Abstract
The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space. In the present paper, we study a generalized sinh-Gordon equation with variable coefficients with the goal of obtaining analytical traveling wave solutions. Our results show that the traveling waves of the variable coefficient sinh-Gordon equation can be derived from the known solutions of the standard sinh-Gordon equation under a specific selection of a choice of the variable coefficients. These solutions include some real single and multi-solitons, periodic waves, breaking kink waves, singular waves, periodic singular waves, and compactons. These solutions might be valuable when scientists model some real-life phenomena using the sinh-Gordon equation where the balance between dispersion and nonlinearity is perturbed.
Highlights
The sinh-Gordon equation in its standard formTroca Cabella and Carlo Cattani ∂2∂2 u x, t u( x, t) + sinh(u( x, t)) = 0, ( ) ∂t2 ∂x2Received: 9 February 2022Accepted: 2 March 2022Published: 4 March 2022
Other types of solutions for the generalized sinh-Gordon Equations (27) and (31) such as single and multi-solitons, periodic waves, singular waves, periodic singular waves, and compactons can be obtained in a similar manner using Equations (28) and (32) and here, for the sake of brevity, we omit the details
We showed that real-valued traveling waves and soliton solutions could be obtained for the generalized sinh-Gordon equation with variable coefficients by utilizing the transformation of variables innovatively and the known solutions of the standard sinh-Gordon equation
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Is a completely nonlinear integrable partial differential equation that is widely used in physics and sciences [1]. This equation has broad-spectrum scientific applications in integrable quantum field theory, fluid dynamics, kink dynamics, differential geometry theory, and statistical mechanics. It can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space [6,7]. It arises in models of interacting charged particles in plasma physics, the interaction of neighboring particles of equal mass in a lattice formation with a crystal, and on effects of weak dislocation potential on nonlinear wave propagation in the anharmonic crystal [1,8,9].
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