The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations

Optical solitons in birefringent fibers with four-wave mixing for quadratic–cubic nonlinearity by F-expansion

A solution for the new Hamiltonian amplitude equation is derived using a property of the reciprocal Weierstrass elliptic function. The Weierstrass elliptic function solution has been subsequently expressed in terms of the Jacobian elliptic function. The solitary wave solution which is the infinite period counterpart of the Jacobian elliptic function solution has also been derived.

This work presents solutions to the strongly coupled nonlinear Wilberforce pendulum model. We begin by presenting three specific cases of the equations of motion of the nonlinear Wilberforce Lagrangian. We find solutions governed by Jacobi elliptic functions, Mathieu functions, Weierstrass elliptic functions and obtaining beating behavior of plane waves when their frequencies are close and the phase is proportional to their amplitudes. Similarly, we find analytical solutions showing the power of these methods using the tanh, Riccati, Jacobi, and generalized Jacobi solitary wave solution techniques, without any restrictions.

All real solutions of the Lane–Emden equation for n = 5 are obtained in terms of Jacobian and Weierstrass elliptic functions. A new family of solutions is found. It is expressed by remarkably simple formulae involving Jacobian elliptic functions only. The general properties and discrete scaling symmetries of these new solutions are discussed. We also comment on their possible applications.

Ocean physics describes the way the ocean surface layer interacts with the atmosphere and extends to the transmission of energy from ocean waves and tides to the seafloor. Hence, for the first time, this article explores the analytical study of a generalized geophysical Korteweg-de Vries equation found in ocean physics with power-law nonlinearity. The model is analyzed using the Lie group theory that ensures mapping of the existing solutions to other solutions. Initially, the calculation of the associated Lie algebra for the model is carried out in a systematic manner, after which one parameter transformation group for the algebra is derived. Besides, a one-dimensional optimal system of subalgebra is derived in a procedural manner. Sequel to this, the subalgebras and combination of the achieved symmetries are invoked in the reduction process, culminating in generating nonlinear ordinary differential equations associated with the model under study. In consequence, analytic soliton solutions, including non-topological soliton solutions and general periodic function solutions of note, in the structure of Weierstrass elliptic and Jacobi elliptic functions are obtained for the model. One of the interesting results also includes an implicit hypergeometric function solution. Additionally, numerical simulations are utilized to develop a basic understanding of the physical phenomena described by the model in ocean physics. Ultimately, conserved vectors are determined for the model by applying Ibragimov’s theorem together with Noether’s theorem.

Using He's homotopy perturbation method (HPM), the modified KdV equation which has not a small parameter is solved. The approximate Jacobi elliptic function solution is obtained. When the modulus of the Jacobian function tends to unit or zero, the corresponding solitary wave solution and trigonometric function solution are obtained. The results reveal that the HPM is very effective, convenient and quite accurate to systems of nonlinear equations.

Oscillators are omnipresent; most of them are inherently nonlinear. Though a nonlinear equation mostly does not yield an exact analytic solution for itself, plethora of elementary yet practical techniques exist for extracting important information about the solution of equation. The purpose of this chapter is to introduce some new techniques for the readers which are carefully illustrated using mainly the examples of Duffing’s oscillator. Using the exact analytical solution to cubic Duffing and cubic-quinbic Duffing oscillators, we describe the way other conservative and some non conservative damped nonlinear oscillators may be studied using analytical techniques described here. We do not make use of perturbation techniques. However, some comparison with such methods are performed. We consider oscillators having the form x¨+fx=0 as well as x¨+2εẋ+fx=Ft, where x=xt and f=fx and Ft are continuous functions. In the present chapter, sometimes we will use f−x=−fx and take the approximation fx≈∑j=1Npjxj, where j=1,3,5,⋯N only odd integer values and x∈−AA. Moreover, we will take the approximation fx≈∑j=0Npjxj, where j=1,2,3,⋯N, and x∈−AA. Arbitrary initial conditions are considered. The main idea is to approximate the function f=fx by means of some suitable cubic or quintic polynomial. The analytical solutions are expressed in terms of the Jacobian and Weierstrass elliptic functions. Applications to plasma physics, electronic circuits, soliton theory, and engineering are provided.

The JEFE method and periodic solutions of two kinds of nonlinear wave equations

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

Based on a Riccati equation and a symbolic computation system-Maple, a generalized Riccati equation expansion method is presented for constructing soliton-like solutions and periodic form solutions for some nonlinear evolution equations (NEEs) or NEEs with variable coefficients. Compared with most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and the generalized hyperbolic-function method, the proposed method is more powerful. We study a (2+1)-dimensional general nonintegrable KdV equation, a KdV equation with variable coefficients. As a result, rich new families of exact solutions, including the non-travelling wave's and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions, are obtained. When setting the arbitrary functions in some solutions be equal to special constants or special functions, the solitary wave solutions can be recovered.

Exact solutions of higher-dimensional nonlinear equations takes a major place in the study of nonlinear phenomena observed in nature. In this article, some new kink type solutions are investigated for the new (3+1)-dimensional Boiti-Leon-Manna-Pempinelli(BLMP) equation. Firstly, a variety of solutions are obtained by Hirota’s bilinear form, which include kink type wave solution, periodic solitary wave solutions and singular solitary wave solutions using extended homoclinic test approach. Secondly, solutions with three wave form are obtained by generalized three wave method. The extended homoclinic test approach is also used to construct solutions with a tail which explain some physical phenomenon. Moreover, some figures of the solutions are shown behind.

This article focuses on the exact periodic solutions of nonlinear wave equations using the well-known Jacobi elliptic function expansion method. This method is more general than the hyperbolic tangent function expansion method. The periodic solutions are found using this method which contains both solitary wave and shock wave solutions. In this paper, the new results are computed using the closed-form solution including solitary or shock wave solutions which are obtained using Jacobi elliptic function method. The corresponding solitary or shock wave solutions are compared with the actual results. The results are visualised and the periodic behaviour of the solution is described in detail. The shock waves are found to break with time, whereas, solitary waves are found to be improved continuously with time.

PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.

An extended Jacobian elliptic function expansion method presented recently by us is applied to the mKdV equation such that thirteen families of Jacobian elliptic function solutions including both new solutions and Fu's all results are obtained. When the modulus or 0, we can find the corresponding six solitary wave solutions and six trigonometric function solutions. This shows that our method is more powerful to construct more exact Jacobian elliptic function solutions and can be applied to other nonlinear differential equations.

This paper is concerned with the lifespan of and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation ∂ t 2 u − ∑ i = 1 2 ∂ i ( c i 2 ( u ) ∂ i u ) \partial _t^2u-\sum _{i=1}^2\partial _i(c_i^2(u)\partial _iu) = 0 =0 , where c i ( u ) ∈ C ∞ ( R n ) c_i(u)\in C^{\infty }(\mathbb {R}^n) , c i ( 0 ) ≠ 0 c_i(0)\neq 0 , and ( c 1 ′ ( 0 ) ) 2 + ( c 2 ′ ( 0 ) ) 2 ≠ 0 (c_1’(0))^2+(c_2’(0))^2\neq 0 . This equation has an interesting physical background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition ( u ( 0 , x ) , ∂ t u ( 0 , x ) ) = ( ε u 0 ( x ) , ε u 1 ( x ) ) (u(0,x), \partial _tu(0,x))=(\varepsilon u_0(x), \varepsilon u_1(x)) with u 0 ( x ) , u 1 ( x ) ∈ C 0 ∞ ( R 2 ) u_0(x), u_1(x)\in C_0^{\infty }(\mathbb {R}^2) , and ε > 0 \varepsilon >0 is small, we will show that the classical solution u ( t , x ) u(t,x) stops to be smooth at some finite time T ε T_{\varepsilon } . Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives ∇ t , x u ( t , x ) \nabla _{t,x}u(t,x) , while u ( t , x ) u(t,x) itself is continuous up to the blowup time T ε T_{\varepsilon } .