doi: 10.3934/dcdss.2020272

Generalized Wronskian solutions of modified Boussinesq equation

College of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou, 450005, China

* Corresponding author: Qian Li

Received  May 2019 Revised  May 2019 Published  February 2020

The generalized Wronskian solutions whose elements satisfy matrix equation of linear problem for the modified Boussinesq equation are obtained through Wronskian technique. Furthermore, rational solutions, Matveev solutions, complexiton solutions and interaction solutions are derived by taking special cases in general solutions.

Citation: Qian Li. Generalized Wronskian solutions of modified Boussinesq equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020272
References:
[1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series, 149. Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511623998.  Google Scholar
[2]

M. J. Ablowitz and J. Satsuma, Solitons and rational solutions of nonlinear evolution equations, J. Math. Phys., 19 (1978), 2180-2186.  doi: 10.1063/1.523550.  Google Scholar

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M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Studies in Applied Mathematics, 4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981.  Google Scholar

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R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line, Mathematical Surveys and Monographs, 28. American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/028.  Google Scholar

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J. Boussinesq, Rational solutions of the Boussinesq equation, Comptes Rendus, 72 (1871), 755-759.   Google Scholar

[6]

D. Y. ChenD. J. Zhang and J. B. Bi, New double wronskian solutions of the AKNS equation, Science in China Series A: Mathematics, 51 (2008), 55-69.  doi: 10.1007/s11425-007-0165-6.  Google Scholar

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P. A. Clarkson, New similarity solutions for the modified Boussinesq equation, J. Phys. A: Math. Gen., 22 (1989), 2355-2367.  doi: 10.1088/0305-4470/22/13/029.  Google Scholar

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S. CuendaN. R. Quintero and A. Sánchez, Sine-Gordon wobbles through Bäcklund transformations, Discrete Contin. Discrete Cont Dyn-S, 4 (2011), 1047-1056.  doi: 10.3934/dcdss.2011.4.1047.  Google Scholar

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H.-H. Dai and X. G. Geng, Finite-dimensional integrable systems through the decomposition of a modified Boussinesq equation, Phys. Lett. A, 317 (2003), 389-400.  doi: 10.1016/j.physleta.2003.08.049.  Google Scholar

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P. DeiftC. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math., 35 (1982), 567-628.  doi: 10.1002/cpa.3160350502.  Google Scholar

[11]

N. C. Freeman and J. J. C. Nimmo, Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: The wronskian technique, Phys. Lett. A, 95 (1983), 1-3.  doi: 10.1016/0375-9601(83)90764-8.  Google Scholar

[12]

Ge genhasi and X.-B. Hu, Integrability of a differential-difference KP equation with self-consistent sources, Mathematics and Computers in Simulation, 74 (2007), 145-158.  doi: 10.1016/j.matcom.2006.10.034.  Google Scholar

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X. G. Geng, Lax pair and Darboux transformation solutions of the modified Boussinesq equation, Acta Math. Appl. Sinica, 11 (1988), 324-328.   Google Scholar

[14]

X. G. Geng, Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations, J. Phys. A, 36 (2003), 2289-2303.  doi: 10.1088/0305-4470/36/9/307.  Google Scholar

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R. Hirota and J. Satsuma, Nonlinear evolution equations generated from the Bäcklund transformation for the Toda Lattice, Prog. Theor. Phys., 57 (1977), 797-807.   Google Scholar

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M. Jaworski, Breather-like solution of the Korteweg-de Vries equation, Phys. Lett. A, 104 (1984), 245-247.  doi: 10.1016/0375-9601(84)90060-4.  Google Scholar

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B. Karaagac, New exact solutions for some fractional order differential equations via improved sub-equation method, Discrete Contin. Discrete Cont. Dyn. S, 12 (2019), 447-454.   Google Scholar

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C. M. KhaliqueO. D. Adeyemo and I. Simbanefayi, On optimal system, exact solutions and conservation laws of the modified equal-width equation, Appl. Math. Nonlinear Sci., 3 (2018), 409-417.  doi: 10.21042/AMNS.2018.2.00031.  Google Scholar

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[22]

Q. LiT.-C. Xia and D.-Y. Chen, $2N+1$-soliton solutions of Boussinesq-Burgers equation, Communications in Mathematical Researarch, 33 (2017), 26-32.   Google Scholar

[23]

X. Lü and F. H. Lin, Soliton excitations and shape-changing collisions in alpha helical proteins with interspine coupling at higher order, Nonlinear Sci. Numer Simul., 32 (2016), 241-261.  doi: 10.1016/j.cnsns.2015.08.008.  Google Scholar

[24]

W. X. Ma, Complexiton solution to the Korteweg-de Vries equation, Phys. Lett. A, 301 (2002), 35-44.  doi: 10.1016/S0375-9601(02)00971-4.  Google Scholar

[25]

W. X. Ma and Wr onskians, Wronskians generalized Wronskians and solutions to the Korteweg-de Vries equation, Chaos, Solitons Fractals, 19 (2004), 163-170.  doi: 10.1016/S0960-0779(03)00087-0.  Google Scholar

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W.-X. Ma and Y. C. You, Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions, Trans. Amer. Math. Soc., 357 (2005), 1753-1778.  doi: 10.1090/S0002-9947-04-03726-2.  Google Scholar

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V. B. Matveev, Generalized Wronskian formula for solutions of the KdV equations: First applications, Phys. Lett. A, 166 (1992), 205-208.  doi: 10.1016/0375-9601(92)90362-P.  Google Scholar

[29]

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-662-00922-2.  Google Scholar

[30]

H. P. McKean, Boussinesq's equation on the circle, Comm. Pure Appl. Math., 34 (1981), 599-691.  doi: 10.1002/cpa.3160340502.  Google Scholar

[31]

L. D. MolelekiT. Motsepa and C. M. Khalique, Solutions and conservation laws of a generalized second extended $(3+1)$-dimensional Jimbo-Miwa equation, Appl. Math. Nonlinear Sci., 3 (2018), 459-474.  doi: 10.2478/AMNS.2018.2.00036.  Google Scholar

[32]

J. J. C. Nimmo, Soliton solution of three differential-difference equations in wronskian form, Phys. Lett. A, 99 (1983), 281-286.  doi: 10.1016/0375-9601(83)90885-X.  Google Scholar

[33]

J. J. C. Nimmo and N. C. Freeman, Rational solutions of the Korteweg-de Vries equation in wronskian form, Phys. Lett. A, 96 (1983), 443-446.  doi: 10.1016/0375-9601(83)90159-7.  Google Scholar

[34]

T. K. NingD. J. ZhangD. Y. Chen and S. F. Deng, Exact solutions and conservation laws for a nonisospectral sine-Gordon equation, Chaos, Solitons Fractals, 25 (2005), 611-620.  doi: 10.1016/j.chaos.2004.11.027.  Google Scholar

[35]

S. Novikov, S. V. Manakov, L. P. Pitaevskiǐ and V. E. Zakharov, Theory of Solitons, The Inverse Scattering Methods, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1984.  Google Scholar

[36]

G. R. W. QuispelF. W. Nijhoff and H. W. Capel, Linearization of the Boussinesq equation and the modified Boussinesq equation, Phys. Lett. A, 91 (1982), 143-145.  doi: 10.1016/0375-9601(82)90817-9.  Google Scholar

[37] C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511606359.  Google Scholar
[38]

B. Abraham-Shrauner, Exact solutions of nonlinear partial differential equations, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 577-582.  doi: 10.3934/dcdss.2018032.  Google Scholar

[39]

S. SirianunpiboonS. D. Howard and S. K. Roy, New interaction solutions of (3+1)-dimensional Zakharov-Kuznetsov equation, Phys. Lett. A, 134 (1988), 31-33.  doi: 10.1016/0375-9601(88)90541-5.  Google Scholar

[40]

A. O. Smirnov, On a class of elliptic solutions of the Boussinesq equations, Theoret. Math. Phys., 109 (1996), 1515-1522.  doi: 10.1007/BF02073868.  Google Scholar

[41]

C. H. TsangB. A. Malomed and K. W. Chow, Exact solutions for periodic and solitary matter waves in nonlinear lattices, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1299-1325.  doi: 10.3934/dcdss.2011.4.1299.  Google Scholar

[42]

H. Wu and D. J. Zhang, Mixed rational-soliton solutions of two differential-difference equations in Casorati determinant form, J. Phys. A, 36 (2003), 4867-4873.  doi: 10.1088/0305-4470/36/17/313.  Google Scholar

[43]

V. E. Zakharov, On Stochastization of one-dimensional chains of nonlinear oscillators, Sov. Phys. JETP, 38 (1974), 108-110.   Google Scholar

[44]

D. J. Zhang, Notes on solutions in Wronskian form to soliton equations: Korteweg-deVries-type, arXiv: nlin.SI/0603008. Google Scholar

[45]

D.-J. Zhang and D.-Y. Chen, The $N$-soliton solutions of the sine-Gordon equation with self-consistent sources, Phys. A, 321 (2003), 467-481.  doi: 10.1016/S0378-4371(02)01742-9.  Google Scholar

[46]

D. J. Zhang and D. Y. Chen, Negatons, positons, rational-like solutions and conservation laws of the Korteweg-de Vries equation with loss and non-uniformity terms, J. Phys. A, 37 (2004), 851-865.  doi: 10.1088/0305-4470/37/3/021.  Google Scholar

[47]

D. J. Zhang and H. Wu, A note on Casoratian solutions to the 2-dimensional Toda lattice, Commun. Theore. Phys., 47 (2007), 390-392.   Google Scholar

[48]

D.-J. Zhang, S.-L. Zhao, Y.-Y. Sun and J. Zhou, Solutions to the modified Korteweg-de Vries equation, Rev. Math. Phys., 26 (2014), 14300064. doi: 10.1142/S0129055X14300064.  Google Scholar

show all references

References:
[1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series, 149. Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511623998.  Google Scholar
[2]

M. J. Ablowitz and J. Satsuma, Solitons and rational solutions of nonlinear evolution equations, J. Math. Phys., 19 (1978), 2180-2186.  doi: 10.1063/1.523550.  Google Scholar

[3]

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Studies in Applied Mathematics, 4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981.  Google Scholar

[4]

R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line, Mathematical Surveys and Monographs, 28. American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/028.  Google Scholar

[5]

J. Boussinesq, Rational solutions of the Boussinesq equation, Comptes Rendus, 72 (1871), 755-759.   Google Scholar

[6]

D. Y. ChenD. J. Zhang and J. B. Bi, New double wronskian solutions of the AKNS equation, Science in China Series A: Mathematics, 51 (2008), 55-69.  doi: 10.1007/s11425-007-0165-6.  Google Scholar

[7]

P. A. Clarkson, New similarity solutions for the modified Boussinesq equation, J. Phys. A: Math. Gen., 22 (1989), 2355-2367.  doi: 10.1088/0305-4470/22/13/029.  Google Scholar

[8]

S. CuendaN. R. Quintero and A. Sánchez, Sine-Gordon wobbles through Bäcklund transformations, Discrete Contin. Discrete Cont Dyn-S, 4 (2011), 1047-1056.  doi: 10.3934/dcdss.2011.4.1047.  Google Scholar

[9]

H.-H. Dai and X. G. Geng, Finite-dimensional integrable systems through the decomposition of a modified Boussinesq equation, Phys. Lett. A, 317 (2003), 389-400.  doi: 10.1016/j.physleta.2003.08.049.  Google Scholar

[10]

P. DeiftC. Tomei and E. Trubowitz, Inverse scattering and the Boussinesq equation, Comm. Pure Appl. Math., 35 (1982), 567-628.  doi: 10.1002/cpa.3160350502.  Google Scholar

[11]

N. C. Freeman and J. J. C. Nimmo, Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: The wronskian technique, Phys. Lett. A, 95 (1983), 1-3.  doi: 10.1016/0375-9601(83)90764-8.  Google Scholar

[12]

Ge genhasi and X.-B. Hu, Integrability of a differential-difference KP equation with self-consistent sources, Mathematics and Computers in Simulation, 74 (2007), 145-158.  doi: 10.1016/j.matcom.2006.10.034.  Google Scholar

[13]

X. G. Geng, Lax pair and Darboux transformation solutions of the modified Boussinesq equation, Acta Math. Appl. Sinica, 11 (1988), 324-328.   Google Scholar

[14]

X. G. Geng, Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations, J. Phys. A, 36 (2003), 2289-2303.  doi: 10.1088/0305-4470/36/9/307.  Google Scholar

[15]

R. Hirota, Exact solution of the Korteweg-deVries equation, for multiple collisions of soliton, Phys. Rev. Lett., 27 (1971), 1192-1194.   Google Scholar

[16] R. Hirota, The Direct Method in Soliton Theory, Cambridge Tracts in Mathematics, 155. Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511543043.  Google Scholar
[17]

R. Hirota and J. Satsuma, Nonlinear evolution equations generated from the Bäcklund transformation for the Toda Lattice, Prog. Theor. Phys., 57 (1977), 797-807.   Google Scholar

[18]

M. Jaworski, Breather-like solution of the Korteweg-de Vries equation, Phys. Lett. A, 104 (1984), 245-247.  doi: 10.1016/0375-9601(84)90060-4.  Google Scholar

[19]

B. Karaagac, New exact solutions for some fractional order differential equations via improved sub-equation method, Discrete Contin. Discrete Cont. Dyn. S, 12 (2019), 447-454.   Google Scholar

[20]

C. M. KhaliqueO. D. Adeyemo and I. Simbanefayi, On optimal system, exact solutions and conservation laws of the modified equal-width equation, Appl. Math. Nonlinear Sci., 3 (2018), 409-417.  doi: 10.21042/AMNS.2018.2.00031.  Google Scholar

[21]

Q. LiT.-C. Xia and D.-Y. Chen, $N$-soliton solutions to the modified Boussinesq equation, J. Shanghai. Univ., 13 (2009), 497-500.  doi: 10.1007/s11741-009-0613-1.  Google Scholar

[22]

Q. LiT.-C. Xia and D.-Y. Chen, $2N+1$-soliton solutions of Boussinesq-Burgers equation, Communications in Mathematical Researarch, 33 (2017), 26-32.   Google Scholar

[23]

X. Lü and F. H. Lin, Soliton excitations and shape-changing collisions in alpha helical proteins with interspine coupling at higher order, Nonlinear Sci. Numer Simul., 32 (2016), 241-261.  doi: 10.1016/j.cnsns.2015.08.008.  Google Scholar

[24]

W. X. Ma, Complexiton solution to the Korteweg-de Vries equation, Phys. Lett. A, 301 (2002), 35-44.  doi: 10.1016/S0375-9601(02)00971-4.  Google Scholar

[25]

W. X. Ma and Wr onskians, Wronskians generalized Wronskians and solutions to the Korteweg-de Vries equation, Chaos, Solitons Fractals, 19 (2004), 163-170.  doi: 10.1016/S0960-0779(03)00087-0.  Google Scholar

[26]

W.-X. Ma and Y. C. You, Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions, Trans. Amer. Math. Soc., 357 (2005), 1753-1778.  doi: 10.1090/S0002-9947-04-03726-2.  Google Scholar

[27] Y. Matsuno, Bilinear Transformation Method, Mathematics in Science and Engineering, 174. Academic Press, Inc., Orlando, FL, 1984.   Google Scholar
[28]

V. B. Matveev, Generalized Wronskian formula for solutions of the KdV equations: First applications, Phys. Lett. A, 166 (1992), 205-208.  doi: 10.1016/0375-9601(92)90362-P.  Google Scholar

[29]

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-662-00922-2.  Google Scholar

[30]

H. P. McKean, Boussinesq's equation on the circle, Comm. Pure Appl. Math., 34 (1981), 599-691.  doi: 10.1002/cpa.3160340502.  Google Scholar

[31]

L. D. MolelekiT. Motsepa and C. M. Khalique, Solutions and conservation laws of a generalized second extended $(3+1)$-dimensional Jimbo-Miwa equation, Appl. Math. Nonlinear Sci., 3 (2018), 459-474.  doi: 10.2478/AMNS.2018.2.00036.  Google Scholar

[32]

J. J. C. Nimmo, Soliton solution of three differential-difference equations in wronskian form, Phys. Lett. A, 99 (1983), 281-286.  doi: 10.1016/0375-9601(83)90885-X.  Google Scholar

[33]

J. J. C. Nimmo and N. C. Freeman, Rational solutions of the Korteweg-de Vries equation in wronskian form, Phys. Lett. A, 96 (1983), 443-446.  doi: 10.1016/0375-9601(83)90159-7.  Google Scholar

[34]

T. K. NingD. J. ZhangD. Y. Chen and S. F. Deng, Exact solutions and conservation laws for a nonisospectral sine-Gordon equation, Chaos, Solitons Fractals, 25 (2005), 611-620.  doi: 10.1016/j.chaos.2004.11.027.  Google Scholar

[35]

S. Novikov, S. V. Manakov, L. P. Pitaevskiǐ and V. E. Zakharov, Theory of Solitons, The Inverse Scattering Methods, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1984.  Google Scholar

[36]

G. R. W. QuispelF. W. Nijhoff and H. W. Capel, Linearization of the Boussinesq equation and the modified Boussinesq equation, Phys. Lett. A, 91 (1982), 143-145.  doi: 10.1016/0375-9601(82)90817-9.  Google Scholar

[37] C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511606359.  Google Scholar
[38]

B. Abraham-Shrauner, Exact solutions of nonlinear partial differential equations, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 577-582.  doi: 10.3934/dcdss.2018032.  Google Scholar

[39]

S. SirianunpiboonS. D. Howard and S. K. Roy, New interaction solutions of (3+1)-dimensional Zakharov-Kuznetsov equation, Phys. Lett. A, 134 (1988), 31-33.  doi: 10.1016/0375-9601(88)90541-5.  Google Scholar

[40]

A. O. Smirnov, On a class of elliptic solutions of the Boussinesq equations, Theoret. Math. Phys., 109 (1996), 1515-1522.  doi: 10.1007/BF02073868.  Google Scholar

[41]

C. H. TsangB. A. Malomed and K. W. Chow, Exact solutions for periodic and solitary matter waves in nonlinear lattices, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 1299-1325.  doi: 10.3934/dcdss.2011.4.1299.  Google Scholar

[42]

H. Wu and D. J. Zhang, Mixed rational-soliton solutions of two differential-difference equations in Casorati determinant form, J. Phys. A, 36 (2003), 4867-4873.  doi: 10.1088/0305-4470/36/17/313.  Google Scholar

[43]

V. E. Zakharov, On Stochastization of one-dimensional chains of nonlinear oscillators, Sov. Phys. JETP, 38 (1974), 108-110.   Google Scholar

[44]

D. J. Zhang, Notes on solutions in Wronskian form to soliton equations: Korteweg-deVries-type, arXiv: nlin.SI/0603008. Google Scholar

[45]

D.-J. Zhang and D.-Y. Chen, The $N$-soliton solutions of the sine-Gordon equation with self-consistent sources, Phys. A, 321 (2003), 467-481.  doi: 10.1016/S0378-4371(02)01742-9.  Google Scholar

[46]

D. J. Zhang and D. Y. Chen, Negatons, positons, rational-like solutions and conservation laws of the Korteweg-de Vries equation with loss and non-uniformity terms, J. Phys. A, 37 (2004), 851-865.  doi: 10.1088/0305-4470/37/3/021.  Google Scholar

[47]

D. J. Zhang and H. Wu, A note on Casoratian solutions to the 2-dimensional Toda lattice, Commun. Theore. Phys., 47 (2007), 390-392.   Google Scholar

[48]

D.-J. Zhang, S.-L. Zhao, Y.-Y. Sun and J. Zhou, Solutions to the modified Korteweg-de Vries equation, Rev. Math. Phys., 26 (2014), 14300064. doi: 10.1142/S0129055X14300064.  Google Scholar

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