Startseite Naturwissenschaften Exact solutions of equal-width equation and its conservation laws
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Exact solutions of equal-width equation and its conservation laws

  • Chaudry Masood Khalique EMAIL logo , Karabo Plaatjie und Innocent Simbanefayi
Veröffentlicht/Copyright: 4. Oktober 2019
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Aus der Zeitschrift Open Physics Band 17 Heft 1

Abstract

In this work we investigate the equal-width equation, which is used for simulation of (1-D) wave propagation in non-linear medium with dispersion process. Firstly, Lie symmetries are determined and then used to establish an optimal system of one-dimensional subalgebras. Thereafter with its aid we perform symmetry reductions and compute new invariant solutions, which are snoidal and cnoidal waves. Additionally, the conservation laws for the aforementioned equation are established by invoking multiplier method and Noether’s theorem.

Keywords: equal-width equation; Lie symmetries; optimal system of one-dimensional subalgebras; cnoidal and snoidal waves; conservation laws
PACS: 02.30.Jr; 02.30.Hq; 04.20.Jb; 11.30-j

1 Introduction

In this work we conduct a study of the third-order equal-width (EW) equation

(1) u t + 2 α u u x − β u t x x = 0 , α , β ≠ 0 ,

where α is the nonlinearity parameter and β is the dispersion parameter. This equation was first introduced in [1] and represents a model equation that describes nonlinear dispersive waves, e.g., waves created in shallow water channel. Several methods have been used to derive solutions of this equation, for instance the authors of [2] used Petrov-Galerkin method by invoking quadratic B-spline finite element. In [3], authors used least-squares technique to construct numerical solutions of this equation. This equation (1) was recently studied in [4], where the researchers employed simple equation technique and also the exp(−φ(ξ)) expansion technique to construct travelling wave solutions.

In our study, using a different technique, we construct new exact travelling wave solutions, namely cnoidal and snoidal waves solutions of EW equation (1). Moreover, by employing two approaches, viz., the multiplier approach and Noether’s theorem we construct conservation laws of (1).

2 Solutions using optimal system

Firstly, we determine Lie symmetries of (1) and then utilize them to establish an optimal system of one-dimensional subalgebras (OSODS). Thereafter, we use this OSODS to performs symmetry reductions (SRs) and invariant solutions of (1) [5, 6, 7, 8, 9, 10, 11, 12].

2.1 Lie symmetries

The infinitesimal generator

(2) G = τ ∂ ∂ t + ξ ∂ ∂ x + η ∂ ∂ u ,

where the functions (τ, ξ, η) depend on t, x and u, is a Lie symmetry of equation (1) if

(3) G ( 3 ) F | F = 0 = 0 ,

where

F ≡ u t + 2 α u u x − β u t x x

and G(3) denotes third prolongation [7] of G and is given by

(4) G ( 3 ) = G + ζ 1 ∂ ∂ u t + ζ 2 ∂ ∂ u x + ζ 12 ∂ ∂ u t x + ζ 122 ∂ ∂ u t x x .

Here the coefficients ζ1, ζ2, ζ12 and ζ122 are determined by

(5) ζ 1 = − u t D t ( τ ) − u x D t ( ξ ) + D t ( η ) , ζ 2 = − u t D x ( τ ) − u x D x ( ξ ) + D x ( η ) , ζ 12 = − u t t D x ( τ ) − u t x D x ( ξ ) + D x ( ζ 1 ) , ζ 122 = − u t x x D x ( τ ) − u x x x D x ( ξ ) + D x ( ζ 12 ) ,

with

(6) D x = ∂ ∂ x + u x ∂ ∂ u + u x x ∂ ∂ u x + u x t ∂ ∂ u t + ⋯ D t = ∂ ∂ t + u t ∂ ∂ u + u t t ∂ ∂ u t + u t x ∂ ∂ u x + ⋯ .

We expand equation (3) and since τ, ξ and η do not depend on derivatives of u, equation (3) decomposes into system of linear homogeneous PDEs. Solving these equations one finds values of τ, ξ and η, which lead to three Lie symmetries of (1) given by

G 1 = ∂ ∂ t , G 2 = ∂ ∂ x , G 3 = t ∂ ∂ t − u ∂ ∂ u .

We note that the first two symmetries represent translations in time and space respectively, whereas the third describes a scaling symmetry of the EW equation.

2.2 Optimal system of one-dimensional subalgebras

We now utilize technique of [7] to calculate an OSODS by using Lie symmetries of (1). We first construct the commutator table. Thereafter we compute adjoint representation using

Ad ( exp ⁡ ( ε G s ) ) G k = ∑ n = 0 ∞ ε n n ! ( ad G s ) n ( G k ) = G k − ε [ G s , G k ] + ε 2 2 ! [ G s , [ G s , G k ] ] − ⋯ ,

where ε ∈ ℜ and commutator [Gs , Gk] is defined as

[ G s , G k ] = G s G k − G k G s .

The commutators of Lie symmetries of (1) and adjoint symmetry group of (1) are presented in Tables 1 and 2, respectively. Consequently, we utilize Tables 1 and 2 to derive the OSODS for (1).

Table 1

Commutator table of (1)

[ , ] G1 G2 G3
G1 0 0 G1
G2 0 0 0
G3 −G1 0 0
Table 2

Adjoint symmetry group of subalgebras

Ad G1 G2 G3
G1 G1 G2 −"G1 + G3
G2 G1 G2 G3
G3 eεG1 G2 G3

Thus following [7] and utilizing Table 1 and Table 2 we can obtain an OSODS {G1 + cG2, G3 + aG2, G2}, with c and a are constants.

2.3 Solutions and symmetry reductions

We now utilise the OSODS obtained above and find symmetry reductions and invariant solutions for equation (1).

Consider the first operator G1 + cG2 of the optimal system. This operator has invariants

ξ = x − c t , U = u ,

which give the invariant solution U = U(ξ). Using ξ as our new independent variable, equation (1) is reduced into nonlinear ODE

(7) c β U ‴ ( ξ ) + 2 α U ( ξ ) U ′ ( ξ ) − c U ′ ( ξ ) = 0.

We now construct travelling wave solutions of our PDE (1) by invoking Jacobi elliptic expansion technique [13]. We suppose solutions of ODE (7) are

(8) U ( ξ ) = ∑ s = − N N A s H ( ξ ) s ,

where positive integer N and As are constants to be determined. The function H is a solution of nonlinear first-order ODE

(9) d H d ξ = − ( 1 − ω + ω H 2 ) ( 1 − H 2 )

or

(10) d H d ξ = ( 1 − ω H 2 ) ( 1 − H 2 ) .

We recall that Jacobi cosine function

(11) H = cn ( ξ | ω )

solves (9), whereas Jacobi sine function

(12) H = sn ( ξ | ω )

satisfies (10) with 0 ≤ ω ≤ 1 [13, 14].

2.3.1 Cnoidal waves

Nonlinear ODE (7) provides us with N = 2 and so (8) leads to

(13) U = A − 2 H − 2 + A − 1 H − 1 + A 0 + A 1 H + A 2 H 2 .

Substitution of U from (13) into (7) and utilizing (9) we obtain

4 H 10 α A 2 2 − 48 β c ω A − 2 − 4 α ω A − 2 2 + 2 H 8 α A 1 2 − 4 H 8 α A 2 2 − 2 H 8 c A 2 − H 7 c A 1 − 2 H 6 α A 1 2 + 2 H 6 c A 2 + H 5 c A − 1 + H 5 c A 1 − 2 H 4 α A − 1 2 + 2 H 4 c A − 2 − H 3 c A − 1 − 4 H 2 α A − 2 2 + 2 H 2 α A − 1 2 + 8 H 4 α ω A − 2 A 0 − 2 H 2 c A − 2 + 12 H 3 α ω A − 2 A − 1 − 2 H 3 α ω A − 2 A 1 − 2 H 3 α ω A − 1 A 0 − 4 H 2 α ω A − 2 A 0 − 6 H ξ α ω A − 2 A − 1 + 6 H 11 α ω A 1 A 2 + 2 H 9 α ω A − 1 A 2 + 2 H 9 α ω A 0 A 1 − 12 H 9 α ω A 1 A 2 − 8 H 8 α ω A 0 A 2 − 2 H 7 α ω A − 2 A 1 − 2 H 7 α ω A − 1 A 0 − 4 H 7 α ω A − 1 A 2 − 4 H 7 α ω A 0 A 1 + 6 H 7 α ω A 1 A 2 − 4 H 6 α ω A − 2 A 0 + 4 H 6 α ω A 0 A 2 − 6 H 5 α ω A − 2 A − 1 + 4 H 5 α ω A − 2 A 1 + 4 H 10 α ω A 0 A 2 + 4 H 5 α ω A − 1 A 0 + 2 H 5 α ω A − 1 A 2 + 2 H 5 α ω A 0 A 1 + 24 β c A − 2 + 2 H 2 c ω A − 2 + 6 H α A − 2 A − 1 + 6 H β c A − 1 + 24 β c ω 2 A − 2 + 4 H 12 α ω A 2 2 + 2 H 10 α ω A 1 2 − 8 H 10 α ω A 2 2 − 2 H 10 c ω A 2 + 6 H 9 α A 1 A 2 − H 9 c ω A 1 − 4 H 8 α ω A 1 2 + 4 H 8 α ω A 2 2 + 4 H 8 α A 0 A 2 + 4 H 8 c ω A 2 + 2 H 7 α A − 1 A 2 + 2 H 7 α A 0 A 1 − 6 H 7 α A 1 A 2 + H 7 c ω A − 1 + 2 H 7 c ω A 1 − 2 H 6 α ω A − 1 2 + 2 H 6 α ω A 1 2 − 4 H 6 α A 0 A 2 + 2 H 6 c ω A − 2 − 2 H 6 c ω A 2 − 2 H 5 α A − 2 A 1 − 2 H 5 α A − 1 A 0 − 2 H 5 α A − 1 A 2 − 2 H 5 α A 0 A 1 − 2 H 5 c ω A − 1 − H 5 c ω A 1 − 4 H 4 α ω A − 2 2 + 4 H 4 α ω A − 1 2 − 4 H 4 α A − 2 A 0 − 4 H 4 c ω A − 2 + 6 H β c ω 2 A − 1 + 2 H 3 α A − 2 A 1 + 2 H 3 α A − 1 A 0 + H 3 c ω A − 1 + 8 H 2 α ω A − 2 2 − 12 H β c ω A − 1 − 2 H 2 α ω A − 1 2 + 4 H 2 α A − 2 A 0 + 4 α A − 2 2 − 8 H 8 β c A 2 − H 7 β c A 1 + H 5 β c A − 1 + H 5 β c A 1 + 8 H 4 β c A − 2 − 7 H 3 β c A − 1 − 32 H 2 β c A − 2 − 7 H 9 β c ω A 1 − 56 H 8 β c ω 2 A 2 + 56 H 8 β c ω A 2 − 2 H 7 β c ω 2 A − 1 − 10 H 7 β c ω 2 A 1 + H 7 β c ω A − 1 + 10 H 7 β c ω A 1 − 16 H 6 β c ω 2 A − 2 + 16 H 6 β c ω 2 A 2 + 8 H 6 β c ω A − 2 − 24 H 6 β c ω A 2 + 10 H 5 β c ω 2 A − 1 + 8 H 6 β c A 2 + 2 H 5 β c ω 2 A 1 − 10 H 5 β c ω A − 1 − 3 H 5 β c ω A 1 + 56 H 4 β c ω 2 A − 2 − 56 H 4 β c ω A − 2 − 14 H 3 β c ω 2 A − 1 + 21 H 3 β c ω A − 1 − 64 H 2 β c ω 2 A − 2 + 96 H 2 β c ω A − 2 − 24 H 12 β c ω 2 A 2 − 6 H 11 β c ω 2 A 1 + 64 H 10 β c ω 2 A 2 − 32 H 10 β c ω A 2 + 14 H 9 β c ω 2 A 1 − 6 H 3 α A − 2 A − 1 = 0.

The above equation decomposes on similar powers of H and leads to following thirteen algebraic equations:

α ω A 2 2 − 6 β c ω 2 A 2 = 0 , α ω A 1 A 2 − β c ω 2 A 1 = 0 , 6 β c ω 2 A − 2 − α ω A − 2 2 − 12 β c ω A − 2 + α A − 2 2 + 6 β c A − 2 = 0 , β c ω 2 A − 1 − α ω A − 2 A − 1 − 2 β c ω A − 1 + α A − 2 A − 1 + β c A − 1 = 0 , 32 β c ω 2 A 2 + 2 α ω A 0 A 2 + α ω A 1 2 − 4 α ω A 2 2 − 16 β c ω A 2 + 2 α A 2 2 − c ω A 2 = 0 , 14 β c ω 2 A 1 + 2 α ω A − 1 A 2 + 2 α ω A 0 A 1 − 12 α ω A 1 A 2 − 7 β c ω A 1 + 6 α A 1 A 2 − c ω A 1 = 0 , 28 β c ω 2 A − 2 − 2 α ω A − 2 2 + 4 α ω A − 2 A 0 + 2 α ω A − 1 2 − 28 β c ω A − 2 − 2 α A − 2 A 0 − α A − 1 2 + 4 β c A − 2 − 2 c ω A − 2 + c A − 2 = 0 , 4 α ω A − 2 2 − 32 β c ω 2 A − 2 − 2 α ω A − 2 A 0 − α ω A − 1 2 + 48 β c ω A − 2 − 2 α A − 2 2 + 2 α A − 2 A 0 + α A − 1 2 − 16 β c A − 2 + c ω A − 2 − c A − 2 = 0 , α A 1 2 − 28 β c ω 2 A 2 − 4 α ω A 0 A 2 − 2 α ω A 1 2 + 2 α ω A 2 2 + 28 β c ω A 2 + 2 α A 0 A 2 − 2 α A 2 2 − 4 β c A 2 + 2 c ω A 2 − c A 2 = 0 , 21 β c ω A − 1 − 14 β c ω 2 A − 1 + 12 α ω A − 2 A − 1 − 2 α ω A − 2 A 1 − 2 α ω A − 1 A 0 − 6 α A − 2 A − 1 + 2 α A − 2 A 1 + 2 α A − 1 A 0 − 7 β c A − 1 + c ω A − 1 − c A − 1 = 0 , α ω A 1 2 − 8 β c ω 2 A − 2 + 8 β c ω 2 A 2 − 2 α ω A − 2 A 0 − α ω A − 1 2 + 2 α ω A 0 A 2 + 4 β c ω A − 2 − 12 β c ω A 2 − 2 α A 0 A 2 − α A 1 2 + 4 β c A 2 + c ω A − 2 − c ω A 2 + c A 2 = 0 , β c ω A − 1 − 2 β c ω 2 A − 1 − 10 β c ω 2 A 1 − 2 α ω A − 2 A 1 − 2 α ω A − 1 A 0 − 4 α ω A − 1 A 2 − 4 α ω A 0 A 1 + 6 α ω A 1 A 2 + 10 β c ω A 1 + 2 α A − 1 A 2 + 2 α A 0 A 1 − 6 α A 1 A 2 − β c A 1 + c ω A − 1 + 2 c ω A 1 − c A 1 = 0 , 10 β c ω 2 A − 1 + 2 β c ω 2 A 1 − 6 α ω A − 2 A − 1 + 4 α ω A − 2 A 1 + 4 α ω A − 1 A 0 + 2 α ω A − 1 A 2 + 2 α ω A 0 A 1 − 10 β c ω A − 1 − 3 β c ω A 1 − 2 α A − 2 A 1 − 2 α A − 1 A 0 − 2 α A − 1 A 2 − 2 α A 0 A 1 + β c A − 1 + β c A 1 − 2 c ω A − 1 − c ω A 1 + c A − 1 + c A 1 = 0.

Solving the above system we get two cases, namely

Case 1.

(14) A − 2 = 0 , A − 1 = 0 , A 0 = − c 8 β ω − 4 β − 1 2 α , A 1 = 0 , A 2 = 6 β c ω α .

Case 2.

(15) A − 2 = 6 α β c ω − 1 , A − 1 = 0 , A 0 = − c 2 α 8 β ω − 4 β − 1 , A 1 = 0 , A 2 = 6 β c ω α .

Thus cnoidal wave solutions of PDE (1) are

u = c 1 + 4 β − 8 β ω 2 α + 6 β c ω α cn 2 ξ ω

and

u ( t , x ) = 6 β c ω − 1 α nc 2 ξ ω − c 8 β ω − 4 β − 1 2 α + 6 β c ω α cn 2 ξ ω

where nc = 1/cn.

2.3.2 Snoidal waves

In this case, again N = 2 and following the same procedure as above but invoking ODE (10) this time, we obtain

8 H 8 β c A 2 + H 7 β c A 1 − 8 H 6 β c A 2 − H 5 β c A − 1 − H 5 β c A 1 − 8 H 4 β c A − 2 + 7 H 3 β c A − 1 + 32 H 2 β c A − 2 + 4 H 2 α ω A − 2 2 − 4 H 2 α A − 2 A 0 − 6 H β c A − 1 − 6 H α A − 2 A − 1 + 4 H 12 α ω A 2 2 + 2 H 10 α ω A 1 2 − 4 H 10 α ω A 2 2 − 2 H 10 c ω A 2 − 6 H 9 α A 1 A 2 − H 9 c ω A 1 − 2 H 8 α ω A 1 2 − 4 H 8 α A 0 A 2 + 2 H 8 c ω A 2 − 2 H 7 α A − 1 A 2 − 2 H 7 α A 0 A 1 + 6 H 7 α A 1 A 2 + H 7 c ω A − 1 + H 7 c ω A 1 − 2 H 6 α ω A − 1 2 + 4 H 6 α A 0 A 2 + 2 H 6 c ω A − 2 + 2 H 5 α A − 2 A 1 + 2 H 5 α A − 1 A 0 + 2 H 5 α A − 1 A 2 + 2 H 5 α A 0 A 1 − H 5 c ω A − 1 − 4 H 4 α ω A − 2 2 + 2 H 4 α ω A − 1 2 + 4 H 4 α A − 2 A 0 − 2 H 4 c ω A − 2 + 6 H 3 α A − 2 A − 1 − 2 H 3 α A − 2 A 1 − 2 H 3 α A − 1 A 0 − 4 α A − 2 2 − 4 H 10 α A 2 2 2 H 8 α A 1 2 + 4 H 8 α A 2 2 + 2 H 8 c A 2 + H 7 c A 1 − H 5 c A 1 + 2 H 6 α A 1 2 2 H 6 c A 2 − H 5 c A − 1 + 2 H 4 α A − 1 2 − 2 H 4 c A − 2 + 4 H 2 α A − 2 2 + H 3 c A − 1 − 2 H 2 α A − 1 2 + 2 H 2 c A − 2 + 6 H 11 α ω A 1 A 2 + 4 H 10 α ω A 0 A 2 + 2 H 9 α ω A − 1 A 2 + 2 H 9 α ω A 0 A 1 − 6 H 9 α ω A 1 A 2 − 4 H 8 α ω A 0 A 2 − 2 H 7 α ω A − 2 A 1 − 2 H 7 α ω A − 1 A 0 − 2 H 7 α ω A − 1 A 2 − 2 H 7 α ω A 0 A 1 − 4 H 6 α ω A − 2 A 0 − 6 H 5 α ω A − 2 A − 1 + 2 H 5 α ω A − 2 A 1 + 2 H 5 α ω A − 1 A 0 + 4 H 4 α ω A − 2 A 0 + 6 H 3 α ω A − 2 A − 1 − 24 β c A − 2 + H 7 β c ω 2 A 1 + H 7 β c ω A − 1 + 8 H 7 β c ω A 1 + 8 H 6 β c ω 2 A − 2 + 8 H 6 β c ω A − 2 − 8 H 6 β c ω A 2 − H 5 β c ω 2 A − 1 − 8 H 5 β c ω A − 1 − 8 H 4 β c ω 2 A − 2 − 40 H 4 β c ω A − 2 + 7 H 3 β c ω A − 1 − H 5 β c ω A 1 + 24 H 12 β c ω 2 A 2 + 32 H 2 β c ω A − 2 + 6 H 11 β c ω 2 A 1 − 32 H 10 β c ω 2 A 2 − 32 H 10 β c ω A 2 − 7 H 9 β c ω 2 A 1 − 7 H 9 β c ω A 1 + 8 H 8 β c ω 2 A 2 + 40 H 8 β c ω A 2 + H 7 β c ω 2 A − 1 = 0.

Splitting on similar powers of H yields algebraic equations

α A − 2 2 + 6 β c A − 2 = 0 , 6 β c ω 2 A 2 + α ω A 2 2 = 0 , α A − 2 A − 1 + β c A − 1 = 0 , β c ω 2 A 1 + α ω A 1 A 2 = 0 , 2 α ω A − 2 2 + 16 β c ω A − 2 + 2 α A − 2 2 − 2 α A − 2 A 0 − α A − 1 2 + 16 β c A − 2 + c A − 2 = 0 , 2 α ω A 0 A 2 − 16 β c ω 2 A 2 + α ω A 1 2 − 2 α ω A 2 2 − 16 β c ω A 2 − 2 α A 2 2 − c ω A 2 = 0 , 6 α ω A − 2 A − 1 + 7 β c ω A − 1 + 6 α A − 2 A − 1 − 2 α A − 2 A 1 − 2 α A − 1 A 0 + 7 β c A − 1 + c A − 1 = 0 , 2 α ω A − 1 A 2 − 7 β c ω 2 A 1 + 2 α ω A 0 A 1 − 6 α ω A 1 A 2 − 7 β c ω A 1 − 6 α A 1 A 2 − c ω A 1 = 0 , 2 α ω A − 2 A 0 − 4 β c ω 2 A − 2 − 2 α ω A − 2 2 + α ω A − 1 2 − 20 β c ω A − 2 + 2 α A − 2 A 0 + α A − 1 2 − 4 β c A − 2 − c ω A − 2 − c A − 2 = 0 , 4 β c ω 2 A 2 − 2 α ω A 0 A 2 − α ω A 1 2 + 20 β c ω A 2 − 2 α A 0 A 2 − α A 1 2 + 2 α A 2 2 + 4 β c A 2 + c ω A 2 + c A 2 = 0 , 4 β c ω 2 A − 2 − α ω A − 2 A 0 − α ω A − 1 2 + 2 β c ω A − 2 − 2 β c ω A 2 + α A 0 A 2 + α A 1 2 − 2 β c A 2 + c ω A − 2 − c A 2 = 0 , 2 α ω A − 2 A 1 − β c ω 2 A − 1 − 6 α ω A − 2 A − 1 + 2 α ω A − 1 A 0 − 8 β c ω A − 1 − β c ω A 1 + 2 α A − 2 A 1 + 2 α A − 1 A 0 + 2 α A − 1 A 2 + 2 α A 0 A 1 − β c A − 1 − β c A 1 − c ω A − 1 − c A − 1 − c A 1 = 0 , β c ω 2 A − 1 + β c ω 2 A 1 − 2 α ω A − 2 A 1 − 2 α ω A − 1 A 0 − 2 α ω A − 1 A 2 − 2 α ω A 0 A 1 + β c ω A − 1 + 8 β c ω A 1 − 2 α A − 1 A 2 − 2 α A 0 A 1 + 6 α A 1 A 2 + β c A 1 + c ω A − 1 + c ω A 1 + c A 1 = 0.

Solving the above system of equations yields the following two cases:

Case 1.

A − 2 = 0 , A − 1 = 0 , A 0 = c 2 α 4 β ω + 4 β + 1 , A 1 = 0 , A 2 = − 6 β c ω α .

Case 2.

A − 2 = − 6 β c α , A − 1 = 0 , A 0 = c 2 α 4 β ω + 4 β + 1 , A 1 = 0 , A 2 = − 6 β c ω α .

Thus we obtain two snoidal wave solutions of the equal-width equation (1) as

(16) u = c 1 + 4 β + 4 β ω ( 1 − 3 sn 2 ξ ω ) 2 α ,

and

(17) u = − 6 β c α ns 2 ξ ω + c 4 β ω + 4 β + 1 2 α − 6 β c ω α sn 2 ξ ω

where ns = 1/sn.

We now consider the second operator X3 + aX2 of the optimal system. This symmetry operator yields the two invariants J 1 = e x t − a / 2 and J 2 = u t 1 / 2 . Thus J2 = f (J1) provides an invariant solution of (1), viz.,

u = 1 t f ( e x t − a / 2 ) .

Substitution of u in PDE (1), gives nonlinear third-order ODE

a β z 3 f ′ ′ ′ ( z ) + β 3 a + 1 z 2 f ′ ′ ( z ) + ( a β − a + β ) z f ′ ( z ) + 6 α z f ( z ) 2 f ′ ( z ) − f ( z ) = 0 ,

where z = e x t − a / 2 .

The third operator X2 provides invariants J1 = t, J2 = u. Thus u = (t) is invariant solution. Substituting this value of u into equation (1) and solving the resultant ODE gives

u = C 1 ,

where C1 is an arbitrary constant.

2.4 Conservation laws

We construct conservations laws [6, 7, 15, 16, 17, 18, 19, 20] of equal-width equation (1) by using two approaches; multiplier approach [21, 22, 23] and Noether’s theorem [24, 25].

2.4.1 Conservation laws by invoking multiplier approach

We seek zeroth-order multiplier Q, which depends on (t, x, u). To determine this multiplier Q we invoke

(18) δ δ u Q u t + 2 α u u x − β u t x x = 0 ,

where

δ δ u = ∂ ∂ u − D t ∂ ∂ u t − D x ∂ ∂ u x − D t D x 2 ∂ ∂ u t x x

and Dt, Dx are as defined in (6). The above equation yields

u t Q u + 2 α u u x Q u − β u t x x Q u + 2 α u x Q − D t ( Q ) − D x ( 2 α u Q ) + D x 2 D t β Q = 0 ,

which on expanding gives

β Q t x x − Q t − 2 α u Q x + 2 β u x Q t x u + β u x 2 Q t u u + β u x x Q t u + β u t Q x x u + 2 β u t u x Q x u u + 2 β u t x Q x u − 2 + β u t u x 2 Q u u + β u t u x x Q u u + 2 β u x u t x Q u u = 0.

Decomposing above equation on derivatives of u, we obtain

Q t u = 0 , Q x u = 0 , Q u u = 0 , Q t + 2 α u Q x − β Q t x x = 0 ,

which on solving gives two multipliers

Q 1 ( t , x , u ) = u

and

Q 2 ( t , x , u ) = 1.

Associated with above multipliers, we acquire two conservation laws whose components are

T 1 t = 1 2 β u x 2 + 1 2 u 2 , T 1 x = 2 3 α u 3 − β u u t x

and

T 2 t = u , T 2 x = α u 2 − β u t x .

2.4.2 Conservation laws using Noether’s theorem

We now invoke Noether’s approach [24, 25] to construct conservation laws for PDE (1). This equation is of third order and as such has no Lagrangian. However, by letting u = vx we can make it variational. Thus (1) now becomes

(19) v t x + 2 α v x v x x − β v t x x x = 0 ,

which has a Lagrangian given by

(20) L = − 1 2 v t v x − 1 3 α v x 3 − 1 2 β v t x v x x

as δL/δv = 0 on the equation (19) where

δ δ v = ∂ ∂ v − D t ∂ ∂ v t − D x ∂ ∂ v x + D x 2 ∂ ∂ v x x + D t D x ∂ ∂ v t x .

Consider the generator

(21) X = τ ∂ ∂ t + ξ ∂ ∂ x + η ∂ ∂ v ,

where , ξ and η are functions of t, x and v. To determine Noether symmetries X of (19) we insert the value of L from (20) in the determining equation

(22) X [ 2 ] ( L ) + L D t τ + D x ξ = D t B t + D x B x ,

where (Bt , Bx) are gauge terms that are functions of (t, x, v) and X[2] is the second prolongation of X defined as

(23) X [ 2 ] = X + ζ 1 ∂ ∂ v t + ζ 2 ∂ ∂ v x + ζ 12 ∂ ∂ v t x + ζ 22 ∂ ∂ v x x

with 1, 2, 12 and 22 as defined in [8]. Expansion of equation (22) and decomposing on derivatives of v yields a system of PDEs. Thereafter solving these PDEs we obtain the Noether symmetries together with their gauge functions as

X 1 = ∂ ∂ t , B t = 0 , B x = 0 , X 2 = ∂ ∂ x , B t = 0 , B x = 0 , X f = f ( t ) ∂ ∂ v , B t = 0 , B x = − 1 2 v f ′ ( t ) .

Next, we use the above results to compute the conserved vectors of the fourth-order equation (19). The formulae for the conserved vector (Tt , Tx) [26]

T k = L τ k + ( ξ α − ψ x j α τ j ) ( ∂ L ∂ ψ x k α − ∑ l = 1 k D x l ( ∂ L ∂ ψ x l x k α ) ) + ∑ l = k n ( η l α − ψ x l x j α τ j ) ∂ L ∂ ψ x k x l α − f k

yield three conservations laws associated with the three Noether symmetries X1, X2 and Xf . Thus the three conserved vectors are

T 1 t = − 1 3 α u 3 , T 1 x = 1 2 ∫ u t d x 2 + α u 2 ∫ u t d x − β u t x ∫ u t d x + 1 2 β u t 2 ; T 2 t = 1 2 u 2 + 1 2 β u x 2 , T 2 x = 2 3 α u 3 − β u u t x ; T f t = − 1 2 f ( t ) u , T f x = β f ( t ) u t x − 1 2 f ( t ) ∫ u t d x − α f ( t ) u 2 + 1 2 f ′ ( t ) ∫ u d x .

Remark: It should be noted that because of arbitrary function f (t) we have infinitely many nonlocal conservation laws.

3 Conclusion

In this work we studied the equal-width equation (1) from symmetry standpoint. Firstly, Lie symmetries were computed and were used to construct OSODS. Thereafter, symmetry reductions were performed on (1). On the ODE obtained, Jacobi elliptic function expansion technique was employed which resulted in exact solutions of (1) in form of cnoidal and snoidal waves. Secondly, conservation laws for this equation were established by invoking both the multiplier approach and the celebrated Noether’s theorem.

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

Acknowledgement

The authors would like to thank T Motsepa for fruitful discussions. CMK thanks the North-West University, Mafikeng Campus for its continued support.

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Received: 2019-03-04
Accepted: 2019-05-23
Published Online: 2019-10-04

© 2019 C. Masood Khalique et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  12. Investigating the impact of dissolved natural gas on the flow characteristics of multicomponent fluid in pipelines
  13. Analysis of impact load on tubing and shock absorption during perforating
  14. Energy characteristics of a nonlinear layer at resonant frequencies of wave scattering and generation
  15. Ion charge separation with new generation of nuclear emulsion films
  16. On the influence of water on fragmentation of the amino acid L-threonine
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  18. Displacement Reliability Analysis of Submerged Multi-body Structure’s Floating Body for Connection Gaps
  19. Deposits of iron oxides in the human globus pallidus
  20. Integrability, exact solutions and nonlinear dynamics of a nonisospectral integral-differential system
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  22. Visual Analysis of Cylindrically Polarized Light Beams’ Focal Characteristics by Path Integral
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  27. Evaluation of the realism of a full-color reflection H2 analog hologram recorded on ultra-fine-grain silver-halide material
  28. Graph cutting and its application to biological data
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  31. MHD and Slip Effect on Two-immiscible Third Grade Fluid on Thin Film Flow over a Vertical Moving Belt
  32. Vibration Analysis of a Three-Layered FGM Cylindrical Shell Including the Effect Of Ring Support
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https://doi.org/10.1515/phys-2019-0052
Schlagwörter für diesen Artikel
equal-width equation; Lie symmetries; optimal system of one-dimensional subalgebras; cnoidal and snoidal waves; conservation laws
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Regular Articles
  2. Non-equilibrium Phase Transitions in 2D Small-World Networks: Competing Dynamics
  3. Harmonic waves solution in dual-phase-lag magneto-thermoelasticity
  4. Multiplicative topological indices of honeycomb derived networks
  5. Zagreb Polynomials and redefined Zagreb indices of nanostar dendrimers
  6. Solar concentrators manufacture and automation
  7. Idea of multi cohesive areas - foundation, current status and perspective
  8. Derivation method of numerous dynamics in the Special Theory of Relativity
  9. An application of Nwogu’s Boussinesq model to analyze the head-on collision process between hydroelastic solitary waves
  10. Competing Risks Model with Partially Step-Stress Accelerate Life Tests in Analyses Lifetime Chen Data under Type-II Censoring Scheme
  11. Group velocity mismatch at ultrashort electromagnetic pulse propagation in nonlinear metamaterials
  12. Investigating the impact of dissolved natural gas on the flow characteristics of multicomponent fluid in pipelines
  13. Analysis of impact load on tubing and shock absorption during perforating
  14. Energy characteristics of a nonlinear layer at resonant frequencies of wave scattering and generation
  15. Ion charge separation with new generation of nuclear emulsion films
  16. On the influence of water on fragmentation of the amino acid L-threonine
  17. Formulation of heat conduction and thermal conductivity of metals
  18. Displacement Reliability Analysis of Submerged Multi-body Structure’s Floating Body for Connection Gaps
  19. Deposits of iron oxides in the human globus pallidus
  20. Integrability, exact solutions and nonlinear dynamics of a nonisospectral integral-differential system
  21. Bounds for partition dimension of M-wheels
  22. Visual Analysis of Cylindrically Polarized Light Beams’ Focal Characteristics by Path Integral
  23. Analysis of repulsive central universal force field on solar and galactic dynamics
  24. Solitary Wave Solution of Nonlinear PDEs Arising in Mathematical Physics
  25. Understanding quantum mechanics: a review and synthesis in precise language
  26. Plane Wave Reflection in a Compressible Half Space with Initial Stress
  27. Evaluation of the realism of a full-color reflection H2 analog hologram recorded on ultra-fine-grain silver-halide material
  28. Graph cutting and its application to biological data
  29. Time fractional modified KdV-type equations: Lie symmetries, exact solutions and conservation laws
  30. Exact solutions of equal-width equation and its conservation laws
  31. MHD and Slip Effect on Two-immiscible Third Grade Fluid on Thin Film Flow over a Vertical Moving Belt
  32. Vibration Analysis of a Three-Layered FGM Cylindrical Shell Including the Effect Of Ring Support
  33. Hybrid censoring samples in assessment the lifetime performance index of Chen distributed products
  34. Study on the law of coal resistivity variation in the process of gas adsorption/desorption
  35. Mapping of Lineament Structures from Aeromagnetic and Landsat Data Over Ankpa Area of Lower Benue Trough, Nigeria
  36. Beta Generalized Exponentiated Frechet Distribution with Applications
  37. INS/gravity gradient aided navigation based on gravitation field particle filter
  38. Electrodynamics in Euclidean Space Time Geometries
  39. Dynamics and Wear Analysis of Hydraulic Turbines in Solid-liquid Two-phase Flow
  40. On Numerical Solution Of The Time Fractional Advection-Diffusion Equation Involving Atangana-Baleanu-Caputo Derivative
  41. New Complex Solutions to the Nonlinear Electrical Transmission Line Model
  42. The effects of quantum spectrum of 4 + n-dimensional water around a DNA on pure water in four dimensional universe
  43. Quantum Phase Estimation Algorithm for Finding Polynomial Roots
  44. Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia
  45. The Errors Recognition and Compensation for the Numerical Control Machine Tools Based on Laser Testing Technology
  46. Evaluation and Decision Making of Organization Quality Specific Immunity Based on MGDM-IPLAO Method
  47. Key Frame Extraction of Multi-Resolution Remote Sensing Images Under Quality Constraint
  48. Influences of Contact Force towards Dressing Contiguous Sense of Linen Clothing
  49. Modeling and optimization of urban rail transit scheduling with adaptive fruit fly optimization algorithm
  50. The pseudo-limit problem existing in electromagnetic radiation transmission and its mathematical physics principle analysis
  51. Chaos synchronization of fractional–order discrete–time systems with different dimensions using two scaling matrices
  52. Stress Characteristics and Overload Failure Analysis of Cemented Sand and Gravel Dam in Naheng Reservoir
  53. A Big Data Analysis Method Based on Modified Collaborative Filtering Recommendation Algorithms
  54. Semi-supervised Classification Based Mixed Sampling for Imbalanced Data
  55. The Influence of Trading Volume, Market Trend, and Monetary Policy on Characteristics of the Chinese Stock Exchange: An Econophysics Perspective
  56. Estimation of sand water content using GPR combined time-frequency analysis in the Ordos Basin, China
  57. Special Issue Applications of Nonlinear Dynamics
  58. Discrete approximate iterative method for fuzzy investment portfolio based on transaction cost threshold constraint
  59. Multi-objective performance optimization of ORC cycle based on improved ant colony algorithm
  60. Information retrieval algorithm of industrial cluster based on vector space
  61. Parametric model updating with frequency and MAC combined objective function of port crane structure based on operational modal analysis
  62. Evacuation simulation of different flow ratios in low-density state
  63. A pointer location algorithm for computer visionbased automatic reading recognition of pointer gauges
  64. A cloud computing separation model based on information flow
  65. Optimizing model and algorithm for railway freight loading problem
  66. Denoising data acquisition algorithm for array pixelated CdZnTe nuclear detector
  67. Radiation effects of nuclear physics rays on hepatoma cells
  68. Special issue: XXVth Symposium on Electromagnetic Phenomena in Nonlinear Circuits (EPNC2018)
  69. A study on numerical integration methods for rendering atmospheric scattering phenomenon
  70. Wave propagation time optimization for geodesic distances calculation using the Heat Method
  71. Analysis of electricity generation efficiency in photovoltaic building systems made of HIT-IBC cells for multi-family residential buildings
  72. A structural quality evaluation model for three-dimensional simulations
  73. WiFi Electromagnetic Field Modelling for Indoor Localization
  74. Modeling Human Pupil Dilation to Decouple the Pupillary Light Reflex
  75. Principal Component Analysis based on data characteristics for dimensionality reduction of ECG recordings in arrhythmia classification
  76. Blinking Extraction in Eye gaze System for Stereoscopy Movies
  77. Optimization of screen-space directional occlusion algorithms
  78. Heuristic based real-time hybrid rendering with the use of rasterization and ray tracing method
  79. Review of muscle modelling methods from the point of view of motion biomechanics with particular emphasis on the shoulder
  80. The use of segmented-shifted grain-oriented sheets in magnetic circuits of small AC motors
  81. High Temperature Permanent Magnet Synchronous Machine Analysis of Thermal Field
  82. Inverse approach for concentrated winding surface permanent magnet synchronous machines noiseless design
  83. An enameled wire with a semi-conductive layer: A solution for a better distibution of the voltage stresses in motor windings
  84. High temperature machines: topologies and preliminary design
  85. Aging monitoring of electrical machines using winding high frequency equivalent circuits
  86. Design of inorganic coils for high temperature electrical machines
  87. A New Concept for Deeper Integration of Converters and Drives in Electrical Machines: Simulation and Experimental Investigations
  88. Special Issue on Energetic Materials and Processes
  89. Investigations into the mechanisms of electrohydrodynamic instability in free surface electrospinning
  90. Effect of Pressure Distribution on the Energy Dissipation of Lap Joints under Equal Pre-tension Force
  91. Research on microstructure and forming mechanism of TiC/1Cr12Ni3Mo2V composite based on laser solid forming
  92. Crystallization of Nano-TiO2 Films based on Glass Fiber Fabric Substrate and Its Impact on Catalytic Performance
  93. Effect of Adding Rare Earth Elements Er and Gd on the Corrosion Residual Strength of Magnesium Alloy
  94. Closed-die Forging Technology and Numerical Simulation of Aluminum Alloy Connecting Rod
  95. Numerical Simulation and Experimental Research on Material Parameters Solution and Shape Control of Sandwich Panels with Aluminum Honeycomb
  96. Research and Analysis of the Effect of Heat Treatment on Damping Properties of Ductile Iron
  97. Effect of austenitising heat treatment on microstructure and properties of a nitrogen bearing martensitic stainless steel
  98. Special Issue on Fundamental Physics of Thermal Transports and Energy Conversions
  99. Numerical simulation of welding distortions in large structures with a simplified engineering approach
  100. Investigation on the effect of electrode tip on formation of metal droplets and temperature profile in a vibrating electrode electroslag remelting process
  101. Effect of North Wall Materials on the Thermal Environment in Chinese Solar Greenhouse (Part A: Experimental Researches)
  102. Three-dimensional optimal design of a cooled turbine considering the coolant-requirement change
  103. Theoretical analysis of particle size re-distribution due to Ostwald ripening in the fuel cell catalyst layer
  104. Effect of phase change materials on heat dissipation of a multiple heat source system
  105. Wetting properties and performance of modified composite collectors in a membrane-based wet electrostatic precipitator
  106. Implementation of the Semi Empirical Kinetic Soot Model Within Chemistry Tabulation Framework for Efficient Emissions Predictions in Diesel Engines
  107. Comparison and analyses of two thermal performance evaluation models for a public building
  108. A Novel Evaluation Method For Particle Deposition Measurement
  109. Effect of the two-phase hybrid mode of effervescent atomizer on the atomization characteristics
  110. Erratum
  111. Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance
  112. Erratum to: Energy converting layers for thin-film flexible photovoltaic structures
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