Abstract
In this paper, we solve the Duffing equation for given initial conditions. We introduce the concept of the discriminant for the Duffing equation and we solve it in three cases depending on sign of the discriminant. We also show the way the Duffing equation is applied in soliton theory.
1. Introduction
The nonlinear equation describing an oscillator with a cubic nonlinearity is called the Duffing equation. Duffing [1], a German engineer, wrote a comprehensive book about this in 1918. Since then there has been a tremendous amount of work done on this equation, including the development of solution methods (both analytical and numerical) and the use of these methods to investigate the dynamic behavior of physical systems that are described by the various forms of the Duffing equation. Because of its apparent and enigmatic simplicity, and because so much is now known about the Duffing equation, it is used by many researchers as an approximate model of many physical systems or as a convenient mathematical model to investigate new solution methods [2–7]. This equation exhibits an enormous range of well-known behavior in nonlinear dynamical systems and is used by many educators and researchers to illustrate such behavior. Since the 1970s, it has become really popular with researchers into chaos, as it is possibly one of the simplest equations that describes chaotic behavior of a system. This equation is also useful in the study of soliton solutions to important physics models such as KdV equation, mKdV equation, sine-Gordon equation, Klein–Gordon equation, nonlinear Schrodinger equation, and shallow water wave equation [8–18].
2. Undamped and Unforced Duffing Equation
Let , , , and be real numbers. The general solution to the undamped and unforced Duffing equation may be expressed in terms of any of the twelve Jacobian elliptic functions, as shown in Table 1.
In this section, we will solve the initial value problem
The numberis called the discriminant for problem (1).
2.1. First Case:
In the case, when , we get so that and the problem reduces to
Its solution is given by
Let . First of all, observe that if is a solution to the ode , then is also a solution for any constant . Secondly, let (c1 = a nonzero constant) be the Jacobi elliptic function cn with modulus and parameter defined by . We have
Therefore, comparing (1) and (5) givesand we conclude that the analytic functionis the general solution to the Duffing equation for arbitrary constants and . The values of these constants are determined from the initial conditions and .
We haveand the value of results from solving the equation , i.e.,
Squaring this last equation and taking into account relations (6) and the identitieswe arrive at the equation
Solving equation (11) for gives
To avoid the ambiguity with plus-minus signs, we define
Making use of the addition formulathe solution may also be written in the formwhere
The solution is periodic and its main period equals
In the case, when , we make use if the identities
The main period will then be
If , we transform the solution by means of the following identities:
Remember that cd = cn/dn, sd = sn/dn, and nd = 1/dn. For reference, Tables 2–4 give useful conversion formulas.
Example 1. Let , , , and . The solution to the i.v.p.is given byThis solution is periodic with main period . See Figure 1 for a comparison with Runge–Kutta numerical solution (dashed curve).
Example 2. Let , , , and . The solution to the i.v.p.,readsAn equivalent expression without the imaginary unit isSee Figure 2 for a comparison with Runge–Kutta numerical solution (dashed curve).
2.2. Second Case:
Define
Since , necessarily . From the equalityit is evident that . We seek a solution to the i.v.p. (1) in the ansatz formwhere is the solution to some Duffing equations
We have
Inserting the ansatz (30) into the ode and taking into account (30) and (31), we get
Equating to zero the coefficients of , and in (32), we obtain an algebraic system. Solving it gives
Observe that
Thus, the Duffing equation (29) has a positive discriminant. The solution to the i.v.p. (3) is then given bywhere
The values of , and are found from (33).
2.3. Third Case:
When the discriminant vanishes, then and the only solution to problem (1) with iswhich may be verified by direct computation.
3. New Trigonometric Jacobian Functions
Define the generalized cosine and sine functions as follows:
Our aim is to find some so that
Define
Observe that when . Let . We have
We will choose so that
Define
The obtained approximations are good. This is seen from Tables 5 and 6.
We now will introduce new Jacobian “trigonometric functions” as follows:
Define
We extend the new functions (44)–(47) and for and and imaginary argument it using Tables 1–3 replacing the with and with and so on.
4. Applications in Physics
Many partial differential equations arising in soliton theory may be reduced to odes or systems of odes by means of a traveling wave transformation. These odes are generally nonlinear and some of them are Duffing type equations. Let us consider some important models of soliton theory.
4.1. The Klein–Gordon–Zakharov (KGZ) Equation in Plasmas
The KGZ equation reads
We transform the KGZ by means of the traveling wave substitutionto obtain the system
We choose so that and integrating the equation (51) twice taking null integration constants, we obtainand the problem reduces to solve a Duffing equation.
4.2. The Sine-Gordon Equation
This is the equation
This important model appears in differential geometry and relativistic field theory. It is denominated following its similar form to the Klein–Gordon equation. The equation, as well as several solution techniques, was known in the 19th century, but the equation grew greatly in importance when it was realized that it led to solutions (“kink” and “antikink”) with the collision properties of solitons.
The sine-Gordon equation is widely applied in physical and engineering applications, including the propagation of fluxons in Josephson junctions (a junction between two superconductors), the motion of rigid pendular attached to a stretched wire, and dislocations in crystals. It also arises in nonlinear optics. We apply the traveling wave transformationand the sine-Gordon equation converts into
It may be easily verified that equation (55) holds for any solution of the Duffing equation
4.3. The Pendulum Equation
This equation reads
Let
Inserting ansatz (58) into (57) gives
Equation (59) is satisfied for any solution to Duffing equation obeying
4.4. The KdV Equation
This equation originated from soliton theory. It reads
Let . The traveling wave substitution gives . Integrating once, we obtainwhere is the constant of integration. We seek a solution to the nonlinear ode (62) in the ansatz formwhere the constants , , , and are to be determined. Since , it is clear that
Inserting the ansatz (63) into (62) and taking into account (63), we obtain
Equating the coefficients of , , and to zero gives an algebraic system. Solving it, we arrive at the expressions
4.5. The Nonlinear Schrodinger Equation
The nonlinear Schrödinger equation is among the most prominent equations in nonlinear physics, especially in nonlinear optics. The nonlinear Schrödinger equation is of particular importance in the description of nonlinear effects in optical fibers. The nonlinear Schrödinger equation is a central model of nonlinear science, applying to hydrodynamics, plasma physics, molecular biology, and optics. It has been studied for more than 40 years, and it is employed in numerous fields well beyond plasma physics and nonlinear optics, where it originally appeared. The nonlinear Schrödinger equation (NLSE) is in the following form:where γ is a nonzero real constants and is a complex valued function of two real variables . The Schrödinger equations occur in various areas of physics, including nonlinear optics, plasma physics, superconductivity, and quantum mechanics. The NLSE (67) exhibits soliton and periodic cnoidal wave solutions. Let
Under this transformation, the NLSE (67) takes the formwhich is a Duffing equation.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.