Abundant Exact Solition-Like Solutions to the Generalized Bretherton Equation with Arbitrary Constants

Abstract

The Riccati equation is employed to construct exact travelling wave solutions to the generalized Bretherton equation. Taking full advantage of the Riccati equation which has more new solutions, abundant new multiple solition-like solutions are obtained for the generalized Bretherton equation.

1. Introduction

Nonlinear partial differential equations (PDE) are widely chosen to describe complex phenomena in physics sciences. Searching for exact solutions to nonlinear differential equations plays more and more important role in nonlinear science. Recently, various direct methods have been proposed, such as the tanh-function method [1, 2], the Jacobi elliptic function expansion method [3, 4], the F-expansion [5–8], sine-cosine method [9, 10], and the homogeneous balance method [11–13]. Among them, the tanh-function method is improved continuously [14–17] as one of the most effectively straightforward methods for constructing exact solutions to PDEs. In the paper an extended tanh-function method is used to solve the generalized Bretherton equation with arbitrary constants.

In [18], Bretherton introduced the partial differential equation𝑢_𝑡𝑡+𝑢_𝑥𝑥+𝑢_{𝑥𝑥𝑥𝑥}+𝑢−𝑢²=0 (1a) in time and one spatial dimension as a model of a dispersive wave system to study the resonant nonlinear interaction between three liner models. The modified Bretherton equation 𝑢_𝑡𝑡+𝑢_𝑥𝑥+𝑢_{𝑥𝑥𝑥𝑥}+𝑢−𝑢³=0 (1b) was studied by Kudryashov [19], Kudryashov et al. [20], and Berloff and Howard [21], and its travelling wave solutions were obtained.

Our aim in this paper is to investigate multiple soliton-like solutions to the generalized Bretherton equation in [22] by using the solutions to the Riccati equation: 𝑢_𝑡𝑡+𝛼𝑢_𝑥𝑥+𝛽𝑢_{𝑥𝑥𝑥𝑥}+𝛿𝑢+𝛾𝑢³=0. (1c)

2. Multiple Soliton-Like Solutions to the Generalized Bretherton Equation

We assume the travelling wave variable 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑥−𝑉𝑡, (2) where 𝑉 is the speed of the travelling wave.

Making use of the travelling wave transformation (2), (1c) is converted into an ordinary differential equation (ODE) for 𝑢=𝑢(𝜉) as follows: 𝑉²+𝛼𝑢^′′+𝛽𝑢⁽⁴⁾+𝛿𝑢+𝛾𝑢³=0. (3) We assume that the solutions to (3) can be expressed in the form 𝑢(𝜉)=^𝑀_𝑖=−𝑀𝑎_𝑖𝜙^𝑖, (4) where 𝜙 is a solution of the Riccati equation, 𝜙^′=𝐴+𝐵𝜙+𝐶𝜙², (5) where 𝑎_𝑖(𝑖=0,±1,±2,…,±𝑀)𝐴,𝐵, and 𝐶 are constants to be determined later, and either 𝑎_−𝑀 or 𝑎_𝑀 can be zero, but they cannot be zero together.

Substituting (4) into (3) together with (5) and considering the homogeneous balance between the highest-order derivative 𝑢⁽⁴⁾ and the nonlinear term 𝑢³, we obtain 𝑀=2. Thus the solution to (3) takes the following form: 𝑢(𝜉)=𝑎₂𝜙²+𝑎₁𝜙+𝑎₀+𝑎₋₁𝜙⁻¹+𝑎₋₂𝜙⁻². (6) Substituting (6) with (5) into (3) and collecting all the terms of the same power of 𝜙, the left-hand side of (3) is converted into another polynomial of 𝜙. Setting the coefficients of 𝜙^𝑖(𝑖=0,±1,±2) to zero yields a set of algebraic equations 𝛾𝑎³₋₂−120𝛽𝑎₋₂𝐴⁴=0,3𝛾𝑎₋₁𝑎²₋₂+24𝛽𝑎₋₁𝐴⁴+96𝛽𝑎₋₂𝐵𝐴³=0,330𝛽𝑎₋₂𝐵²𝐴²+60𝛽𝑎₋₁𝐵𝐴³+3𝛾𝑎²₋₁𝑎₋₂+6𝛼𝑎₋₂𝐴²+3𝛾𝑎₀𝑎²₋₂+6𝑉²𝑎₋₂𝐴²=0,40𝛽𝑎₋₁𝐴³𝐶+3𝛾𝑎₁𝑎²₋₂+2𝑉²𝑎₋₁𝐴²+10𝑉²𝑎₋₂𝐵𝐴+2𝛼𝑎₋₁𝐴²+50𝛽𝑎₋₁𝐵²𝐴²+6𝛾𝑎₋₁𝑎₀𝑎₋₂+130𝛽𝐵³𝐴+𝛾𝑎³₋₁+440𝛽𝑎₋₂𝐶𝐵𝐴²+10𝛼𝑎₋₂𝐵𝐴=0,6𝛾𝑎₁𝑎₋₁𝑎₋₂+60𝛽𝑎₋₁𝐴²𝐶𝐵+3𝛾𝑎₂𝑎²₋₂+𝛿𝑎₋₂+3𝛾𝑎²₋₁𝑎₀+3𝛼𝑎₋₁𝐵𝐴+8𝑉²𝑎₋₂𝐶𝐴+8𝛼𝑎₋₂𝐶𝐴+136𝛽𝑎₋₂𝐶²𝐴²+3𝛾𝑎²₀𝑎₋₂+4𝛼𝑎₋₂𝐵²+4𝑉²𝑎₋₂𝐵²+16𝛽𝑎₋₂𝐵⁴+3𝑉²𝑎₋₁𝐵𝐴+15𝛽𝑎₋₁𝐵³𝐴+232𝛽𝑎₋₂𝐶𝐵²𝐴=0,16𝛽𝑎₋₁𝐴²𝐶²+6𝑉²𝑎₋₂𝐶𝐵+2𝛼𝑎₋₁𝐴𝐶+𝛽𝑎₋₁𝐵⁴+𝛿𝑎₋₁+6𝛾𝑎₂𝑎₋₁𝑎₋₂+30𝛽𝑎₋₂𝐶𝐵³+3𝛾𝑎₋₁𝑎²₀+2𝑉²𝑎₋₁𝐴𝐶+6𝛾𝑎₁𝑎₀𝑎₋₂+𝛼𝑎₋₁𝐵²+120𝛽𝑎₋₂𝐶²𝐴𝐵+6𝛼𝑎₋₂𝐶𝐵+22𝛽𝑎₋₁𝐴𝐶𝐵²+𝑎₋₁𝑉²𝐵²+3𝛾𝑎₁𝑎²₋₁=0,2𝛼𝑎₋₂𝐶²+2𝛼𝑎₂𝐴²+2𝑎₂𝑉²𝐴²+2𝑎₋₂𝐶²+3𝛾𝑎²₁𝑎₋₂+3𝛾𝑎₂𝑎²₋₁+8𝛽𝑎₋₁𝐴𝐶²𝐵+𝛿𝑎₀+𝑎₁𝑉²𝐵𝐴+𝑉²𝑎₋₁𝐵𝐶+𝛼𝑎₁𝐵𝐴+𝛼𝑎₋₁𝐵𝐶+16𝛽𝑎₂𝐶𝐴³+𝛽𝑎₁𝐵³𝐴+14𝛽𝑎₂𝐵²𝐴²+16𝛽𝑎₋₂𝐶³𝐴+𝛽𝑎₋₁𝐵³𝐶+14𝛽𝑎₋₂𝐶²𝐵²+6𝛾𝑎₂𝑎₀𝑎₋₂+6𝛾𝑎₁𝑎₋₁𝑎₀+8𝛽𝛼₁𝐶𝐵𝐴²+𝛾𝑎³₀=0,6𝛾𝑎₂𝑎₋₁𝑎₀+𝛼𝑎₋₁𝐵²+6𝑎₂𝑉²𝐵𝐴+𝑎₁𝑉²𝐵²+3𝛾𝑎₁𝑎²₀+2𝑎₁𝑉²𝐶𝐴+2𝛼𝑎₁𝐶𝐴+3𝛾𝑎²₁𝑎₋₁+𝛿𝑎₁+6𝛼𝑎₂𝐵𝐴+22𝛽𝑎₁𝐶𝐵²𝐴+30𝛽𝑎₂𝐵³𝐴+16𝛽𝑎₁𝐶²𝐴²+𝛽𝑎₁𝐵⁴+6𝛾𝑎₂𝑎₁𝑎₋₂+120𝛽𝑎₂𝐶𝐵𝐴²=0,4𝛼𝑎₂𝐵²+8𝛼𝑎₂𝐶𝐴+136𝛽𝑎₂𝐶²𝐴²+16𝛽𝑎₂𝐵⁴+3𝛾𝑎₂𝑎²₀+𝛿𝑎₂+3𝑉²𝐵𝐶+3𝛼𝑎₁𝐶𝐵+15𝛽𝑎₁𝐶𝐵³+4𝑎₂𝑉²𝐵²+60𝛽𝑎₁𝐴𝐵𝐶²+8𝑎₂𝑉²𝐶𝐴+3𝛾𝑎²₁𝑎₀+23𝛽𝑎₂𝐶𝐵²𝐴+3𝛾𝑎²₂𝑎₋₂+6𝛾𝑎₂𝑎₁𝑎₋₁=0,40𝛽𝑎₁𝐶³𝐴+2𝛼𝑎₁𝐶²+50𝛽𝑎₁𝐶²𝐵²+10𝛼𝑎₂𝐶𝐵+6𝛾𝑎₂𝑎₁𝑎₀+𝛾𝑎³₁+10𝑎₂𝑉²𝐵𝐶+3𝛾𝑎²₂𝑎₋₁+130𝛽𝑎₂𝐶𝐵³+440𝛽𝑎₂𝐶²𝐵𝐴+2𝑎₁𝑉²𝐶²=0,3𝛾𝑎²₂𝑎₀+3𝛾𝑎²₁𝑎₂+6𝑎₂𝑉²𝐶²+60𝛽𝑎₁𝐶³𝐵+240𝛽𝑎₂𝐶³𝐴+330𝛽𝑎₂𝐶²𝐵²+6𝛼𝑎₂𝐶²=0,3𝛾𝑎²₂𝑎₁+24𝛽𝑎₁𝐶⁴+336𝛽𝑎₂𝐶³𝐵=0,𝛾𝑎³₂+120𝛽𝑎₂𝐶⁴=0. (7) Solving (7) with the help of the symbolic computation software Maple, we obtain the following.

Case 1. One has 𝑎₋₁=0,𝑎₋₂=0,𝑎₂=2𝐶²−30𝛽𝛾,𝑎₁=2𝐵𝐶−30𝛽𝛾,𝑎₀=2𝐴𝐶−30𝛽𝛾,𝑉=±−𝛼−5𝐵²𝛽+20𝐶𝐴𝛽,𝛿=4𝛽−8𝐶𝐴𝐵²+𝐵⁴+16𝐴²𝐶². (8a)

Case 2. One has 𝑎₋₁=0,𝑎₋₂=0,𝑎₂=−2𝐶²−30𝛽𝛾,𝑎₁=−2𝐵𝐶−30𝛽𝛾,𝑎₀=−2𝐴𝐶−30𝛽𝛾,𝑉=±−𝛼−5𝐵²𝛽+20𝐶𝐴𝛽,𝛿=4𝛽−8𝐶𝐴𝐵²+𝐵⁴+16𝐴²𝐶², (8b) where 𝐴, 𝐵, and 𝐶 are arbitrary constants, but 𝐶 cannot be zero.

Case 3. One has 𝑎₋₁=0,𝑎₂=0,𝑎₁=0,𝑎₋₂=2𝐴²30𝛽𝛾,𝑎₀=2𝐴𝐶30𝛽𝛾,𝐵=0,𝑉=±−𝛼−60𝐶𝐴𝛽,𝛿=−16𝛽𝐴²𝐶². (8c)

Case 4. One has 𝑎₋₁=0,𝑎₂=0,𝑎₁=0,𝑎₋₂=2𝐴²30𝛽𝛾,𝑎₀√=−−15±16515𝐶𝐴30𝛽𝛾,𝐵=0,𝑉=𝜀−𝛼+−30±2√165𝐶𝐴𝛽,𝛿=−4𝐴²𝐶²𝛽13±√165, (8d) where 𝐴,𝐵, and 𝐶 are arbitrary constants, but 𝐴 cannot be zero.

Case 5. One has 𝑎₋₁=0,𝑎₂=0,𝑎₁=0,𝑎₋₂=−2𝐴²30𝛽𝛾,𝑎₀=−2𝐴𝐶30𝛽𝛾,𝐵=0,𝑉=±−𝛼−60𝐶𝐴𝛽,𝛿=−16𝐵𝐴²𝐶². (8e)

Case 6. One has 𝑎₋₁=0,𝑎₂=0,𝑎₁=0,𝑎₋₂=−2𝐴²30𝛽𝛾,𝑎₀=−15±√165𝐶𝐴1530𝛽𝛾,𝐵=0,𝑉=𝜀−𝛼+−30±2√165𝐶𝐴𝛽,𝛿=−4𝐴²𝐶²𝛽13±√165, (8f) where 𝐴,𝐵, and 𝐶 are arbitrary constants, while 𝐶 cannot be zero in Cases 1 and 2 and 𝐴 cannot be zero in Cases 3–6. 𝜀 is an arbitrary element of {−1,1}.

Substituting ((8a), (8b), (8c), (8d), (8e), (8f)) into (6) respectively and taking advantage of solitions to (5), we can find the following solutions which contain multiple solition-like and triangular periodic solutions for the generalized Bretherton equation.

When 𝛿=4𝛽,𝑢_1𝑎=𝜃−15𝛽2𝛾coth𝑥+𝜀−𝛼−5𝛽𝑡±csch𝑥+𝜀−𝛼−5𝛽𝑡²−𝜃−15𝛽,𝑢2𝛾_1𝑏=𝜃−15𝛽2𝛾tanh𝑥+𝜀−𝛼−5𝛽𝑡1±sech𝑥+𝜀−𝛼−5𝛽𝑡²−𝜃−15𝛽,𝑢2𝛾_1𝑐=𝜃−15𝛽2𝛾tan𝑥+𝜀−𝛼+5𝛽𝑡±sec𝑥+𝜀−𝛼+5𝛽𝑡²+𝜃−15𝛽,𝑢2𝛾_1𝑑=𝜃−15𝛽2𝛾csc𝑥+𝜀−𝛼+5𝛽𝑡−cot𝑥+𝜀−𝛼+5𝛽𝑡²+𝜃−15𝛽,𝑢2𝛾_1𝑒=𝜃−15𝛽2𝛾tan𝑥+𝜀−𝛼+5𝛽𝑡1±sec𝑥+𝜀−𝛼+5𝛽𝑡²+𝜃−15𝛽,𝑢2𝛾_1𝑓=𝜃−15𝛽2𝛾cot𝑥+𝜀−𝛼+5𝛽𝑡±csc𝑥+𝜀−𝛼+5𝛽𝑡²+𝜃−15𝛽,𝑢2𝛾_1𝑔=𝜃−15𝛽2𝛾sec𝑥+𝜀−𝛼+5𝛽𝑡−tan𝑥+𝜀−𝛼+5𝛽𝑡²+𝜃−15𝛽,𝑢2𝛾_1ℎ=𝜃−15𝛽2𝛾cot𝑥+𝜀−𝛼+5𝛽𝑡1±csc𝑥+𝜀−𝛼+5𝛽𝑡²+𝜃−15𝛽,2𝛾 (9a)(9b)(9c)(9d)(9e)(9f)(9g)(9h) when 𝛿=64𝛽,

𝑢_2𝑎=2𝜃−30𝛽𝛾tanh²𝑥±−𝛼−20𝛽𝑡−2𝜃−30𝛽𝛾,𝑢_2𝑏=2𝜃−30𝛽𝛾coth²𝑥±−𝛼−20𝛽𝑡−2𝜃−30𝛽𝛾,𝑢_2𝑐=2𝜃−30𝛽𝛾tan²𝑥±−𝛼+20𝛽𝑡+2𝜃−30𝛽𝛾,𝑢_2𝑑=2𝜃−30𝛽𝛾cot²𝑥±−𝛼+20𝛽𝑡+2𝜃−30𝛽𝛾,𝑢_2𝑒=8𝜃−30𝛽𝛾tan𝑥+𝜀−𝛼+20𝛽𝑡1±tan𝑥+𝜀−𝛼+20𝛽𝑡²∓8𝜃−30𝛽𝛾tan𝑥+𝜀−𝛼+20𝛽𝑡1±tan𝑥+𝜀−𝛼+20𝛽𝑡+4𝜃−30𝛽𝛾,𝑢_2𝑓=8𝜃−30𝛽𝛾cot𝑥+𝜀−𝛼+20𝛽𝑡1±cot𝑥+𝜀−𝛼+20𝛽𝑡²∓8𝜃−30𝛽𝛾cot𝑥+𝜀−𝛼+20𝛽𝑡1±cot𝑥+𝜀−𝛼+20𝛽𝑡+4𝜃−30𝛽𝛾, (10a)(10b)(10c)(10d)(10e)(10f) when 𝛿=1024𝛽,

𝑢_3𝑎=32𝜃−30𝛽𝛾tanh𝑥±−𝛼−80𝛽𝑡1+tanh²𝑥±−𝛼−80𝛽𝑡²−8𝜃−30𝛽𝛾,𝑢_3𝑏=32𝜃−30𝛽𝛾tan𝑥±−𝛼+80𝛽𝑡1−tan²𝑥±−𝛼+80𝛽𝑡²+8𝜃−30𝛽𝛾,𝑢_3𝑐=32𝜃−30𝛽𝛾cot𝑥±−𝛼+80𝛽𝑡1−cot²𝑥±−𝛼+80𝛽𝑡²+8𝜃−30𝛽𝛾, (11a)(11b)(11c) when 𝛿=0,𝑢₄=±2−30𝛽/𝛾𝑥+𝜀√−𝛼𝑡²+𝑐₀². (12) When 𝛿=−𝛽,𝑢_5𝑎=𝜃15𝛽2𝛾coth𝑥+𝜀−𝛼+15𝛽𝑡±csch𝑥+𝜀−𝛼+15𝛽𝑡⁻²−𝜃15𝛽,𝑢2𝛾_5𝑏=𝜃15𝛽2𝛾tanh𝑥+𝜀−𝛼+15𝛽𝑡1±sech𝑥+𝜀−𝛼+15𝛽𝑡⁻²−𝜃15𝛽,𝑢2𝛾_5𝑐=𝜃15𝛽2𝛾tan𝑥+𝜀−𝛼−15𝛽𝑡±sec𝑥+𝜀−𝛼−15𝛽𝑡⁻²+𝜃15𝛽,𝑢2𝛾_5𝑑=𝜃15𝛽2𝛾tan𝑥+𝜀−𝛼−15𝛽𝑡1±sec𝑥+𝜀−𝛼−15𝛽𝑡⁻²+𝜃15𝛽,𝑢2𝛾_5𝑒=𝜃15𝛽2𝛾cot𝑥+𝜀−𝛼−15𝛽𝑡±csc𝑥+𝜀−𝛼−15𝛽𝑡⁻²+𝜃15𝛽,𝑢2𝛾_5𝑓=𝜃15𝛽2𝛾sec𝑥+𝜀−𝛼−15𝛽𝑡−tan𝑥+𝜀−𝛼−15𝛽𝑡⁻²+𝜃15𝛽,𝑢2𝛾_5𝑔=𝜃15𝛽2𝛾cot𝑥+𝜀−𝛼−15𝛽𝑡1±csc𝑥+𝜀−𝛼−15𝛽𝑡⁻²+𝜃−15𝛽,2𝛾 (13a)(13b)(13c)(13d)(13e)(13f)(13g) when 𝛿=−16𝛽,

𝑢_6𝑎=2𝜃30𝛽𝛾tanh⁻²𝑥±−𝛼+60𝛽𝑡−2𝜃30𝛽𝛾,𝑢_6𝑏=2𝜃30𝛽𝛾coth⁻²𝑥±−𝛼+60𝛽𝑡−2𝜃30𝛽𝛾,𝑢_6𝑐=2𝜃30𝛽𝛾tan⁻²𝑥±−𝛼−60𝛽𝑡+2𝜃30𝛽𝛾,𝑢_6𝑑=2𝜃30𝛽𝛾cot⁻²𝑥±−𝛼−60𝛽𝑡+2𝜃30𝛽𝛾, (14a)(14b)(14c)(14d) when 𝛿=−64𝛽,

𝑢_7𝑎=2𝜃30𝛽𝛾tanh𝑥±−𝛼−240𝛽𝑡1−tanh²𝑥±−𝛼−240𝛽𝑡⁻²+8𝜃30𝛽𝛾,𝑢_7𝑏=2𝜃30𝛽𝛾tanh𝑥±−𝛼+240𝛽𝑡1+tanh²𝑥±−𝛼+240𝛽𝑡⁻²−8𝜃30𝛽𝛾,𝑢_7𝑐=2𝜃30𝛽𝛾cot𝑥±−𝛼−240𝛽𝑡1−cot²𝑥±−𝛼−240𝛽𝑡⁻²+8𝜃30𝛽𝛾, (15a)(15b)(15c) when 𝛿=−(13±√165)𝛽,

𝑢_8𝑎=𝜃15𝛽×2𝛾coth𝑥+𝜀1−𝛼−4−30+2𝜎√165𝛽𝑡±csch𝑥+𝜀1−𝛼−4−30+2𝜎√165𝛽𝑡⁻²+14𝜃30𝛽𝛾√−1+𝜎165,𝑢15_8𝑏=𝜃15𝛽×2𝛾tanh𝑥+𝜀−𝛼−(1/4)−30+2𝜎√165𝛽𝑡1±sech𝑥+𝜀−𝛼−(1/4)−30+2𝜎√165𝛽𝑡⁻²+14𝜃30𝛽𝛾√−1+𝜎165,𝑢15_8𝑐=𝜃15𝛽2𝛾tan𝑥+𝜀1−𝛼+4−30+2𝜎√165𝛽𝑡±sec𝑥+𝜀1−𝛼+4−30+2𝜎√165𝛽𝑡⁻²−14𝜃30𝛽𝛾√−1+𝜎165,𝑢15_8𝑑=𝜃15𝛽×2𝛾csc𝑥+𝜀1−𝛼+4−30+2𝜎√165𝛽𝑡−cot𝑥+𝜀1−𝛼+4−30+2𝜎√165𝛽𝑡⁻²−14𝜃30𝛽𝛾√−1+𝜎165,𝑢15_8𝑒=𝜃15𝛽×2𝛾tan𝑥+𝜀−𝛼+(1/4)−30+2𝜎√165𝛽𝑡1±sec𝑥+𝜀−𝛼+(1/4)−30+2𝜎√165𝛽𝑡⁻²−14𝜃30𝛽𝛾√−1+𝜎165,𝑢15_8𝑓=𝜃15𝛽×2𝛾cot𝑥+𝜀1−𝛼+4−30+2𝜎√165𝛽𝑡±csc𝑥+𝜀1−𝛼+4−30+2𝜎√165𝛽𝑡⁻²−14𝜃30𝛽𝛾√−1+𝜎165,𝑢15_8𝑔=𝜃15𝛽×2𝛾sec𝑥+𝜀1−𝛼+4−30+2𝜎√165𝛽𝑡−tan𝑥+𝜀1−𝛼+4−30+2𝜎√165𝛽𝑡⁻²−14𝜃30𝛽𝛾√−1+𝜎165,𝑢15_8ℎ=𝜃15𝛽×2𝛾cot𝑥+𝜀−𝛼+(1/4)−30+2𝜎√165𝛽𝑡1±csc𝑥+𝜀−𝛼+(1/4)−30+2𝜎√165𝛽𝑡⁻²−14𝜃30𝛽𝛾√−1+𝜎165,15 (16a)(16b)(16c)(16d)(16e)(16f)(16g)(16h) when 𝛿=−4(13±√165)𝛽,

𝑢_9𝑎=2𝜃30𝛽𝛾tanh⁻²𝑥+𝜀−𝛼−−30+2𝜎√165𝛽𝑡+𝜃30𝛽𝛾√−1+𝜎165,𝑢15_9𝑏=2𝜃30𝛽𝛾coth⁻²𝑥+𝜀−𝛼−−30+2𝜎√165𝛽𝑡+𝜃30𝛽𝛾√−1+𝜎165,𝑢15_9𝑐=2𝜃30𝛽𝛾tan⁻²𝑥+𝜀−𝛼+−30+2𝜎√165𝛽𝑡−𝜃30𝛽𝛾√−1+𝜎165,𝑢15_9𝑑=2𝜃30𝛽𝛾cot⁻²𝑥+𝜀−𝛼+−30+2𝜎√165𝛽𝑡−𝜃30𝛽𝛾√−1+𝜎165,15 (17a)(17b)(17c)(17d) when 𝛿=−64(13±√165)𝛽,

𝑢_10𝑎=2𝜃30𝛽𝛾×tanh𝑥+𝜀−𝛼−4−30+2𝜎√165𝛽𝑡1+tanh²𝑥+𝜀−𝛼−4−30+2𝜎√165𝛽𝑡⁻²+4𝜃30𝛽𝛾√−1+𝜎165,𝑢15_10𝑏=2𝜃30𝛽𝛾×tanh𝑥+𝜀−𝛼+4−30+2𝜎√165𝛽𝑡1−tanh²𝑥+𝜀−𝛼+4−30+2𝜎√165𝛽𝑡⁻²−4𝜃30𝛽𝛾√−1+𝜎165,𝑢15_10𝑐=2𝜃30𝛽𝛾×cot𝑥+𝜀−𝛼+4−30+2𝜎√165𝛽𝑡1−cot²𝑥+𝜀−𝛼+4−30+2𝜎√165𝛽𝑡⁻²−4𝜃30𝛽𝛾√−1+𝜎165,15 (18a)(18b)(18c) where 𝜀,𝜃, and 𝜎 are arbitrary elements of {−1,1}.

3. Conclusion

In this paper, we have used solutions to the Riccati equation to solve the generalized Bretherton equation with arbitrary constants and obtained abundant new multiple solition-like and triangular periodic solutions. It is significant to observe practical denotation of the obtained solutions, so the obtained solutions involving arbitrary constants in this paper have potential applications in dispersive wave systems to research for resonant nonlinear interactions.

Acknowledgment

This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2010AL019).