Generalized Hyperbolic Function Solution to a Class of Nonlinear Schrödinger-Type Equations

Abstract

With the help of the generalized hyperbolic function, the subsidiary ordinary differential equation method is improved and proposed to construct exact traveling wave solutions of the nonlinear partial differential equations in a unified way. A class of nonlinear Schrödinger-type equations including the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Liu equation are investigated and the exact solutions are derived with the aid of the homogenous balance principle and generalized hyperbolic functions. We study the effect of the generalized hyperbolic function parameters p and q in the obtained solutions by using the computer simulation.

1. Introduction

The nonlinear Schrödinger (NLS) equation is a ubiquitous and significant model that naturally arises in many fields of physics, such as nonlinear optical systems, plasmas, fluid dynamics, and Bose-Einstein condensation. In the last three decades, great progress has been made on the construction of exact solutions of NLS equation. Many significant methods have been established by mathematicians and physicists to obtain special solutions of nonlinear partial differential equations (NLPDEs), including the inverse scattering method, Darboux transformation, Hirota’s bilinear method, homogeneous balance method, Jacobi elliptic function method, variational iteration method, the sine-cosine method, the (𝐺′/𝐺) expansion method, tanh-function method, F-expansion method, Lucas Riccati method, auxiliary equation method, algebraic method, and others [1–36]. The last five methods mentioned above belong to a class of method called subsidiary ordinary differential equation (sub-ODE) method. The key ideas of the sub-ODE method are that the travelling wave solutions of the complicated NLPDE can be expressed as a polynomial, the variable of which is one of the solutions of simple and solvable ODE that called the sub-ODE. The sub-ODEs which were often used are the Riccati equation, Jacobi elliptic equation, projective Riccati equations, and so forth. With the development of computer science, recently, the sub-ODEs with nonlinear terms of high order have attracted much attention [34, 35]. This is due to the availability of symbolic computation systems like Mathematica or Maple which enable us to perform the complex and tedious computation on computers. Recently, Sirendaoreji [21–23] introduced the auxiliary equation method by using the six degree first-order nonlinear differential equation, Wang et al. [24] proposed new sub-ODE equation involving an arbitrary positive power to construct exact travelling wave solutions of NLPDEs in a unified way. In this study, we improve the method presented by Wang et al. [24] and introduce some solutions for the sub-ODE equation in terms of the generalized hyperbolic functions (GHFs) [36].

The rest of this paper is organized as follows: in the following section, we introduce the improved sub-ODE method to construct exact solutions of some NLPDEs in terms of the GHFs. In Section 3, we apply this method to the generalized Zakharov system, the Rangwala-Rao equation and the Chen-Lee-Liu equation. Finally, we conclude the paper and give some futures and comments.

2. The Improved Subsidiary Ordinary Differential Equation Method

The main idea of this method is to express the solutions of NLPDEs as polynomials in the solution of sub-ODE involving an arbitrary positive power of dependent variable that the GHFs satisfy (see the appendix). Consider a given NLPDE𝐻𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑡,𝑢_𝑡𝑥,𝑢_𝑥𝑥,…=0. (2.1) The improved sub-ODE method for solving NLPDE (2.1) proceeds in the following four steps [36].

Step 1. We seek its traveling wave solution of (2.1) in the form 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑘𝑥−𝜔𝑡, (2.2) where 𝑘 and 𝜔 are constants to be determined later. Substituting (2.2) into (2.1) yields an ordinary differential equation 𝐻𝑢,𝑢^′,𝑢^′′,𝑢^′′′,…=0,𝑢′=𝑑𝑢𝑑𝜉,…, (2.3) where 𝐻 is a polynomial of 𝑢 and its various derivatives. If 𝐻 is not a polynomial of 𝑢 and its various derivatives, then we may use new variables 𝑣=𝑣(𝜉) which makes 𝐻 become polynomial of 𝑣 and its various derivatives.

Step 2. Suppose that 𝑢(𝜉) can be expressed by a finite power series of 𝐹(𝜉)𝑢(𝜉)=𝑎₀+^𝑛_𝑗=1𝑎_𝑗𝐹^𝑗(𝜉),𝑎_𝑛≠0, (2.4) where 𝑛 is a positive integer which can be determined by balancing the highest derivative term with the highest nonlinear term in (2.3) and 𝑎_𝑗(𝑗=0,1,…,𝑛)are some parameters to be determined. The function 𝐹(𝜉) satisfies the sub-ODE with an arbitrary positive power 𝐹^′2(𝜉)=𝐴𝐹²(𝜉)+𝐵𝐹^2+𝑟(𝜉)+𝐶𝐹^2+2𝑟(𝜉),𝑟>0, (2.5) where 𝐴,𝐵, and 𝐶 are parameters to be determined.

Step 3. Substituting (2.4) and (2.5) into the ODE (2.3), then the left-hand side of (2.3) can be converted into a polynomial in 𝐹(𝜉). Setting all coefficients of the polynomial to zero yields system of algebraic equations for 𝑎_𝑗(𝑗=0,1,…,𝑛), 𝑘,and 𝜔.

Step 4. Solving this system obtained in Step 3, then 𝑎_𝑗(𝑗=0,1,…,𝑛), 𝑘, and 𝜔 can be expressed by 𝐴,𝐵, and 𝐶. Substituting these results into (2.4), then the general formulae of the travelling wave solution of (2.1) can be obtained. Selecting the values of 𝐴,𝐵,𝐶, and the corresponding GHF solution 𝐹(𝜉) of (2.5) given bellow to obtain the exact solutions of (2.1). The definition and proprieties of the GHFs are given in the appendix.
We list various types of exact solutions of (2.5) as follows.

Case 1. If 𝐴>0,𝐵=2𝜎𝐴,and 𝐶=(𝜎²−𝑝𝑞)𝐴, then (2.5) admits the following positive solution: 𝐹(𝜉)=1cosh_𝑝𝑞𝑟√𝐴𝜉−𝜎^1/𝑟,𝜎<1. (2.6)

Case 2. If 𝐴>0,𝐵=2𝜎𝐴,and 𝐶=(𝜎²+𝑝𝑞)𝐴, then (2.5) admits the following positive solution: 𝐹(𝜉)=1sinh_𝑝𝑞𝑟√𝐴𝜉−𝜎^1/𝑟,𝜎<1. (2.7)

Case 3. If 𝐴>0,𝐵=−2√𝐴𝐶,and 𝐶>0, then (2.5) admits the following positive solution: 𝐹(𝜉)=12𝐴𝐶1±tanh_𝑝𝑞𝑟2√𝐴𝜉^1/𝑟,𝐹(𝜉)=12𝐴𝐶1±coth_𝑝𝑞𝑟2√𝐴𝜉^1/𝑟. (2.8)

Case 4. If 𝐴=0,𝐵=4/𝑟², and 𝐶=−4𝜎/𝑟², then (2.5) admits the following positive solution: 𝐹(𝜉)=1𝜉²+𝜎^1/𝑟,𝜎>0. (2.9)

3. Applications

In the following we use the improved sub-ODE method to seek exact traveling wave solutions of the class of nonlinear Schrödinger-type equations which are of interest in plasma physics, wave propagation in nonlinear optical fibers, Ginzburg-Landau theory of superconductivity, and so forth.

3.1. Generalized Zakharov System

In the interaction of laser-plasma the system of Zakharov equation plays an important role. This system has wide interest and attention for many scientists. Let us consider the generalized Zakharov system [18]:𝑢_𝑡𝑡−𝑐²_𝑠𝑢_𝑥𝑥=𝛽|𝐸|²_𝑥𝑥,𝑖𝐸_𝑡+𝛼𝐸_𝑥𝑥−𝛿₁𝑢𝐸+𝛿₂|𝐸|²𝐸+𝛿₃|𝐸|⁴𝐸=0. (3.1) When 𝛿₂=𝛿₃=0, the generalized Zakharov system reduce to the famous Zakharov system which describe the propagation Langmuir waves in plasmas. The real unknown function 𝑢(𝑥,𝑡) is the fluctuation in the ion density about its equilibrium value, and the complex unknown function 𝐸(𝑥,𝑡) is the slowly varying envelope of highly oscillatory electron field. The parameters 𝛿₁,𝛿₂,𝛿₃and𝑐_𝑠are real numbers, where𝑐_𝑠is proportional to the ion acoustic speed (or electron sound speed). Here we seek its traveling wave solution in the form𝐸(𝑥,𝑡)=𝐻(𝜉)𝑒^{𝑖(𝑘𝑥−𝜔𝑡)},𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑥−𝑐𝑡, (3.2) where 𝑘,𝜔, and 𝑐are constants and 𝐻(𝜉) is real function. Therefore, system (3.1) is reduced to𝑐²−𝑐²_𝑠𝑢^′′=𝛽(𝐻²)^′′,𝛼𝐻^′′+𝑖(2𝛼𝑘−𝑐)𝐻^′+𝜔−𝛼𝑘²𝐻−𝛿₁𝑢𝐻+𝛿₂𝐻³+𝛿₃𝐻⁵=0. (3.3) Integrating the first equation of (3.3) with respect to 𝜉 and taking the integration constants to zero yields𝛽𝑢=𝑐²−𝑐²_𝑠𝐻²,𝑐²−𝑐²_𝑠≠0. (3.4) Substituting (3.4) into (3.3) results in𝐻^′′+1𝛼𝜔−𝛼𝑘²𝐻+𝛿₂−𝛿₁𝛽𝑐²−𝑐²_𝑠𝐻³+𝛿₃𝐻⁵=0,𝑐=2𝛼𝑘,𝛼≠0. (3.5) Now, by balancing the higher order derivative term 𝐻^′′ to the higher power nonlinear term 𝐻⁵in (3.5), we have 𝑛+2𝑟=5𝑛 which gives2𝑟=4𝑛. Since 𝑛 must be a positive integer, thus 𝑟=2,4,6,… this gives 𝑛=1,2,3,…. For simplicity, we take 𝑟=2 and 𝑛=1. We suppose that𝐻(𝜉)=𝑎₀+𝑎₁𝐹(𝜉). (3.6) Substituting (3.6) with (2.5) into (3.5) and equating each of the coefficients of 𝐹(𝜉) to zero, we obtain system of algebraic equations. Solving this system by using Maple, yields𝑎₀=0,𝑐=2𝛼𝑘,𝜔=𝛼𝑘²−𝐴,𝑎₁=±3𝐶2𝛿₃𝐵𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠. (3.7) Now based on the solutions of (2.5), one can obtain new types of solitary wave solution of the generalized Zakharov system. The general formulae take the form𝑢(𝑥,𝑡)=3𝐶𝛽2𝛿₃𝐵4𝛼²𝑘²−𝑐²_𝑠𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠𝐹²(),𝐸()𝑥−2𝛼𝑘𝑡𝑥,𝑡=±3𝐶2𝛿₃𝐵𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠𝐹(𝑥−2𝛼𝑘𝑡)𝑒^{𝑖(𝑘𝑥−𝜔𝑡)}. (3.8) By selecting the special values of the 𝐴,𝐵,𝐶and the corresponding function𝐹(𝜉), we have the following solutions of the generalized Zakharov system (3.1):𝑢₁=3𝛽𝜎²−𝑝𝑞4𝛿₃𝜎4𝛼²𝑘²−𝑐²_𝑠𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠1cosh_𝑝𝑞2√𝐴(),𝐸𝑥−2𝛼𝑘𝑡−𝜎₁=±3𝜎²−𝑝𝑞4𝛿₃𝜎𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠1cosh_𝑝𝑞2√𝐴()𝑒𝑥−2𝛼𝑘𝑡−𝜎^{𝑖(𝑘𝑥−𝜔𝑡)},𝑢₂=3𝛽𝜎²+𝑝𝑞4𝛿₃𝜎4𝛼²𝑘²−𝑐²_𝑠𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠1sinh_𝑝𝑞2√𝐴(,𝐸𝑥−2𝛼𝑘𝑡)−𝜎₂=±3𝜎²+𝑝𝑞4𝛿₃𝜎𝛿₂−𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠1sinh_𝑝𝑞2√𝐴(𝑥−2𝛼𝑘𝑡)−𝜎^1/2𝑒^{𝑖(𝑘𝑥−𝜔𝑡)},𝑢₃=3𝛽𝐶8𝛿₃𝐴4𝛼²𝑘²−𝑐²_𝑠𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠−𝛿₂1±tanh_𝑝𝑞√𝐴(),𝐸𝑥−2𝛼𝑘𝑡₃1=±23𝐶2𝛿₃𝐴𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠−𝛿₂1±tanh_𝑝𝑞√𝐴(𝑥−2𝛼𝑘𝑡)𝑒^{𝑖(𝑘𝑥−𝜔𝑡)},𝑢₄=3𝛽𝐶8𝛿₃𝐴4𝛼²𝑘²−𝑐²_𝑠𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠−𝛿₂1±coth_𝑝𝑞√𝐴(),𝐸𝑥−2𝛼𝑘𝑡₄1=±23𝐶2𝛿₃𝐴𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠−𝛿₂1±coth_𝑝𝑞√𝐴(𝑥−2𝛼𝑘𝑡)𝑒^{𝑖(𝑘𝑥−𝜔𝑡)},𝑢₅=3𝜎𝛽2𝛿₃4𝛼²𝑘²−𝑐²_𝑠𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠−𝛿₂1(𝑥−2𝛼𝑘𝑡)²,𝐸+𝜎₅=±32𝛿₃𝛽𝛿₁4𝛼²𝑘²−𝑐²_𝑠−𝛿₂1(𝑥−2𝛼𝑘𝑡)²𝑒+𝜎^{𝑖(𝑘𝑥−𝜔𝑡)}. (3.9)(3.10)(3.11)(3.12)(3.13) In order to understand the significance of these solutions expressed by (3.9)–(3.13), the main features of them are investigated by using direct computer simulations with the accuracy as high as 10⁻⁹. We study the effect of the parameters 𝑝 and 𝑞 of the GHF in the solution 𝑢₁ given by the first equation of (3.9) by choosing different values of the parameters 𝑝 and 𝑞, for 𝐴=𝛿₁=𝛿₂=𝛿₃=𝛼=𝛽=𝑘=1, 𝜎=0.9, and 𝑐_𝑠=0.4, by means of Figures 1, 2, and 3 over same region {(𝑥,𝑡)∶‖𝑥‖≤5,‖𝑡‖≤5}. From Figures 1, 2 and 3, we see that the solution 𝑢₁ describes dark soliton solution (increase, decrease, and have singularity with the change of the two parameters 𝑝 and 𝑞). Also, we discuss 𝑢₂ given by the first equation of (3.10), by taking the same value of the parameters by means of Figures 4, 5, and 6 over the region {(𝑥,𝑡)∶‖𝑥‖≤5,‖𝑡‖≤5}. From Figures 4, 5, and 6, we see that the solution 𝑢₂ describes singular wave solution changed with different value of the two parameters 𝑝 and 𝑞. Of course, we can plot the other figures of the exact solutions of (3.1), we omit them here for convenience. As a result, we find that the parameters 𝑝 and 𝑞 affect the solution structure.

(a)

(b)

(c)

(d)

(e)

Figure 1 

Evolutional behaviour of 𝑢1 describes dark soliton: (a) 𝑝=𝑞=1;(b) 𝑝=1/32,𝑞=1/33, (singular wave solution); (c) 𝑝=23,𝑞=32; (d) 𝑝=1/13,𝑞=11; (e) 𝑝=13,𝑞=1/11.

(a)

(b)

(c)

Figure 2 

Evolutional behaviour of 𝑢1 describes dark soliton: (a) 𝑝=1,𝑞=1/32 (singular wave solution); (b) 𝑝=1,𝑞=3; (c) 𝑝=1,𝑞=27.

(a)

(b)

(c)

Figure 3 

Evolutional behaviour of 𝑢1 describes dark soliton: (a) 𝑝=1/32,𝑞=1; (singular wave solution); (b) 𝑝=3,𝑞=1; (c) 𝑝=27,𝑞=1.

(a)

(b)

(c)

(d)

(e)

Figure 4 

Evolutional behaviour of 𝑢2 describes singular wave solution: (a) 𝑝=𝑞=1;(b) 𝑝=1/32,𝑞=1/33; (c) 𝑝=23,𝑞=32; (d) 𝑝=1/13,𝑞=11; (e) 𝑝=13,𝑞=1/11.

(a)

(b)

(c)

Figure 5 

Evolutional behaviour of 𝑢2describes singular wave solution: (a) 𝑝=1,𝑞=1/32; (b) 𝑝=1,𝑞=3;(c) 𝑝=1,𝑞=27.

(a)

(b)

(c)

Figure 6 

Evolutional behaviour of 𝑢2describes singular wave solution: (a) 𝑝=1/32,𝑞=1;(b) 𝑝=3,𝑞=1; (c) 𝑝=27,𝑞=1.

3.2. Rangwala-Rao Equation

The Rangwala-Rao equation [19]𝑢_𝑥𝑡−𝛽₁𝑢_𝑥𝑥+𝑢+𝑖𝑇𝛽₂|𝑢|²𝑢_𝑥=0,𝑇=±1, (3.14) where 𝛽₁and 𝛽₂ are real constants. Rangwala and Rao introduced (3.14) as the integrability condition when they studied the mixed derivative nonlinear Schrödinger equations and looked for the Bäckland transformation and solitary wave solutions.

Suppose the exact solutions of (3.14) is of the form𝑢(𝑥,𝑡)=𝑒^−𝜔𝑡𝑒^{𝑖𝜓(𝑥−𝑐𝑡)}𝐻(𝑥−𝑐𝑡), (3.15) where 𝜔, 𝑐 are constants determined later and 𝜓,𝐻 are undetermined real functions with one variable only. Set the relation of 𝜓,𝐻 as𝜓^′(𝜉)=𝜔2𝑐+𝛽₁+𝑇𝛽₂4𝑐+𝛽₁𝐻²(𝜉)𝑑,′=𝑑𝜉,𝜉=𝑥−𝑐𝑡, (3.16) substituting (3.15) with (3.16) into (3.14) simultaneously yields𝐻^′′−4𝑐+𝛽₁−𝜔²4𝑐+𝛽₁²𝐻−𝑇𝛽₂𝜔2𝑐+𝛽₁²𝐻³+3𝑇²𝛽²₂16𝑐+𝛽₁²𝐻⁵=0. (3.17) Applying the homogeneous balance principle, to (3.17) we have 𝑟=2,4,6,… this gives 𝑛=1,2,3,…. For simplicity, we suppose𝐻(𝜉)=𝑎₀+𝑎₁𝐹(𝜉). (3.18) Substituting (3.18) with (2.5) into (3.17) and equating each of the coefficients of 𝐹(𝜉) to zero, we obtain system of algebraic equations. Solving this system with the aid of Maple, we obtain the following solution:𝑎₀=0,𝜔=±2𝑐+𝛽₁1−𝑐+𝛽₁𝐴,𝑎₁=±−8𝐶𝜔3𝑇𝐵𝛽₂. (3.19) The general formulae of the solution of Rangwala-Rao equation𝑢(𝑥,𝑡)=±−8𝐶𝜔3𝑇𝐵𝛽₂𝐹(𝑥−𝑐𝑡)𝑒^{−𝑖𝜔𝑡}𝑒^{𝑖𝜓(𝑥−𝑐𝑡)}, (3.20) with 𝜓(𝜉)=𝜔/(6𝐵(𝑐+𝛽₁))∫[3𝐵+4𝐶𝐹²(𝜉)]𝑑𝜉,𝜔=±2(𝑐+𝛽₁)[1−(𝑐+𝛽₁)𝐴]. By selecting the special values of the 𝐴,𝐵,𝐶and the corresponding function𝐹(𝜉)yields𝑢₁(𝑥,𝑡)²4𝜔=−𝜎−𝑝𝑞3𝑇𝜎𝛽₂cosh_𝑝𝑞2√𝐴(𝑥−𝑐𝑡)−𝜎, (3.21)𝑢₂(𝑥,𝑡)²4𝜔=−𝜎+𝑝𝑞3𝑇𝜎𝛽₂sinh_𝑝𝑞2√𝐴(𝑥−𝑐𝑡)−𝜎. (3.22) We omitted the reminder solutions for simplicity. Now, we study the affect of the parameters 𝑝 and 𝑞 of the GHF in the intensity 𝑢₁ given by the (3.21) by choosing different values of the parameters 𝑝 and 𝑞, for 𝐴=𝛿=𝑐=𝛽₂=𝑇=1,𝜔=0.4, and 𝜎=0.9, by means of Figure 7 over the region {(𝑥,𝑡)∶‖𝑥‖≤5,‖𝑡‖≤5}. We have bright soliton which are increase, decrease and have singularity with the different choose of the two parameters 𝑝 and 𝑞. This means that the parameters 𝑝 and 𝑞 affect in the solution structure.

(a)

(b)

(c)

Figure 7 

Evolutional behaviour of the intensity |𝑢1|2describes the bright soliton: (a) 𝑝=1,𝑞=1;(b) 𝑝=1/332,𝑞=1/333 (singular wave solution); (c) 𝑝=2.132,𝑞=1.723.

3.3. Chen-Lee-Liu Equation

The Chen-Lee-Liu equation [19]𝑖𝑢_𝑡+𝑢_𝑥𝑥+𝑖𝛿|𝑢|²𝑢_𝑥=0, (3.23) were 𝛿is a real constant. Similar as before, we suppose that the exact solution of (3.23)is of the form𝑢(𝑥,𝑡)=𝑒^−𝜔𝑡𝑒^{𝑖𝜓(𝑥−𝑐𝑡)}𝐻(𝑥−𝑐𝑡). (3.24) Set the relation of 𝜓,𝐻 as𝜓^′(𝜉)=𝑐2+𝛿4𝐻²(𝜉)𝑑,′=𝑑𝜉,𝜉=𝑥−𝑐𝑡, (3.25) substituting (3.25) with (3.24) into (3.23) simultaneously yields𝐻^′′+144𝜔+𝑐²𝐻−𝑐𝛿2𝐻³+3𝛿²𝐻16⁵=0. (3.26) According the homogeneous balance principle, we have 𝑟=2,4,6,… and 𝑛=1,2,3,…, we let𝐻(𝜉)=𝑎₀+𝑎₁𝐹(𝜉). (3.27) Substituting (3.27) with (2.5) into (3.26) and equating each of the coefficients of 𝐹(𝜉) to zero, we obtain system of algebraic equations. Solving this system with the aid of Maple, we obtain the following solution:𝑎₀𝑐=0,𝜔=𝐴−²4,𝑎₁=±2−𝑐𝐶𝛿𝛽₂. (3.28) The general formulae of the solution of Chen-Lee-Liu equation𝑢(𝑥,𝑡)=±2−𝑐𝐶𝛿𝛽₂𝐹(𝑥−𝑐𝑡)𝑒^{−𝑖𝜔𝑡}𝑒^{𝑖𝜓(𝑥−𝑐𝑡)}, (3.29) with 𝜓(𝜉)=(𝑐/2𝐵)∫[𝐵+2𝐶𝐹²(𝜉)]𝑑𝜉,𝜔=𝐴−(𝑐²/4). By selecting the special values of the 𝐴,𝐵,𝐶and the corresponding function 𝐹(𝜉)yields𝑢₁(𝑥,𝑡)²=−2𝑐𝜎−𝑝𝑞𝛿𝜎cosh_𝑝𝑞2√𝐴(𝑥−𝑐𝑡)−𝜎, (3.30)𝑢₂(𝑥,𝑡)²=−2𝑐𝜎+𝑝𝑞𝛿𝜎sinh_𝑝𝑞2√𝐴(𝑥−𝑐𝑡)−𝜎. (3.31) We omitted the reminder solutions for simplicity. Now, we show the effect of the parameters 𝑝 and 𝑞 of the GHF in the intensity 𝑢₁ given by the (3.30) by choosing different values of the parameters 𝑝 and 𝑞, for 𝐴=𝛿=𝑐=1, and 𝜎=0.9, by means of Figure 8 over the region {(𝑥,𝑡)∶|𝑥|≤5,|𝑡|≤5}. We have bright soliton which are increase, decrease and have singularity with the different choose of the two parameters 𝑝 and 𝑞. This means that the parameters 𝑝 and 𝑞 affect in the solution structure.

(a)

(b)

(c)

Figure 8 

Evolutional behaviour of the intensity |𝑢1|2describes the bright soliton: (a) 𝑝=1,𝑞=1; (b) 𝑝=1/332,𝑞=1/333 (singular wave solution); (c) 𝑝=2.132,𝑞=1.723.

Besides the solutions obtained above, the ODE equation (2.5), albeit with different parameters, has been studied in the different context [20–24]. It has been shown that this equation possesses abundant solutions, including Weierstrass function solutions, kink solutions and periodic solutions, and so forth. To the best of our knowledge, some of our explicit solutions are new.

Notice that the GHFs are generalization of the hyperbolic functions as stated in the appendix. Also, the two parameters 𝑝 and 𝑞 describe the degree of the wave energy localization in the obtained solutions. Referring to [18, 19], it is easy to see that the obtained results in this paper are new and the improved method is valuable.

4. Summary and Discussion

In this paper, a set of sub-ODEs are introduced by using GHFs. Also, we have obtained many families of exact traveling wave solutions of a class of nonlinear Schrödinger equations including the generalized Zakharov system, the Rangwala-Rao equation, and the chen-Lee-Liu equation. We study and analyze the properties of these solutions by taking different parameter values of the generalized hyperbolic functions 𝑝 and 𝑞. As a result, we find that these parameter values affect the solution structure. These solutions include the GHF solution, hyperbolic function solution, 𝑞deformed hyperbolic function solution, and others. The obtained solutions which depend on 𝑝 and 𝑞 may be of important significance for the explanation of some practical physical problems. We believe that one can apply this method to many other nonlinear differential equations in mathematical physics.

Appendix

The generalized hyperbolic sine, cosine, and tangent functions aresinh_𝑝𝑞(𝜉)=𝑝𝑒^𝜉−𝑞𝑒^−𝜉2,cosh_𝑝𝑞(𝜉)=𝑝𝑒^𝜉+𝑞𝑒^−𝜉2,tanh_𝑝𝑞(𝜉)=𝑝𝑒^𝜉−𝑞𝑒^−𝜉𝑝𝑒^𝜉+𝑞𝑒^−𝜉, (A.1) where 𝜉 is an independent variable, 𝑝 and 𝑞 are arbitrary constants greater than zero and called deformation parameters. The generalized hyperbolic cotangent function is coth_𝑝𝑞(𝜉)=1/tanh_𝑝𝑞(𝜉), the generalized hyperbolic secant function is sech_𝑝𝑞(𝜉)=1/cosh_𝑝𝑞(𝜉), the generalized hyperbolic cosecant function is csch_𝑝𝑞(𝜉)=1/sinh_𝑝𝑞(𝜉), the above six kinds of functions are said GHFs. These functions satisfy the following relations [9, 12]:cosh²_𝑝𝑞(𝜉)−sinh²_𝑝𝑞(𝜉)=𝑝𝑞,cosh_𝑝𝑞(𝜉)=√1𝑝𝑞cosh𝜉−2𝑞ln𝑝,sinh_𝑝𝑞(𝜉)^′=cosh_𝑝𝑞(𝜉),cosh_𝑝𝑞(𝜉)^′=sinh_𝑝𝑞(𝜉). (A.2)

Note that if 𝑝,𝑞≠1 then sinh_𝑝𝑞(𝜉)is not odd and cosh_𝑝𝑞(𝜉) is not even:sinh_𝑝𝑞(−𝜉)=−𝑝𝑞sinh_{(1/𝑝)(1/𝑞)}(𝜉),cosh_𝑝𝑞(−𝜉)=𝑝𝑞cosh_{(1/𝑝)(1/𝑞)}(𝜉). (A.3)

Acknowledgment

The authors would like to express their sincere thanks to the referees for their valuable suggestions. This work has been supported by the Deanship of Scientific Research at Qassim University.