Fractional quasi AKNS-technique for nonlinear space-time fractional evolution equations. (English) Zbl 1431.37052
Summary: This paper aims to formulate the fractional quasi-inverse scattering method. Also, we give a positive answer to the following question: can the Ablowitz-Kaup-Newell-Segur (AKNS) method be applied to the space-time fractional nonlinear differential equations? Besides, we derive the Bäcklund transformations for the fractional systems under study. Also, we construct the fractional quasi-conservation laws for the considered fractional equations from the defined fractional quasi AKNS-like system. The nonlinear fractional differential equations to be studied are the space-time fractional versions of the Kortweg-de Vries equation, modified Kortweg-de Vries equation, the sine-Gordon equation, the sinh-Gordon equation, the Liouville equation, the cosh-Gordon equation, the short pulse equation, and the nonlinear Schrödinger equation.
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35Q51 | Soliton equations |
| 35R11 | Fractional partial differential equations |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 26A33 | Fractional derivatives and integrals |
Keywords:
Bäcklund transformations; conformable fractional derivative; fractional partial differential equation; fractional quasi AKNS method; fractional quasi-conservation lawsReferences:
| [1] | LoutsenkoI, RoubtsovD. Critical velocities in exciton superfluidity. Phys Rev Lett. 1997;78(15):3011‐3014.; TajiriM, MaesonoH. Resonant interactions of drift vortex solitons in a convective motion of a plasma. Phys Rev E. 1997;55(3):3351‐3357.; KivsharYS, MelomendBA. Dynamics of solitons in nearly integrable systems. Rev Mod Phys. 1989;61:765. |
| [2] | AblowitzMJ, KaupDJ, NewellAC, SegurH. Nonlinear‐evolution equations of physical significance. Phys Rev Lett. 1973;31(2):125‐127.; The inverse scattering transform‐Fourier analysis for nonlinear problems. Stud Appl Math. 1974;53(4):249‐315. · Zbl 0408.35068 |
| [3] | AblowitzMJ, ClarksonPA. Solitons, nonlinear evolution equations and inverse scattering. Cambridge: Cambridge University Press; 1991. · Zbl 0762.35001 |
| [4] | LuBQ, QuBZ, PanZL, JiangXF. Exact traveling wave solution of one class of nonlinear diffusion equations. Phys Lett A. 1993;175(2):113‐115.; YangZJ. Travelling wave solutions to nonlinear evolution and wave equations. J Phys A Math Gen. 1994;27(8):2837‐2855.; WangML, ZhouYB, LiZB. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys Lett A. 1996;216(1‐5):67‐75.; YanCT. A simple transformation for nonlinear waves. Phys Lett A. 1996;224(1‐2):77‐84. |
| [5] | MiuraRM. Bäcklund Transformation. Berlin: Springer Verlag; 1979.; RogersC, ShadwickWE. Bäcklund Transformations and their Applications. New York: Academic Press; 1982. · Zbl 0492.58002 |
| [6] | AblowitzMJ, SegurH. Solitons and the Inverse Scattering Transform. Philadelphia: SIAM; 1981. · Zbl 0472.35002 |
| [7] | KonnoK, WadatiM. Simple derivation of Backlund transformation from Riccati form of inverse method. Prog Theor Phys. 1975;53(6):1652‐1656. · Zbl 1079.35505 |
| [8] | NucciMC. Pseudopotentials, Lax equations and Backlund transformations for nonlinear evolution equations. J Phys A Math Gen. 1988;21(1):73‐79. · Zbl 0697.35134 |
| [9] | MatveevVB, SalleMA. Darboux Transformations and Solitons. Berlin: Springer Verlag; 1991. · Zbl 0744.35045 |
| [10] | DunajskiM. Soliton, Instantons, and Twistors. New York: Oxford University Press Inc.; 2010. · Zbl 1197.35209 |
| [11] | FaddeevLLD, TakhtajanLA. Hamiltonian Methods in the Theory of Solitons, reprinted 1987. Berlin Heielberg: Springser‐Verlag; 2007. · Zbl 0632.58004 |
| [12] | LambLB. Elements of Solitons. New York: Wiley‐Interscience; 1980. · Zbl 0445.35001 |
| [13] | WadatiM, SanukiH, KonnoK. Relationships among inverse method, Backlund transformation and an infinite number of conservation laws. Prog Theor Phys. 1975;53(2):419‐436. · Zbl 1079.35506 |
| [14] | KonnoK, SanukiH, IchikawaYH. Conservation laws of nonlinear‐evolution equations. Prog Theor Phys. 1974;52(3):886‐889. |
| [15] | ZakharovV, ShabatA. Exact theory of two‐dimensional self‐focusing and one‐dimensional self‐modulation of waves in nonlinear media. Sov Phys JETP. 1972;34:62. |
| [16] | MouradMF, SasakiR. Nonlinear sigma models on a half plane. Int J Mod Phys A. 1996;11(17):3127‐3143. · Zbl 1044.81724 |
| [17] | WadatiM. Transformation theories for nonlinear discrete systems. Prog Theor Phys Suppl. 1976;59:36‐63.; WadatiM, WaatanabeM. Conservation laws of a Volterra system and nonlinear self‐dual network equation. Prog Theor Phys. 1977;57(3):808‐811.; TsuchidaT, UjinoH, WadatiM. Integrable semi‐discretization of the coupled modified KdV equations. J Math Phys. 1998;39(9):4785‐4813.; Integrable semi‐discretization of the coupled nonlinear Schrödinger equations. J Phys A Math Gen. 1999;32(11):2239‐2262. |
| [18] | GardnerCS, GreeneJM, KruskalMD, MiuraRM. Method for solving the Korteweg‐deVries equation. Phys Rev Lett. 1967;19(19):1095‐1097. · Zbl 1061.35520 |
| [19] | LaxPD. Integrals of nonlinear equations of evolution and solitary waves. Commun Pure Appl Math. 1968;21(5):467‐490. · Zbl 0162.41103 |
| [20] | ZaitRA. Bäcklund transformations and solutions for some evolution equations. Phys Scripta. 1998;57(5):545‐548. · Zbl 1063.35537 |
| [21] | FujiiK, KoikawaT, SasakiR. Classical solutions for the supersymmetric Grassmannian sigma models in two dimensions. I. Prog Theor Phys. 1984;71(2):388‐394.; FujiiK, SasakiR. Classical solutions for the supersymmetric Grassmannian sigma models in two dimensions. II. Prog Theor Phys. 1984;71(4):831‐839.; El‐SabbaghMF, ZaitRA. Nonlocal conserved currents for the supersymmetric U(N) sigma model. Phys Scripta. 1993;47(1):9‐12.; ZaitRA. Baecklund transformations for the U (N) σ models and the SUSY U (N) σ models. Helv Phys Acta. 1996;69:105.; ZaitRA. On the Bäcklund transformations for integrable models in two dimensions. Nonlinearity. 1998;11(3):631‐640.; ZaitRA. Bäcklund transformations, cnoidal wave and travelling wave solutions of the SK and KK equations. Chaos, Solitons Fractals. 2003;15(4):673‐678.; HassanienIA, ZaitRA, Abdel‐SalamEA‐B. Multicnoidal and multitravelling wave solutions for some nonlinear equations of mathematical physics. Phys Scripta. 2003;67(6):457‐463. |
| [22] | SakovichA, SakovichS. The short pulse equation is integrable. J Physical Soc Japan. 2005;74(1):239‐241. · Zbl 1067.35115 |
| [23] | KonnoK. Integrable coupled dispersionless equations. Appl Anal. 1995;57(1‐2):209‐220.; KonnoK, KakuhataH. Interaction among growing, decaying and stationary solitons for coupled integrable dispersionless equations. J Physical Soc Japan. 1995;64(8):2707‐2709.; KonnoK, KakuhataH. Novel solitonic evolutions in a coupled integrable, dispersionless system. J Physical Soc Japan. 1996;65(3):713‐721.; KakuhataH, KonnoK. Lagrangian, Hamiltonian and conserved quantities for coupled integrable, dispersionless equations. J Physical Soc Japan. 1996;65(1):1‐2. |
| [24] | KhaterAH, El‐KalaawyOH, HelalMA. Two new classes of exact solutions for the KdV equation via Bäcklund transformations. Chaos, Solitons Fractals. 1997;8(12):1901‐1909;; KhaterAH, IbrahimRS, El‐KalaawyOH, CallebautDK. Bäcklund transformations and exact soliton solutions for some nonlinear evolution equations of the zs/akns system. Chaos, Solitons Fractals. 1998;9(11):1847‐1855.; KhaterAH, CallebautDK, AbdallaAA, SayedSM. Exact solutions for self‐dual Yang-Mills equations. Chaos, Solitons Fractals. 1999;10(8):1309‐1320;; KhaterAH, CallebautDK, SayedSM. Conservation laws for some nonlinear evolution equations which describe pseudo‐spherical surfaces. J Geom Phys. 51(2004):332‐352. |
| [25] | FaddeevLD. Two‐dimensional integrable models in quantum field theory. Phys Scripta. 1981;24(5):832‐835. · Zbl 1063.81643 |
| [26] | PodlubnyI. Fractional Differential Equations. New York: Academic Press; 1999.; KilbasAA, SrivastavaHM, TrujilloJJ. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier B.V.; 2006.; HilferR. Applications of Fractional Calculus in Physics. Singapore: World Sci.; 2000.; MillerKS, RossB. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York, NY, USA: John Wiley and Sons; 1993.; SamkoSG, KilbasAA, MarichevOI. Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science. Switzerland: Yverdon; 1993. · Zbl 0789.26002 |
| [27] | Abdel‐SalamEA‐B, YousifEA, El‐AasserMA. Analytical Solution of the Space‐Time Fractional Nonlinear Schrödinger Equation. Rep Math Phys. 2016;77(19):19‐34.; OliveiraEC, CostaFS, VazJJr. The fractional Schrödinger equation for delta potentials. J Math Phys. 2010;51(12):123517.; AmoreP, FernndezFM, HofmannCP, SenzRA. Collocation method for fractional quantum mechanics. J Math Phys. 2010;51(12):122101.; BaleanuD, VacaruSI. Fractional curve flows and solitonic hierarchies in gravity and geometric mechanics. J Math Phys. 2011;52(5):053514. |
| [28] | EslamiM, RezazadehH. The first integral method for Wu‐Zhang system with conformable time‐fractional derivative. CAL. 2016;53(3):475‐485. · Zbl 1434.35273 |
| [29] | AminikhahH, SheikhaniAR, RezazadehH. Sub‐equation method for the fractional regularized long‐wave equations with conformable fractional derivatives. Scientia Iranica Trans B, Mech Eng. 2016;23(3):1048‐1054. |
| [30] | JaradF, AbdeljawadT, AlzabutJ. Generalized fractional derivatives generated by a class of local proportional derivatives. Eur Phys J Spec Top. 2017;226(16‐18):3457‐3471. |
| [31] | AbdeljawadT, JaradF, AlzabutJ. Fractional proportional differences with memory. Eur Phys J Spec Top. 2017;226(16‐18):3333‐3354. |
| [32] | AbdeljawadT, Al‐MdallalQM, JaradF. Fractional logistic models in the frame of fractional operators generated by conformable derivatives. Chaos, Solitons Fractals. 2019;119:94‐101. · Zbl 1448.34006 |
| [33] | JaradF, UğurluE, AbdeljawadT, BaleanuD. On a new class of fractional operators. Advances in Difference Equations. 2017;2017(1):247. · Zbl 1422.26004 |
| [34] | AbdeljawadT. On conformable fractional calculus. J Comput Appl Math. 2015;279:57‐66. · Zbl 1304.26004 |
| [35] | AbdeljawadT, AlzabutJ, JaradF. A generalized Lyapunov‐type inequality in the frame of conformable derivatives. Adv Difference Equ. 2017;2017(1):321. · Zbl 1444.34045 |
| [36] | Al‐RifaeM, AbdeljawadT. Fundamental Results of Conformable Sturm‐Liouville Eigenvalue Problems. Complexity. 2017;2017:3720471, 7 pages, https://doi.org/10.1155/2017/3720471. · Zbl 1373.34046 · doi:10.1155/2017/3720471 |
| [37] | KhalilR, Al‐HoraniM, YousefA, SababhehM. A new definition of fractional derivative. J Comput Appl Math. 2014;264:65‐70. · Zbl 1297.26013 |
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