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doi: 10.3934/dcds.2020215

Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations

1. 

University of Illinois at Chicago, Department of Mathematics, Statistics and Computer Science, 851 South Morgan Street, Chicago, IL 60607, USA

2. 

University of Valladolid, Applied Mathematics Department, P/ Belen 15, 47011, Valladolid, Spain

3. 

Victoria University of Wellington, School of Mathematics and Statistics, PO Box 600, Wellington 6140, New Zealand

* Corresponding author: Jerry L. Bona

Received  October 2019 Revised  February 2020 Published  May 2020

Considered here are systems of partial differential equations arising in internal wave theory. The systems are asymptotic models describing the two-way propagation of long-crested interfacial waves in the Benjamin-Ono and the Intermediate Long-Wave regimes. Of particular interest will be solitary-wave solutions of these systems. Several methods of numerically approximating these solitary waves are put forward and their performance compared. The output of these schemes is then used to better understand some of the fundamental properties of these solitary waves.

The spatial structure of the systems of equations is non-local, like that of their one-dimensional, unidirectional relatives, the Benjamin-Ono and the Intermediate Long-Wave equations. As the non-local aspect is comprised of Fourier multiplier operators, this suggests the use of spectral methods for the discretization in space. Three iterative methods are proposed and implemented for approximating traveling-wave solutions. In addition to Newton-type and Petviashvili iterations, an interesting wrinkle on the usual Petviashvili method is put forward which appears to offer advantages over the other two techniques. The performance of these methods is checked in several ways, including using the approximations they generate as initial data in time-dependent codes for obtaining solutions of the Cauchy problem.

Attention is then turned to determining speed versus amplitude relations of these families of waves and their dependence upon parameters in the models. There are also provided comparisons between the unidirectional and bidirectional solitary waves. It deserves remark that while small-amplitude solitary-wave solutions of these systems are known to exist, our results suggest the amplitude restriction in the theory is artificial.

Citation: Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020215
References:
[1]

L. AbdelouhabJ. L. BonaM. Felland and J.-C. Saut, Nonlocal models for nonlinear dispersive waves, Physica D, 40 (1989), 360-392.  doi: 10.1016/0167-2789(89)90050-X.  Google Scholar

[2]

A. A. AlazmanJ. P. AlbertJ. L. BonaM. Chen and J. Wu, Comparisons between the BBM equation and a Boussinesq system, Advances Differential Eq., 11 (2006), 121-166.   Google Scholar

[3]

J. P. Albert and J. L. Bona, Comparisons between model equations for long waves, J. Nonlinear Sci., 1 (1991), 345-374.  doi: 10.1007/BF01238818.  Google Scholar

[4]

J. P. Albert and J. L. Bona, Total positivity and the stability of internal waves in fluids of finite depth, IMA J. Applied Math., 46 (1991), 1-19.  doi: 10.1093/imamat/46.1-2.1.  Google Scholar

[5]

J. P. AlbertJ. L. Bona and J.-M. Restrepo, Solitary-wave solutions of the Benjamin equation, SIAM J. Appl. Math., 59 (1999), 2139-2161.  doi: 10.1137/S0036139997321682.  Google Scholar

[6]

J. P. AlbertJ. L. Bona and J.-C. Saut, Model equations for waves in stratified fluids, Proc. Royal Soc. London, Series A, 453 (1997), 1233-1260.  doi: 10.1098/rspa.1997.0068.  Google Scholar

[7]

J. P. Albert and J. F. Toland, On the exact solutions of the intermediate long-wave equation, Differential Integral Eq., 7 (1994), 601-612.   Google Scholar

[8]

M. H. Alford et al., The formation and fate of internal waves in the South China Sea, Nature, 521 (2015), 65-69.   Google Scholar

[9]

J. Álvarez and A. Durán, An extended Petviashvili method for the numerical generation of traveling and localized waves, Comm. Nonlinear Sci. Numer. Simul., 19 (2014), 2272-2283.  doi: 10.1016/j.cnsns.2013.12.004.  Google Scholar

[10]

J. Álvarez and A. Durán, Petviashvili type methods for traveling wave computations: Ⅰ. Analysis of convergence, J. Comput. Appl. Math., 266 (2014), 39-51.  doi: 10.1016/j.cam.2014.01.015.  Google Scholar

[11]

J. Álvarez and A. Durán, Petviashvili type methods for traveling wave computations: Ⅱ. Acceleration with vector extrapolation methods, Math. Comput. Simul., 123 (2016), 19-36.  doi: 10.1016/j.matcom.2015.10.015.  Google Scholar

[12]

D. M. Ambrose, J. L. Bona and T. Milgrom, Global solutions and ill-posedness for the Kaup system and related Boussinesq systems, Indiana U. Math. J, 68 (2019), 1173–1198. doi: 10.1512/iumj.2019.68.7721.  Google Scholar

[13]

C. J. Amick and J. F. Toland, Uniqueness of Benjamin's solitary wave solution of the Benjamin-Ono equation, IMA J. Appl. Math., 46 (1991), 21-28.  doi: 10.1093/imamat/46.1-2.21.  Google Scholar

[14]

J. Angulo-Pava and J.-C. Saut, Existence of solitary waves solutions for internal waves in two-layers systems, Quart. Appl. Math., 78 (2020), 75-105.  doi: 10.1090/qam/1546.  Google Scholar

[15]

C. T. Anh, Influence of surface tension and bottom topography on internal waves, Math. Models Methods Appl. Sci., 19 (2009), 2145-2175.  doi: 10.1142/S0218202509004078.  Google Scholar

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T. B. BenjaminJ. L. Bona and D. K. Bose, Solitary-wave solutions of nonlinear problems, Phil. Trans. Royal Soc. London, Series A, 331 (1990), 195-244.  doi: 10.1098/rsta.1990.0065.  Google Scholar

[18]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear, dispersive media, Philos. Trans. Royal Soc. London, Series A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[19]

D. P. BennettR. W. BrownS. E. StansfieldJ. D. Stroughair and J. L. Bona, The stability of internal solitary waves in stratified fluids, Math. Proc. Cambridge Philos. Soc., 94 (1983), 351-379.  doi: 10.1017/S0305004100061193.  Google Scholar

[20]

J. L. Bona, Convergence of periodic wave trains in the limit of large wavelength, Appl. Sci. Res., 37 (1981), 21-30.  doi: 10.1007/BF00382614.  Google Scholar

[21]

J. L. BonaX. CarvajalM. Panthee and M. Scialom, Higher-order Hamiltonian model for unidirectional water waves, J. Nonlinear Sci., 28 (2018), 543-577.  doi: 10.1007/s00332-017-9417-y.  Google Scholar

[22]

J. L. Bona and H. Chen, Solitary waves in nonlinear dispersive systems, Discrete Cont. Dynamical Sys. B, 2 (2002), 313-378.  doi: 10.3934/dcdsb.2002.2.313.  Google Scholar

[23]

J. L. BonaM. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅰ: Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4.  Google Scholar

[24]

J. L. BonaV. A. DougalisO. A. Karakashian and W. R. McKinney, Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation, Philos. Trans. Royal Soc. London, Series A, 351 (1995), 107-164.  doi: 10.1098/rsta.1995.0027.  Google Scholar

[25]

J. L. Bona, A. Durán and D. Mitsotakis, Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. Ⅱ. Dynamics, In preparation. Google Scholar

[26]

J. L. BonaD. Lannes and J.-C. Saut, Asymptotic models for internal waves, J. Math. Pures. Appl., 89 (2008), 538-566.  doi: 10.1016/j.matpur.2008.02.003.  Google Scholar

[27]

J. L. Bona and Y. A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430.  doi: 10.1016/S0021-7824(97)89957-6.  Google Scholar

[28]

J. L. Bona and M. Scialom, The effect of change in the nonlinearity and the dispersion relation of model equations for long waves, Canadian Appl. Math. Quart., 3 (1995), 1-41.   Google Scholar

[29]

J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta Methods and General Linear Methods, John Wiley & Sons, Ltd., Chichester, 1987.  Google Scholar

[30]

R. CamassaW. ChoiH. MichalletP.-O. Rusas and J. K. Sveen, On the realm of validity of strongly nonlinear asymptotic approximations for internal waves, J. Fluid Mech., 549 (2006), 1-23.  doi: 10.1017/S0022112005007226.  Google Scholar

[31]

C. Canuto, M. Y. Hussaini, A. Quarteroni and A. T. Zang, Spectral Methods in Fluid Dynamics, Springer; New York, 1985. Google Scholar

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H. Chen, Long-period limit of nonlinear, dispersive waves: The BBM equation, Differential Integral Eq., 19 (2006), 463-480.   Google Scholar

[33]

H. H. Chen and Y. C. Lee, Internal-wave solitons of fluids with finite depth, Phys. Rev. Lett., 43 (1979), 264-266.  doi: 10.1103/PhysRevLett.43.264.  Google Scholar

[34]

W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluid system, J. Fluid Mech., 313 (1996), 83-103.  doi: 10.1017/S0022112096002133.  Google Scholar

[35]

W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system, J. Fluid Mech., 396 (1999), 1-36.  doi: 10.1017/S0022112099005820.  Google Scholar

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W. CraigP. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Comm. Pure Appl. Math., 58 (2005), 1587-1641.  doi: 10.1002/cpa.20098.  Google Scholar

[37]

V. A. DougalisA. DuránM. A. Lopez-Marcos and D. Mitsotakis, A numerical study of the stability of solitary waves of the Bona-Smith family of Boussinesq systems, J. Nonlinear Sci., 17 (2007), 569-607.  doi: 10.1007/s00332-007-9004-8.  Google Scholar

[38]

V. A. Dougalis, A. Durán and D. Mitsotakis, Numerical approximation of solitary waves of the Benjamin equation, Math. Comput. Simul., 127 (2016), 56–79. doi: 10.1016/j.matcom.2012.07.008.  Google Scholar

[39]

V. Duchene, Asymptotic shallow water models for internal waves in a two-fluid system with a free surface, SIAM J. Math. Anal., 42 (2010), 2229-2260.  doi: 10.1137/090761100.  Google Scholar

[40]

V. Duchene, Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation, M2AN Math. Model. Numer. Anal., 46 (2012), 145-185.  doi: 10.1051/m2an/2011037.  Google Scholar

[41]

M. Frigo and S. G. Johnson, The design and implementation of fftw3, Proc. IEEE, 93 (2005), 216-231.  doi: 10.1109/JPROC.2004.840301.  Google Scholar

[42]

P. GuyenneD. Lannes and J.-C. Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves, Nonlinearity, 23 (2010), 237-275.  doi: 10.1088/0951-7715/23/2/003.  Google Scholar

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K. R. Helfrich and W. K. Melville, Long nonlinear internal waves, Annual Review of Fluid Mechanics, 38 (2006), 395-425.  doi: 10.1146/annurev.fluid.38.050304.092129.  Google Scholar

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H. Kalisch, Error analysis of spectral projections of the regularized Benjamin-Ono equation, BIT, 45 (2005), 69-89.  doi: 10.1007/s10543-005-2636-x.  Google Scholar

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H. Kalisch and J. L. Bona, Models for internal waves in deep water, Discrete Contin. Dynamical Syst., 6 (2000), 1-20.  doi: 10.3934/dcds.2000.6.1.  Google Scholar

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show all references

References:
[1]

L. AbdelouhabJ. L. BonaM. Felland and J.-C. Saut, Nonlocal models for nonlinear dispersive waves, Physica D, 40 (1989), 360-392.  doi: 10.1016/0167-2789(89)90050-X.  Google Scholar

[2]

A. A. AlazmanJ. P. AlbertJ. L. BonaM. Chen and J. Wu, Comparisons between the BBM equation and a Boussinesq system, Advances Differential Eq., 11 (2006), 121-166.   Google Scholar

[3]

J. P. Albert and J. L. Bona, Comparisons between model equations for long waves, J. Nonlinear Sci., 1 (1991), 345-374.  doi: 10.1007/BF01238818.  Google Scholar

[4]

J. P. Albert and J. L. Bona, Total positivity and the stability of internal waves in fluids of finite depth, IMA J. Applied Math., 46 (1991), 1-19.  doi: 10.1093/imamat/46.1-2.1.  Google Scholar

[5]

J. P. AlbertJ. L. Bona and J.-M. Restrepo, Solitary-wave solutions of the Benjamin equation, SIAM J. Appl. Math., 59 (1999), 2139-2161.  doi: 10.1137/S0036139997321682.  Google Scholar

[6]

J. P. AlbertJ. L. Bona and J.-C. Saut, Model equations for waves in stratified fluids, Proc. Royal Soc. London, Series A, 453 (1997), 1233-1260.  doi: 10.1098/rspa.1997.0068.  Google Scholar

[7]

J. P. Albert and J. F. Toland, On the exact solutions of the intermediate long-wave equation, Differential Integral Eq., 7 (1994), 601-612.   Google Scholar

[8]

M. H. Alford et al., The formation and fate of internal waves in the South China Sea, Nature, 521 (2015), 65-69.   Google Scholar

[9]

J. Álvarez and A. Durán, An extended Petviashvili method for the numerical generation of traveling and localized waves, Comm. Nonlinear Sci. Numer. Simul., 19 (2014), 2272-2283.  doi: 10.1016/j.cnsns.2013.12.004.  Google Scholar

[10]

J. Álvarez and A. Durán, Petviashvili type methods for traveling wave computations: Ⅰ. Analysis of convergence, J. Comput. Appl. Math., 266 (2014), 39-51.  doi: 10.1016/j.cam.2014.01.015.  Google Scholar

[11]

J. Álvarez and A. Durán, Petviashvili type methods for traveling wave computations: Ⅱ. Acceleration with vector extrapolation methods, Math. Comput. Simul., 123 (2016), 19-36.  doi: 10.1016/j.matcom.2015.10.015.  Google Scholar

[12]

D. M. Ambrose, J. L. Bona and T. Milgrom, Global solutions and ill-posedness for the Kaup system and related Boussinesq systems, Indiana U. Math. J, 68 (2019), 1173–1198. doi: 10.1512/iumj.2019.68.7721.  Google Scholar

[13]

C. J. Amick and J. F. Toland, Uniqueness of Benjamin's solitary wave solution of the Benjamin-Ono equation, IMA J. Appl. Math., 46 (1991), 21-28.  doi: 10.1093/imamat/46.1-2.21.  Google Scholar

[14]

J. Angulo-Pava and J.-C. Saut, Existence of solitary waves solutions for internal waves in two-layers systems, Quart. Appl. Math., 78 (2020), 75-105.  doi: 10.1090/qam/1546.  Google Scholar

[15]

C. T. Anh, Influence of surface tension and bottom topography on internal waves, Math. Models Methods Appl. Sci., 19 (2009), 2145-2175.  doi: 10.1142/S0218202509004078.  Google Scholar

[16]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.  doi: 10.1017/S002211206700103X.  Google Scholar

[17]

T. B. BenjaminJ. L. Bona and D. K. Bose, Solitary-wave solutions of nonlinear problems, Phil. Trans. Royal Soc. London, Series A, 331 (1990), 195-244.  doi: 10.1098/rsta.1990.0065.  Google Scholar

[18]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear, dispersive media, Philos. Trans. Royal Soc. London, Series A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[19]

D. P. BennettR. W. BrownS. E. StansfieldJ. D. Stroughair and J. L. Bona, The stability of internal solitary waves in stratified fluids, Math. Proc. Cambridge Philos. Soc., 94 (1983), 351-379.  doi: 10.1017/S0305004100061193.  Google Scholar

[20]

J. L. Bona, Convergence of periodic wave trains in the limit of large wavelength, Appl. Sci. Res., 37 (1981), 21-30.  doi: 10.1007/BF00382614.  Google Scholar

[21]

J. L. BonaX. CarvajalM. Panthee and M. Scialom, Higher-order Hamiltonian model for unidirectional water waves, J. Nonlinear Sci., 28 (2018), 543-577.  doi: 10.1007/s00332-017-9417-y.  Google Scholar

[22]

J. L. Bona and H. Chen, Solitary waves in nonlinear dispersive systems, Discrete Cont. Dynamical Sys. B, 2 (2002), 313-378.  doi: 10.3934/dcdsb.2002.2.313.  Google Scholar

[23]

J. L. BonaM. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅰ: Derivation and linear theory, J. Nonlinear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4.  Google Scholar

[24]

J. L. BonaV. A. DougalisO. A. Karakashian and W. R. McKinney, Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation, Philos. Trans. Royal Soc. London, Series A, 351 (1995), 107-164.  doi: 10.1098/rsta.1995.0027.  Google Scholar

[25]

J. L. Bona, A. Durán and D. Mitsotakis, Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. Ⅱ. Dynamics, In preparation. Google Scholar

[26]

J. L. BonaD. Lannes and J.-C. Saut, Asymptotic models for internal waves, J. Math. Pures. Appl., 89 (2008), 538-566.  doi: 10.1016/j.matpur.2008.02.003.  Google Scholar

[27]

J. L. Bona and Y. A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430.  doi: 10.1016/S0021-7824(97)89957-6.  Google Scholar

[28]

J. L. Bona and M. Scialom, The effect of change in the nonlinearity and the dispersion relation of model equations for long waves, Canadian Appl. Math. Quart., 3 (1995), 1-41.   Google Scholar

[29]

J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta Methods and General Linear Methods, John Wiley & Sons, Ltd., Chichester, 1987.  Google Scholar

[30]

R. CamassaW. ChoiH. MichalletP.-O. Rusas and J. K. Sveen, On the realm of validity of strongly nonlinear asymptotic approximations for internal waves, J. Fluid Mech., 549 (2006), 1-23.  doi: 10.1017/S0022112005007226.  Google Scholar

[31]

C. Canuto, M. Y. Hussaini, A. Quarteroni and A. T. Zang, Spectral Methods in Fluid Dynamics, Springer; New York, 1985. Google Scholar

[32]

H. Chen, Long-period limit of nonlinear, dispersive waves: The BBM equation, Differential Integral Eq., 19 (2006), 463-480.   Google Scholar

[33]

H. H. Chen and Y. C. Lee, Internal-wave solitons of fluids with finite depth, Phys. Rev. Lett., 43 (1979), 264-266.  doi: 10.1103/PhysRevLett.43.264.  Google Scholar

[34]

W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluid system, J. Fluid Mech., 313 (1996), 83-103.  doi: 10.1017/S0022112096002133.  Google Scholar

[35]

W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system, J. Fluid Mech., 396 (1999), 1-36.  doi: 10.1017/S0022112099005820.  Google Scholar

[36]

W. CraigP. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Comm. Pure Appl. Math., 58 (2005), 1587-1641.  doi: 10.1002/cpa.20098.  Google Scholar

[37]

V. A. DougalisA. DuránM. A. Lopez-Marcos and D. Mitsotakis, A numerical study of the stability of solitary waves of the Bona-Smith family of Boussinesq systems, J. Nonlinear Sci., 17 (2007), 569-607.  doi: 10.1007/s00332-007-9004-8.  Google Scholar

[38]

V. A. Dougalis, A. Durán and D. Mitsotakis, Numerical approximation of solitary waves of the Benjamin equation, Math. Comput. Simul., 127 (2016), 56–79. doi: 10.1016/j.matcom.2012.07.008.  Google Scholar

[39]

V. Duchene, Asymptotic shallow water models for internal waves in a two-fluid system with a free surface, SIAM J. Math. Anal., 42 (2010), 2229-2260.  doi: 10.1137/090761100.  Google Scholar

[40]

V. Duchene, Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation, M2AN Math. Model. Numer. Anal., 46 (2012), 145-185.  doi: 10.1051/m2an/2011037.  Google Scholar

[41]

M. Frigo and S. G. Johnson, The design and implementation of fftw3, Proc. IEEE, 93 (2005), 216-231.  doi: 10.1109/JPROC.2004.840301.  Google Scholar

[42]

P. GuyenneD. Lannes and J.-C. Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves, Nonlinearity, 23 (2010), 237-275.  doi: 10.1088/0951-7715/23/2/003.  Google Scholar

[43]

K. R. Helfrich and W. K. Melville, Long nonlinear internal waves, Annual Review of Fluid Mechanics, 38 (2006), 395-425.  doi: 10.1146/annurev.fluid.38.050304.092129.  Google Scholar

[44]

R. I. Joseph, Solitary waves in a finite depth fluid, J. Phys. A, 10 (1977), L225–L227. doi: 10.1088/0305-4470/10/12/002.  Google Scholar

[45]

R. I. Joseph, Comment on "internal-wave solitons of fluids with finite depth", Phys. Rev. A, 21 (1980), 691-692.  doi: 10.1103/PhysRevA.21.691.  Google Scholar

[46]

H. Kalisch, Error analysis of spectral projections of the regularized Benjamin-Ono equation, BIT, 45 (2005), 69-89.  doi: 10.1007/s10543-005-2636-x.  Google Scholar

[47]

H. Kalisch and J. L. Bona, Models for internal waves in deep water, Discrete Contin. Dynamical Syst., 6 (2000), 1-20.  doi: 10.3934/dcds.2000.6.1.  Google Scholar

[48]

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Figure 1.  Sketch of an ideal fluid system for internal wave propagation. The fluid layers are homogeneous with densities $ \rho_{1}<\rho_{2} $
Figure 2.  Solitary-wave solutions of the BO system. Approximate $ \zeta $ and $ u $ profiles generated using the Petviashvili method (18) with $ \gamma = 0.8, \alpha = 1.2 $ and $ c_s = 0.57 $
Figure 3.  Discrepancy $ |1-M_n| $ of the stabilizing factor vs. number of iterations for the BO system using the Petviashvili-type methods (18) and (20)
Figure 4.  Logarithm of residual errors (25) vs. the number $ N_{iter} $ of iterations and vs. CPU time for the BO system and for the three iterative methods
Figure 5.  Graph of an approximate solitary-wave solution of the ILW system. The approximate profiles $ \zeta,u $ were generated using the Petviashvili method (18) with $ \gamma = 0.8 $
Figure 6.  Propagation of an approximate solitary-wave solution for the BO system. The profile was generated using the Petviashvili method (18) with $ c_s = 0.57 $
Figure 7.  Solitary waves of the BO system (left) and the rBO equation (right) for various values of $ \gamma $ and $ c_s $
Figure 8.  Peak amplitude of the computed solitary waves as a function of the speed $ c_s $ f or BO system and rBO equation for various values of $ \gamma $
Figure 9.  Comparison of solitary-wave solutions of the BO system and the rBO equation with $ \gamma = 0.8 $
Figure 10.  Comparison of some solitary waves of the BO system and the rBO equation with $ \gamma = 0.1 $
Figure 11.  Convergence of the solitary waves of the ILW system to a solitary wave solution of the BO system for large values of the lower depth $ d_2 $; $ \alpha = 1.2 $, $ \gamma = 0.8 $, $ \varepsilon = \sqrt{\mu} = 0.1 $
Table 1.  The six eigenvalues largest in magnitude of the iteration matrices evaluated at the profiles shown in Figure 2: (left) classical fixed point algorithm (17) and (right) the Petviashvili method (18)
Classical fixed point method Petviashvili method
1.9999999 0.9999999
0.9999999 0.8192378
0.8192378 0.6840761
0.6840761 0.6220421
0.6220421 0.5686637
0.5686637 0.5454789
Classical fixed point method Petviashvili method
1.9999999 0.9999999
0.9999999 0.8192378
0.8192378 0.6840761
0.6840761 0.6220421
0.6220421 0.5686637
0.5686637 0.5454789
Table 2.  Normalized amplitude error $ AE_n $, shape error $ SE_n $, phase error $ PE_n $ and speed error $ CE_n $ in the case of a solitary wave with $ \gamma = 0.8, \alpha = 1.2 $ and $ c_s = 0.57 $ for the BO system
$ t^n $ $ AE_n $ $ SE_n $ $ PE_n $ $ CE_n $
$ 10 $ $ 0.1091\times 10^{-6} $ $ 0.0768\times 10^{-6} $ $ -0.0437\times 10^{-6} $ $ -0.4905\times 10^{-8} $
$ 20 $ $ 0.1256\times 10^{-6} $ $ 0.0871\times 10^{-6} $ $ -0.0750\times 10^{-6} $ $ -0.5884\times 10^{-8} $
$ 30 $ $ 0.1356\times 10^{-6} $ $ 0.0932\times 10^{-6} $ $ -0.1106\times 10^{-6} $ $ -0.6482\times 10^{-8} $
$ 40 $ $ 0.1431\times 10^{-6} $ $ 0.0974\times 10^{-6} $ $ -0.1494\times 10^{-6} $ $ -0.7026\times 10^{-8} $
$ 50 $ $ 0.1492\times 10^{-6} $ $ 0.1007\times 10^{-6} $ $ -0.1910\times 10^{-6} $ $ -0.7536\times 10^{-8} $
$ 60 $ $ 0.1546\times 10^{-6} $ $ 0.1035\times 10^{-6} $ $ -0.2354\times 10^{-6} $ $ -0.8017\times 10^{-8} $
$ 70 $ $ 0.1596\times 10^{-6} $ $ 0.1059\times 10^{-6} $ $ -0.2825\times 10^{-6} $ $ -0.8461\times 10^{-8} $
$ 80 $ $ 0.1642\times 10^{-6} $ $ 0.1081\times 10^{-6} $ $ -0.3322\times 10^{-6} $ $ -0.8982\times 10^{-8} $
$ 90 $ $ 0.1686\times 10^{-6} $ $ 0.1101\times 10^{-6} $ $ -0.3844\times 10^{-6} $ $ -0.9366\times 10^{-8} $
$ 100 $ $ 0.1728\times 10^{-6} $ $ 0.1121\times 10^{-6} $ $ -0.4392\times 10^{-6} $ $ -0.9815\times 10^{-8} $
$ t^n $ $ AE_n $ $ SE_n $ $ PE_n $ $ CE_n $
$ 10 $ $ 0.1091\times 10^{-6} $ $ 0.0768\times 10^{-6} $ $ -0.0437\times 10^{-6} $ $ -0.4905\times 10^{-8} $
$ 20 $ $ 0.1256\times 10^{-6} $ $ 0.0871\times 10^{-6} $ $ -0.0750\times 10^{-6} $ $ -0.5884\times 10^{-8} $
$ 30 $ $ 0.1356\times 10^{-6} $ $ 0.0932\times 10^{-6} $ $ -0.1106\times 10^{-6} $ $ -0.6482\times 10^{-8} $
$ 40 $ $ 0.1431\times 10^{-6} $ $ 0.0974\times 10^{-6} $ $ -0.1494\times 10^{-6} $ $ -0.7026\times 10^{-8} $
$ 50 $ $ 0.1492\times 10^{-6} $ $ 0.1007\times 10^{-6} $ $ -0.1910\times 10^{-6} $ $ -0.7536\times 10^{-8} $
$ 60 $ $ 0.1546\times 10^{-6} $ $ 0.1035\times 10^{-6} $ $ -0.2354\times 10^{-6} $ $ -0.8017\times 10^{-8} $
$ 70 $ $ 0.1596\times 10^{-6} $ $ 0.1059\times 10^{-6} $ $ -0.2825\times 10^{-6} $ $ -0.8461\times 10^{-8} $
$ 80 $ $ 0.1642\times 10^{-6} $ $ 0.1081\times 10^{-6} $ $ -0.3322\times 10^{-6} $ $ -0.8982\times 10^{-8} $
$ 90 $ $ 0.1686\times 10^{-6} $ $ 0.1101\times 10^{-6} $ $ -0.3844\times 10^{-6} $ $ -0.9366\times 10^{-8} $
$ 100 $ $ 0.1728\times 10^{-6} $ $ 0.1121\times 10^{-6} $ $ -0.4392\times 10^{-6} $ $ -0.9815\times 10^{-8} $
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