Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions

Abstract

In this article, we investigate two types of double dispersion equations in two different dimensions, which arise in several physical applications. Double dispersion equations are derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Firstly, we obtain conservation laws for both these equations. To do this, we employ the multiplier method, which is an efficient method to derive conservation laws as it does not require the PDEs to admit a variational principle. Secondly, we obtain travelling waves and line travelling waves for these two equations. In this process, the conservation laws are used to obtain a triple reduction. Finally, a line soliton solution is found for the double dispersion equation in two dimensions.

An optimal System of Lie Subalgebras and Group-Invariant Solutions with Conserved Currents of a (3+1)-D Fifth-Order Nonlinear Model I with Applications in Electrical Electronics, Chemical Engineering and Pharmacy

“…Moreover, studies that bother on gaining soliton solutions to the nonlinear equations within a variety of fields have attracted a huge number of works. Towards this goal, powerful methods in their varieties for constructing multiple soliton solutions have been established in diverse fields of mathematical physics, other nonlinear sciences and engineering [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

“…In recent studies, physicists, engineers, as well as mathematicians mostly focus their studies on (1?1)-dimensional alongside (2?1)-dimensional integrable models. In other words, investigations of various solitary waves in these dimensions have extensively been studied both in theoretical as well as experimental aspects [3,[6][7][8].…”

“…Moreover, in [4] the vector bright solitons, as well as their interaction characteristics of the coupled Fokas-Lenells system, which models the femtosecond optical pulses in a birefringent optical fibre, were investigated. In addition, the Boussinesq-Burgers-type system of equations which delineates shallow water waves appearing close to lakes or ocean beaches was studied in [5], the list continues, see also [5][6][7][8][9][10]. However, there had been in general no methodical theories that can be infused into NLPDEs so that their closed-form solutions can be found.…”

The solutions of dissipation-dispersive models arising in material science

“…The various solutions of Eq. ( 2) have been proposed via F-expansion method (Silambarasan et al, 2019), sine-Gordon expansion method (Yel, 2020;Yazgan et al, 2022Yazgan et al, , 2023, Jacobi elliptic function (Yazgan et al, 2022), conservation laws (Gandarias et al, 2020), invariant preserving schemes (Kolkovska and Vucheva, 2019), Sardar sub-equation method (Rehman et al, 2022), etc. Additionally, asymptotic profile of solutions (Wang and Chen, 2016), instability and stability properties (Erbay et al, 2016) and non-uniqueness (Yang et al, 2015) are studied in the literature.…”

Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

“…In addition, the Boussinesq-Burgers-type system of equations, which delineates shallow water waves appearing close to lakes or ocean beaches, was studied in [4]. The list continues; see also [5][6][7][8][9][10][11][12][13][14][15].…”