3.22) and $X_{7}$ given by

(3.23)

\begin{matrix} X_{7} = & h (t, x, y) \frac{\partial}{\partial u} + k (t, x, y) \frac{\partial}{\partial v}, B^{1} = - h_{t} v - k_{t} u + n, \\ B^{2} = k_{x} u + h_{x} v + r, B^{3} = k_{y} u + h_{y} v + r, n_{t} + r_{x} + s_{y} = 0, \end{matrix}

where

h (t, x, y), k (t, x, y)

(arbitrary functions) satisfy

k_{tt} - k_{xx} - k_{yy} + γ h = 0, h_{tt} - h_{xx} - h_{yy} + α k = 0

. Consequently, Theorem 1 yields seven conserved vectors given by

\begin{matrix} T_{1}^{1} = & u_{y} v_{y} + u_{x} v_{x} + u_{t} v_{t} + \frac{α}{2} v^{2} + β v + \frac{γ}{2} u^{2} + λ u - n, \\ T_{1}^{2} = & - u_{t} v_{x} - u_{x} v_{t} - r, \\ T_{1}^{3} = & - u_{t} v_{y} - u_{y} v_{t} - s; \end{matrix}

\begin{matrix} T_{2}^{1} = & u_{x} v_{t} + u_{t} v_{x} - n, \\ T_{2}^{2} = & u_{y} v_{y} - u_{x} v_{x} - u_{t} v_{t} + \frac{α}{2} v^{2} + β v + \frac{γ}{2} u^{2} + λ u - r, \\ T_{2}^{3} = & - u_{x} v_{y} - u_{y} v_{x} - s; \end{matrix}

\begin{matrix} T_{3}^{1} = & u_{y} v_{t} + u_{t} v_{y} - n, \\ T_{3}^{2} = & - u_{y} v_{x} - u_{x} v_{y} - r, \\ T_{3}^{3} = & u_{x} v_{x} - u_{y} v_{y} - u_{t} v_{t} + \frac{α}{2} v^{2} + β v + \frac{γ}{2} u^{2} + λ u - s; \end{matrix}

\begin{matrix} T_{4}^{1} = & {yu}_{x} v_{x} + {yu}_{y} v_{y} + {tu}_{t} v_{y} + {yu}_{t} v_{t} + {tu}_{y} v_{t} + \frac{α}{2} {yv}^{2} + β yv + \frac{γ}{2} {yu}^{2} + λ yu - n, \\ T_{4}^{2} = & - {yu}_{t} v_{x} - {tu}_{y} v_{x} - {yu}_{x} v_{t} - {tu}_{x} v_{y} - r, \\ T_{4}^{3} = & {tu}_{x} v_{x} - {tu}_{t} v_{t} - {yu}_{t} v_{y} - {tu}_{y} v_{y} - {yu}_{y} v_{t} + \frac{α}{2} {tv}^{2} + β tv + \frac{γ}{2} {tu}^{2} + λ tu - s; \end{matrix}

\begin{matrix} T_{5}^{1} = & {xu}_{x} v_{x} + {xu}_{y} v_{y} + {xu}_{t} v_{t} + {tu}_{x} v_{t} + {tu}_{t} v_{x} + x \frac{α}{2} v^{2} + β xv + x \frac{γ}{2} u^{2} + λ xu - n, \\ T_{5}^{2} = & {tu}_{y} v_{y} - {tu}_{t} v_{t} - {xu}_{x} v_{t} - {tu}_{x} v_{x} - {xu}_{t} v_{x} + \frac{α}{2} {tv}^{2} + β tv + \frac{γ}{2} {tu}^{2} + λ tu - r, \\ T_{5}^{3} = & - {xu}_{t} v_{y} - {tu}_{x} v_{y} - {xu}_{y} v_{t} - {tv}_{x} u_{y} - s; \end{matrix}

\begin{matrix} T_{6}^{1} = & {yu}_{x} v_{t} - {xu}_{y} v_{t} + {yu}_{t} v_{x} - {xv}_{y} u_{t} - n, \\ T_{6}^{2} = & {yu}_{y} v_{y} - {yu}_{t} v_{t} + {xu}_{x} v_{y} - {yu}_{x} v_{x} + {xu}_{y} v_{x} + \frac{α}{2} {yv}^{2} + β yv + \frac{γ}{2} {yu}^{2} + λ yu - r, \\ T_{6}^{3} = & {xu}_{t} v_{t} - {xu}_{x} v_{x} - {yu}_{y} v_{x} - {yu}_{x} v_{y} + {xu}_{y} v_{y} - \frac{α}{2} {xv}^{2} + β xv - \frac{γ}{2} {xu}^{2} + λ xu - s; \end{matrix}

\begin{matrix} T_{7}^{1} = & - {hv}_{t} - {ku}_{t} + h_{t} v + k_{t} u - n, \\ T_{7}^{2} = & {hv}_{x} + {ku}_{x} - k_{x} u - h_{x} v - r, \\ T_{7}^{3} = & {hv}_{y} + {ku}_{y} - k_{y} u - h_{y} v - s . \end{matrix}

Case 3

$P = α v^{q} + β, Q = γ u^{p} + λ$ , with $q, p, α, β, γ, λ$ constants, $α, γ \neq 0$ .

This case has two subcases.

3.1. $β, λ \neq 0, p, q \neq - 1, 2 p + 2 q + 5 - pq \neq 0$ .

Six Noether operators $X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}$ given by generators (3.17)-(3.22) are obtained. The invocation of Theorem 1 yields the following corresponding conserved vectors: $\begin{matrix} T_{1}^{1} = & u_{y} v_{y} + u_{x} v_{x} + u_{t} v_{t} + \frac{α}{q + 1} v^{q + 1} + β v + \frac{γ}{p + 1} u^{p + 1} + λ u - n, \\ T_{2}^{1} = & - u_{t} v_{x} - u_{x} v_{t} - r, \\ T_{3}^{1} = & - u_{t} v_{y} - u_{y} v_{t} - s; \end{matrix}$ $\begin{matrix} T_{1}^{2} = & u_{x} v_{t} + u_{t} v_{x} - n, \\ T_{2}^{2} = & u_{y} v_{y} - u_{x} v_{x} - u_{t} v_{t} + \frac{α}{q + 1} v^{q + 1} + β v + \frac{γ}{p + 1} u^{p + 1} + λ u - r, \\ T_{3}^{2} = & - u_{x} v_{y} - u_{y} v_{x} - s; \end{matrix}$ $\begin{matrix} T_{1}^{3} = & u_{y} v_{t} + u_{t} v_{y} - n, \\ T_{2}^{3} = & - u_{y} v_{x} - u_{x} v_{y} - r, \\ T_{3}^{3} = & u_{x} v_{x} - u_{y} v_{y} - u_{t} v_{t} + \frac{α}{q + 1} v^{q + 1} + β v + \frac{γ}{p + 1} u^{p + 1} + λ u - s; \end{matrix}$ $\begin{matrix} T_{1}^{4} = & {yu}_{x} v_{x} + {yu}_{y} v_{y} + {tu}_{t} v_{y} + {yu}_{t} v_{t} + {tu}_{y} v_{t} + \frac{α}{q + 1} {yv}^{q + 1} + β yv + \frac{β}{p + 1} {yu}^{p + 1} + λ yu - n, \\ T_{2}^{4} = & - {yu}_{t} v_{x} - {tu}_{y} v_{x} - {yu}_{x} v_{t} - {tu}_{x} v_{y} - r, \\ T_{3}^{4} = & {tu}_{x} v_{x} - {tu}_{t} v_{t} - {yu}_{t} v_{y} - {tu}_{y} v_{y} - {yu}_{y} v_{t} + \frac{α}{q + 1} {tv}^{q + 1} + β tv + \frac{γ}{p + 1} {tu}^{p + 1} + λ tu - s; \end{matrix}$ $\begin{matrix} T_{1}^{5} = & {xu}_{x} v_{x} + {xu}_{y} v_{y} + {xu}_{t} v_{t} + {tu}_{x} v_{t} + {tu}_{t} v_{x} + \frac{α}{q + 1} {xv}^{q + 1} + β xv + \frac{γ}{p + 1} {xu}^{p + 1} + λ xu - n, \\ T_{2}^{5} = & {tu}_{y} v_{y} - {tu}_{t} v_{t} - {xu}_{x} v_{t} - {tu}_{x} v_{x} - {xu}_{t} v_{x} + \frac{α}{q + 1} {tv}^{q + 1} + β tv + \frac{γ}{p + 1} {tu}^{p + 1} + λ tu - r, \\ T_{3}^{5} = & - {xu}_{t} v_{y} - {tu}_{x} v_{y} - {xu}_{y} v_{t} - {tv}_{x} u_{y} - s; \end{matrix}$ $\begin{matrix} T_{1}^{6} = & {yu}_{x} v_{t} - {xu}_{y} v_{t} + {yu}_{t} v_{x} - {xv}_{y} u_{t} - n, \\ T_{2}^{6} = & {yu}_{y} v_{y} - {yu}_{t} v_{t} + {xu}_{x} v_{y} - {yu}_{x} v_{x} + {xu}_{y} v_{x} + \frac{α}{q + 1} {yv}^{q + 1} + β yv + \frac{γ}{p + 1} {yu}^{p + 1} + β yu - r, \\ T_{3}^{6} = & {xu}_{t} v_{t} - {xu}_{x} v_{x} - {yu}_{y} v_{x} - {yu}_{x} v_{y} + {xu}_{y} v_{y} - \frac{α}{q + 1} {xv}^{q + 1} + β xv - \frac{γ}{p + 1} {xu}^{p + 1} + β xu - s . \end{matrix}$

3.2. $β, λ = 0, p = 5, q = 5$ .

In this case, we obtain four extra Noether symmetries namely $\begin{matrix} X_{7} = & (y^{2} + x^{2} + t^{2}) \frac{\partial}{\partial t} + 2 tx \frac{\partial}{\partial x} + 2 ty \frac{\partial}{\partial y} - ut \frac{\partial}{\partial u} - vt \frac{\partial}{\partial v}, B^{1} = uv, B^{2} = 0, B^{3} = 0, \\ X_{8} = & - 2 xt \frac{\partial}{\partial t} + (y^{2} - x^{2} - t^{2}) \frac{\partial}{\partial x} - 2 xy \frac{\partial}{\partial y} + ux \frac{\partial}{\partial u} + vx \frac{\partial}{\partial v}, B^{1} = 0, B^{2} = uv, B^{3} = 0, \\ X_{9} = & - 2 yt \frac{\partial}{\partial t} - 2 yx \frac{\partial}{\partial x} + (x^{2} - t^{2} - y^{2}) \frac{\partial}{\partial y} + uy \frac{\partial}{\partial u} + vy \frac{\partial}{\partial v}, B^{1} = 0, B^{2} = 0, B^{3} = uv, \\ X_{10} = & 2 t \frac{\partial}{\partial t} + 2 x \frac{\partial}{\partial x} + 2 y \frac{\partial}{\partial x} - u \frac{\partial}{\partial u} - v \frac{\partial}{\partial v}, B^{1} = 0, B^{2} = 0, B^{3} = 0 . \end{matrix}$

Employing the Noether theorem we obtain the following new extra four nontrivial conserved vectors corresponding to these extra four Noether point symmetries: $\begin{matrix} T_{7}^{1} = & \frac{1}{6} {t^{2} β u^{6} + x^{2} β u^{6} + y^{2} β u^{6} + 6 {tv}_{t} u + t^{2} α v^{6} + x^{2} α v^{6} + y^{2} α v^{6} - 6 uv + 6 t^{2} u_{y} v_{y} + \\ 6 x^{2} u_{y} v_{y} + 6 y^{2} u_{y} v_{y} + 6 t^{2} u_{x} v_{x} + 6 x^{2} u_{x} v_{x} + 6 y^{2} u_{x} v_{x} + 6 {tvu}_{t} + 12 {tyv}_{y} u_{t} + 12 {txv}_{x} u_{t} \\ + 12 {tyu}_{y} v_{t} + 12 {txu}_{x} v_{t} + 6 t^{2} u_{t} v_{t} + 6 x^{2} u_{t} v_{t} + 6 y^{2} u_{t} v_{t}}, \\ T_{7}^{2} = & \frac{1}{3} {tx β u^{6} - 3 {tv}_{x} u + tx α v^{6} + 6 {txu}_{y} v_{y} - 3 {tvu}_{x} - 6 {tyv}_{y} u_{x} - 6 {tyu}_{y} v_{x} - 6 {txu}_{x} v_{x} \\ - 3 t^{2} v_{x} u_{t} - 3 x^{2} v_{x} u_{t} - 3 y^{2} v_{x} u_{t} - 3 t^{2} u_{x} v_{t} - 3 x^{2} u_{x} v_{t} - 3 y^{2} u_{x} v_{t} - 6 {txu}_{t} v_{t}}, \\ T_{7}^{3} = & \frac{1}{3} {ty β u^{6} - 3 {tv}_{y} u + ty α v^{6} - 3 {tvu}_{y} - 6 {tyu}_{y} v_{y} - 6 {txv}_{y} u_{x} - 6 {txu}_{y} v_{x} + 6 {tyu}_{x} v_{x} \\ - 3 t^{2} v_{y} u_{t} - 3 x^{2} v_{y} u_{t} - 3 y^{2} v_{y} u_{t} - 3 t^{2} u_{y} v_{t} - 3 x^{2} u_{y} v_{t} - 3 y^{2} u_{y} v_{t} - 6 {tyu}_{t} v_{t}}; \end{matrix}$ $\begin{matrix} T_{8}^{1} = & \frac{1}{3} {- tx β u^{6} - 3 {xv}_{t} u - tx α v^{6} - 6 {txu}_{y} v_{y} - 6 {txu}_{x} v_{x} - 3 {xvu}_{t} - 6 {xyv}_{y} u_{t} - 3 t^{2} v_{x} u_{t} \\ - 3 x^{2} v_{x} u_{t} + 3 y^{2} v_{x} u_{t} - 6 {xyu}_{y} v_{t} - 3 t^{2} u_{x} v_{t} - 3 x^{2} u_{x} v_{t} + 3 y^{2} u_{x} v_{t} - 6 {txu}_{t} v_{t}}, \\ T_{8}^{2} = & \frac{1}{6} {- t^{2} β u^{6} - x^{2} β u^{6} + y^{2} β u^{6} + 6 {xv}_{x} u - t^{2} α v^{6} - x^{2} α v^{6} + y^{2} α v^{6} - 6 uv - 6 t^{2} u_{y} v_{y} \\ - 6 x^{2} u_{y} v_{y} + 6 y^{2} u_{y} v_{y} + 6 {xvu}_{x} + 12 {xyv}_{y} u_{x} + 12 {xyu}_{y} v_{x} + 6 t^{2} u_{x} v_{x} + 6 x^{2} u_{x} v_{x} \\ - 6 y^{2} u_{x} v_{x} + 12 {txv}_{x} u_{t} + 12 {txu}_{x} v_{t} + 6 t^{2} u_{t} v_{t} + 6 x^{2} u_{t} v_{t} - 6 y^{2} u_{t} v_{t}}, \\ T_{8}^{3} = & \frac{1}{3} {- xy β u^{6} + 3 {xv}_{y} u - xy α v^{6} + 3 {xvu}_{y} + 6 {xyu}_{y} v_{y} + 3 t^{2} v_{y} u_{x} + 3 x^{2} v_{y} u_{x} - 3 y^{2} v_{y} u_{x} \\ + 3 t^{2} u_{y} v_{x} + 3 x^{2} u_{y} v_{x} - 3 y^{2} u_{y} v_{x} - 6 {xyu}_{x} v_{x} + 6 {txv}_{y} u_{t} + 6 {txu}_{y} v_{t} + 6 {xyu}_{t} v_{t}}; \end{matrix}$ $\begin{matrix} T_{9}^{1} = & \frac{1}{3} {- ty β u^{6} - 3 {yv}_{t} u - ty α v^{6} - 6 {tyu}_{y} v_{y} - 6 {tyu}_{x} v_{x} - 3 {yvu}_{t} - 3 t^{2} v_{y} u_{t} + 3 x^{2} v_{y} u_{t} \\ - 3 y^{2} v_{y} u_{t} - 6 {xyv}_{x} u_{t} - 3 t^{2} u_{y} v_{t} + 3 x^{2} u_{y} v_{t} - 3 y^{2} u_{y} v_{t} - 6 {xyu}_{x} v_{t} - 6 {tyu}_{t} v_{t}}, \\ T_{9}^{2} = & \frac{1}{3} {- xy β u^{6} + 3 {yv}_{x} u - xy α v^{6} - 6 {xyu}_{y} v_{y} + 3 {yvu}_{x} + 3 t^{2} v_{y} u_{x} - 3 x^{2} v_{y} u_{x} + 3 y^{2} v_{y} u_{x} \\ + 3 t^{2} u_{y} v_{x} - 3 x^{2} u_{y} v_{x} + 3 y^{2} u_{y} v_{x} + 6 {xyu}_{x} v_{x} + 6 {tyv}_{x} u_{t} + 6 {tyu}_{x} v_{t} + 6 {xyu}_{t} v_{t}}, \\ T_{9}^{3} = & \frac{1}{6} {- t^{2} β u^{6} + x^{2} β u^{6} - y^{2} β u^{6} + 6 {yv}_{y} u - t^{2} α v^{6} + x^{2} α v^{6} - y^{2} α v^{6} - 6 uv + 6 {yvu}_{y} \\ + 6 t^{2} u_{y} v_{y} - 6 x^{2} u_{y} v_{y} + 6 y^{2} u_{y} v_{y} + 12 {xyv}_{y} u_{x} + 12 {xyu}_{y} v_{x} - 6 t^{2} u_{x} v_{x} + 6 x^{2} u_{x} v_{x} \\ - 6 y^{2} u_{x} v_{x} + 12 {tyv}_{y} u_{t} + 12 {tyu}_{y} v_{t} + 6 t^{2} u_{t} v_{t} - 6 x^{2} u_{t} v_{t} + 6 y^{2} u_{t} v_{t}}; \end{matrix}$ $\begin{matrix} T_{10}^{1} = & \frac{1}{3} {3 v_{t} u + 3 u_{t} v + β {tu}^{6} + α {tv}^{6} + 6 {tu}_{x} v_{x} + 6 {xu}_{t} v_{x} + 6 {xv}_{t} u_{x} + 6 {tu}_{y} v_{y} \\ + 6 {yu}_{t} v_{y} + 6 {yv}_{t} u_{y} + 6 {tu}_{t} v_{t}}, \\ T_{10}^{2} = & \frac{1}{3} {- 3 v_{x} u - 3 u_{x} v + β {xu}^{6} + α {xv}^{6} - 6 {tu}_{t} v_{x} - 6 {tv}_{t} u_{x} - 6 {xu}_{t} v_{t} \\ + 6 {xu}_{y} v_{y} - 6 {yu}_{x} v_{y} - 6 {yu}_{y} v_{x} - 6 {xu}_{x} v_{x}}, \\ T_{10}^{3} = & \frac{1}{3} {- 3 v_{y} u - 3 u_{y} v + β {yu}^{6} + α {yv}^{6} - 6 {tu}_{t} v_{y} - 6 {tv}_{t} u_{y} - 6 {yu}_{t} v_{t} \\ - 6 {xu}_{x} v_{y} - 6 {xu}_{y} v_{x} + 6 {yu}_{x} v_{x} - 6 {yu}_{y} v_{y}} . \end{matrix}$

3.3. $β, λ = 0$ .

We get five subcases. See Muatjetjeja and Khalique (2015).

Case 4

$P = α e^{β v} + γ$ , $Q = δ e^{λ u} + σ$ , $α, β, γ, δ, λ, σ$ constants, $α, δ \neq 0$ .

This case gives six Noether operators $X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}$ given by generators (3.17)-(3.22) and corresponding six conserved vectors are $\begin{matrix} T_{1}^{1} = & u_{y} v_{y} + u_{x} v_{x} + u_{t} v_{t} + \frac{α}{β} e^{β v} + \frac{δ}{λ} e^{λ u} + γ v + σ u - n, \\ T_{1}^{2} = & - u_{t} v_{x} - u_{x} v_{t} - r, \\ T_{1}^{3} = & - u_{t} v_{y} - u_{y} v_{t} - s; \end{matrix}$ $\begin{matrix} T_{2}^{1} = & u_{x} v_{t} + u_{t} v_{x} - n, \\ T_{2}^{2} = & u_{y} v_{y} - u_{x} v_{x} - u_{t} v_{t} + \frac{α}{β} e^{β v} + \frac{δ}{λ} e^{λ u} + γ v + σ u - r, \\ T_{2}^{3} = & - u_{x} v_{y} - u_{y} v_{x} - s; \end{matrix}$ $\begin{matrix} T_{3}^{1} = & u_{y} v_{t} + u_{t} v_{y} - n, \\ T_{3}^{2} = & - u_{y} v_{x} - u_{x} v_{y} - r, \\ T_{3}^{3} = & u_{x} v_{x} - u_{y} v_{y} - u_{t} v_{t} + \frac{α}{β} e^{β v} + \frac{δ}{λ} e^{λ u} + γ v + σ u - s; \end{matrix}$ $\begin{matrix} T_{4}^{1} = & {yu}_{x} v_{x} + {yu}_{y} v_{y} + {tu}_{t} v_{y} + {yu}_{t} v_{t} + {tu}_{y} v_{t} + \frac{α}{β} e^{β v} y + \frac{δ}{λ} e^{λ u} y + γ yv + σ yu - n, \\ T_{4}^{2} = & - {yu}_{t} v_{x} - {tu}_{y} v_{x} - {yu}_{x} v_{t} - {tu}_{x} v_{y} - r, \\ T_{4}^{3} = & {tu}_{x} v_{x} - {tu}_{t} v_{t} - {yu}_{t} v_{y} - {tu}_{y} v_{y} - {yu}_{y} v_{t} + \frac{α}{β} e^{β v} t + \frac{δ}{λ} e^{λ u} t + γ tv + σ tu - s; \end{matrix}$ $\begin{matrix} T_{5}^{1} = & {xu}_{x} v_{x} + {xu}_{y} v_{y} + {xu}_{t} v_{t} + {tu}_{x} v_{t} + {tu}_{t} v_{x} + \frac{α}{β} e^{β v} x + \frac{δ}{λ} e^{λ u} x + γ xv + σ xu - n, \\ T_{5}^{2} = & {tu}_{y} v_{y} - {tu}_{t} v_{t} - {xu}_{x} v_{t} - {tu}_{x} v_{x} - {xu}_{t} v_{x} + \frac{α}{β} e^{β v} t + \frac{δ}{λ} e^{λ u} t + γ tv + σ tu - r, \\ T_{5}^{3} = & - {xu}_{t} v_{y} - {tu}_{x} v_{y} - {xu}_{y} v_{t} - {tv}_{x} u_{y} - s; \end{matrix}$ $\begin{matrix} T_{6}^{1} = & {yu}_{x} v_{t} - {xu}_{y} v_{t} + {yu}_{t} v_{x} - {xv}_{y} u_{t} - n, \\ T_{6}^{2} = & {yu}_{y} v_{y} - {yu}_{t} v_{t} + {xu}_{x} v_{y} - {yu}_{x} v_{x} + {xu}_{y} v_{x} + \frac{α}{β} e^{β v} y + \frac{δ}{λ} e^{λ u} y + γ yv + σ yu - r, \\ T_{6}^{3} = & {xu}_{t} v_{t} - {xu}_{x} v_{x} - {yu}_{y} v_{x} - {yu}_{x} v_{y} + {xu}_{y} v_{y} - \frac{α}{β} e^{β v} x + \frac{δ}{λ} e^{λ u} x + γ xv + σ xu - s . \end{matrix}$

Case 5

$P = α \ln v + β, Q = γ \ln u + λ, α, β, γ, λ$ constants with $α, γ \neq 0$ .

For this case we obtain six Noether operators given by (3.17)-(3.22). The invocation of Noether theorem yields $\begin{matrix} T_{1}^{1} = & u_{y} v_{y} + u_{x} v_{x} + u_{t} v_{t} + α v \ln v + γ u \ln u + β v + γ u - α v - γ u - n, \\ T_{1}^{2} = & - u_{t} v_{x} - u_{x} v_{t} - r, \\ T_{1}^{3} = & - u_{t} v_{y} - u_{y} v_{t} - s; \end{matrix}$ $\begin{matrix} T_{2}^{1} = & u_{x} v_{t} + u_{t} v_{x} - n, \\ T_{2}^{2} = & u_{y} v_{y} - u_{x} v_{x} - u_{t} v_{t} + α v \ln v + γ u \ln u + β v + γ u - α v - γ u - r, \\ T_{2}^{3} = & - u_{x} v_{y} - u_{y} v_{x} - s; \end{matrix}$ $\begin{matrix} T_{3}^{1} = & u_{y} v_{t} + u_{t} v_{y} - n, \\ T_{3}^{2} = & - u_{y} v_{x} - u_{x} v_{y} - r, \\ T_{3}^{3} = & u_{x} v_{x} - u_{y} v_{y} - u_{t} v_{t} + α v \ln v + γ u \ln u + β v + γ u - α v - γ u - s; \end{matrix}$ $\begin{matrix} T_{4}^{1} = & {yu}_{x} v_{x} + {yu}_{y} v_{y} + {tu}_{t} v_{y} + {yu}_{t} v_{t} + {tu}_{y} v_{t} + y (α v \ln v + γ u \ln u + β v + γ u - α v - γ u) - n, \\ T_{4}^{2} = & - {yu}_{t} v_{x} - {tu}_{y} v_{x} - {yu}_{x} v_{t} - {tu}_{x} v_{y} - r, \\ T_{4}^{3} = & {tu}_{x} v_{x} - {tu}_{t} v_{t} - {yu}_{t} v_{y} - {tu}_{y} v_{y} - {yu}_{y} v_{t} + t (α v \ln v + γ u \ln u + β v + γ u - α v - γ u) - s; \end{matrix}$ $\begin{matrix} T_{5}^{1} = & {xu}_{x} v_{x} + {xu}_{y} v_{y} + {xu}_{t} v_{t} + {tu}_{x} v_{t} + {tu}_{t} v_{x} + x (α v \ln v + γ u \ln u + β v + γ u - α v - γ u) - n, \\ T_{5}^{2} = & {tu}_{y} v_{y} - {tu}_{t} v_{t} - {xu}_{x} v_{t} - {tu}_{x} v_{x} - {xu}_{t} v_{x} + t (α v \ln v + γ u \ln u + β v + γ u - α v - γ u) - r, \\ T_{5}^{3} = & - {xu}_{t} v_{y} - {tu}_{x} v_{y} - {xu}_{y} v_{t} - {tv}_{x} u_{y} - s; \end{matrix}$ $\begin{matrix} T_{6}^{1} = & {yu}_{x} v_{t} - {xu}_{y} v_{t} + {yu}_{t} v_{x} - {xv}_{y} u_{t} - n, \\ T_{6}^{2} = & {yu}_{y} v_{y} - {yu}_{t} v_{t} + {xu}_{x} v_{y} - {yu}_{x} v_{x} + {xu}_{y} v_{x} + y (α v \ln v + γ u \ln u + β v + γ u - α v - γ u) - r, \\ T_{6}^{3} = & {xu}_{t} v_{t} - {xu}_{x} v_{x} - {yu}_{y} v_{x} - {yu}_{x} v_{y} + {xu}_{y} v_{y} - x (α v \ln v + γ u \ln u + β v + γ u - α v - γ u) - s . \end{matrix}$

Remark. It should be pointed out that Noether symmetry classification of (1.15)–(1.16) prompted five main cases and few subcases for the functions $P (v)$ and $Q (u)$ . So altogether we obtained eight different cases for the functions $P (v)$ and $Q (u)$ . It can be clearly seen that only one case, namely 3.3, was studied in Muatjetjeja and Khalique (2015). All the other seven cases are new results in the paper. .

4 Concluding remarks

In this work, a complete Noether symmetry classification for the generalised system of coupled (2 + 1)-dimensional hyperbolic Eqs. (1.15)–(1.16) was performed with respect to the first order Lagrangian. This resulted in obtaining five cases and several sub-cases for functions $P (v)$ and $Q (u)$ which gave Noether symmetries. Thereafter, corresponding to each of these Noether operators found, we presented the conservation laws. The work on the underlying problem was motivated by the recent work done in Muatjetjeja and Khalique (2015). However, the results obtained therein were not complete because the work in Muatjetjeja and Khalique (2015) was restricted to self-interaction power functions. Thus, in the present work no essential restriction on non-zero arbitrary self-interaction functions was placed. Also, one can see that all the results related to Noether classification obtained in Muatjetjeja and Khalique (2015) can be derived from the present work. Therefore, the results of the present work are new and more general.

Acknowledgement

CMK thanks the North-West University, Mafikeng Campus, South Africa, for its continued support. The authors thank the anonymous referees whose comments helped to improve the paper.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

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Introduction

Preliminaries and notations

Noether symmetries and conservation laws of the system (1.15)–(1.16)

Concluding remarks

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[1] Butt M.M., Ahmad N.S.. Modeling and analysis of debonding in a smart beam in sensing mode, using variational formulation. J. King Saud Univ.-Sci.. 2020;32(1):29-43.
[Google Scholar]

[2] Escobedo M., Herrero A.M.. Boundedness and blow-up for a semilinear reaction-diffusion system. J. Differ Equ.. 1991;89:176-202.
[Google Scholar]

[3] Escobedo M., Herrero A.M.. A semilinear parabolic system in bounded domain. Ann. di Mat. Pura ed Appl.. 1993;169:315-336.
[Google Scholar]

[4] Gao Y., Gao W.. Study of solutions to initial and boundary value problem for certain systems with variable exponents. Bound. Value Probl. 2013
[CrossRef] [Google Scholar]

[5] Ibragimov, N.H., 1994-1996. (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations. Vol 1–3, CRC Press, Boca Raton, FL.

[6] Muatjetjeja B., Khalique C.M.. Conservation laws for a generalised coupled bidimensional Lane-Emden system. Commun. Nonlinear Sci. Numer. Simulat.. 2013;18:851-857.
[Google Scholar]

[7] Muatjetjeja B., Khalique C.M.. Symmetry analysis and conservation laws for a coupled (2+1)-dimensional hyperbolic system. Commun. Nonlinear Sci. Numer. Simulat.. 2015;22:1252-1262.
[Google Scholar]

[8] Noether E.. Invariante Variationsprobleme. König Gesell Wissen Göttingen, Math-Phys Kl Heft.. 1918;2:235-257.
[Google Scholar]

[9] Ovsiannikov, L.V., 1982. Group analysis of differential equations (English translation by W.F. Ames), Academy Press, New York.

[10] Wang J.Z., Zheng B.C., Li H.G.. Generalized variational formulations for extended exponentially fractional integral. J. King Saud Univ.-Sci.. 2016;28:37-40.
[Google Scholar]

[11] Yildirim A., Mohyud-Din S.T.. A variational approach for soliton solutions of good Boussinesq equation. J. King Saud Univ.-Sci.. 2010;22:205-208.
[Google Scholar]

Lagrangian formulation of a generalised coupled hyperbolic system

Abstract

Keywords

Noether operators

Lagrangian

Conservation laws

Generalised system of coupled (2 + 1)-D hyperbolic equations

Potential function

1 Introduction

2 Preliminaries and notations

3 Noether symmetries and conservation laws of the system (1.15)–(1.16)

4 Concluding remarks

Acknowledgement

Declaration of Competing Interest

References

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