# Solitons and Other Exact Solutions for Two Nonlinear PDEs in Mathematical Physics Using the Generalized Projective Riccati Equations Method

Authors: A. M. Shahoot; K. A. E. Alurrfi; I. M. Hassan; A. M. Almsri
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We apply the generalized projective Riccati equations method with the aid of Maple software to construct many new soliton and periodic solutions with parameters for two higher-order nonlinear partial differential equations (PDEs), namely, the nonlinear Schrödinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity and the nonlinear quantum Zakharov-Kuznetsov (QZK) equation. The obtained exact solutions include kink and antikink solitons, bell (bright) and antibell (dark) solitary wave solutions, and periodic solutions. The given nonlinear PDEs have been derived and can be reduced to nonlinear ordinary differential equations (ODEs) using a simple transformation. A comparison of our new results with the well-known results is made. Also, we drew some graphs of the exact solutions using Maple. The given method in this article is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics.


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## Body

## 1. Introduction

The investigation of exact traveling wave solutions to nonlinear PDEs plays an important role in the study of nonlinear physical phenomena. Nonlinear waves appear in various scientific fields, especially in physics such as fluid mechanics, plasma physics, optical fibers, and solid state physics. In recent years, many powerful tools have been established to determine soliton and periodic wave solutions of nonlinear PDEs, such as the(G′/G)-expansion method [1–6], the extended auxiliary equation method [7, 8], the new mapping method [9–11], the generalized projective Riccati equations method [12–17], and the (G′/G,1/G)-expansion method [18]. Conte and Musette [12] presented an indirect method to find solitary wave solutions of some nonlinear PDEs that can be expressed as polynomials in two elementary functions which satisfy a projective Riccati equation [19]. This method has been applied to many nonlinear PDEs and the solitary wave solutions of these equations can be found in [13–17]. Recently, Yan [16] has been given a generalization of Conte and Musette’s method.The objective of this article is to use the generalized projective Riccati equations method [13–17] to construct the soliton and periodic solutions of the following two higher-order nonlinear PDEs.(1) The nonlinear Schrödinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity [5] is(1)iqt+aqxx-bqxxxx+cq2m+k1q4mq=0,which describes the propagation of optical pulse in a medium, and q(x,t) is the slowly varying envelope of the electromagnetic field, where a,b,c are real numbers. If b=0, (1) reduces to the NLS. In addition if m=1, (1) reduces to parabolic law nonlinearity, which has been discussed in [20] using two direct algebraic methods. The coefficient of a represents the group velocity dispersion (GVD), while the coefficient of c represents the self-phase modulation (SPM) with dual power law nonlinearity. The constant k1 binds the two nonlinear terms and the exponent m governs the power law. Also, the coefficients of b are the fourth-order dispersion terms. Equation (1) has been studied in [5] using five different techniques, namely, the (G′/G)-expansion method, the improved Sub-ODE method, the extended auxiliary equation method, the new mapping method, and the Jacobi elliptic function method.(2) The nonlinear quantum Zakharov-Kuznetsov (QZK) equation [6, 21–23] is(2)∂U∂T+AU∂U∂Z+B∂3U∂Z3+C∂∂Z∂2∂X2+∂2∂Y2U=0,which arises in quantum magnetoplasma, where A, B, C are constants. Equation (2) has been derived in [21] using the reductive perturbation technique and in [22] using a series of transformations method. Here U is the electrostatic potential, which X, Y, Z, T are the stretched space-time coordinates which are defined in [21]. Moslem et al. [21] have derived (2) for electron-ion quantum plasma and solitary explosive and periodic solutions are presented. In [23], the authors applied the auxiliary equation method and Hirota bilinear method to study (2) and some types of exact solutions are obtained. Recently Zayed and Alurrfi have discussed (2) in [6] using the extended generalized (G′/G)-expansion method with the Jacobi elliptic equation and determined its exact traveling wave solutions.This paper is organized as follows: in Section2, the description of the well-known generalized projective Riccati equations method is given. In Section 3, we use the given method described in Section 2, to find new soliton and periodic solutions of the NLS equation (1) and the QZK equation (2). In Section 4, we draw some figures for some solutions of (1). In Section 5, some conclusions are obtained.


## 2. Description of the Generalized Projective Riccati Equations Method

Consider a nonlinear PDE in the following form:(3)Pu,ux,ut,uxx,utt,…=0,where u=u(x,t) is an unknown function, P is a polynomial in u(x,t), and its partial derivatives in which the highest-order derivatives and nonlinear terms are involved. Let us now give the main steps of the generalized projective Riccati equations method [13–17].Step 1.
We use the following transformation:(4)ux,t=uξ,ξ=x-vt,to reduce (3) to the following nonlinear ODE:(5)Hu,u′,u′′,…=0,where v is velocity of the propagation, H is a polynomial of u(ξ) and its total derivatives u′(ξ),u′′(ξ),… and ′=d/dξ.Step 2.
We suppose that the solution of (5) has the following form: (6)uξ=A0+∑i=1Nσi-1ξAiσξ+Biτξ,where A0, Ai, and Bi are constants to be determined. The functions σ(ξ) and τ(ξ) satisfy the ODEs:(7)σ′ξ=εσξτξ,τ′ξ=R+ετ2ξ-μσξ,ε=±1,where(8)τ2ξ=-εR-2μσξ+μ2+rRσ2ξ,and here r=±1 and R, μ are nonzero constants.
IfR=μ=0, (5) has the formal solution:(9)uξ=∑i=0NAiτiξ,where τ(ξ) satisfies the nonlinear ODE:(10)τ′ξ=τ2ξ.Step 3.
The positive integer numberN in (6) must be determined by using the homogeneous balance between the highest-order derivatives and the highest nonlinear terms in (5).Step 4.
Substitute (6) along with (7) and (8) into (5). Collecting all terms of the same order of σj(ξ)τi(ξ)(j=0,1,;i=0,1). Setting each coefficient to zero yields a set of algebraic equations which can be solved to find the values of A0, Ai, Bi, v, μ, and R.Step 5.
It is well-known [13–17] that (7) admits the following solutions.
Case 1. When ε=-1,r=-1,R>0,(11)σ1ξ=Rsech⁡Rξμsech⁡Rξ+1,τ1ξ=Rtanh⁡Rξμsech⁡Rξ+1.
Case 2. When ε=-1,r=1,R>0,(12)σ2ξ=Rcsch⁡Rξμcsch⁡Rξ+1,τ2ξ=Rcoth⁡Rξμcsch⁡Rξ+1.
Case 3. When ε=1,r=-1,R>0,(13)σ3ξ=Rsec⁡Rξμsec⁡Rξ+1,τ3ξ=Rtan⁡Rξμsec⁡Rξ+1,σ4ξ=Rcsc⁡Rξμcsc⁡Rξ+1,τ4ξ=-Rcot⁡Rξμcsc⁡Rξ+1.
Case 4. When R=μ=0,(14)σ5ξ=Cξ,τ5ξ=1εξ,where C is a nonzero constant.Step 6.
Substituting the values ofA0, Ai, Bi, v, μ, and R as well as the solutions (11)–(14) into (6), we obtain the exact solutions of (3).


## 3. Applications

In this section, we apply the generalized projective Riccati equations method described in Section2 to find many new soliton and periodic solutions of (1) and (2) in the following subsections.


### 3.1. On Solving (1) Using the Method of Section 2

In order to solve (1), we assume that the solution of (1) has the following form:(15)qx,t=ϕξeiQx,t,i=-1,where ϕξ is the amplitude portion which is a real function of ξ, while Q(x,t) is the phase portion of the soliton. It is assumed that ξ and Q(x,t) are given by(16)ξ=x-vt,Qx,t=-kx+ωt+θ,where v,k,ω, and θ are constants, such that v is the velocity of the soliton, ω is the frequency of the soliton, k is the wave number, and θ is a phase constant.Substituting (15) into (1) and separating the real and imaginary parts, we get(17)Re:-ω+ak2+bk4ϕ+a+6bk2ϕ′′-bϕ′′′′+cϕ2m+1+k1ϕ4m+1=0,(18)Im:-v+2ak+4bk3ϕ′+4bkϕ′′′=0,and, differentiating (18) and substituting the resulting equation in (17), we have the following nonlinear ODE:(19)a1ϕ′′-b1ϕ+c1ϕ2m+1+k1ϕ4m+1=0,where a1, b1, and c1 are given by(20)a1=2ak+20bk3-v,b1=-4kω+ak2+bk4,c1=4kc.Balancing ϕ′′ with ϕ4m+1 in (19), then the following relation is attained:(21)N+2=N4m+1⟹N=12m.Since the balance number N is not integer, then we use the following new transformation:(22)ϕξ=uξ1/2m,where u is a new function of ξ. Substituting (22) into (19), we have the new equation(23)2a1muu′′+a11-2mu′2+4b1m2u2+4c1m2u3+4k1c1m2u4=0.Balancing uu′′ and u4 in (23), then the following relation is obtained:(24)N+N+2=4N⟹N=1.From (6) the formal solution of (23) has the following form:(25)uξ=A0+A1σξ+B1τξ,where A0, A1, and B1 are constants to be determined such that A1≠0 or B1≠0.Substituting (25) into (23) and using (7) and (8), the left-hand side of (23) becomes a polynomial in σ(ξ) and τ(ξ). Setting the coefficients of this polynomial to be zero yields the following system of algebraic equations:(26)σ4:4m2A14c1k1-1Rεμ2+r24c1k1m2A12B12+4a1mε2A12+1-2ma1ε2A12+1R2ε2μ2+r24c1k1m2B14+4a1mε2B12+1-2ma1ε2B12=0,σ3:-1Rεμ2+r12m2A1B12c1-21-2mμεB12a1-6mμεB12a1+4mε2A0A1a1+48m2A0A1B12c1k1+4m2A13c1+16m2A0A13c1k1-2mμεA12a1-4Rμε2μ2+r4c1k1m2B14+4a1mε2B12+1-2ma1ε2B12+2με24c1k1m2A12B12+4a1mε2A12+1-2ma1ε2A12=0,σ3τ:1Rεμ2+r16A1c1k1m2B13+8A1a1mε2B1+21-2mA1a1ε2B1-16m2A13B1c1k1=0,σ2:4m2A12b1+ε26μ2+2r4c1k1m2B14+4a1mε2B12+1-2ma1ε2B12+1-2mμ2B12a1+2με12m2A1B12c1-21-2mμεB12a1-6mμεB12a1+4mε2A0A1a1+48m2A0A1B12c1k1-1Rεμ2+r24c1k1m2A02B12+12c1m2A0B12+4b1m2B12+4Rεa1mB12+21-2mRεa1B12+24m2A02A12c1k1-2mμεA0A1a1-εR24c1k1m2A12B12+4a1mε2A12+1-2ma1ε2A12+12m2A0A12c1+2RmεA12a1=0,σ2τ:2εμ16A1c1k1m2B13+8A1a1mε2B1+21-2mA1a1ε2B1+12m2A12B1c1-21-2mμεA1B1a1-1Rεμ2+r4m2B13c1+16m2A0B13c1k1+4mε2A0B1a1+48m2A0A12B1c1k1-8mμεA1B1a1=0,σ:-εR12m2A1B12c1-21-2mμεB12a1-6mμεB12a1+4mε2A0A1a1+48m2A0A1B12c1k1+2εμ24c1k1m2A02B12+12c1m2A0B12+4b1m2B12+4Rεa1mB12+21-2mRεa1B12+8m2A0A1b1+12m2A02A1c1-4Rμε24c1k1m2B14+4a1mε2B12+1-2ma1ε2B12+16m2A03A1c1k1-21-2mRμB12a1+2RmεA0A1a1=0,στ:8m2A1B1b1-εR16A1c1k1m2B13+8A1a1mε2B1+21-2mA1a1ε2B1+2εμ4m2B13c1+16m2A0B13c1k1+4mε2A0B1a1+24m2A0A1B1c1+21-2mRεA1B1a1+6RmεA1B1a1-6mμεA0B1a1+48m2A02A1B1c1k1=0,τ:8m2A0B1b1-Rε4m2B13c1+16m2A0B13c1k1+4mε2A0B1a1+12m2A02B1c1+16m2A03B1c1k1+4RmεA0B1a1=0,σ0:R2ε24c1k1m2B14+4a1mε2B12+1-2ma1ε2B12+4m2A02b1+4m2A03c1+1-2mR2B12a1-Rε24c1k1m2A02B12+12c1m2A0B12+4b1m2B12+4Rεa1mB12+21-2mRεa1B12+4m2A04c1k1=0.According to Step 5, of Section 2, there are three cases of solutions of the algebraic equations (26) to be discussed as follows:Case 1 (ε=-1, r=-1). It leads to the following results.Result 1. We have(27)A0=-b1m+1c1,A1=0,B1=±m+1-b1a1c1m,μ=0,R=-b1m2a1,k1=c12m+14b1m+12.From (11), (25), and (27), we deduce the kink-shaped soliton solutions of (1) as follows:(28)q11x,t=-b1m+1c11±tanh⁡-b1m2a1ξ1/2mexp⁡i-kx+ωt+θ,where a1b1<0 and c1b1<0.Result 2. We have(29)A0=0,A1=a1μm+12c1m2,B1=0,R=-4b1m2a1,k1=c1μ2-12m+14b1μ2m+12.In this result, we deduce the bell-shaped soliton solutions of (1) as follows:(30)q12x,t=-2b1μm+1c1sech⁡-4b1m2/a1ξμsech⁡-4b1m2/a1ξ+11/2mexp⁡i-kx+ωt+θ,where a1b1<0 and c1b1μ<0.Result 3. We have(31)A0=-b1m+1c1,A1=±a1m+1μ2-14c1m2,B1=±m+1-a1b12c1m,R=-4b1m2a1,k1=c12m+14b1m+12.In this result, we deduce the bell-kink-shaped soliton solutions of (1) as follows:(32)q13x,t=-b1m+1c11±μ2-1sech⁡-4b1m2/a1ξμsech⁡-4b1m2/a1ξ+1±tanh⁡-4b1m2/a1ξμsech⁡-4b1m2/a1ξ+11/2mexp⁡i-kx+ωt+θ,where a1b1<0,c1b1<0 and μ2>1.Case 2 (ε=-1, r=1). It leads to the following results.Result 1. We have the same result (27).From (12), (25), and (27), we deduce the singular kink-shaped soliton solutions of (1) as follows:(33)q21x,t=-b1m+1c11±coth⁡-b1m2a1ξ1/2mexp⁡i-kx+ωt+θ,where a1b1<0 and c1b1<0.Result 2. We have(34)A0=0,A1=a1μm+12c1m2,B1=0,R=-4b1m2a1,k1=c1μ2+12m+14b1μ2m+12.In this result, we deduce the singular bell-shaped soliton solutions of (1) as follows:(35)q22x,t=-2b1μm+1c1csch⁡-4b1m2/a1ξμcsch⁡-4b1m2/a1ξ+11/2mexp⁡i-kx+ωt+θ,where a1b1<0 and c1b1μ<0.Result 3. We have(36)A0=-b1m+1c1,A1=±a1m+1μ2+14c1m2,B1=±m+1-a1b12c1m,R=-4b1m2a1,k1=c12m+14b1m+12.In this result, we deduce the singular bell-kink-shaped soliton solutions of (1) as follows:(37)q23x,t=-b1m+1c11±μ2+1csch⁡-4b1m2/a1ξμcsch⁡-4b1m2/a1ξ+1±coth⁡-4b1m2/a1ξμcsch⁡-4b1m2/a1ξ+11/2mexp⁡i-kx+ωt+θ,where a1b1<0 and c1b1<0.Case 3 (ε=1, r=-1). It leads to the following results.Result 1. We have(38)A0=0,A1=-a1μm+12c1m2,B1=0,R=4b1m2a1,k1=c1μ2-12m+14b1μ2m+12.From (13), (25), and (38), we deduce the periodic solutions of (1) as follows:(39)q31x,t=-2b1μm+1c1sec⁡4b1m2/a1ξμsec⁡4b1m2/a1ξ+11/2mexp⁡i-kx+ωt+θ,(40)q32x,t=-2b1μm+1c1csc⁡4b1m2/a1ξμcsc⁡4b1m2/a1ξ+11/2mexp⁡i-kx+ωt+θ,where a1b1>0 and c1b1μ<0.Remark 1.
Note that our results (28) and (33) are in agreement with the results obtained in [5], while the other results are new, which are not found elsewhere.Remark 2.
Note that, ifR=μ=0, then we have the trivial solution.


### 3.2. On Solving (2) Using the Method of Section 2

In this subsection, we apply the generalized projective Riccati equations method of Section2 to find new soliton and periodic solutions of (2). To this aim, we use the following wave transformation:(41)UX,Y,Z,T=Uξ,ξ=αX+βY+γZ+υT,to reduce (2) to the following nonlinear ODE:(42)VU′-AUU′-Bγ2+Cα2+β2U′′′=0,where α,β,γ are the direction cosines, and υ is the quantum ion-acoustic wave speed and V=υ/γ. Integrating (42) with respect to ξ once and vanishing the constant of integration, we find the following nonlinear ODE:(43)VU-12AU2-Bγ2+Cα2+β2U′′=0.Balancing U′′ with U2 gives N=2. Therefore, (6) reduces to(44)Uξ=A0+A1σξ+A2σ2ξ+B1τξ+B2σξτξ,where Ai(i=0,1,2), B1 and B2 are constants to be determined such that A2≠0 or B2≠0.Substituting (44) and using (7) and (8) into (43), the left-hand side of (43) becomes a polynomial in σ(ξ) and τ(ξ). Setting the coefficients of this polynomial to be zero yields the following system of algebraic equations:(45)σ4:12ARεB22μ2+r-12AA22=0,σ3:2Bγ2+Cα2+β2μεA2-AA1A2-12Aε2μB22-2RB1B2μ2+r=0,σ3τ:AA2B2-4Rε3B2μ2+rBγ2+Cα2+β2=0,σ2:VA2-12AA12+2A0A2+12AεRB22+1RB12μ2+r-4μB1B2=0,σ2τ:AA1B2+A2B1+Bγ2+Cα2+β2ε2A1-6μεB2+8Bγ2+Cα2+β2με3B2=0,σ:VA1-12Aε2μB12-2RB1B2-AA0A1=0,στ:VB2-Bγ2+Cα2+β25RεB2-3μεB1-AA0B2+A1B1+4Bγ2+Cα2+β2Rε3B2=0,τ:AA0B1-VB1+2Bγ2+Cα2+β2RεB1=0,σ0:VA0-12AA02-Bγ2+Cα2+β2εB2+12ARεB12=0.According to Step 5 of Section 2, there are three cases of solutions of the algebraic equations (45) to be discussed as follows.Case 1 (ε=-1, r=-1). It leads to the following results.Result 1. We have (46)A0=0,A1=6μBγ2+Cα2+β2A,A2=-6μ2-1Bγ2+Cα2+β2AR,B1=0,B2=±6μ2-1Bγ2+Cα2+β2AR,V=RBγ2+Cα2+β2.From (11), (44), and (46), we deduce the bell-kink-shaped soliton solution of (2) as follows:(47)U11X,Y,Z,T=6Bγ2+Cα2+β2ARsech⁡Rξμsech⁡Rξ+11-μ2sech⁡Rξμsech⁡Rξ+1±μ2-1tanh⁡Rξμsech⁡Rξ+1+μ,where R>0,μ2>1 and ξ=αX+βY+γZ+RγBγ2+C(α2+β2)T.Result 2. We have(48)A0=-2RBγ2+Cα2+β2A,A1=6μBγ2+Cα2+β2A,A2=-6μ2-1Bγ2+Cα2+β2AR,B1=0,B2=±6μ2-1Bγ2+Cα2+β2AR,V=-RBγ2+Cα2+β2.In this result, we deduce the bell-kink-shaped soliton solution of (2) as follows:(49)U12X,Y,Z,T=6Bγ2+Cα2+β2ARsech⁡Rξμsech⁡Rξ+11-μ2sech⁡Rξμsech⁡Rξ+1±μ2-1tanh⁡Rξμsech⁡Rξ+1+μ-2RBγ2+Cα2+β2A,where R>0,μ2>1 and ξ=αX+βY+γZ-RγBγ2+C(α2+β2)T.Case 2 (ε=-1, r=1). It leads to the following results:Result 1. We have(50)A0=0,A1=6μBγ2+Cα2+β2A,A2=-6μ2+1Bγ2+Cα2+β2AR,B1=0,B2=±6μ2+1Bγ2+Cα2+β2AR,V=RBγ2+Cα2+β2.From (12), (44), and (50), we deduce the singular bell-kink-shaped soliton solution of (2) as follows:(51)U21X,Y,Z,T=6Bγ2+Cα2+β2ARcsch⁡Rξμcsch⁡Rξ+1-μ2+1csch⁡Rξμcsch⁡Rξ+1±μ2+1coth⁡Rξμcsch⁡Rξ+1+μ,where R>0 and ξ=αX+βY+γZ+RγBγ2+Cα2+β2T.Result 2. We have(52)A0=-2RBγ2+Cα2+β2A,A1=6μBγ2+Cα2+β2A,A2=-6μ2+1Bγ2+Cα2+β2AR,B1=0,B2=±6μ2+1Bγ2+Cα2+β2AR,V=-RBγ2+Cα2+β2.In this result, we deduce the singular bell-kink-shaped soliton solution of (2) as follows:(53)U22X,Y,Z,T=6Bγ2+Cα2+β2ARcsch⁡Rξμcsch⁡Rξ+1-μ2+1csch⁡Rξμcsch⁡Rξ+1±μ2+1coth⁡Rξμcsch⁡Rξ+1+μ-2RBγ2+Cα2+β2A,where R>0 and ξ=αX+βY+γZ-RγBγ2+C(α2+β2)T.Case 3 (ε=1, r=-1). It leads to the following results.Result 1. We have(54)A0=0,A1=-6μBγ2+Cα2+β2A,A2=6μ2-1Bγ2+Cα2+β2AR,B1=0,B2=±61-μ2Bγ2+Cα2+β2AR,V=-RBγ2+Cα2+β2.From (13), (44), and (54), we deduce the periodic solutions of (2) as follows:(55)U31X,Y,Z,T=6Bγ2+Cα2+β2ARsec⁡Rξμsec⁡Rξ+1μ2-1sec⁡Rξμsec⁡Rξ+1±1-μ2tan⁡Rξμsec⁡Rξ+1-μ,U32X,Y,Z,T=6Bγ2+Cα2+β2ARcsc⁡Rξμcsc⁡Rξ+1μ2-1csc⁡Rξμcsc⁡Rξ+1±1-μ2cot⁡Rξμcsc⁡Rξ+1-μ,where R>0,μ2<1 and ξ=αX+βY+γZ-RγBγ2+C(α2+β2)T.Result 2. We have(56)A0=2RBγ2+Cα2+β2A,A1=-6μBγ2+Cα2+β2A,A2=6μ2-1Bγ2+Cα2+β2AR,B1=0,B2=±61-μ2Bγ2+Cα2+β2AR,V=RBγ2+Cα2+β2.In this result, we deduce the periodic solutions of (2) as follows:(57)U33X,Y,Z,T=6Bγ2+Cα2+β2ARsec⁡Rξμsec⁡Rξ+1μ2-1sec⁡Rξμsec⁡Rξ+1±1-μ2tan⁡Rξμsec⁡Rξ+1-μ+2RBγ2+Cα2+β2A,U34X,Y,Z,T=6Bγ2+Cα2+β2ARcsc⁡Rξμcsc⁡Rξ+1μ2-1csc⁡Rξμcsc⁡Rξ+1±1-μ2cot⁡Rξμcsc⁡Rξ+1-μ+2RBγ2+Cα2+β2A,where R>0,μ2<1 and ξ=αX+βY+γZ+RγBγ2+C(α2+β2)T.


## 3.1. On Solving (1) Using the Method of Section 2

In order to solve (1), we assume that the solution of (1) has the following form:(15)qx,t=ϕξeiQx,t,i=-1,where ϕξ is the amplitude portion which is a real function of ξ, while Q(x,t) is the phase portion of the soliton. It is assumed that ξ and Q(x,t) are given by(16)ξ=x-vt,Qx,t=-kx+ωt+θ,where v,k,ω, and θ are constants, such that v is the velocity of the soliton, ω is the frequency of the soliton, k is the wave number, and θ is a phase constant.Substituting (15) into (1) and separating the real and imaginary parts, we get(17)Re:-ω+ak2+bk4ϕ+a+6bk2ϕ′′-bϕ′′′′+cϕ2m+1+k1ϕ4m+1=0,(18)Im:-v+2ak+4bk3ϕ′+4bkϕ′′′=0,and, differentiating (18) and substituting the resulting equation in (17), we have the following nonlinear ODE:(19)a1ϕ′′-b1ϕ+c1ϕ2m+1+k1ϕ4m+1=0,where a1, b1, and c1 are given by(20)a1=2ak+20bk3-v,b1=-4kω+ak2+bk4,c1=4kc.Balancing ϕ′′ with ϕ4m+1 in (19), then the following relation is attained:(21)N+2=N4m+1⟹N=12m.Since the balance number N is not integer, then we use the following new transformation:(22)ϕξ=uξ1/2m,where u is a new function of ξ. Substituting (22) into (19), we have the new equation(23)2a1muu′′+a11-2mu′2+4b1m2u2+4c1m2u3+4k1c1m2u4=0.Balancing uu′′ and u4 in (23), then the following relation is obtained:(24)N+N+2=4N⟹N=1.From (6) the formal solution of (23) has the following form:(25)uξ=A0+A1σξ+B1τξ,where A0, A1, and B1 are constants to be determined such that A1≠0 or B1≠0.Substituting (25) into (23) and using (7) and (8), the left-hand side of (23) becomes a polynomial in σ(ξ) and τ(ξ). Setting the coefficients of this polynomial to be zero yields the following system of algebraic equations:(26)σ4:4m2A14c1k1-1Rεμ2+r24c1k1m2A12B12+4a1mε2A12+1-2ma1ε2A12+1R2ε2μ2+r24c1k1m2B14+4a1mε2B12+1-2ma1ε2B12=0,σ3:-1Rεμ2+r12m2A1B12c1-21-2mμεB12a1-6mμεB12a1+4mε2A0A1a1+48m2A0A1B12c1k1+4m2A13c1+16m2A0A13c1k1-2mμεA12a1-4Rμε2μ2+r4c1k1m2B14+4a1mε2B12+1-2ma1ε2B12+2με24c1k1m2A12B12+4a1mε2A12+1-2ma1ε2A12=0,σ3τ:1Rεμ2+r16A1c1k1m2B13+8A1a1mε2B1+21-2mA1a1ε2B1-16m2A13B1c1k1=0,σ2:4m2A12b1+ε26μ2+2r4c1k1m2B14+4a1mε2B12+1-2ma1ε2B12+1-2mμ2B12a1+2με12m2A1B12c1-21-2mμεB12a1-6mμεB12a1+4mε2A0A1a1+48m2A0A1B12c1k1-1Rεμ2+r24c1k1m2A02B12+12c1m2A0B12+4b1m2B12+4Rεa1mB12+21-2mRεa1B12+24m2A02A12c1k1-2mμεA0A1a1-εR24c1k1m2A12B12+4a1mε2A12+1-2ma1ε2A12+12m2A0A12c1+2RmεA12a1=0,σ2τ:2εμ16A1c1k1m2B13+8A1a1mε2B1+21-2mA1a1ε2B1+12m2A12B1c1-21-2mμεA1B1a1-1Rεμ2+r4m2B13c1+16m2A0B13c1k1+4mε2A0B1a1+48m2A0A12B1c1k1-8mμεA1B1a1=0,σ:-εR12m2A1B12c1-21-2mμεB12a1-6mμεB12a1+4mε2A0A1a1+48m2A0A1B12c1k1+2εμ24c1k1m2A02B12+12c1m2A0B12+4b1m2B12+4Rεa1mB12+21-2mRεa1B12+8m2A0A1b1+12m2A02A1c1-4Rμε24c1k1m2B14+4a1mε2B12+1-2ma1ε2B12+16m2A03A1c1k1-21-2mRμB12a1+2RmεA0A1a1=0,στ:8m2A1B1b1-εR16A1c1k1m2B13+8A1a1mε2B1+21-2mA1a1ε2B1+2εμ4m2B13c1+16m2A0B13c1k1+4mε2A0B1a1+24m2A0A1B1c1+21-2mRεA1B1a1+6RmεA1B1a1-6mμεA0B1a1+48m2A02A1B1c1k1=0,τ:8m2A0B1b1-Rε4m2B13c1+16m2A0B13c1k1+4mε2A0B1a1+12m2A02B1c1+16m2A03B1c1k1+4RmεA0B1a1=0,σ0:R2ε24c1k1m2B14+4a1mε2B12+1-2ma1ε2B12+4m2A02b1+4m2A03c1+1-2mR2B12a1-Rε24c1k1m2A02B12+12c1m2A0B12+4b1m2B12+4Rεa1mB12+21-2mRεa1B12+4m2A04c1k1=0.According to Step 5, of Section 2, there are three cases of solutions of the algebraic equations (26) to be discussed as follows:Case 1 (ε=-1, r=-1). It leads to the following results.Result 1. We have(27)A0=-b1m+1c1,A1=0,B1=±m+1-b1a1c1m,μ=0,R=-b1m2a1,k1=c12m+14b1m+12.From (11), (25), and (27), we deduce the kink-shaped soliton solutions of (1) as follows:(28)q11x,t=-b1m+1c11±tanh⁡-b1m2a1ξ1/2mexp⁡i-kx+ωt+θ,where a1b1<0 and c1b1<0.Result 2. We have(29)A0=0,A1=a1μm+12c1m2,B1=0,R=-4b1m2a1,k1=c1μ2-12m+14b1μ2m+12.In this result, we deduce the bell-shaped soliton solutions of (1) as follows:(30)q12x,t=-2b1μm+1c1sech⁡-4b1m2/a1ξμsech⁡-4b1m2/a1ξ+11/2mexp⁡i-kx+ωt+θ,where a1b1<0 and c1b1μ<0.Result 3. We have(31)A0=-b1m+1c1,A1=±a1m+1μ2-14c1m2,B1=±m+1-a1b12c1m,R=-4b1m2a1,k1=c12m+14b1m+12.In this result, we deduce the bell-kink-shaped soliton solutions of (1) as follows:(32)q13x,t=-b1m+1c11±μ2-1sech⁡-4b1m2/a1ξμsech⁡-4b1m2/a1ξ+1±tanh⁡-4b1m2/a1ξμsech⁡-4b1m2/a1ξ+11/2mexp⁡i-kx+ωt+θ,where a1b1<0,c1b1<0 and μ2>1.Case 2 (ε=-1, r=1). It leads to the following results.Result 1. We have the same result (27).From (12), (25), and (27), we deduce the singular kink-shaped soliton solutions of (1) as follows:(33)q21x,t=-b1m+1c11±coth⁡-b1m2a1ξ1/2mexp⁡i-kx+ωt+θ,where a1b1<0 and c1b1<0.Result 2. We have(34)A0=0,A1=a1μm+12c1m2,B1=0,R=-4b1m2a1,k1=c1μ2+12m+14b1μ2m+12.In this result, we deduce the singular bell-shaped soliton solutions of (1) as follows:(35)q22x,t=-2b1μm+1c1csch⁡-4b1m2/a1ξμcsch⁡-4b1m2/a1ξ+11/2mexp⁡i-kx+ωt+θ,where a1b1<0 and c1b1μ<0.Result 3. We have(36)A0=-b1m+1c1,A1=±a1m+1μ2+14c1m2,B1=±m+1-a1b12c1m,R=-4b1m2a1,k1=c12m+14b1m+12.In this result, we deduce the singular bell-kink-shaped soliton solutions of (1) as follows:(37)q23x,t=-b1m+1c11±μ2+1csch⁡-4b1m2/a1ξμcsch⁡-4b1m2/a1ξ+1±coth⁡-4b1m2/a1ξμcsch⁡-4b1m2/a1ξ+11/2mexp⁡i-kx+ωt+θ,where a1b1<0 and c1b1<0.Case 3 (ε=1, r=-1). It leads to the following results.Result 1. We have(38)A0=0,A1=-a1μm+12c1m2,B1=0,R=4b1m2a1,k1=c1μ2-12m+14b1μ2m+12.From (13), (25), and (38), we deduce the periodic solutions of (1) as follows:(39)q31x,t=-2b1μm+1c1sec⁡4b1m2/a1ξμsec⁡4b1m2/a1ξ+11/2mexp⁡i-kx+ωt+θ,(40)q32x,t=-2b1μm+1c1csc⁡4b1m2/a1ξμcsc⁡4b1m2/a1ξ+11/2mexp⁡i-kx+ωt+θ,where a1b1>0 and c1b1μ<0.Remark 1.
Note that our results (28) and (33) are in agreement with the results obtained in [5], while the other results are new, which are not found elsewhere.Remark 2.
Note that, ifR=μ=0, then we have the trivial solution.


## 3.2. On Solving (2) Using the Method of Section 2

In this subsection, we apply the generalized projective Riccati equations method of Section2 to find new soliton and periodic solutions of (2). To this aim, we use the following wave transformation:(41)UX,Y,Z,T=Uξ,ξ=αX+βY+γZ+υT,to reduce (2) to the following nonlinear ODE:(42)VU′-AUU′-Bγ2+Cα2+β2U′′′=0,where α,β,γ are the direction cosines, and υ is the quantum ion-acoustic wave speed and V=υ/γ. Integrating (42) with respect to ξ once and vanishing the constant of integration, we find the following nonlinear ODE:(43)VU-12AU2-Bγ2+Cα2+β2U′′=0.Balancing U′′ with U2 gives N=2. Therefore, (6) reduces to(44)Uξ=A0+A1σξ+A2σ2ξ+B1τξ+B2σξτξ,where Ai(i=0,1,2), B1 and B2 are constants to be determined such that A2≠0 or B2≠0.Substituting (44) and using (7) and (8) into (43), the left-hand side of (43) becomes a polynomial in σ(ξ) and τ(ξ). Setting the coefficients of this polynomial to be zero yields the following system of algebraic equations:(45)σ4:12ARεB22μ2+r-12AA22=0,σ3:2Bγ2+Cα2+β2μεA2-AA1A2-12Aε2μB22-2RB1B2μ2+r=0,σ3τ:AA2B2-4Rε3B2μ2+rBγ2+Cα2+β2=0,σ2:VA2-12AA12+2A0A2+12AεRB22+1RB12μ2+r-4μB1B2=0,σ2τ:AA1B2+A2B1+Bγ2+Cα2+β2ε2A1-6μεB2+8Bγ2+Cα2+β2με3B2=0,σ:VA1-12Aε2μB12-2RB1B2-AA0A1=0,στ:VB2-Bγ2+Cα2+β25RεB2-3μεB1-AA0B2+A1B1+4Bγ2+Cα2+β2Rε3B2=0,τ:AA0B1-VB1+2Bγ2+Cα2+β2RεB1=0,σ0:VA0-12AA02-Bγ2+Cα2+β2εB2+12ARεB12=0.According to Step 5 of Section 2, there are three cases of solutions of the algebraic equations (45) to be discussed as follows.Case 1 (ε=-1, r=-1). It leads to the following results.Result 1. We have (46)A0=0,A1=6μBγ2+Cα2+β2A,A2=-6μ2-1Bγ2+Cα2+β2AR,B1=0,B2=±6μ2-1Bγ2+Cα2+β2AR,V=RBγ2+Cα2+β2.From (11), (44), and (46), we deduce the bell-kink-shaped soliton solution of (2) as follows:(47)U11X,Y,Z,T=6Bγ2+Cα2+β2ARsech⁡Rξμsech⁡Rξ+11-μ2sech⁡Rξμsech⁡Rξ+1±μ2-1tanh⁡Rξμsech⁡Rξ+1+μ,where R>0,μ2>1 and ξ=αX+βY+γZ+RγBγ2+C(α2+β2)T.Result 2. We have(48)A0=-2RBγ2+Cα2+β2A,A1=6μBγ2+Cα2+β2A,A2=-6μ2-1Bγ2+Cα2+β2AR,B1=0,B2=±6μ2-1Bγ2+Cα2+β2AR,V=-RBγ2+Cα2+β2.In this result, we deduce the bell-kink-shaped soliton solution of (2) as follows:(49)U12X,Y,Z,T=6Bγ2+Cα2+β2ARsech⁡Rξμsech⁡Rξ+11-μ2sech⁡Rξμsech⁡Rξ+1±μ2-1tanh⁡Rξμsech⁡Rξ+1+μ-2RBγ2+Cα2+β2A,where R>0,μ2>1 and ξ=αX+βY+γZ-RγBγ2+C(α2+β2)T.Case 2 (ε=-1, r=1). It leads to the following results:Result 1. We have(50)A0=0,A1=6μBγ2+Cα2+β2A,A2=-6μ2+1Bγ2+Cα2+β2AR,B1=0,B2=±6μ2+1Bγ2+Cα2+β2AR,V=RBγ2+Cα2+β2.From (12), (44), and (50), we deduce the singular bell-kink-shaped soliton solution of (2) as follows:(51)U21X,Y,Z,T=6Bγ2+Cα2+β2ARcsch⁡Rξμcsch⁡Rξ+1-μ2+1csch⁡Rξμcsch⁡Rξ+1±μ2+1coth⁡Rξμcsch⁡Rξ+1+μ,where R>0 and ξ=αX+βY+γZ+RγBγ2+Cα2+β2T.Result 2. We have(52)A0=-2RBγ2+Cα2+β2A,A1=6μBγ2+Cα2+β2A,A2=-6μ2+1Bγ2+Cα2+β2AR,B1=0,B2=±6μ2+1Bγ2+Cα2+β2AR,V=-RBγ2+Cα2+β2.In this result, we deduce the singular bell-kink-shaped soliton solution of (2) as follows:(53)U22X,Y,Z,T=6Bγ2+Cα2+β2ARcsch⁡Rξμcsch⁡Rξ+1-μ2+1csch⁡Rξμcsch⁡Rξ+1±μ2+1coth⁡Rξμcsch⁡Rξ+1+μ-2RBγ2+Cα2+β2A,where R>0 and ξ=αX+βY+γZ-RγBγ2+C(α2+β2)T.Case 3 (ε=1, r=-1). It leads to the following results.Result 1. We have(54)A0=0,A1=-6μBγ2+Cα2+β2A,A2=6μ2-1Bγ2+Cα2+β2AR,B1=0,B2=±61-μ2Bγ2+Cα2+β2AR,V=-RBγ2+Cα2+β2.From (13), (44), and (54), we deduce the periodic solutions of (2) as follows:(55)U31X,Y,Z,T=6Bγ2+Cα2+β2ARsec⁡Rξμsec⁡Rξ+1μ2-1sec⁡Rξμsec⁡Rξ+1±1-μ2tan⁡Rξμsec⁡Rξ+1-μ,U32X,Y,Z,T=6Bγ2+Cα2+β2ARcsc⁡Rξμcsc⁡Rξ+1μ2-1csc⁡Rξμcsc⁡Rξ+1±1-μ2cot⁡Rξμcsc⁡Rξ+1-μ,where R>0,μ2<1 and ξ=αX+βY+γZ-RγBγ2+C(α2+β2)T.Result 2. We have(56)A0=2RBγ2+Cα2+β2A,A1=-6μBγ2+Cα2+β2A,A2=6μ2-1Bγ2+Cα2+β2AR,B1=0,B2=±61-μ2Bγ2+Cα2+β2AR,V=RBγ2+Cα2+β2.In this result, we deduce the periodic solutions of (2) as follows:(57)U33X,Y,Z,T=6Bγ2+Cα2+β2ARsec⁡Rξμsec⁡Rξ+1μ2-1sec⁡Rξμsec⁡Rξ+1±1-μ2tan⁡Rξμsec⁡Rξ+1-μ+2RBγ2+Cα2+β2A,U34X,Y,Z,T=6Bγ2+Cα2+β2ARcsc⁡Rξμcsc⁡Rξ+1μ2-1csc⁡Rξμcsc⁡Rξ+1±1-μ2cot⁡Rξμcsc⁡Rξ+1-μ+2RBγ2+Cα2+β2A,where R>0,μ2<1 and ξ=αX+βY+γZ+RγBγ2+C(α2+β2)T.


## 4. Graphical Representations of Some Solutions

In this section, we draw graphs of some exact solutions. The obtained soliton and periodic solutions are kink and antikink solitons, bell (bright) and antibell (dark) solitary wave solutions, and trigonometric solutions.Let us now examine Figures1–3 as they illustrate some of our solutions obtained in this article. To this aim, we select some special values of the parameters obtained, for example, in some of the solutions (28), (37), and (39) of the NLS equation with fourth-order dispersion and dual power law nonlinearity (1). For more convenience the graphical representations of these solutions are shown in Figures 1, 2, and 3.Figure 1
Plot solutionq11x,t of (28) with a1=-1,b1=11/5,c1=-2,m=v=1.Figure 2
Plot solutionq23x,t of (37) with a1=23,b1=-12,c1=168,μ=1,v=-1,m=2.Figure 3
Plot solutionq31x,t of (39) with a1=-19,b1=-4,c1=20,μ=3,v=1,m=2.From Figures1, 2, and 3, one can see that the obtained solutions possess the solitary wave solutions and periodic wave solution. Also, these figures express the behavior of these solutions which give some perspective to readers on how the behavior solutions are produced.


## 5. Conclusions

In this article, based on the generalized projective Riccati equations method described in Section2, we have obtained many new types of soliton and periodic solutions for the two higher-order nonlinear PDEs, namely, the NLS equation (1) and the nonlinear QZK equation (2). The exact solutions of these equations include kink and antikink solitons, bell (bright) and antibell (dark) solitary wave solutions, and periodic solutions. Comparing our results in this article with the well-known results of [5, 6, 21–23], we conclude that all our results are new and not found elsewhere except for results (28) and (33) which are in agreement with results (19) and (20) obtained in [5]. The proposed method of this article is effective and can be applied to many other nonlinear PDEs. Finally, all solutions obtained in this article have been checked with the Maple 2015 by putting them back into the original equations (1) and (2).



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