Exact solutions and dynamics of generalized AKNS equations associated with the nonisospectral depending on exponential function
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Authors
Sheng Zhang
- School of Mathematics and Physics, Bohai University, Jinzhou 121013, China.
Xudong Gao
- School of Mathematics and Statistics, Kashgar University, Kashgar 844000, China.
Abstract
No matter constructing or solving nonlinear evolution equations (NLEEs), it is important and interesting
in the field of nonlinear science. In this paper, generalized Ablowitz-Kaup-Newell{Segur (AKNS) equations
are constructed and solved exactly. To be specific, the famous AKNS spectral problem is first generalized by
embedding a nonisospectral parameter whose varying with time obeys the exponential function of spectral
parameter. Based on the generalized AKNS spectral problem and its corresponding time evolution equation,
we then derive a generalized AKNS equation with infinite number of terms. Furthermore, exact solutions of
the generalized AKNS equations are formulated through the inverse scattering transform method. Finally, in
the case of reflectionless potentials, the obtained exact solutions are reduced to explicit n-soliton solutions.
It is shown that the dynamical evolutions of such soliton solutions possess not only time-varying speeds and
amplitudes but also singular points in the process of propagations.
Share and Cite
ISRP Style
Sheng Zhang, Xudong Gao, Exact solutions and dynamics of generalized AKNS equations associated with the nonisospectral depending on exponential function, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 6, 4529--4541
AMA Style
Zhang Sheng, Gao Xudong, Exact solutions and dynamics of generalized AKNS equations associated with the nonisospectral depending on exponential function. J. Nonlinear Sci. Appl. (2016); 9(6):4529--4541
Chicago/Turabian Style
Zhang, Sheng, Gao, Xudong. "Exact solutions and dynamics of generalized AKNS equations associated with the nonisospectral depending on exponential function." Journal of Nonlinear Sciences and Applications, 9, no. 6 (2016): 4529--4541
Keywords
- Exact solution
- n-soliton solution
- dynamical evolution
- generalized AKNS equations
- inverse scattering transform.
MSC
References
-
[1]
M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, Method for solving the sine-Gordon equation, Phys. Rev. Lett., 30 (1973), 1262-1264.
-
[2]
M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Sekur, The scattering transform Fourier analysis for nonlinear problems , Stud. Appl. Math., 53 (1974), 249-315.
-
[3]
M. J. Ablowitz, J. F. Ladik, Nonlinear differential-difference equations, J. Math. Phys., 16 (1975), 598-603.
-
[4]
D. Baleanu, B. Kilic, I. Mustafa, The first integral method for Wu-Zhang nonlinear system with time-dependent coefficients, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci., 16 (2015), 160-167.
-
[5]
D. Baleanu, R. L. Magin, S. Bhalekar, V. Daftardar-Gejji, Chaos in the fractional order nonlinear Bloch equation with delay, Commun. Nonlinear Sci. Numer. Simul., 25 (2015), 41-49.
-
[6]
G. Biondini, G. Kovacic, Inverse scattering transform for the focusing nonlinear Schrodinger equation with nonzero boundary conditions, J. Math. Phys., 55 (2014), 22 pages.
-
[7]
F. Calogero, A. Degasperis, Coupled nonlinear evolution equations solvable via the inverse spectral transform, and solitons that come back: the boomeron, Lett. Nuovo Cimento, 16 (1976), 425-433.
-
[8]
F. Calogreo, A. Degasperis, Extension of the spectral transform method for solving nonlinear evolutions equations, Lett. Nuovo Cimento, 22 (1978), 131-137.
-
[9]
F. Calogreo, A. Degasperis, Extension of the spectral transform method for solving nonlinear evolutions equations II, Lett. Nuovo Cimento, 22 (1978), 263-269.
-
[10]
F. Calogreo, A. Degasperis, Exact solution via the spectral transform of a generalization with linearly x-dependent coefficients of the modified Korteweg-de Vries equation, Lett. Nuovo Cimento, 22 (1978), 270-273.
-
[11]
S. Chakravarty, B. Prinari, M. J. Ablowitz, Inverse scattering transform for 3-level coupled Maxwell-Bloch equations with inhomogeneous broadening, Phys. D, 278-279 (2014), 58-78.
-
[12]
W. L. Chan, K.-S. Li , Nonpropagating solitons of the variable coefficient and nonisospectral Korteweg-de Vries equation , J. Math. Phys., 30 (1989), 2521-2526.
-
[13]
W. L. Chan, Y.-K. Zheng, Non-isospectral variable-coefficient higher-order Korteweg{de Vries equations, Inverse Probl., 7 (1991), 63-75.
-
[14]
D. Y. Chen, Introduction of Soliton, (in Chinese), Science Press, Beijing (2006)
-
[15]
H. H. Chen, C. S. Liu , Solitons in nonuniform media, Phys. Rev. Lett., 37 (1976), 693-697.
-
[16]
E. G. Fan, Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems, Phys. Lett. A, 300 (2002), 243-249.
-
[17]
H. Flaschka, On the Toda Lattice, II, Prog. Theor. Phys., 51 (1974), 703-716.
-
[18]
C. S. Garder, J. M. Greene, M. D. Kruskal, R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1965), 1095-1097.
-
[19]
C. S. Garder, J. M. Greene, M. D. Kruskal, R. M. Miura, Korteweg-de Vries equation and generalizations, VI. Method for exact solution, Comm. Pure Appl. Math., 27 (1974), 97-133.
-
[20]
R. Hirota, J. Satsuma, N-soliton solutions of the K-dV equation with loss and nonuniformity terms, J. Phys. Soc. Jpn., 41 (1976), 2141-2142.
-
[21]
M. Inc, B. Kilic, D. Baleanu, Optical soliton solutions of the pulse propagation generalized equation in parabolic-law media with space-modulated coefficients, OPTIK, 127 (2016), 1056-1058.
-
[22]
M. Inc, Z. S. Korpinar, M. M. A. Qurashi, D. Baleanu, A new method for approximate solutions of some nonlinear equations: Residual power series method, Adv. Mech. Eng., 8 (2016), 7 pages.
-
[23]
P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490.
-
[24]
Y. S. Li, A class of evolution equations and the spectral deformation, Sci. Sinica Ser. A, 25 (1982), 911-917.
-
[25]
W. X. Ma , An approach for constructing nonisospectral hierarchies of evolution equations, J. Phys. A, 25 (1992), 719-726.
-
[26]
W.-X. Ma, T. W. Huang, Y. Zhang, A multiple exp-function method for nonlinear differential equations and its application, Phys. Scr., 82 (2010), 12 pages.
-
[27]
M. R. Miurs, Bäcklund Transformation, Springer-Verlag, Berlin (1978)
-
[28]
A. I. Nachman, M. J. Ablowitz, A multidimensional inverse scattering method, Stud. Appl. Math., 71 (1984), 243-250.
-
[29]
V. N. Serkin, T. L. Belyaeva, The Lax representation in the problem of soliton management, Quant. Electron, 31 (2001), 1007-1015.
-
[30]
V. N. Serkin, A. Hasegawa, Novel soliton solutions of the nonlinear Schrödinger equation model, Phys. Rev. Lett., 85 (2000), 4502-4505.
-
[31]
V. N. Serkin, A. Hasegawa, T. L. Belyaeva, Nonautonomous solitons in external potentials, Phys. Rev. Lett., 98 (2007), 74-102.
-
[32]
V. N. Serkin, A. Hasegawa, T. L. Belyaeva, Solitary waves in nonautonomous nonlinear and dispersive systems: nonautonomous solitons, J. Mod. Opt., 57 (2010), 1456-1472.
-
[33]
V. N. Serkin, A. Hasegawa, T. L. Belyaeva, Nonautonomous matter-wave solitons near the Feshbach resonance, Phys. Rev. A, 81, 023610 (2010)
-
[34]
Z. D. Shan, H. W. Yang, B. S. Yin , Nonlinear integrable couplings of Levi hierarchy and WKI hierarchy , Abstr. Appl. Anal., 2014 (2014), 7 pages.
-
[35]
G. Z. Tu , The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys., 30 (1989), 330-338.
-
[36]
M. Wadati, The modified Korteweg-de Vries equation, J. Phys. Soc. Japan, 34 (1973), 1289-1296.
-
[37]
M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A, 213 (1996), 279-287.
-
[38]
J. Weiss, M. Tabor, G. Carnevale, The Painlevé property for partial differential equations, J. Math. Phys., 24 (1983), 522-526.
-
[39]
B. Z. Xu, S. Q. Zhao, Inverse scattering transformation for the variable coefficient sine-Gordon type equation, Appl. Math. J. Chinese Univ. Ser., 9 (1994), 331-337.
-
[40]
X. J. Yang, D. Baleanu, P. Lazarević Mihailo, S. Cajić Milan, Fractal boundary value problems for integral and differential equations with local fractional operators, Therm. Sci., 19 (2015), 959-966.
-
[41]
X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional similarity solution for the diffusion equation defined on Cantor sets, Appl. Math. Lett., 47 (2015), 54-60.
-
[42]
X.-J. Yang, D. Baleanu, H. M. Srivastava, Local Fractional Integral Transforms and Their Applications, Academic Press, New York (2015)
-
[43]
X.-J. Yang, H. M. Srivastava, An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives, Commun. Nonlinear Sci., 29 (2015), 499-504.
-
[44]
X.-J. Yang, H. M. Srivastava, C. Cattani , Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Rom. Rep. Phys., 67 (2015), 752-761.
-
[45]
F. J. Yu, L. Li, A new method to construct the integrable coupling system for discrete soliton equation with the Kronecker product, Phys. Lett. A, 372 (2008), 3548-3554.
-
[46]
V. E. Zakharov, A. B. Shabat, Interaction between solitons in a stable medium, Sov. Phys. JETP, 34 (1972), 62-69.
-
[47]
Y. B. Zeng, W.-X. Ma, R. L. Lin, Integration of the soliton hierarchy with self consistent sources, J. Math. Phys., 41 (2000), 5453-5489.
-
[48]
Y. F. Zhang, B. L. Feng, A few Lie algebras and their applications for generating integrable hierarchies of evolution types, Commun. Nonlinear Sci., 16 (2011), 3045-3061.
-
[49]
S. Zhang, X. Guo, A KN-like hierarchy with variable coefficients and its Hamiltonian structure and exact solutions, Int. J. Appl. Math., 46 (2016), 82-86.
-
[50]
S. Zhang, B. Xu, H.-Q. Zhang, Exact solutions of a KdV equation hierarchy with variable coefficients, Int. J. Comput. Math., 91 (2014), 1601-1616.