Group Invariant Solutions and Conservation Laws of the Fornberg– Whitham Equation

Abstract

In this paper, we utilize the Lie symmetry analysis method to calculate new solutions for the Fornberg-Whitham equation (FWE). Applying a reduction method introduced by M. C. Nucci, exact solutions and first integrals of reduced ordinary differential equations (ODEs) are considered. Nonlinear self-adjointness of the FWE is proved and conserved vectors are computed

On the construction of conservation laws: A mixed approach

“…In a recent paper [23], using the IM, it was proved that (43) is nonlinear selfadjoint with ψ = k 0 (k 0 constant), even if only trivial conservation laws have been obtained [24]. Now we show that the method proposed in the previous section is able to provide nontrivial conservation laws for the Fornberg-Whitham equation ( 43).…”

A modified formal Lagrangian formulation for general differential equations

“…It was realised that many systems are not self-adjoint through a substitution v = h(u). In recent years, there have been generalisations to, for instance, weak self-adjointness and nonlinear self-adjointness (e.g., [10,17]), that, however, have been found restricted in deriving nontrivial conservation laws (e.g., [13]); such an example is studied in Section 4.1. Here, we only introduce the simplest case of self-adjointness.…”

“…Symmetry analysis of a bigger family of nonlinear partial differential equations was conducted in [7]. It was shown in [13] (see also [18]) that the FW equation is neither quasi self-adjoint nor weak self-adjoint through the formal Lagrangian approach; although it is nonlinearly self-adjoint but only trivial conservation laws could be obtained. In this subsection, we will study its modified formal Lagrangian formulation to derive conservation laws.…”

Impact of Multiplicative Noise on the Exact Solutions of the Fractional‐Stochastic Boussinesq‐Burger System

“…Recently, the searching of exact soliton solutions to NLPDEs has become an enthralling research topic in engineering and applied sciences. Many techniques have been used to determine exact solutions for NLPDE including tanh-sech [14,15], Darboux transformation [16], sinecosine [17,18], exp ð−ϕðςÞÞ-expansion [19], ðG ′ /GÞ-expan-sion [20][21][22], Lie symmetry analysis method [23], improved F-expansion method [24,25], Hirota's function [26], the Jacobi elliptic function [27,28], and perturbation [29,30].…”