COMPLEXITY Complexity 1099-0526 1076-2787 Hindawi 10.1155/2019/7952871 7952871 Research Article Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example https://orcid.org/0000-0002-4631-7033 Zhang Sheng 1 2 3 Wei Yuanyuan 1 Xu Bo 4 Boutayeb Mohamed 1 School of Mathematics and Physics Bohai University Jinzhou 121013 China bhu.edu.cn 2 Department of Mathematics Hohhot Minzu College Hohhot 010051 China 3 School of Mathematical Sciences Qufu Normal University Qufu 273165 China qfnu.edu.cn 4 School of Education and Sports Bohai University Jinzhou 121013 China bhu.edu.cn 2019 682019 2019 25 03 2019 19 06 2019 682019 2019 Copyright © 2019 Sheng Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, the spectral transform with the reputation of nonlinear Fourier transform is extended for the first time to a local time-fractional Korteweg-de vries (tfKdV) equation. More specifically, a linear spectral problem associated with the KdV equation of integer order is first equipped with local time-fractional derivative. Based on the spectral problem with the equipped local time-fractional derivative, the local tfKdV equation with Lax integrability is then derived and solved by extending the spectral transform. As a result, a formula of exact solution with Mittag-Leffler functions is obtained. Finally, in the case of reflectionless potential the obtained exact solution is reduced to fractional n-soliton solution. In order to gain more insights into the fractional n-soliton dynamics, the dynamical evolutions of the reduced fractional one-, two-, and three-soliton solutions are simulated. It is shown that the velocities of the reduced fractional one-, two-, and three-soliton solutions change with the fractional order.

National Natural Science Foundation of China 11547005 Natural Science Foundation of Liaoning Province 20170540007 Natural Science Foundation of Education Department of Liaoning Province of China LZ2017002 Innovative Talents Support Program in Colleges and Universities of Liaoning Province LR2016021
1. Introduction

Since the increasing interest on fractional calculus and its applications, dynamical processes and dynamical systems of fractional orders have attracted much attention. In 2010, Fujioka et al.  investigated soliton propagation of an extended nonlinear Schröinger equation with fractional dispersion term and fractional nonlinearity term. In 2014, Yang et al.  used a local fractional KdV equation to model fractal waves on shallow water surfaces.

In the field of nonlinear mathematical physics, the spectral transform  with the reputation of nonlinear Fourier transform is a famous analytical method for constructing exact and explicit n-soliton solutions of nonlinear partial differential equations (PDEs). Since put forward by Gardner et al. in 1967, the spectral transform method has achieved considerable developments . With the close attentions of fractional calculus and its applications , some of the natural questions are whether the existing methods like those in  in soliton theory can be extended to nonlinear PDEs of fractional orders and what about the fractional soliton dynamics and integrability of fractional PDEs. As far as we know there are no research reports on the spectral transform for nonlinear PDEs of fractional orders. This paper is motivated by the desire to extend the spectral transform to nonlinear fractional PDEs and then gain more insights into the fractional soliton dynamics of the obtained solutions. For such a purpose, we consider the following local tfKdV equation:(1)Dtαu+6uux+uxxx=0,0<α1.Here we note that if α=1 then eq. (1) becomes the celebrated KdV equation ut+6uux+uxxx=0. In eq. (1), the local time-fractional derivative Dtαu at the point t=t0 is defined as (2)Dtαux,t0=αux,ttαt=t0=limtt0Δαux,t-ux,t0t-t0α,where Δα(u(x,t)-u(x,t0))Γ(1+α)(u(x,t)-u(x,t0)); some useful properties  of the local time-fractional derivative have been used in this paper.

The rest of this paper is organized as follows. In Section 2, we derive the local tfKdV eq. (1) by introducing a linear spectral problem equipped with local time-fractional derivative. In Section 3, we construct fractional n-soliton solution of the local tfKdV eq. (1) by extending the spectral transform method. In Section 4, we investigate the dynamical evolutions of the obtained fractional one-soliton solution, two-soliton solution, and three-soliton solution. In Section 5, we conclude this paper.

2. Derivation of the Local tfKdV Equation

For the local tfKdV eq. (1), we have the following Theorem 1.

Theorem 1.

The local tfKdV eq. (1) is a Lax system, which can be derived from the linear spectral problem equipped with a local time-fractional evolution equation:(3)ϕxx=η-uϕ,η=-k2,(4)Dtαϕ=ux+ϑϕ+4k2-2uϕx,where ϕ=ϕ(x,t) and u=u(x,t) are all differentiable functions with respect to x and t, the spectral parameter η is independent of t, and ϑ is an arbitrary constant.

Proof.

Taking the time-fractional derivative of eq. (3) yields(5)Dtαϕxx=-ϕDtαu-u+k2Dtαϕ,Substituting eq. (4) into eq. (5), we have(6)Dtαϕxx=-Dtαu+ux+ϑu+k2ϕ-4k2-2uu+k2ϕx,Taking the derivative of eq. (4) with respect to x twice gives(7)Dtαϕxx=uxxxϕ-3ux-ϑϕxx+4k2-2uϕxxx,With the help of eq. (3), from eq. (7) we have (8)Dtαϕxx=uxxx+5u-k2ux-ϑu+k2ϕ-4k2-2uu+k2ϕx.

On the other hand, at the aribitrary point t=t0 we have(9)Dtαϕxxx,t0=limtt0Δαϕxxx,t-ϕxxx,t0t-t0α,Dtαϕx,t0xx=limtt0Δαϕx,t-ϕx,t0xxt-t0α.

Finally, using eqs. (6), (8), and (9) we arrive at eq. (1). Thus, we finish the proof. The process of proof shows that eq. (1) is a Lax integrable system.

3. Local Fractional Spectral Transform

Since the local tfKdV eq. (1) is a local time-fractional system, the orders of the derivatives with respect to space variable x are integers. So, all the existing results about the spectral problem (3), a part of the Lax pair for the classical KdV equation, can be translated to the local tfKdV eq. (1).

For the direct scattering problem, we translate some necessary results and definitions  to the local tfKdV eq. (1).

Lemma 2.

If the real potential u(x) defined in the whole real axis -<x< and its various derivatives are differentiable functions which vanishes rapidly as x± and satisfies(10)-+xjuxdx<+,j=0,1,2,then the linear spectral problem (3) has a set of basic solutions called Jost solutions ϕ+(x,k) and ϕ-(x,k), and they are not only bounded for all values of x but also analytic for Imk>0 and continuous for Imk0 and have the following asymptotic properties:(11)ϕ+x,keikx,x+,(12)ϕ-x,ke-ikx,x-.

Lemma 3.

Define the Wronskian(13)Wϕ+x,k,ϕ-x,k=ϕx+x,kϕ-x,k-ϕ+x,kϕx-x,k,and let (14)ϕ-x,k=akϕ+x,-k+bkϕ+x,k.Then(15)ak=12ikWϕ-x,k,ϕ+x,k,bk=12ikWϕ+x,-k,ϕ-x,k,where a(k) is analytic for Imk>0 and continuous for Imk0, b(k) is defined only on the real axis Imk=0, and the analytic function a(k) has a finite number of simple zeros km=iκm(κm>0,m=1,2,,n).

Lemma 4.

For the linear spectral problem (3), there exists a constant bm, such that(16)ϕ-x,iκm=bmϕ+x,iκm,(17)-+cm2ϕ+2x,iκmdx=1,cm2=-ibmakiκm.

Definition 5.

The constant cm satisfying eq. (17) is named the normalization constant for the eigenfunction ϕ+(x,iκm), and cmϕ+(x,iκm) is named normalization eigenfunction.

Definition 6.

The set(18)kImk=0,Rk=bkak,iκm,cm,m=1,2,,nis named the scattering data of the linear spectral problem (3).

Lemma 7.

If the eigenfunction ϕ(x,k) satisfies the linear spectral problem (3), then (19)P=Dtαϕ-ux+ϑϕ-4k2-2uϕxsolves eq. (3) as well.

Proof.

A direct computation on eq. (19) tells that(20)Pxx=Dtαϕxx-uxxxϕ-4k2-2uϕxxx+3ux-ϑϕxx.With the help of eqs. (3) and (20), we have(21)Pxx+k2+uP=-ϕDtαu+6uux+uxxx=0.Thus, the proof is finished.

For the time dependence of the scattering data, we have the following Theorem 8.

Theorem 8.

If the time evolution of u(x,t) obeys the local tfKdV eq. (1), then the scattering data (18) for the linear spectral problem (3) possess the following time dependences: (22)Dtακmt=0,Dtαcmt=4κm3tcmt,(23)Dtαak=0,Dtαbk=8ik3bk.

Proof.

Substituting eqs. (12) and (14) into eq. (3) and using the asymptotic properties of eqs. (11) and (12) as x+ and x-, respectively, we have(24)ϑ-4ik3e-ikx=0,(25)Dtαake-ikx+Dtαbkeikx=ϑake-ikx+bkeikx+4k2-ikake-ikx+ikbkeikx.Namely,(26)ϑ=4ik3,(27)Dtαak=akϑ-4ik3=0,(28)Dtαbk=bkϑ+4ik3=8ik3bk.It is easy to see that all the zeros of a(k) are independent of t because of Dt(α)a(k)=0. Therefore, we arrive at Dt(α)κm(t)=0.

Similarly, substituting eq. (16) into eq. (3) and using the asymptotic property of eq. (11) as x+, we have (29)Dtαbmt=ϑ+4κm3tbmt=8κm3tbmt.In view of eqs. (17) and (28), we obtain(30)Dtαcm2t=-Dtαibmtakiκm=8κm3tcm2t,which can be finally reduced to the second term of eq. (22). Then we finish the proof.

For the inverse scattering problem, we have the following Theorem 9.

Theorem 9.

The local tfKdV eq. (1) has an exact solution of the form(31)ux,t=2ddxKx,x,t,where K(x,y,t) satisfies the Gel’fand-Levitan-Marchenko (GLM) integral equation:(32)Kx,y,t+Fx+y,t+xKx,z,tFx+z,tdz=0,with (33)Fx,t=12π-Rk,teikxdk+m=1ncm2eiκmx,and R(k,t), cm, and κm are determined by eqs. (22) and (23).

Proof.

The process of the proof of Theorem 9 is similar to that of the classical KdV equation  with integer order, and the only differences are the scattering data. To avoid unnecessary repetition, we omit it here.

For the fractional n-soliton solution, we have the following Theorem 10.

Theorem 10.

In the case of reflectionless potential, the local tfKdV eq. (1) has fractional n-soliton solution of the form(34)ux,t=2d2dx2lndetDx,t,where(35)Dx,t=djmx,tn×n,djmx,t=δjm+cjtcmtκj0+κm0e-κj0+κm0x,(36)cjt=cj0Eα4κj30tα,cmt=cm0Eα4κm30tα.In eq. (36), Eα(·) is the Mittag-Leffler function .

Proof.

Firstly, we further determine the scattering data. Solving eqs. (22) and (23) yields (37)κmt=κm0,cmt=cm0Eα4κm30tα,(38)ak,t=ak,0,bk,t=bk,0Eα8iκ30tα.

Secondly, we let R(k,t)=0. In this case of reflectionless, eq. (32) reduces to(39)Kx,y,t+m=1ncm2te-κm0x+y+m=1ncm2te-κm0yxKx,z,te-κm0zdz=0.

Suppose that eq. (39) has a separation solution(40)Kx,y,t=j=1ncjthjxe-κj0y,where hj(x) is an undetermined function which can be determined by the substitution of eq. (40) into eq. (39). With the determined function hj(x), we have(41)Kx,y,t=ddxlndetDx,t.Finally, from eqs. (31), (37), (38), and (41) we obtain eq. (34). Therefore, the proof is over.

4. Fractional Soliton Dynamics

In order to gain more insights into the soliton dynamics of the obtained fractional n-soliton solution (34), we consider the cases of n=1,2,3.

When n=1, we have(42)detDx,t=1+c120Eα8κ130tα2κ10e-2κ10x,and we, hence, obtain, from eq. (34), the fractional one-soliton solution:(43)u=2κ120sech2κ10x-12lnc120Eα8κ130tα2κ10.

Similarly, when n=2 we obtain the fractional two-soliton solution:(44)u=2ln1+c120Eα8κ130tα2κ10e-2κ10x+c220Eα8κ230tα2κ20e-2κ20x+c120c220κ10-κ202Eα8κ10tαEα8κ230tα4κ10κ20κ10+κ202e-2κ10+κ20xxx.

When n=3, we obtain the fractional three-soliton solution:(45)u=2ln1+c120Eα8κ130tα2κ10e-2κ10x+c220Eα8κ230tα2κ20e-2κ20x+c320Eα8κ330tα2κ30e-2κ30x+c120c220κ10-κ202Eα8κ10tαEα8κ230tα4κ10κ20κ10+κ202e-2κ10+κ20x+c120c320κ10-κ302Eα8κ10tαEα8κ330tα4κ10κ30κ10+κ302e-2κ10+κ30x+c220c320κ20-κ302Eα8κ20tαEα8κ330tα4κ20κ30κ20+κ302e-2κ20+κ30x+c120c220c320κ10-κ202κ10-κ302κ20-κ3028κ10κ20κ30κ10+κ202κ10+κ302κ20+κ302Eα8κ10tαEα8κ20tαEα8κ330tαe-2κ10+κ20+κ30xxx.

In Figure 1, we simulate the fractional one-soliton solution (43) with different values of α, where the parameters are selected as κ1(0)=0.8 and c1(0)=0.2. With the help of velocity images in Figure 2 and the formula of velocity(46)v=32ακ150tα-1,we can see that the bell-shaped solitons have different velocities depending on the values of α. At the initial stage, the smaller the value of α is selected, the faster the soliton propagates. But soon it was the opposite; for more details see Figures 3 and 4.

Fractional one-solitons determined by solution (43) at time t=1.

Velocity images of fractional one-solitons determined by solution (43).

Fractional one-solitons determined by solution (43) at time t=80.

Fractional one-solitons determined by solution (43) at time t=195.

For the fractional two-solitons and three-solitons determined, respectively, by solutions (44) and (45), similar features shown in Figures 57 are observed. In Figures 5 and 6, we select the parameters as κ1(0)=0.5, c1(0)=1.5, κ2(0)=1, and c2(0)=-0.2. While the parameters in Figure 7 are selected as κ1(0)=1, c1(0)=1, κ2(0)=1.1, c2(0)=0.5, κ3(0)=1.3, c3(0)=2.

Dynamical evolutions of fractional two-solitons determined by solution (44) at different times.

t = 0

t = 0.1

t = 0.5

t = 0.7

Fractional two-solitons determined by solution (44) at t=3.

Fractional three-solitons determined by solution (45) at t=1.

5. Conclusion

In summary, we have derived and solved the local tfKdV eq. (1) in the fractional framework of the spectral transform method. This is due to the linear spectral problem (3) equipped with the local time-fractional evolution (4). As for the fractional derivatives, there are many definitions  except the local fractional derivative, such as Grünwald–Letnikov fractional derivative, Riemann-Liouville fractional derivative and Caputo’s fractional derivative. Generally speaking, whether or not the spectral transform can be extended to some other nonlinear evolution equations with another type of fractional derivative depends on whether fractional derivative has the good properties required by the spectral transform method. To the best of our knowledge, combined with the Mittag-Leffler functions the obtained exact solution (31), the fractional n-soliton solution (34), and its special cases, the fractional one-, two-, and three-soliton solutions (43)-(45), are all new, and they have not been reported in literature. It is graphically shown that the fractional order of the local tfKdV eq. (1) influences the velocity of the fractional one-soliton solution (43) with Mittag-Leffler function in the process of propagations. More importantly, the fractional scheme of the spectral transform presented in this paper for constructing n-soliton solution of the local tfKdV eq. (1) can be extended to some other integrable local time-fractional PDEs.

Data Availability

The data in the paper are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This work was supported by the Natural Science Foundation of China (11547005), the Natural Science Foundation of Liaoning Province of China (20170540007), the Natural Science Foundation of Education Department of Liaoning Province of China (LZ2017002), and Innovative Talents Support Program in Colleges and Universities of Liaoning Province (LR2016021).

Fujioka J. Espinosa A. Rodríguez R. Fractional optical solitons Physics Letters A 2010 374 9 1126 1134 10.1016/j.physleta.2009.12.051 Yang X.-J. Hristov J. Srivastava H. M. Ahmad B. Modelling fractal waves on shallow water surfaces via local fractional korteweg-de vries equation Abstract and Applied Analysis 2014 2014 10 278672 10.1155/2014/278672 Gardner C. S. Greene J. M. Kruskal M. D. Miura R. M. Method for solving the Korteweg-deVries equation Physical Review Letters 1967 19 19 1095 1097 10.1103/PhysRevLett.19.1095 2-s2.0-36049057587 Ablowitz M. J. Kaup D. J. Newell A. C. Segur H. The inverse scattering transform-Fourier analysis for nonlinear problems Studies in Applied Mathematics 1974 53 4 249 315 10.1137/1015113 MR450815 Zbl0408.35068 Chen H. H. Liu C. S. Solitons in nonuniform media Physical Review Letters 1976 37 11 693 697 10.1103/PhysRevLett.37.693 MR0411488 2-s2.0-0000354152 Hirota R. Satsuma J. N-soliton solutions of the K-dV equation with loss and nonuniformity terms Journal of the Physical Society of Japan 1976 41 6 2141 2142 10.1143/JPSJ.41.2141 2-s2.0-68049127184 Calogero F. Degasperis A. Coupled nonlinear evolution equations solvable via the inverse spectral transform, and solitons that come back: the boomeron Lettere Al Nuovo Cimento 1976 16 14 425 433 10.1007/BF02751683 Chan W. L. Li K.-S. Nonpropagating solitons of the variable coefficient and nonisospectral Korteweg-de Vries equation Journal of Mathematical Physics 1989 30 11 2521 2526 10.1063/1.528533 MR1018997 Zbl0698.35141 2-s2.0-0001595864 Ablowitz M. J. Clarkson P. A. Solitons, Nonlinear Evolution Equations and Inverse Scattering 1991 Cambridge, UK Cambridge University Press 10.1017/CBO9780511623998 MR1149378 Xu B. Z. Zhao S. Q. Inverse scattering transformation for the variable coefficient sine-Gordon type equation Applied Mathematics-A Journal of Chinese Universities 1994 9 4 331 337 10.1007/BF02665219 MR1333489 Zbl0819.35127 2-s2.0-51649136056 Zeng Y. Ma W. Lin R. Integration of the soliton hierachy with selfconsistent sources Journal of Mathematical Physics 2000 41 8 5453 5489 10.1063/1.533420 MR1770966 Serkin V. N. Hasegawa A. Novel soliton solutions of the nonlinear Schrodinger equation model Physical Review Letters 2000 85 21 4502 4505 2-s2.0-0034319701 10.1103/PhysRevLett.85.4502 Serkin V. N. Belyaeva T. L. Optimal control of optical soliton parameters: Part 1. The Lax representation in the problem of soliton management Quantum Electronics 2001 31 11 1007 1015 2-s2.0-0035511333 10.1070/QE2001v031n11ABEH002093 Ning T.-k. Chen D.-y. Zhang D.-j. The exact solutions for the nonisospectral AKNS hierarchy through the inverse scattering transform Physica A: Statistical Mechanics and its Applications 2004 339 3-4 248 266 10.1016/j.physa.2004.03.021 MR2091374 Serkin V. N. Hasegawa A. Belyaeva T. L. Nonautonomous solitons in external potentials Physical Review Letters 2007 98 7 2-s2.0-33847012957 10.1103/PhysRevLett.98.074102 074102 Guo B. Ling L. Riemann-Hilbert approach and N-soliton formula for coupled derivative Schrödinger equation Journal of Mathematical Physics 2012 53 7 20 073506 10.1063/1.4732464 MR2985246 Zhang S. Xu B. Zhang H.-Q. Exact solutions of a KdV equation hierarchy with variable coefficients International Journal of Computer Mathematics 2014 91 7 1601 1616 10.1080/00207160.2013.855730 MR3252137 2-s2.0-84906781087 Zhang S. Wang D. Variable-coefficient nonisospectral Toda lattice hierarchy and its exact solutions Pramana—Journal of Physics 2015 85 6 1143 1156 2-s2.0-84949740339 10.1007/s12043-014-0918-z Zhang S. Gao X.-D. Mixed spectral AKNS hierarchy from linear isospectral problem and its exact solutions Open Physics 2015 13 1 310 322 2-s2.0-84949222057 10.1515/phys-2015-0040 Randoux S. Suret P. El G. Inverse scattering transform analysis of rogue waves using local periodization procedure Scientific Reports 2016 6 10 29238 2-s2.0-84978175083 Zhang S. Gao X. Exact solutions and dynamics of a generalized AKNS equations associated with the nonisospectral depending on exponential function Journal of Nonlinear Sciences and Applications 2016 9 6 4529 4541 MR3543481 10.22436/jnsa.009.06.91 Zhang S. Li J. Soliton solutions and dynamical evolutions of a generalized AKNS system in the framework of inverse scattering transform Optik - International Journal for Light and Electron Optics 2017 137 228 237 10.1016/j.ijleo.2017.02.104 Zhang S. Hong S. Lax integrability and soliton solutions for a nonisospectral integro-differential system Complexity 2017 2017 10 9457078 10.1155/2017/9457078 Zbl1377.45006 Zhang S. Hong S. On a generalized Ablowitz–Kaup–Newell–Segur hierarchy in inhomogeneities of media: soliton solutions and wave propagation influenced from coefficient functions and scattering data Waves in Random and Complex Media 2017 28 3 435 452 10.1080/17455030.2017.1362134 Zhang S. Hong S. Lax Integrability and Exact Solutions of a Variable-Coefficient and Nonisospectral AKNS Hierarchy International Journal of Nonlinear Sciences and Numerical Simulation 2018 19 3-4 251 262 10.1515/ijnsns-2016-0191 Zbl1401.35279 Kang Z. Xia T. Ma X. Multi-solitons for the coupled Fokas–Lenells system via Riemann–Hilbert approach Chinese Physics Letters 2018 35 7 5 070201 10.1088/0256-307X/35/7/070201 Zhang S. Zhang H. Q. Fractional sub-equation method and its applications to nonlinear fractional PDEs Physics Letters A 2011 375 7 1069 1073 10.1016/j.physleta.2011.01.029 MR2765013 Zbl1242.35217 2-s2.0-79251635229 Aslan İ. Analytic investigation of a reaction-diffusion brusselator model with the time-space fractional derivative International Journal of Nonlinear Sciences and Numerical Simulation 2014 15 2 149 155 10.1515/ijnsns-2013-0077 Aslan İ. An analytic approach to a class of fractional differential-difference equations of rational type via symbolic computation Mathematical Methods in the Applied Sciences 2015 38 1 27 36 10.1002/mma.3047 Zbl1323.34002 Aslan İ. Symbolic computation of exact solutions for fractional differential-difference equation models Nonlinear Analysis: Modelling and Control 2015 20 1 132 144 10.15388/NA.2015.1.9 Zbl07079594 Wang Y. Liu L. Zhang X. Wu Y. Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection Applied Mathematics and Computation 2015 258 312 324 10.1016/j.amc.2015.01.080 MR3323070 Yang X. Srivastava H. An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives Communications in Nonlinear Science and Numerical Simulation 2015 29 1-3 499 504 10.1016/j.cnsns.2015.06.006 Yang X.-J. Baleanu D. Srivastava H. M. Local Fractional Integral Transforms and Their Applications 2015 London, UK Academic Press 10.1016/B978-0-12-804002-7.00001-2 MR3430825 Wu G.-C. Baleanu D. Deng Z.-G. Zeng S.-D. Lattice fractional diffusion equation in terms of a Riesz-Caputa difference Physica A: Statistical Mechanics and its Applications 2015 438 335 339 10.1016/j.physa.2015.06.024 MR3384291 Hu Y. He J. On fractal space-time and fractional calculus THERMAL SCIENCE 2016 20 3 773 777 10.2298/TSCI1603773H Guo L. Liu L. Wu Y. Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions Nonlinear Analysis, Modelling and Control 2016 21 5 635 650 10.15388/NA.2016.5.5 MR3533290 Zhang S. Liu M. Zhang L. Variable separation for time fractional advection-dispersion equation with initial and boundary conditions THERMAL SCIENCE 2016 20 3 789 792 10.2298/TSCI1603789Z Liu L. L. Zhang X. Liu L. Wu Y. Iterative positive solutions for singular nonlinear fractional differential equation with integral boundary conditions Advances in Difference Equations 2016 Paper No. 154, 13 10.1186/s13662-016-0876-5 MR3510702 Zhu B. Liu L. Wu Y. Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay Applied Mathematics Letters 2016 61 73 79 10.1016/j.aml.2016.05.010 2-s2.0-84973535695 Liu H. Meng F. Some new generalized Volterra-Fredholm type discrete fractional sum inequalities and their applications Journal of Inequalities and Applications 2016 2016 1, article no. 213 7 10.1186/s13660-016-1152-7 2-s2.0-84986222174 Aslan İ. Exact Solutions for a Local Fractional DDE Associated with a Nonlinear Transmission Line Communications in Theoretical Physics 2016 66 3 315 320 10.1088/0253-6102/66/3/315 Aslan İ. Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform Mathematical Methods in the Applied Sciences 2016 39 18 5619 5625 10.1002/mma.3946 Wu J. Zhang X. Liu L. Wu Y. Twin iterative solutions for a fractional differential turbulent flow model Boundary Value Problems 2016 9 98 10.1186/s13661-016-0604-9 MR3500488 Xu R. Meng F. Some new weakly singular integral inequalities and their applications to fractional differential equations Journal of Inequalities and Applications 2016 2016 16 78 10.1186/s13660-016-1015-2 MR3465080 Yang X. Tenreiro Machado J. A. Baleanu D. Cattani C. On exact traveling-wave solutions for local fractional Korteweg-de Vries equation Chaos: An Interdisciplinary Journal of Nonlinear Science 2016 26 8 5 084312 10.1063/1.4960543 Yang X. Gao F. Srivastava H. Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations Computers & Mathematics with Applications 2017 73 2 203 210 10.1016/j.camwa.2016.11.012 Zbl1386.35460 Yang X.-J. Machado J. A. A new fractional operator of variable order: application in the description of anomalous diffusion Physica A: Statistical Mechanics and its Applications 2017 481 276 283 10.1016/j.physa.2017.04.054 MR3648947 Wang Y. Jiang J. Existence and nonexistence of positive solutions for the fractional coupled system involving generalized p-Laplacian Advances in Difference Equations 2017 Paper No. 337, 19 10.1186/s13662-017-1385-x MR3714782 Zhang K. M. On a sign-changing solution for some fractional differential equations Boundary Value Problems 2017 2017 59 8 MR3638419 Du X. Mao A. Existence and multiplicity of nontrivial solutions for a class of semilinear fractional schrödinger equations Journal of Function Spaces 2017 2017 7 3793872 10.1155/2017/3793872 Wang Y. Liu L. Positive solutions for a class of fractional 3-point boundary value problems at resonance Advances in Difference Equations 2017 Paper No. 7, 13 10.1186/s13662-016-1062-5 MR3592890 Zhang X. Liu L. Wu Y. Wiwatanapataphee B. Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion Applied Mathematics Letters 2017 66 1 8 10.1016/j.aml.2016.10.015 MR3583852 He J.-H. Fractal calculus and its geometrical explanation Results in Physics 2018 10 272 276 10.1016/j.rinp.2018.06.011 2-s2.0-85048543160 Hirota R. Exact solution of the korteweg—de vries equation for multiple Collisions of solitons Physical Review Letters 1971 27 18 1192 1194 10.1103/PhysRevLett.27.1192 Zbl1168.35423 Wang M. Exact solutions for a compound KdV-Burgers equation Physics Letters A 1996 213 5-6 279 287 10.1016/0375-9601(96)00103-X MR1390282 Fan E. G. Extended tanh-function method and its applications to nonlinear equations Physics Letters A 2000 277 4-5 212 218 10.1016/S0375-9601(00)00725-8 MR1827770 Yan Z. Zhang H. New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water Physics Letters A 2001 285 5-6 355 362 10.1016/S0375-9601(01)00376-0 MR1851579 Fan E. G. Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems Physics Letters A 2002 300 2-3 243 249 10.1016/s0375-9601(02)00776-4 MR1928028 Zhang S. Xia T. A generalized F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equations Applied Mathematics and Computation 2006 183 2 1190 1200 10.1016/j.amc.2006.06.043 MR2294076 He J. Wu X. Exp-function method for nonlinear wave equations Chaos, Solitons & Fractals 2006 30 3 700 708 10.1016/j.chaos.2006.03.020 MR2238695 Aslan I. Multi-wave and rational solutions for nonlinear evolution equations International Journal of Nonlinear Sciences and Numerical Simulation 2010 11 8 619 623 10.1515/IJNSNS.2010.11.8.619 Aslan I. Analytic investigation of the (2 + 1)-dimensional Schwarzian Korteweg–de Vries equation for traveling wave solutions Applied Mathematics and Computation 2011 217 12 6013 6017 10.1016/j.amc.2010.12.115 MR2770225 Wang Y. Chen Y. Binary Bell polynomial manipulations on the integrability of a generalized (2+1)-dimensional Korteweg–de Vries equation Journal of Mathematical Analysis and Applications 2013 400 2 624 634 Lou S. Y. Consistent Riccati expansion for integrable systems Studies in Applied Mathematics 2015 134 3 372 402 Tian S. F. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval Communications on Pure and Applied Analysis 2018 17 3 923 957 Zhao P. Fan E. On quasiperiodic solutions of the modified Kadomtsev–Petviashvili hierarchy Applied Mathematics Letters 2019 97 27 33 Yan Z. The NLS(n, n) equation: Multi-hump compactons and their stability and interaction scenarios Chaos, Solitons & Fractals 2019 112 25 31 Wazwaz A.-M. Compacton solutions of higher order nonlinear dispersive KdV-like equations Applied Mathematics and Computation 2004 147 2 449 460 Kong L.-Q. Liu J. Jin D.-Q. Ding D.-J. Dai C.-Q. Soliton dynamics in the three-spine α-helical protein with inhomogeneous effect Nonlinear Dynamics 2017 87 1 83 92 10.1007/s11071-016-3027-3 2-s2.0-84983483199 Zhang B. Zhang X.-L. Dai C.-Q. Discussions on localized structures based on equivalent solution with different forms of breaking soliton model Nonlinear Dynamics 2017 87 4 2385 2393 10.1007/s11071-016-3197-z