Exact traveling wave solutions of the perturbed Klein-Gordon equation with quadratic nonlinearity in (1+1)-dimension, Part I-without local inductance and dissipation effect

Abstract

Abstract:In this paper, the auxiliary ordinary differential equation is employed to solve the perturbed Klein-Gordon equation with quadratic nonlinearity in the (1+1)-dimension without local inductance and dissipation effect. By using this method, we obtain abundant new types of exact traveling wave solutions.

Study on Date–Jimbo–Kashiwara–Miwa Equation with Conformable Derivative Dependent on Time Parameter to Find the Exact Dynamic Wave Solutions

“…where l (coefficient of spatial variable x), p (coefficient of spatial variable y), and c (velocity of wave) are nonzero arbitrary constants. Using Equation (12) in Equation (11), we obtain the following converted nonlinear ordinary equation:…”

“…To explain the natural incidents occurred in many scientific fields-such as the field of engineering, solid-state physics, plasma physics, applied physics, applied mathematics, plasma waves, fluid dynamics, quantum physics, electrodynamics, magneto hydrodynamics and turbulence, biology, fiber optics, chemistry, chemical physics, astrophysics, general theory of relativity, cosmology, medical science, and so on -we need to look for the exact solutions of the NLEEs. Many researchers have built up various types of methods to obtain the exact solutions for the NLEEs, including the following: the homogeneous balance method [1,2], the first integral method [3,4], the Jacobi elliptic function expansion method [5,6], the exponential function method [7,8], the tanh-function method [9,10], the auxiliary ordinary differential equation method [11], the homotopy analysis method [12], the tanh-coth method [13], the sine-Gordon expansion method [14], the (G /G 2 )-expansion method [15], the unified method [16], the variational iteration method [17], the extended direct algebraic method [18], the exp(−φ(ξ))-expansion method [19,20], Lie group method [21], the Hirota's bilinear method [22,23], the generalized Kudryshov method [24][25][26], the Backlund transform method [27], the Cole-Hopf transformation method [28], the Riccati equation method [29], the (G /G)-expansion method [30,31], the new generalized (G /G)expansion method [32], the modified (G /G)expansion method [33], and so on.…”

New Exact Solutions for the Klein-Gordon Equation

“…where , , , , are constants and is the perturbation parameter. Zhang [7] investigated this equation without local inductance and dissipation effect:…”

New exact traveling wave solutions to the (1+1)-dimensional Klein-Gordon-Zakharov equation for wave propagation in plasma using the exp(-Φ(ξ))-expansion method

“…In particular, the traveling wave solutions play a very important role in the study of these physical models arising from various natural phenomena for the field of applied sciences and engineering. Researchers have used diverse methods to get solutions of nonlinear PDEs, such as, inverse scattering transform [1], the Hirota's bilinear method [2], the tanh method [3], the extended tanh-method [4,5], the modified extended tanh-function method [6,7], the Jacobi elliptic function expansion method [8], the expfunction method [9,10], the improved F-expansion method [11], the exp(-Φ(ξ))-expansion method [12,13], the ðG 0 =GÞ-expansion method [14][15][16][17], the trigonometric function series method [18,19], the modified mapping and extended mapping method [20], the modified trigonometric function series method [21,22], the dynamical system approach [23][24][25], the multiple exp-function method [26], the transformed rational function method [27], the symmetry algebra method (consisting of Lie point symmetries) [28], the Wronskian technique [29], the homogeneous balance method [30], the infinite series and Jacobi elliptic function method [31][32], the first integral method [33], the auxiliary ordinary differential equation method [34] and so on.…”