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October  2020, 13(10): 2655-2665. doi: 10.3934/dcdss.2020225

Conservation laws and line soliton solutions of a family of modified KP equations

1. 

Department of Mathematics and Statistics, Brock University, St. Catharines, ON L2S3A1, Canada

2. 

Department of Mathematics, Faculty of Sciences, University of Cádiz, Puerto Real, Cádiz, 11510, Spain

Received  February 2019 Revised  July 2019 Published  October 2020 Early access  December 2019

A family of modified Kadomtsev-Petviashvili equations (mKP) in 2+1 dimensions is studied. This family includes the integrable mKP equation when the coefficients of the nonlinear terms and the transverse dispersion term satisfy an algebraic condition. The explicit line soliton solution and all conservation laws of low order are derived for all equations in the family and compared to their counterparts in the integrable case.

Citation: Stephen C. Anco, Maria Luz Gandarias, Elena Recio. Conservation laws and line soliton solutions of a family of modified KP equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2655-2665. doi: 10.3934/dcdss.2020225
References:
[1]

M. J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech., 92 (1979), 691-715.  doi: 10.1017/S0022112079000835.

[2]

S. C. Anco, Generalization of Noether's theorem in modern form to non-variational partial differential equations, Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, Fields Inst. Commun., Springer, New York, 79 (2017), 119-182. 

[3]

S. C. Anco, Conservation laws of scaling-invariant field equations, J. Phys. A: Math. and Gen., 36 (2003), 8623-8638.  doi: 10.1088/0305-4470/36/32/305.

[4]

S. C. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations Part II: General treatment, Euro. J. Appl. Math., 41 (2002), 567-585.  doi: 10.1017/S0956792501004661.

[5]

S. C. AncoM. L. Gandarias and E. Recio, Conservation laws, symmetries, and line soliton solutions of generalized KP and Boussinesq equations with $p$-power nonlinearities in two dimensions, Theor. Math. Phys., 197 (2018), 1393-1411.  doi: 10.4213/tmf9483.

[6]

G. W. Bluman, A. F. Cheviakov and S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, 168. Springer, New York, 2010. doi: 10.1007/978-0-387-68028-6.

[7]

Y. Z. Chen and P. L.-F. Liu, A generalized modified Kadomtsev-Petviashvili equation for interfacial wave propagation near the critical depth level, Wave Motion, 27 (1998), 321-339.  doi: 10.1016/S0165-2125(97)00057-7.

[8]

G. C. Das and J. Sarma, Evolution of solitary wave in multicomponent plasmas, Chaos, Solitons and Fractals, 9 (1998), 901-911.  doi: 10.1016/S0960-0779(97)00170-7.

[9]

F. GesztesyH. HoldenE. Saab and B. Simon, Explicit construction of solutions of the modified Kadomtsev-Petviashvili equation, J. Funct. Anal., 98 (1991), 211-228.  doi: 10.1016/0022-1236(91)90096-N.

[10]

B. B. Kadomstev and V. I. Petviashvili, On the stability of waves in weakly dispersive media, Sov. Phys. Dokl., 15 (1970), 539-541. 

[11]

B. Konopel'chenko and V. G. Dubrovsky, Some new integrable nonlinear evolution equations in $2+1$ dimensions, Phys. Lett. A, 102 (1984), 15-17.  doi: 10.1016/0375-9601(84)90442-0.

[12]

B. G. Konopel'chenko and V. G. Dubrovsky, Inverse spectral transform for the modified Kadomtsev-Petviashvili equation, Studies in Applied Math., 86 (1992), 219-268.  doi: 10.1002/sapm1992863219.

[13]

R. Naz, Z. Ali and I. Naeem, Reductions and new exact solutions of ZK, Gardner KP, and modified KP equations via generalized double reduction theorem, Abstract and Applied Analysis, (2013), Art. ID 340564, 11 pp. doi: 10.1155/2013/340564.

[14]

P. J. Olver, Applications of Lie Groups to Differential Equations, Second edition. Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.

[15]

E. Recio and S. C. Anco, Conservation laws and symmetries of radial generalized nonlinear $p$-Laplacian evolution equations, J. Math. Anal. Appl., 452 (2017), 1229-1261.  doi: 10.1016/j.jmaa.2017.03.050.

[16]

V. Veerakumar and M. Daniel, Modified Kadomtsev-Petviashvili (MKP) equation and electromagnetic soliton, Math. Comput. Simulat., 62 (2003), 163-169.  doi: 10.1016/S0378-4754(02)00176-3.

[17]

T. Wolf, A comparison of four approaches to the calculation of conservation laws, Euro. J. Appl. Math., 13 (2002), 129-152.  doi: 10.1017/S0956792501004715.

[18]

X. S. ZhaoW. XuH. B. Jia and H. X. Zhou, Solitary wave solutions for the modified Kadomtsev-Petviashvili equation, Chaos, Solitons and Fractals, 34 (2007), 465-475.  doi: 10.1016/j.chaos.2006.03.046.

show all references

References:
[1]

M. J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech., 92 (1979), 691-715.  doi: 10.1017/S0022112079000835.

[2]

S. C. Anco, Generalization of Noether's theorem in modern form to non-variational partial differential equations, Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, Fields Inst. Commun., Springer, New York, 79 (2017), 119-182. 

[3]

S. C. Anco, Conservation laws of scaling-invariant field equations, J. Phys. A: Math. and Gen., 36 (2003), 8623-8638.  doi: 10.1088/0305-4470/36/32/305.

[4]

S. C. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations Part II: General treatment, Euro. J. Appl. Math., 41 (2002), 567-585.  doi: 10.1017/S0956792501004661.

[5]

S. C. AncoM. L. Gandarias and E. Recio, Conservation laws, symmetries, and line soliton solutions of generalized KP and Boussinesq equations with $p$-power nonlinearities in two dimensions, Theor. Math. Phys., 197 (2018), 1393-1411.  doi: 10.4213/tmf9483.

[6]

G. W. Bluman, A. F. Cheviakov and S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, 168. Springer, New York, 2010. doi: 10.1007/978-0-387-68028-6.

[7]

Y. Z. Chen and P. L.-F. Liu, A generalized modified Kadomtsev-Petviashvili equation for interfacial wave propagation near the critical depth level, Wave Motion, 27 (1998), 321-339.  doi: 10.1016/S0165-2125(97)00057-7.

[8]

G. C. Das and J. Sarma, Evolution of solitary wave in multicomponent plasmas, Chaos, Solitons and Fractals, 9 (1998), 901-911.  doi: 10.1016/S0960-0779(97)00170-7.

[9]

F. GesztesyH. HoldenE. Saab and B. Simon, Explicit construction of solutions of the modified Kadomtsev-Petviashvili equation, J. Funct. Anal., 98 (1991), 211-228.  doi: 10.1016/0022-1236(91)90096-N.

[10]

B. B. Kadomstev and V. I. Petviashvili, On the stability of waves in weakly dispersive media, Sov. Phys. Dokl., 15 (1970), 539-541. 

[11]

B. Konopel'chenko and V. G. Dubrovsky, Some new integrable nonlinear evolution equations in $2+1$ dimensions, Phys. Lett. A, 102 (1984), 15-17.  doi: 10.1016/0375-9601(84)90442-0.

[12]

B. G. Konopel'chenko and V. G. Dubrovsky, Inverse spectral transform for the modified Kadomtsev-Petviashvili equation, Studies in Applied Math., 86 (1992), 219-268.  doi: 10.1002/sapm1992863219.

[13]

R. Naz, Z. Ali and I. Naeem, Reductions and new exact solutions of ZK, Gardner KP, and modified KP equations via generalized double reduction theorem, Abstract and Applied Analysis, (2013), Art. ID 340564, 11 pp. doi: 10.1155/2013/340564.

[14]

P. J. Olver, Applications of Lie Groups to Differential Equations, Second edition. Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.

[15]

E. Recio and S. C. Anco, Conservation laws and symmetries of radial generalized nonlinear $p$-Laplacian evolution equations, J. Math. Anal. Appl., 452 (2017), 1229-1261.  doi: 10.1016/j.jmaa.2017.03.050.

[16]

V. Veerakumar and M. Daniel, Modified Kadomtsev-Petviashvili (MKP) equation and electromagnetic soliton, Math. Comput. Simulat., 62 (2003), 163-169.  doi: 10.1016/S0378-4754(02)00176-3.

[17]

T. Wolf, A comparison of four approaches to the calculation of conservation laws, Euro. J. Appl. Math., 13 (2002), 129-152.  doi: 10.1017/S0956792501004715.

[18]

X. S. ZhaoW. XuH. B. Jia and H. X. Zhou, Solitary wave solutions for the modified Kadomtsev-Petviashvili equation, Chaos, Solitons and Fractals, 34 (2007), 465-475.  doi: 10.1016/j.chaos.2006.03.046.

Figure 1.  Kinematically allowed region in $ (c,\theta) $ for the mKP family line soliton (40)
Figure 2.  Kinematically allowed region in $ (c,\theta) $ for the mKP family line soliton (40) in the defocussing case ($ \sigma_1 = -1 $)
Figure 3.  Profile of the mKP line soliton (39) and mKP family line soliton (40) and (45) for $ (h,w) = $ $ (1,1) $ (solid); $ (4,1) $ (long dash); $ (\tfrac{1}{5},1) $ (dash dot); $ (1,4) $ (dash); $ (1,\tfrac{1}{2}) $ (dot)
Figure 4.  Kinematically allowed region in $ (c,\theta) $ for the mKP family line soliton (45) in the defocussing case ($ \sigma_1 = -1 $)
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