May  2018, 17(3): 823-848. doi: 10.3934/cpaa.2018042

The approximate solution for Benjamin-Bona-Mahony equation under slowly varying medium

1. 

Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu, 212013, China

2. 

School of Mathematical Science, Nanjing Normal University, Nanjing, Jiangsu, 210023, China

* Corresponding author: Wenxia Chen

Received  June 2017 Revised  October 2017 Published  January 2018

In this paper, we investigate the soliton dynamics under slowly varying medium for the BBM equation, that is how the solution of this equation evolve when the time goes. We construct the approximate solution of this equation and prove that the error term due to the approximate solution can be controlled. By using the method of Lyapunov and Weinstein functions, we prove that the approximate solution is stable.

Citation: Wenxia Chen, Ping Yang, Weiwei Gao, Lixin Tian. The approximate solution for Benjamin-Bona-Mahony equation under slowly varying medium. Communications on Pure & Applied Analysis, 2018, 17 (3) : 823-848. doi: 10.3934/cpaa.2018042
References:
[1]

N. Asano, Wave propagation in non-uniform media, Progr. Theoret, Phys. Suppl., 55 (1974), 52-79.   Google Scholar

[2]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersion systems, Philo. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.   Google Scholar

[3]

J. L. Bona, W. G. Pritchard and L. R. Scott, Solitary-wave interaction, Phys. Fluids, 23 (1980), 438-441. Google Scholar

[4]

J. L. BonaP. E. Souganidis and W. A. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. London Ser. A, 411 (1987), 395-412.   Google Scholar

[5]

E. FermiJ. Pasta and S. Ulam, Studies of nonlinear problems Ⅰ, AMS, Editors: A. C. Newell, (1974), 143-156.   Google Scholar

[6]

R. Grimshaw, Slowly varying solitary waves. Ⅰ. Korteweg -de Vries equation, Proc. Roy. Soc. London Ser. A, 368 (1979), 359-375.   Google Scholar

[7]

R. Grimshaw, Slowly varying solitary waves. Ⅱ. Nonlinear Schrödinger equation, Proc. Roy. Soc. London Ser.A, 368 (1979), 377-388.   Google Scholar

[8]

R. H. J. Grimshaw and S. R. Pudjaprasetya, Generation of secondary solitary waves in the variable-coefficient Korteweg-de Vries equation, Stud. Appl. Math., 112 (2004), 271-279.   Google Scholar

[9]

V. I. Karpman and E. M. Maslov, Perturbation theory for solitons, Z. Eksper. Teoret. Fiz., 73 (1977), 281-291.   Google Scholar

[10]

D. J. Kaup and A. C. Newell, Soliton as particles, oscillators, and in slowly changing media a singular perturbation theory, Proc. R. Soc. Lond., 361 (1978), 413-446.   Google Scholar

[11]

K. Ko and H. H. Kuehl, Korteweg-de Vries soliton in a slowly varying medium, Phys. Rev. Lett., 40 (1978), 233-236.   Google Scholar

[12]

R. LeVeque, On the interaction of nearly equal solitons in the Kdv equation, SIAM. J. Appl. Math., 47 (1987), 254-262.   Google Scholar

[13]

Y. Martel, Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math., 127 (2005), 1103-1140.   Google Scholar

[14]

Y. Martel and F. Merle, Inelastic interaction of nearly equal solitons for the bbm equation, Discrete. Cont. Dyn., 27 (2010), 487-532.   Google Scholar

[15]

Y. Martel and F. Merle, Description of the Inelastic Collision of Two Solitary Waves for the BBM Equation, Arch. Ration. Mech. An, 196 (2010), 517-574.   Google Scholar

[16]

Y. Martel and F. Merle, Inelastic interaction of nearly equal solitonsfor the quartic gKdV equation, Invent. Math., 183 (2011), 563-648.   Google Scholar

[17]

Y. Martel and F. Merle, Description of two soliton collision for the quartic gKdV equation, Ann. Math., 174 (2011), 757-857.   Google Scholar

[18]

J. Miller and M. Weinstein, Asymptotic stability of solitary waves for the regularized long wave equation, Comm. Pure Appl. Math., 49 (1996), 399-441.   Google Scholar

[19]

T. Mizumachi, Asymptotic stability of solitary wave solutions to the regularized long wave equation, J. Differ. Equations, 200 (2004), 312-341.   Google Scholar

[20]

C. Muñoz, On the soliton dynamics under a slowly varying medium for generalized KdV equations, Anal. PDE., 4 (2011), 573-638.   Google Scholar

[21]

C. Muñoz, Dynamics of soliton-like solutions for slowly varying, generalized gKdV equations: refraction vs. Reflection, SIAM J. Math. Anal., 44 (2012), 1-60.   Google Scholar

[22]

C. Muñoz, Inelastic character of solitons of slowly varying gKdV equations, Commun. Math. Phys., 314 (2012), 817-852.   Google Scholar

[23]

C. Muñoz, On the soliton dynamics under slowly varying medium for Nonlinear Schrodinger equations, Math. Ann., 353 (2012), 867-943.   Google Scholar

[24]

C. Muñoz, Sharp inelastic character of slowly varying NLS solitons, preprint, arXiv: 1202.5807v2. Google Scholar

[25]

A. C. Newell, Solitons in mathematics and physics, Society for Industrial and Applied Mathematics, (1985). doi: 10.1137/1.9781611970227.  Google Scholar

[26]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-68.   Google Scholar

[27]

M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagations, Commun. Part. Diff. Eq., 12 (1987), 1133-1173.   Google Scholar

[28]

J. Wright, Soliton production and solutions to perturbed Korteweg-de Vries equations, Phys. Rev. A, 21 (1980), 335-339.   Google Scholar

[29]

N. J. Zabusky and M. D. Kruskal, Interaction of "solitons" in a collisionless plasma and recurrence of initial states, Phys. Rev. Lett, 15 (1965), 240-243.   Google Scholar

show all references

References:
[1]

N. Asano, Wave propagation in non-uniform media, Progr. Theoret, Phys. Suppl., 55 (1974), 52-79.   Google Scholar

[2]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersion systems, Philo. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.   Google Scholar

[3]

J. L. Bona, W. G. Pritchard and L. R. Scott, Solitary-wave interaction, Phys. Fluids, 23 (1980), 438-441. Google Scholar

[4]

J. L. BonaP. E. Souganidis and W. A. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. London Ser. A, 411 (1987), 395-412.   Google Scholar

[5]

E. FermiJ. Pasta and S. Ulam, Studies of nonlinear problems Ⅰ, AMS, Editors: A. C. Newell, (1974), 143-156.   Google Scholar

[6]

R. Grimshaw, Slowly varying solitary waves. Ⅰ. Korteweg -de Vries equation, Proc. Roy. Soc. London Ser. A, 368 (1979), 359-375.   Google Scholar

[7]

R. Grimshaw, Slowly varying solitary waves. Ⅱ. Nonlinear Schrödinger equation, Proc. Roy. Soc. London Ser.A, 368 (1979), 377-388.   Google Scholar

[8]

R. H. J. Grimshaw and S. R. Pudjaprasetya, Generation of secondary solitary waves in the variable-coefficient Korteweg-de Vries equation, Stud. Appl. Math., 112 (2004), 271-279.   Google Scholar

[9]

V. I. Karpman and E. M. Maslov, Perturbation theory for solitons, Z. Eksper. Teoret. Fiz., 73 (1977), 281-291.   Google Scholar

[10]

D. J. Kaup and A. C. Newell, Soliton as particles, oscillators, and in slowly changing media a singular perturbation theory, Proc. R. Soc. Lond., 361 (1978), 413-446.   Google Scholar

[11]

K. Ko and H. H. Kuehl, Korteweg-de Vries soliton in a slowly varying medium, Phys. Rev. Lett., 40 (1978), 233-236.   Google Scholar

[12]

R. LeVeque, On the interaction of nearly equal solitons in the Kdv equation, SIAM. J. Appl. Math., 47 (1987), 254-262.   Google Scholar

[13]

Y. Martel, Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math., 127 (2005), 1103-1140.   Google Scholar

[14]

Y. Martel and F. Merle, Inelastic interaction of nearly equal solitons for the bbm equation, Discrete. Cont. Dyn., 27 (2010), 487-532.   Google Scholar

[15]

Y. Martel and F. Merle, Description of the Inelastic Collision of Two Solitary Waves for the BBM Equation, Arch. Ration. Mech. An, 196 (2010), 517-574.   Google Scholar

[16]

Y. Martel and F. Merle, Inelastic interaction of nearly equal solitonsfor the quartic gKdV equation, Invent. Math., 183 (2011), 563-648.   Google Scholar

[17]

Y. Martel and F. Merle, Description of two soliton collision for the quartic gKdV equation, Ann. Math., 174 (2011), 757-857.   Google Scholar

[18]

J. Miller and M. Weinstein, Asymptotic stability of solitary waves for the regularized long wave equation, Comm. Pure Appl. Math., 49 (1996), 399-441.   Google Scholar

[19]

T. Mizumachi, Asymptotic stability of solitary wave solutions to the regularized long wave equation, J. Differ. Equations, 200 (2004), 312-341.   Google Scholar

[20]

C. Muñoz, On the soliton dynamics under a slowly varying medium for generalized KdV equations, Anal. PDE., 4 (2011), 573-638.   Google Scholar

[21]

C. Muñoz, Dynamics of soliton-like solutions for slowly varying, generalized gKdV equations: refraction vs. Reflection, SIAM J. Math. Anal., 44 (2012), 1-60.   Google Scholar

[22]

C. Muñoz, Inelastic character of solitons of slowly varying gKdV equations, Commun. Math. Phys., 314 (2012), 817-852.   Google Scholar

[23]

C. Muñoz, On the soliton dynamics under slowly varying medium for Nonlinear Schrodinger equations, Math. Ann., 353 (2012), 867-943.   Google Scholar

[24]

C. Muñoz, Sharp inelastic character of slowly varying NLS solitons, preprint, arXiv: 1202.5807v2. Google Scholar

[25]

A. C. Newell, Solitons in mathematics and physics, Society for Industrial and Applied Mathematics, (1985). doi: 10.1137/1.9781611970227.  Google Scholar

[26]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-68.   Google Scholar

[27]

M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagations, Commun. Part. Diff. Eq., 12 (1987), 1133-1173.   Google Scholar

[28]

J. Wright, Soliton production and solutions to perturbed Korteweg-de Vries equations, Phys. Rev. A, 21 (1980), 335-339.   Google Scholar

[29]

N. J. Zabusky and M. D. Kruskal, Interaction of "solitons" in a collisionless plasma and recurrence of initial states, Phys. Rev. Lett, 15 (1965), 240-243.   Google Scholar

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