New Generalized Hyperbolic Functions to Find New Exact Solutions of the Nonlinear Partial Differential Equations

Abstract

We firstly give some new functions called generalized hyperbolic functions. By the using of the generalized hyperbolic functions, new kinds of transformations are defined to discover the exact approximate solutions of nonlinear partial differential equations. Based on the generalized hyperbolic function transformation of the generalized KdV equation and the coupled equal width wave equations (CEWE), we find new exact solutions of two equations and analyze the properties of them by taking different parameter values of the generalized hyperbolic functions. We think that these solutions are very important to explain some physical phenomena.

1. Introduction

To find exact solutions to nonlinear evolution equations, some new techniques have been introduced in recent years such as the Hirota method, the extended Jacobi elliptic function expansion method, the exponential function method, (𝐺^′/𝐺)-expansion method, the simplest equation method, the trial equation method, Kudryashov’s method, and solitary ansatz method [1–19]. Also some authors define new functions which are named generalized hyperbolic functions for constructing new solutions [20–22]. There are a lot of nonlinear evolution equations that are integrated using the various mathematical methods. Soliton solutions, compactons, singular solitons, and other solutions have been found by using these approaches. These types of solutions are very important and appear in various areas of applied mathematics.

In Section 2, we give the definition and properties of generalized hyperbolic functions. In Section 3, as applications, we obtain exact solutions of the generalized KdV equation and the coupled equal width wave equations.

2. The Definition and Properties of the Symmetrical Hyperbolic Fibonacci and Lucas Functions

In this section, we will define new functions which are named the symmetrical hyperbolic Fibonacci and Lucas functions for constructing new exact solutions of NPDEs and then study the properties of these functions.

Definition 1. Suppose that 𝜉 is an independent variable; 𝑝, 𝑞, and 𝑘 are all constants. The generalized hyperbolic sine function is sinh_𝑎(𝜉)=𝑝𝑎^𝑘𝜉−𝑞𝑎^−𝑘𝜉2, (1) generalized hyperbolic cosine function is cosh_𝑎(𝜉)=𝑝𝑎^𝑘𝜉+𝑞𝑎^−𝑘𝜉2, (2) generalized hyperbolic tangent function is tanh_𝑎(𝜉)=𝑝𝑎^𝑘𝜉−𝑞𝑎^−𝑘𝜉𝑝𝑎^𝑘𝜉+𝑞𝑎^−𝑘𝜉, (3) generalized hyperbolic cotangent function is coth_𝑎(𝜉)=𝑝𝑎^𝑘𝜉+𝑞𝑎^−𝑘𝜉𝑝𝑎^𝑘𝜉−𝑞𝑎^−𝑘𝜉, (4) generalized hyperbolic secant function is sech_𝑎(𝜉)=2𝑝𝑎^𝑘𝜉+𝑞𝑎^−𝑘𝜉, (5) generalized hyperbolic cosecant function is cosech_𝑎(𝜉)=2𝑝𝑎^𝑘𝜉−𝑞𝑎^−𝑘𝜉, (6) and the above six kinds of functions are said to be generalized new hyperbolic functions. Thus we can prove the following theory of generalized hyperbolic functions on the basis of Definition 1.

Theorem 2. The generalized hyperbolic functions satisfy the following relations: cosh²_𝑎(𝜉)−sinh²_𝑎(𝜉)=𝑝𝑞,1−tanh²_𝑎(𝜉)=𝑝𝑞sech²_𝑎(𝜉),1−coth²_𝑎(𝜉)=−𝑝𝑞cosech²_𝑎(𝜉),sech_𝑎(𝜉)=1cosh_𝑎(𝜉),cosech_𝑎(𝜉)=1sinh_𝑎(𝜉),tanh_𝑎(𝜉)=sinh_𝑎(𝜉)cosh_𝑎(𝜉),coth_𝑎(𝜉)=cosh_𝑎(𝜉)sinh_𝑎(𝜉). (7) The following just part of them are proved here for simplification.

Theorem 3. The derivative formulae of generalized hyperbolic functions are as follows: 𝑑sinh_𝑎(𝜉)𝑑𝜉=𝑘ln𝑎cosh_𝑎(𝜉),𝑑cosh_𝑎(𝜉)𝑑𝜉=𝑘ln𝑎sinh_𝑎(𝜉),𝑑tanh_𝑎(𝜉)𝑑𝜉=𝑘𝑝𝑞ln𝑎sech²_𝑎(𝜉),𝑑coth_𝑎(𝜉)𝑑𝜉=−𝑘𝑝𝑞ln𝑎cosech²_𝑎(𝜉),𝑑sech_𝑎(𝜉)𝑑𝜉=−𝑘ln𝑎sech_𝑎(𝜉)tanh_𝑎(𝜉),𝑑cosech_𝑎(𝜉)𝑑𝜉=−𝑘ln𝑎cosech_𝑎(𝜉)coth_𝑎(𝜉). (8)

Proof of (8). According to (7) and (8), we can get 𝑑tanh_𝑎(𝜉)=𝑑𝜉sinh_𝑎(𝜉)cosh_𝑎(𝜉)^′=sinh_𝑎(𝜉)^′cosh_𝑎(𝜉)−cosh_𝑎(𝜉)^′sinh_𝑎(𝜉)cosh²_𝑎(𝜉)=𝑘ln𝑎cosh²_𝑎(𝜉)−𝑘ln𝑎sinh²_𝑎(𝜉)cosh²_𝑎(𝜉)=𝑘𝑝𝑞ln𝑎sech²_𝑎(𝜉). (9) Similarly, we can prove other differential coefficient formulae in Theorem 3.

Remark 4. We see that when 𝑝=1, 𝑞=1, 𝑘=1, and 𝑎=𝑒 in (1)–(6), new generalized hyperbolic functions sinh_𝑎(𝜉), cosh_𝑎(𝜉), tanh_𝑎(𝜉), coth_𝑎(𝜉), sech_𝑎(𝜉), and cosech_𝑎(𝜉) degenerate as hyperbolic functions sinh(𝜉), cosh(𝜉), tanh(𝜉), coth(𝜉), sech(𝜉), and cosech(𝜉), respectively. In addition, when 𝑝=0 or 𝑞=0 in (1)–(6), sinh_𝑎(𝜉), cosh_𝑎(𝜉), tanh_𝑎(𝜉), coth_𝑎(𝜉), sech_𝑎(𝜉), and cosech_𝑎(𝜉) degenerate as exponential function (1/2)𝑝𝑎^𝑘(𝜉), ±(1/2)𝑞𝑎^{−𝑘(𝜉)}, 2𝑝𝑎^{−𝑘(𝜉)}, ±2𝑞𝑎^𝑘(𝜉), and ±1, respectively.

3. Applications

Example 5. Application to the generalized form of KdV equation is presented.
The generalized KdV equation that will be studied in this paper are given by [23] 𝑢^𝑙_𝑡+𝛼𝑢^𝑛𝑢_𝑥+𝛽𝑢^𝑛𝑢_𝑥𝑥𝑥+𝛾𝑢𝑢^𝑛_𝑥𝑥𝑥=0, (10) where the first term represents the generalized evolution. The special case with 𝑙=1 is the regular evolution term. The coefficients of 𝛼 are the nonlinear terms while the coefficients of 𝛽 and 𝛾 are the nonlinear dispersion terms. This equation with 𝑙=1 has been already studied by [18], where, in addition to soliton solution, compactons and periodic solutions were also obtained. The hypothesis for solving this equation is 𝑢()=𝐴𝑥,𝑡cosh^𝑠_𝑎𝜂, (11) where 𝜂=𝐵(𝑥−𝑣𝑡). (12) Here, in (11) and (12), 𝐴 represents the amplitude of the soliton while 𝐵 is the inverse width of the soliton and 𝑣 is the velocity of the soliton. The exponent 𝑠 is unknown at this point and will be evaluated during the course of the derivation of the solutions to (10). From (11), it is possible to obtain 𝑢^𝑙_𝑡=𝑘𝑙𝑠𝑣𝐴^𝑙𝐵ln𝑎tanh_𝑎𝜂cosh_𝑎^𝑠𝑙𝜂,𝑢^𝑛𝑢_𝑥=−𝑘𝑠𝐴^𝑛+1𝐵ln𝑎tanh_𝑎𝜂cosh_𝑎^𝑠(𝑛+1)𝜂,𝑢^𝑛𝑢_𝑥𝑥𝑥=𝑘³𝑠𝐴^𝑛+1𝐵³𝑝𝑞(ln𝑎)³(𝑠+1)(𝑠𝑛+𝑠+2)tanh_𝑎𝜂cosh_𝑎^{𝑠(𝑛+1)+2}𝜂−𝑘³𝑠³(𝑛+1)𝐴^𝑛+1𝐵³(ln𝑎)³tanh_𝑎𝜂cosh_𝑎^𝑠(𝑛+1)𝜂,𝑢𝑢^𝑛_𝑥𝑥𝑥=−𝑘³𝑠³𝑛³𝐴^𝑛+1𝐵³(ln𝑎)³tanh_𝑎𝜂cosh_𝑎^𝑠(𝑛+1)𝜂+𝑘³𝑠𝑛𝑝𝑞𝐴^𝑛+1𝐵³(ln𝑎)³(𝑠𝑛+1)(𝑠𝑛+2)tanh_𝑎𝜂cosh_𝑎^{𝑠(𝑛+1)+2}𝜂. (13) These results will now be substituted in (10) to obtain the 1-soliton solution of the generalized KdV equation. Equation (10) by virtue of (13) reduces to 𝑘𝑙𝑠𝑣𝐴^𝑙𝐵ln𝑎tanh_𝑎𝜂cosh_𝑎^𝑠𝑙𝜂−𝛼𝑘𝑠𝐴^𝑛+1𝐵ln𝑎tanh_𝑎𝜂cosh_𝑎^𝑠(𝑛+1)𝜂+𝛽𝑘³𝑠𝑝𝑞𝐴^𝑛+1𝐵³(ln𝑎)³(𝑠+1)(𝑠𝑛+𝑠+2)tanh_𝑎𝜂cosh_𝑎^{𝑠(𝑛+1)+2}𝜂−𝛽𝑘³𝑠³(𝑛+1)𝐴^𝑛+1𝐵³(ln𝑎)³tanh_𝑎𝜂cosh_𝑎^𝑠(𝑛+1)𝜂+𝛾𝑘³𝑠𝑛𝑝𝑞𝐴^𝑛+1𝐵³(ln𝑎)³(𝑠𝑛+1)(𝑠𝑛+2)tanh_𝑎𝜂cosh_𝑎^{𝑠(𝑛+1)+2}𝜂−𝛾𝑘³𝑠³𝑛³𝐴^𝑛+1𝐵³(ln𝑎)³tanh_𝑎𝜂cosh_𝑎^𝑠(𝑛+1)𝜂=0. (14) From (14), equating the exponents 𝑠𝑙 and 𝑠𝑛+𝑠+2 gives 𝑠𝑙=𝑠𝑛+𝑠+2, (15) that leads to 2𝑠=.𝑙−𝑛−1 (16) Now from (14), the two linearly independent functions are 1/cosh^{𝑛𝑠+𝑠+𝑗} for 𝑗=0,2. Thus setting their coefficients to zero gives ()()𝑣=𝛼𝑝𝑞𝛽𝑙𝑙+1−𝑛+𝑛𝛾𝑙−1)(𝑙+𝑛−12𝑙𝐴^{𝑙−𝑛−1}𝛽(𝑛+1)+𝑛³𝛾,√𝐵=−𝛼(𝑛+1−𝑙)2𝑘ln𝑎𝛽(𝑛+1)+𝑛³𝛾. (17)(18) Thus the 1-soliton solution of the generalized KdV equation with generalized evolution is given by 𝑢()=𝐴𝑥,𝑡cosh_𝑎^{2/(𝑙−𝑛−1)}[𝐵(.𝑥−𝑣𝑡)] (19) This shows that the restriction on the nonlinear exponents of (10) must be 𝑙>𝑛+1. (20)

Remark 6. If we take the corresponding values for some parameters, the solution (19) can be reduced to the solution (17) mentioned in [23].

Example 7. Application to the coupled equal width wave equations is presented.

We consider the coupled equal width wave equations [24] 𝑢_𝑡+𝑢𝑢_𝑥−𝑢_𝑥𝑥𝑡+𝑟𝑟_𝑥𝑟=0,_𝑡+𝑟𝑟_𝑥−𝑟_𝑥𝑥𝑡=0. (21) The hypothesis for solving these equations is 𝑟()=𝐴𝑥,𝑡¹cosh^𝑠1_𝑎𝜂,𝑢()=𝐴𝑥,𝑡²cosh^𝑠2_𝑎𝜂, (22) where 𝜂=𝐵(𝑥−𝑣𝑡). (23) Here in (22)–(23) 𝐴₁ and 𝐴₂ are the amplitudes of the 𝑟-soliton and 𝑢-soliton, respectively, while 𝑣₁ and 𝑣₂ are the velocity of the soliton and 𝐵₁ and 𝐵₂ are the inverse widths of the solitons. The exponents 𝑠₁ and 𝑠₂ are unknown at this point and their values will fall out in the process of deriving the solution of this equation. From (22), it is possible to obtain 𝑢_𝑡=𝑘𝑠₂𝑣𝐴₂𝐵ln𝑎tanh_𝑎𝜂cosh^𝑠2_𝑎𝜂,𝑟_𝑡=𝑘𝑠₁𝑣𝐴₁𝐵ln𝑎tanh_𝑎𝜂cosh^𝑠1_𝑎𝜂,𝑢_𝑥=−𝑘𝑠₂𝐴₂𝐵ln𝑎tanh_𝑎𝜂cosh^𝑠2_𝑎𝜂,𝑟_𝑥=−𝑘𝑠₁𝐴₁𝐵ln𝑎tanh_𝑎𝜂cosh^𝑠1_𝑎𝜂,𝑢_𝑥𝑥𝑡=𝑘³𝑠³₂𝑣𝐴₂𝐵³(ln𝑎)³tanh_𝑎𝜂cosh^𝑠2_𝑎𝜂−𝑘³𝑠₂𝑝𝑞𝑠₂+1𝑣𝐴₂𝐵³(ln𝑎)³𝑠₂+2tanh_𝑎𝜂cosh^𝑠2_𝑎⁺²𝜂,𝑟_𝑥𝑥𝑡=𝑘³𝑠³₁𝑣𝐴₁𝐵³(ln𝑎)³tanh_𝑎𝜂cosh^𝑠1_𝑎𝜂−𝑘³𝑠₁𝑝𝑞𝑠₁+1𝑣𝐴₁𝐵³(ln𝑎)³𝑠₁+2tanh_𝑎𝜂cosh^𝑠1_𝑎⁺²𝜂. (24)(25)(26)(27) Substituting these into (21) yields to 𝑘𝑠₂𝑣𝐴₂𝐵ln𝑎tanh_𝑎𝜂cosh^𝑠2_𝑎𝜂−𝑘𝑠₂𝐴²₂𝐵ln𝑎tanh_𝑎𝜂cosh^2𝑠2_𝑎𝜂−𝑘³𝑠³₂𝑣𝐴₂𝐵³(ln𝑎)³tanh_𝑎𝜂cosh^𝑠2_𝑎𝜂+𝑘³𝑠₂𝑠₂+1𝑠₂+2𝑝𝑞𝑣𝐴₂𝐵³(ln𝑎)³tanh_𝑎𝜂cosh^𝑠2_𝑎⁺²𝜂−𝑘𝑠₁𝐴²₁𝐵ln𝑎tanh_𝑎𝜂cosh^2𝑠1_𝑎𝜂=0,𝑘𝑠₁𝑣𝐴₁𝐵ln𝑎tanh_𝑎𝜂cosh^𝑠1_𝑎𝜂−𝑘𝑠₁𝐴²₁𝐵ln𝑎tanh_𝑎𝜂cosh^2𝑠1_𝑎𝜂−𝑘³𝑠³₁𝑣𝐴₁𝐵³(ln𝑎)³tanh_𝑎𝜂cosh^𝑠1_𝑎𝜂+𝑘³𝑠₁𝑠₁+1𝑠₁+2𝑣𝑝𝑞𝐴₁𝐵³(ln𝑎)³tanh_𝑎𝜂cosh^𝑠1_𝑎⁺²𝜂=0. (28)(29) Now from (28) and (29), equating the exponents 2𝑠₁ and 𝑠₁+2 and also 2𝑠₂ and 𝑠₂+2 gives, respectively, 2𝑠₁=𝑠₁+2,2𝑠₂=𝑠₂+2, (30) that leads to 𝑠₁=𝑠₂=2. (31) So from (28) and (29), the four linearly independent functions are 1/cosh^𝑠1+𝑗 and 1/cosh^𝑠2+𝑗 for 𝑗=0,2. Therefore, setting their respective coefficients to zero we obtain 𝐴₂, 𝐵, and 𝑣 as follow.

Case 1. One has 𝐴₂=𝐴₁21±𝑖√3,𝐵=−1𝐴2𝑘ln𝑎,𝑣=¹,3𝑝𝑞 (32) where 𝐴₁, 𝑘, 𝑝, and 𝑞 are arbitrary constants. Substituting (32) into (22), we obtain new exact solution to (21), 𝑟()=𝐴𝑥,𝑡¹cosh²_𝑎𝜂()=𝐴,𝑢𝑥,𝑡²cosh²_𝑎𝜂, (33) where 𝜂=−(1/2𝑘ln𝑎)(𝑥−(𝐴₁/3𝑝𝑞)𝑡).

Case 2. One has 𝐴₂=𝐴₁21±𝑖√3,𝐵=1𝐴2𝑘ln𝑎,𝑣=¹,3𝑝𝑞 (34) where 𝐴₁, 𝑘, 𝑝, and 𝑞 are arbitrary constants. Substituting (34) into (22), we obtain new exact solution to (21), 𝑟()=𝐴𝑥,𝑡¹cosh²_𝑎𝜂()=𝐴,𝑢𝑥,𝑡²cosh²_𝑎𝜂, (35) where 𝜂=(1/2𝑘ln𝑎)(𝑥−(𝐴₁/3𝑝𝑞)𝑡).

Remark 8. If we search the corresponding values for some parameters and make some transformations, in particular the solution (35) can be reduced to the solution (32) mentioned in [24].

4. Conclusions and Remarks

We consider generalized hyperbolic functions and new kinds of generalized hyperbolic function transformation to construct new exact solutions of nonlinear partial differential equations. This paper obtains 1-soliton solution to the generalized KdV equation and the coupled equal width wave equations. Our methods also can be applied to construct new exact solutions of other nonlinear partial differential equations.