New Travelling-Wave Solutions for Dodd-Bullough Equation

Abstract

A new method, which assumes 𝜙^′ with a function form of 𝑓(𝑒^𝜙), is applied to solve the Dodd-Bullough equation 𝜙_𝑢𝑣=𝑒^𝜙−𝑒^−2𝜙 through travelling-wave transformation. A new family of explicit travelling wave solutions is derived. The proposed method works efficiently to be applied to solve other forms of Dodd-Bullough equations.

1. Introduction

The Dodd-Bullough equation 𝜙_𝑢𝑣=𝑒^𝜙−𝑒^−2𝜙, (1) initiated by Dodd and Bullough [1] and Žiber and Šabat [2], plays a significant role in many scientific applications such as solid state physics, nonlinear optics, and quantum field theory.

There are many research works for Dodd-Bullough equation in the last decades. It is shown that Dodd-Bullough determines the intrinsic geometry of the two-dimensional affine sphere in the three-dimensional unimodular affine space [3]. It has also been shown that the Dodd-Bullough equation is related to the nonlinear two-dimensional SL(3,R) sigma model in [4]. To solve the Dodd-Bullough equation, many mathematicians have put forward some methods, and different forms of solution formulas have been retrieved. Reference [5] develops two-soliton solution formula and N-soliton solution formula by means of dressing the trivial one 𝜙≡0. The dressing procedure for (1) is also suggested in [6]. By using three Weierstrass functions, [7] proposes some explicit solution formulas. Reference [8, 9] obtain some general ineffective formulas for solutions of (1), but no concrete solutions. The MSE (modified simple equation) method has been proposed in [10–12] to explore the exact traveling solution of nonlinear physical systems, then [13] applies the MSE method to the Tzitzeica-Dodd-Bullough equation for some new solutions.

Tanh method is efficient to deal with many kinds of non-linear equations. Applying the tanh method to the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations, [14] derives some solitons and periodic solutions. By developing an extended tanh method, [15] builds some new explicit solutions for Dodd-Bullough equation.

This paper aims to propose a new method to deal with the Dodd-Bullough equation, which is based on an assumption that the first order differential of 𝜙 on the travelling wave variable 𝜉=𝑢+𝑐𝑣. A new family of explicit solutions for (1) is derived. The proposed method works efficiently to solve other forms of Dodd-Bullough equations. We also compare some of our results to some previous solutions and find that the solution of the Dodd-Bullough-Mikhailov equation given in [14] is only a special case of ours. Moreover, our solutions for Tzitzeica-Dodd-Bullough equations are different from previous results in [13] and other works.

2. The Proposed Method

Our method aims to solve some nonlinear equations with the form (2), after the travelling-wave transformation, 𝐹𝜙^′′=𝐺𝑔𝜙, (2) where 𝐹, 𝐺, and 𝑔 might be any functions. The technique is based on a priori assumption: the traveling-wave solutions of (2) satisfy 𝜙^′ which can be expressed in a form of some function of 𝑔(𝜙).

The main steps of our proposed method are as follows.(1) We unite the independent variables 𝑢 and 𝑣 into one wave variable 𝜉=𝑢+𝑐𝑣 to carry out the PDE in two variables, 𝑃𝜙_𝑢𝑣,𝜙^′_𝑢𝑣,𝜙^′′_𝑢𝑣,…=0, (3)into an ODE, 𝑂𝜙(𝜉),𝜙^′(𝜉),𝜙^′′(𝜉),…=0. (4) The precondition for our method is (4) which meets a form of (2) after some transformation.(2) To find the travelling-wave solutions of (2), we take an assumption that 𝜙^′=𝑓𝑔𝜙, (5)then we get 𝜙^′′=𝑓^′𝑔𝜙𝑔^′𝜙𝜙^′. (6) According to the assumption, obviously, 𝜙^′′=𝑓𝑔𝜙𝑓^′𝑔𝜙𝑔^′𝜙, (7) then, substitute (7) into (2), 𝐹𝑓𝑔𝜙𝑓^′𝑔𝜙𝑔^′𝜙=𝐺𝑔𝜙. (8) Let 𝜓=𝑔(𝜙), then we obtain that𝐹𝑓𝜓𝑓^′𝜓𝜓^′=𝐺𝜓. (9)(3) Try to retrieve the form of the function 𝑓 by solving (9). In some cases, (9) is a variable separated ODE. (4) Once the form of the function 𝑓 is retrieved, return to (5), and we can get the solutions of 𝜙 by integration because (5) is also a variable separated ODE of the solutions, actually; 𝑑𝜙𝑓𝑔𝜙=𝑑𝜉,𝑑𝜙𝑓𝑔𝜙=𝑑𝜉=𝜉+𝑃, (10)(11)where 𝑃 is an integration constant.

In some cases, the explicit solutions can be retrieved by seeking the inverse function. Otherwise, only the implicit can be derived if the integration of the left side of (11) is much complex.

3. Solutions for Dodd-Bullough Equation

For the Dodd-Bullough given by (1), after travelling-wave transformation via 𝜉=𝑢+𝑐𝑣, we can get that 𝑐𝜙^′′=𝑒^𝜙−𝑒^−2𝜙. (12) Next, we assume that 𝜙^′(𝜉) meets an ODE form of some functions of 𝑒^𝜙 for 𝜙^′′(𝜉) which satisfies a simple function form of 𝑒^𝜙: 𝜙^′=𝑓𝑒^𝜙. (13) Following (6), we can get that 𝜙^′′=𝑓𝑒^𝜙𝑓^′𝑒^𝜙𝑒^𝜙. (14) Consequently, by substituting (14) into (12), we obtain that𝑐𝑓𝑒^𝜙𝑓^′𝑒^𝜙𝑒^𝜙=𝑒^𝜙−𝑒^−2𝜙. (15) Considering 𝜙^′ is a function of 𝑒^𝜙, let 𝑒^𝜙=𝜓, then 𝑓 is a function of 𝜓.

Now, we try to get the form of the function 𝑓: 𝑐𝑓𝜓𝑓^′𝜓𝜓=𝜓−𝜓⁻²,𝑐𝑓𝜓𝑓^′𝜓=1−𝜓⁻³. (16)(17) Equation (17) is an ODE of variable separated:𝑐2𝑓²𝜓1=𝜓+2𝜓⁻²+𝑃, (18) where 𝑃 is a constant of integration, 𝑓𝜓=±2𝑐1𝜓+2𝜓⁻²+𝑃. (19)

Remembering that 𝑒^𝜙=𝜓, then 𝜙=ln𝜓, 𝜙^′=𝜓^′𝜓. (20)

From (13), obviously, 𝑓𝜓=𝜓^′𝜓. (21)

By comparing (19) and (21), we get 𝜓^′𝜓=±2𝑐1𝜓+2𝜓⁻²+𝑃, (22) that is, 𝑑𝜓𝑑𝜉=𝜓^′=±𝜓2𝑐1𝜓+2𝜓⁻²+𝑃. (23)

Equation (23) is a variable separated ODE; using a symbol computation software program, such as MATHEMATICA, we obtain that𝜉+𝑄=±2Elliptic𝐹arcsin−𝜓+𝑟₃−𝑟₂+𝑟₃,𝑟₂−𝑟₃𝑟₁−𝑟₃×√𝜓−𝑟₃𝜓−𝑟₁−𝑟₁+𝑟₃𝜓−𝑟₂−𝑟₂+𝑟₃×𝜓1+2𝑃𝜓²+2𝜓³𝑐𝜓²𝜓−𝑟₃𝑟₂−𝑟₃⁻¹=±2𝑐𝜓−𝑟₃𝑟₂−𝑟₃×𝜓−𝑟₁𝜓−𝑟₂−𝑟₁+𝑟₃−𝑟₂+𝑟₃×1+2𝑃𝜓²+2𝜓³⁻¹×Elliptic𝐹arcsin−𝜓+𝑟₃−𝑟₂+𝑟₃,𝑟₂−𝑟₃𝑟₁−𝑟₃=±2𝑐𝑟₂−𝑟₃𝜓−𝑟₃𝜓−𝑟₁𝜓−𝑟₂−𝑟₁+𝑟₃−𝑟₂+𝑟₃×1+2𝑃𝜓²+2𝜓³⁻¹×Elliptic𝐹arcsin−𝜓+𝑟₃−𝑟₂+𝑟₃,𝑟₂−𝑟₃𝑟₁−𝑟₃, (24) where 𝑄 is a constant of integration, 𝑟₁,𝑟₂, and 𝑟₃ are the roots of the equation 1+2𝑃𝑥²+2𝑥³=0, and 𝑟₁𝑃=−3+2^2/3𝑃²3³−27−4𝑃³+381+24𝑃³+³−27−4𝑃³+381+24𝑃³3⋅2^2/3𝑟₂𝑃=−3−1+√3𝑖𝑃²3⋅2^1/33−27−4𝑃³+381+24𝑃³+1−√3𝑖³−27−4𝑃³+381+24𝑃³6⋅2^2/3𝑟₃𝑃=−3−1−√3𝑖𝑃²3⋅2^1/33−27−4𝑃³+381+24𝑃³+1+√3𝑖³−27−4𝑃³+381+24𝑃³26⋅23. (25) So, obviously, 2(𝜓−𝑟₃)(𝜓−𝑟₁)(𝜓−𝑟₂)=1+2𝑃𝜓²+2𝜓³. Then, (24) can be simplified as 𝜉+𝑄=±2𝑐𝑟₁−𝑟₃Elliptic𝐹arcsin−𝜓+𝑟₃−𝑟₂+𝑟₃,𝑟₂−𝑟₃𝑟₁−𝑟₃, (26) where the Elliptic𝐹 function, the elliptic integral of the first kind, is the inverse function of JacobiAmplitude function. Then, we get 𝜓=𝑟₂−𝑟₃sin²×±JacobiAmplitude𝑟₁−𝑟₃(),𝑟2𝑐𝜉+𝑄²−𝑟₃𝑟₁−𝑟₃+𝑟₃. (27)

Replace 𝜓 with 𝑒^𝜙, 𝜉 with 𝑢+𝑐𝑣 in (27), and then it is easy to obtain a family of explicit solutions for (1) as follows: 𝜙=ln𝑟₂−𝑟₃sin²×±JacobiAmplitude𝑟₁−𝑟₃×(),𝑟2𝑐𝑢+𝑐𝑣+𝑄²−𝑟₃𝑟₁−𝑟₃+𝑟₃, (28) where 𝑐 and 𝑄 are two arbitrary constants, and 𝑟₁, 𝑟₂, and 𝑟₃, as defined in (25), are the roots of the equation 1+2𝑃𝑥²+2𝑥³=0; 𝑃 is also an arbitrary constant.

4. Discussion and Comparison

4.1. Solutions for Other Forms of Dodd-Bullough Equation

Other forms of Dodd-Bullough equation can also be dealt with using the same method. As examples, we propose the travelling-wave solutions for Dodd-Bullough-Mikhailov equation given by 𝜙_𝑢𝑣+𝑒^𝜙+𝑒^−2𝜙=0, (29) and the Tzitzeica-Dodd-Bullough equation given by 𝜙_𝑢𝑣−𝑒^−𝜙−𝑒^−2𝜙=0. (30) The two equations appear in problems varying from fluid flow to quantum field theory. For (29), the solution retrieved using our method is 𝜙=ln𝑟₂−𝑟₃sin²×±JacobiAmplitude𝑟₃−𝑟₁×(),𝑟2𝑐𝑢+𝑐𝑣+𝑄²−𝑟₃𝑟₁−𝑟₃+𝑟₃, (31) where 𝑐 and 𝑄 are two arbitrary constants and 𝑟₁, 𝑟₂, and 𝑟₃ are the roots of the equation −1−2𝑃𝑥²+2𝑥³=0; 𝑃 is also an arbitrary constant.

And for (30), taking an assumption that 𝜙^′=𝑓(𝑒^−𝜙), and using a similar process, we can get a family of travelling-wave solution as follows: 1𝜙=ln8)𝑒(1+4𝑃^{2𝑃/𝑐(𝑢+𝑐𝑣+𝑄)}𝑃^3/2+𝑒^{−2𝑃/𝑐(𝑢+𝑐𝑣+𝑄)}√𝑃+2𝑃, (32) where 𝑃, 𝑄, and 𝑐 are all arbitrary constants.

Actually, the method proposed in this lecture can be used to solve some more general forms of the Dodd-Bullough equation given by 𝜙_𝑢𝑣+𝛼𝑒^𝜙+𝛽𝑒^−2𝜙=0 (33) or 𝜙_𝑢𝑢±𝜙_𝑣𝑣+𝛼𝑒^−𝜙+𝛽𝑒^−2𝜙=0. (34) It is easy to derive the solutions for these two general Dodd-Bullough equations following the above listed steps. We would not go into the details.

4.2. Comparing of Previous Solutions

Compared to the previous works, we present new traveling-wave solutions for Dodd-Bullough.

Actually, in [14], Wazwaz proposes some travelling solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations using the tanh method. It can be shown that the solutions for the Dodd-Bullough-Mikhailov given in [14] are only some special cases of our results.

For solutions of the Dodd-Bullough-Mikhailov equation given by (29), while we fix the constant 𝑃 to meet the parameter (𝑟₂−𝑟₃)/(𝑟₁−𝑟₃)=1, the JacobiAmplitude function will degenerate to an Arctan function. Then, we get 𝜙=ln𝑟₂−𝑟₃sin²×−𝜋2+2arctan𝑒^{±(𝑟3−𝑟1)/2𝑐(𝑢+𝑐𝑣+𝑄)}+𝑟₃. (35) That is, 𝜙=ln𝑟₂−𝑟₃tanh²±𝑟₃−𝑟₁()2𝑐𝑢+𝑐𝑣+𝑄+𝑟₃. (36)

Remembering that 𝑟₁, 𝑟₂, and 𝑟₃ are the roots of the equation −1−2𝑃𝑥²+2𝑥³=0, and (𝑟₂−𝑟₃)/(𝑟₁−𝑟₃)=1, we can easily find that 𝑟₁=𝑟₂=−1 and 𝑟₃=1/2; then, (36) changes into 1𝜙=21−3tanh²±123𝑐(𝑢+𝑐𝑣+𝑄). (37) Then, it is easy to see that the solution of the Dodd-Bullough-Mikhailov equation given in [14] is a special case of ours.

Meanwhile, the solutions for Tzitzeica-Dodd-Bullough equation given by (30) are also different from previous results in [7, 13, 14].

5. Conclusion

In this paper, we propose a new method to solve Dodd-Bullough equations under the assumption that 𝜙^′=𝑓(𝑒^𝜙). A new family of new explicit travelling-wave solutions for 𝜙_𝑢𝑣=𝑒^𝜙−𝑒^−2𝜙 is derived. Using the proposed method, many other forms of Dodd-Bullough equations, Dodd-Bullough-Mikhailov, Tzitzeica-Dodd-Bullough, and some other general forms can be dealt with to get a new family of new explicit travelling-wave solutions. Compared with previous works, we can find that our solutions for Dodd-Bullough equations are different, and some previous results are special cases of ours.

Acknowledgments

This research is supported by National Natural Science Foundation of China (61171014, 61202436) and the Fundamental Research Funds for the Central Universities.