# A Novel Approach to a Time-Dependent-Coefficient WBK System: Doubly Periodic Waves and Singular Nonlinear Dynamics

Authors: Bo Xu; Sheng Zhang
Journal: Complexity
 (2018)

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## Abstract

Under investigation in this paper is a more general time-dependent-coefficient Whitham-Broer-Kaup (tdcWBK) system, which includes some important models as special cases, such as the approximate equations for long water waves, the WBK equations in shallow water, the Boussinesq-Burgers equations, and the variant Boussinesq equations. To construct doubly periodic wave solutions, we extend the generalizedF-expansion method for the first time to the tdcWBK system. As a result, many new Jacobi elliptic doubly periodic solutions are obtained; the limit forms of which are the hyperbolic function solutions and trigonometric function solutions. It is shown that the original F-expansion method cannot derive Jacobi elliptic doubly periodic solutions of the tdcWBK system, but the novel approach of this paper is valid. To gain more insight into the doubly periodic waves contained in the tdcWBK system, we simulate the dynamical evolutions of some obtained Jacobi elliptic doubly periodic solutions. The simulations show that the doubly periodic waves possess time-varying amplitudes and velocities as well as singularities in the process of propagations.


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## 1. Introduction

Nonlinear complex phenomena in natural world, for example, solitons first observed by Russell in 1834 [1], are often described by nonlinear PDEs. Usually, people restore to exact solutions of nonlinear PDEs to gain more insight into the essence behind these nonlinear phenomena for further applications. In the past several decades, many effective methods for exactly solving nonlinear PDEs have been presented like those in [2–22]. In 2003, Zhou et al. proposed the so-called F-expansion method [22] to construct different Jacobi elliptic doubly periodic solutions of nonlinear PDEs in a uniform way, which can be thought of as an overall generalization of the Jacobi elliptic function expansion method [23]. The F-expansion method has been widely used to a great many of nonlinear PDEs [24–26] and was improved in different manners [27–30]. In 2006, Zhang and Xia [30] generalized the F-expansion method by introducing a new and more general ansätz. The present paper is motivated by the desire to extend the generalized F-expansion method [30] to the new and more general tcdWBK system [31, 32]:
(1)ut+γ1uux+γ2vx+γ3uxx=0,vt+γ4uxv+γ4uvx−γ5vxx+γ6uxxx=0,where γii=1,2,…,6 are arbitrary smooth functions of t, which represent different dispersion and dissipation forces. Clearly, (1) includes some existing well-known important equations as special cases; they are the approximate equations for long water waves [33]:
(2)ut−uux−vx+12uxx=0,vt−uvx−12vxx=0,the WBK equations in shallow water [34]:
(3)ut+uux+vx+βuxx=0,vt+uvx+αuxxx−βvxx=0,the Boussinesq-Burgers (BB) equations [35]:
(4)ut+2uux−12vx=0,vt+2uvx−12uxxx=0,the variant Boussinesq equations [34]:
(5)ut+uux+vx=0,vt+uvx+uxxx=0.It should be pointed out that theF-expansion method [22] cannot derive Jacobi elliptic doubly periodic solutions of (1). To be specific, according to the F-expansion method [22], we first suppose that (1) has exact solutions of the forms:
(6)u=a0+∑i=1naiFiξ+biF−iξ,v=A0+∑i=1mAiFiξ+BiF−iξ,where ξ=kx+η, the integers n and m and the constant k are to be determined, while a0=a0t, ai=ait, bi=bit, A0=A0t, Ai=Ait, Bi=Bit, and η=ηt are all undetermined functions of the indicated variables; Fξ satisfies
(7)F′2ξ=PF4ξ+QF2ξ+R,and hence holds
(8)F″ξ=2PF3ξ+QFξ,F3ξ=6PF2ξ+QF′ξ,F4ξ=24P2F5ξ+20PQF3ξ+Q2+12PRFξ,…,where P, Q, and R are parameters. In [30], Jacobi elliptic function solutions and their degenerated solutions of (7) are listed, which depend on the values of parameters P, Q, and R. Secondly, substituting (6) along with (7) and (8) into (1) and then balancing the highest order partial derivative uux and the highest order nonlinear term uxx yield the integer 2n+1=n+2 which gives n=1. Similarly, we determine the integer m=2 by balancing uvx and uxxx. Thirdly, we substitute (6) given the values of n=1 and m=2 along with (7) and (8) into (1) and collect all terms with the same order of FjξF′sξj=0,±1,±2,…,s=0,1 together, we have
(9)−kb12γ1+2B2γ2F′ξF−3ξ−ka0b1γ1+kB1γ2+b1η′F′ξF−2ξ+ka0a1γ1+kA1γ2+a1η′F′ξ+ka12γ1+2A2γ2F′ξFξ+2k2Rb1γ3F−3ξ+k2Qb1γ3+b1′F−1ξ+a0′+k2Qa1γ3+a1′Fξ+2k2Pa1γ3F3ξ=0,−3kb1B2γ4+2k2Rγ6F′ξF−4ξ−2kb1B1γ4+ka0B2γ4+B2η′F′ξF−3ξ−kA0b1γ4+ka0B1γ4+ka1B2γ4+k3Qb1γ6+B1η′F′ξF−2ξ+kA0a1γ4+ka0A1γ4+kA2b1γ4+k3Qa1γ6+A1η′F′ξ+2ka1A1γ4+ka0A2γ4+A2η′F′ξFξ+3ka1A2γ4+2k2Pγ6F′ξF2ξ−6k2RB2γ5F−4ξ−2k2RB1γ5F−3ξ−4k2QB2γ5−B2′F−2ξ−k2QB1γ5−B1′F−1ξ−2k2RA2γ5−2k2PB2γ5+A0′−k2QA1γ5−A1′Fξ−4k2QA2γ5−A2′F2ξ−2k2PA1γ5F3ξ−6k2PA2γ5F4ξ=0.SincePQR≠0 is the necessary condition that (7) exists Jacobi elliptic doubly periodic, without loss of generality, we have
(10)a0′=a1=b1=A0′=A1=A2=B1=B2=0,when setting each coefficient of FjξF′sξ of (9) to zeros. This tells that (6) let (1) has only constant solutions but not Jacobi elliptic doubly periodic solutions as expected.The present paper is motivated by the desire to investigate Jacobi elliptic doubly periodic solutions of (1). The rest of the paper is organized as follows. In Section 2, the generalized F-expansion method [30] is extended to (1) for constructing Jacobi elliptic doubly periodic solutions. In Section 3, we simulate the dynamical evolutions of some obtained Jacobi elliptic doubly periodic solutions to gain more insight into the doubly periodic waves contained in (1). In Section 4, we conclude this paper.


## 2. Doubly Periodic Waves

To extend the generalizedF-expansion method [30] to (1), in this section, we suppose that (1) has Jacobi elliptic doubly periodic solutions of the forms:
(11)u=a0+a1Fξ+b1F−1ξ+c1F′ξ+d1F−1ξF′ξ,v=A0+A1Fξ+A2F2ξ+B1F−1ξ+B2F−2ξ+C1F′ξ+C2FξF′ξ+D1F−1ξF′ξ+D2F−2ξF′ξ.Substituting (11) along with (7) and (8) into (1) and collecting all terms with the same order of FjξF′sξj=0,±1,±2,…,s=0,1 together, we derive a set of nonlinear PDEs for a0, a1, b1, c1, d1, A0, A1, A2, B1, B2, C1, C2, D1, D2, η, and k. Solving this set of nonlinear PDEs, we have three cases.Case 1.
(12)a1=±2kPγ32+c0γ1γ6γ1,b1=±2kRγ32+c0γ1γ6γ1,a0=A0=const.,A2=−2k2Pγ3γ32+c0γ1γ6c0γ12,B2=−2k2Rγ3γ32+c0γ1γ6c0γ12,C1=∓2k2γ3Pγ32+c0γ1γ6c0γ12,D2=±2k2γ3Rγ32+c0γ1γ6c0γ12,c1=d1=A1=B1=C2=D1=0,η=−ka0∫γ1dt,where k is an arbitrary constant; the signs “±” and “∓” in (12) and (13) mean that all possible combinations of “+” and “−” can be taken. If it is taken the same sign in a1 and b1, then it must be taken “−” in γ3′ and γ6′. If it is taken the different signs in a1 and b1, then it must be taken “+” in γ3′ and γ6′. At the same time, it must be taken the different signs in a1 and C1 and the same sign in b1 and D2. While γii=1,3,6 in (12) satisfy the constraints:
(13)γ4=γ1,γ5=γ3,γ2=c0γ1,γ3′=c0A0γ12+k2Q∓2Pγ32+c0γ1γ6+γ3γ1′γ1,γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12+k2Q∓2Pγ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.Case 2.
(14)a1=±2kPγ32+c0γ1γ6γ1,a0=A0=const.,(15)A2=−2k2Pγ3γ32+c0γ1γ6c0γ12,C1=∓2k2γ3Pγ32+c0γ1γ6c0γ12,(16)b1=c1=d1=A1=B1=B2=C2=D1=D2=0,η=−ka0∫γ1dt,where k is an arbitrary constant; the signs “±” and “∓” in (14) and (15) mean that it must be taken the different signs in a1 and C1, while γii=1,3,6 in (14), (15), and (16) satisfy the constraints:
(17)γ4=γ1,γ5=γ3,γ2=c0γ1,γ3′=c0A0γ12+k2Qγ32+c0γ1γ6+γ3γ1′γ1,γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12+k2Qγ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.Case 3.
(18)b1=±2kRγ32+c0γ1γ6γ1,a0=A0=const.,(19)B2=−2k2Rγ3γ32+c0γ1γ6c0γ12,D2=∓2k2γ3Rγ32+c0γ1γ6c0γ12,(20)a1=c1=d1=A1=A2=B1=C1=C2=D1=0,η=−ka0∫γ1dt,where k is an arbitrary constant; the signs “±” and “∓” in (18) and (19) mean that it must be taken the same sign in b1 and D2, while γii=1,3,…,6 in (18), (19), and (20) satisfy the same constraints as Case 2 in (17).From Cases1–3, we obtain three formulae of fundamental solutions of (1) as follows:
(21)u=a0±2kPγ32+c0γ1γ6γ1Fξ±2kRγ32+c0γ1γ6γ1F−1ξ,v=A0−2k2Pγ32+c0γ1γ6c0γ12F2ξ−2k2Rγ32+c0γ1γ6c0γ12F−2ξ∓2k2γ3Pγ32+c0γ1γ6c0γ12F′ξ±2k2γ3Rγ32+c0γ1γ6c0γ12F−2ξF′ξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,2,3,6 in (21) satisfy the constraints in (13).
(22)u=a0±2kPγ32+c0γ1γ6γ1Fξ,v=A0−2k2Pγ32+c0γ1γ6c0γ12F2ξ∓2k2γ3Pγ32+c0γ1γ6c0γ12F′ξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,3,6 in (22) satisfy the constraints in (17).
(23)u=a0±2kRγ32+c0γ1γ6γ1F−1ξ,v=A0−2k2Rγ32+c0γ1γ6c0γ12F−2ξ±2k2γ3Rγ32+c0γ1γ6c0γ12F−2ξF′ξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,2,3,6 in (24) and (25) satisfy the constraints in (17).With the help of (22), (23), (24), and (25) and (Appendices A, B, and C [30]), we obtain many exact solutions of (1). For example, selecting P=1, Q=−1+m2, R=m2, and Fξ=nsξ, from (21), we obtain new Jacobi elliptic doubly periodic solutions of (1):
(24)u=a0±2kγ32+c0γ1γ6γ1nsξ±2kmγ32+c0γ1γ6γ1snξ,(25)v=A0−2k2γ32+c0γ1γ6c0γ12ns2ξ−2k2m2γ32+c0γ1γ6c0γ12sn2ξ±2k2γ3γ32+c0γ1γ6c0γ12csξdsξ∓2k2mγ3γ32+c0γ1γ6c0γ12sn2ξcsξdsξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,3,6 in (24) and (25) satisfy the constraints γ4=γ1, γ5=γ3, γ2=c0γ1, and
(26)γ3′=c0A0γ1′+k2−1−m2∓2γ3′+c0γ1γ6+γ3γ1′γ1,γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12+k2−1−m2∓2γ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.SelectingP=−1, Q=2−m2, R=m2−1, and Fξ=dnξ, from (22), we obtain new Jacobi elliptic doubly periodic solutions of (1):
(27)u=a0±2k−γ32+c0γ1γ6γ1dnξ,v=A0+2k2γ32+c0γ1γ6c0γ12dn2ξ±2k2m2γ3−γ32+c0γ1γ6c0γ12snξcnξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,3,6 in (27) satisfy the constraints γ4=γ1, γ5=γ3, γ2=c0γ1, and
(28)γ3′=c0A0γ12+k22−m2γ32+c0γ1γ6+γ3γ1′γ1,γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12+k22−m2γ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.SelectingP=1, Q=−1+m2, R=m2, and Fξ=nsξ, from (22), we obtain new Jacobi elliptic doubly periodic solutions of (1):
(29)u=a0±2kγ32+c0γ1γ6γ1nsξ,v=A0−2k2γ32+c0γ1γ6c0γ12ns2ξ±2k2γ3γ32+c0γ1γ6c0γ12csξdsξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,3,6 in (29) satisfy the constraints γ4=γ1, γ5=γ3, γ2=c0γ1, and
(30)γ3′=c0A0γ12−k21+m2γ32+c0γ1γ6+γ3γ1′γ1,γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12−k21+m2γ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.SelectingP=1−m2, Q=2−m2, R=1, and Fξ=scξ, from (22), we obtain new Jacobi elliptic doubly periodic solutions of (1):
(31)u=a0±2k1−m2γ32+c0γ1γ6γ1scξ,v=A0−2k21−m2γ32+c0γ1γ6c0γ12sc2ξ∓2k2γ31−m2γ32+c0γ1γ6c0γ12dcξncξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,3,6 in (31) satisfy the constraints γ4=γ1, γ5=γ3, γ2=c0γ1, and
(32)γ3′=c0A0γ12+k22−m2γ32+c0γ1γ6+γ3γ1′γ1,γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12+k22−m2γ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.SelectingP=1, Q=2m2−1, R=−m21−m2, and Fξ=dsξ, from (22), we obtain new Jacobi elliptic doubly periodic solutions of (1):
(33)u=a0±2kγ32+c0γ1γ6γ1dsξ,v=A0−2k2γ32+c0γ1γ6c0γ12ds2ξ±2k2γ3γ32+c0γ1γ6c0γ12csξnsξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,3,6 in (33) satisfy the constraints γ4=γ1, γ5=γ3, γ2=c0γ1, and
(34)γ3′=c0A0γ12+k22m2−1γ32+c0γ1γ6+γ3γ1′γ1,(35)γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12+k22m2−1γ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.In the limits atm→1 and m→0, the above obtained Jacobi elliptic doubly periodic solutions degenerate into hyperbolic function solutions and trigonometric function solutions, respectively. When m→1, the Jacobi elliptic doubly periodic solutions (24) and (25) degenerate into hyperbolic function solutions:
(36)u=a0±2kγ32+c0γ1γ6γ1cothξ±2kγ32+c0γ1γ6γ1tanhξ,v=A0−2k2γ32+c0γ1γ6c0γ12coth2ξ−2k2γ32+c0γ1γ6c0γ12tanh2ξ±2k2γ3γ32+c0γ1γ6c0γ12csch2ξ∓2k2γ3γ32+c0γ1γ6c0γ12sech2ξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,3,6 in (36) satisfy the constraints γ4=γ1, γ5=γ3, γ2=c0γ1, and
(37)γ3′=c0A0γ12+k2−2∓2γ32+c0γ1γ6+γ3γ1′γ1,γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12+k2−2∓2γ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.Whenm→0, the Jacobi elliptic doubly periodic solutions (24) and (25) degenerate into trigonometric function solutions:
(38)u=a0±2kγ32+c0γ1γ6γ1cscξ,v=A0−2k2γ32+c0γ1γ6c0γ12csc2ξ±2k2γ3γ32+c0γ1γ6c0γ12cotξcscξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,3,6 in (38) satisfy the constraints γ4=γ1, γ5=γ3, γ2=c0γ1, and
(39)γ3′=c0A0γ12+k2−1∓2γ32+c0γ1γ6+γ3γ1′γ1,γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12+k2−1∓2γ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.Whenm→1, the Jacobi elliptic doubly periodic solutions (27) degenerate into hyperbolic function solutions:
(40)u=a0±2k−γ32+c0γ1γ6γ1sechξ,v=A0−2k2γ32+c0γ1γ6c0γ12sech2ξ±2k2γ3−γ32+c0γ1γ6c0γ12tanhξsechξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,3,6 in (40) satisfy the constraints γ4=γ1, γ5=γ3, and
(41)γ3′=c0A0γ12+k2γ32+c0γ1γ6+γ3γ1′γ1,γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12+k2γ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.Whenm→1, the Jacobi elliptic doubly periodic solutions (29) degenerate into hyperbolic function solutions:
(42)u=a0±2kγ32+c0γ1γ6γ1cothξ,(43)v=A0−2k2γ32+c0γ1γ6c0γ12coth2ξ±2k2γ3γ32+c0γ1γ6c0γ12csch2ξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,3,6 in (42) and (43) satisfy the constraints γ4=γ1, γ5=γ3, γ2=c0γ1, and
(44)γ3′=c0A0γ12−2k2γ32+c0γ1γ6+γ3γ1′γ1,γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12−2k2γ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.Whenm→0, the Jacobi elliptic doubly periodic solutions (31) degenerate into trigonometric function solutions:
(45)u=a0±2kγ32+c0γ1γ6γ1tanξ,v=A0−2k2γ32+c0γ1γ6c0γ12tan2ξ∓2k2γ3γ32+c0γ1γ6c0γ12sec2ξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,3,6 in (45) satisfy the constraints γ4=γ1, γ5=γ3, γ2=c0γ1, and
(46)γ3′=c0A0γ12+2k2γ32+c0γ1γ6+γ3γ1′γ1,γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12+2k2γ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.Whenm→1, the Jacobi elliptic doubly periodic solutions (33) degenerate into trigonometric function solutions:
(47)u=a0±2kγ32+c0γ1γ6γ1cschξ,v=A0−2k2γ32+c0γ1γ6c0γ12csch2ξ±2k2γ3γ32+c0γ1γ6c0γ12cothξcschξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,3,6 in (47) satisfy the constraints γ4=γ1, γ5=γ3, γ2=c0γ1, and
(48)γ3′=c0A0γ12+k2γ32+c0γ1γ6+γ3γ1′γ1,γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12+k2γ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.Whenm→0, the Jacobi elliptic doubly periodic solutions (33) degenerate into the following trigonometric function solutions which have the same expressions as solutions (38) but with different constraints (50):
(49)u=a0±2kγ32+c0γ1γ6γ1cscξ,v=A0−2k2γ32+c0γ1γ6c0γ12csc2ξ±2k2γ3γ32+c0γ1γ6c0γ12cotξcscξ,where a0, A0, and k are arbitrary constants, ξ=kx−ka0∫γ1dt, and γii=1,3,6 in (49) satisfy the constraints γ4=γ1, γ5=γ3, γ2=c0γ1, and
(50)γ3′=c0A0γ12−k2γ32+c0γ1γ6+γ3γ1′γ1,γ6′=2c0γ1γ1′γ32+c02γ12γ1′γ6−2c0γ12γ3c0A0γ12−k2γ32+c0γ1γ6−2c0γ1γ1′γ3c02γ13.


## 3. Singular Nonlinear Dynamics

In this section, we further investigate the nonlinear dynamics of (1) by means of Jacobi elliptic doubly periodic solutions. Firstly, we consider solutions (24) and (25). To determine γ3 and γ6 with the sign “−” in (26), we select γ1=et and then have
(51)γ3=e−e−2t/2A0c0c1∫ee−2t/2A0c0dt+c2,γ6=−e−e−2t/2A0c0−tA0c0c1−1+m22ee−2t/2A0c0+2t+k2−1+m22+k21−m2c1∫ee−2t/2A0c0dt+c22c0k2−1+m22,where c1 and c2 are two integration constants.In Figures1 and 2, the spatial structures and contour lines of solutions (24) and (25) determined by (51) are shown by selecting the parameters as a0=2, A0=0.5, c0=−4, c1=3, c2=−1, k=1.5, and m=0.8, respectively. We shown the nonlinear dynamical evolutions of solutions (24) and (25) in Figures 3 and 4. It is easy to see from Figures 1–4 that the doubly periodic waves determined by solutions (24) and (25) possess time-varying amplitudes and velocities as well as singularities in the process of propagations.Figure 1
Spatial structures of solutions (24) and (25) determined by (51).Figure 2
Contour lines of solutions (24) and (25) determined by (51).Figure 3
Nonlinear dynamical evolutions of solution (24) determined by (51).
(a)
t=−3
(b)
t=0
(c)
t=3Figure 4
Nonlinear dynamical evolutions of solution (25) determined by (51).
(a)
t=−3
(b)
t=0
(c)
t=3Secondly, we consider solutions (27). To determine γ3 and γ6 in (28), we let γ1=et and then have
(52)γ3=c1et+c2e2t,γ6=et−A0c0+c2−k22−m2c1+c2et2c0k22−m2,where c1 and c2 are two integration constants.In Figure5, the spatial structures of solutions (27) determined by (52) are shown by selecting the parameters as a0=2, A0=0.5, c0=−4, c1=3, c2=−1, k=1.5, and m=0.8, respectively. We shown the spatial structures of solutions (40) in Figure 6. It is easy to see from Figures 5 and 6 that both the doubly periodic waves determined by solutions (27) and the hyperbolic function solutions (40) possess time-varying amplitudes and velocities as well as singularities in the process of propagations.Figure 5
Spatial structures of solutions (27) determined by (52).Figure 6
Spatial structures of solutions (40) determined by (52) given m=1.Finally, we consider solutions (33). To determine γ3 and γ6 in (34), (35), we let γ1=et and then have
(53)γ3=c1et+c2e2t,γ6=et−A0c0−c2−k21−m2c1+c2et2c0k21−m2,where c1 and c2 are two integration constants.In Figure7, the spatial structures of solutions (33) determined by (53) are shown by selecting the parameters as a0=2, A0=0.5, c0=−4, c1=3, c2=−1, k=1.5, and m=0.8, respectively. We can see from Figure 7 that the doubly periodic waves determined by solutions (33) possess time-varying amplitudes and velocities as well as singularities in the process of propagations.Figure 7
Spatial structures of solutions (33) determined by (52) given m=1.


## 4. Conclusion

In summary, new and more general Jacobi elliptic doubly periodic solutions of the tdcWBK system have been obtained, which degenerate into the hyperbolic function solutions and trigonometric function solutions in the limit cases. To the best of our knowledge, the obtained Jacobi elliptic doubly periodic solutions have not been reported in literatures. It is shown that the originalF-expansion method cannot derive Jacobi elliptic doubly periodic solutions of the tdcWBK system but the novel approach of this paper is valid. In this sense, we would like to conclude that a novel approach of the generalized F-expansion method is extended to the tdcWBK system. The simulations show that the doubly periodic waves possess time-varying amplitudes and velocities as well as singularities in the process of propagations. Recently, fractional-order differential calculus and its applications have attached much attention [36–51]. Constructing Jacobi elliptic doubly periodic solutions of nonlinear PDEs with fractional derivatives is worthy of the study. At the same time, constructing multisoliton solutions via the Riemann-Hilbert approach, for example, see Kang et al.’s meaningful work [52, 53], has been a hot topic. Dealing with initial-boundary problems of nonlinear PDEs by means of the Riemann-Hilbert approach is also worthy of the study.



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