# American Institute of Mathematical Sciences

January  2021, 41(1): 87-111. 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, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215
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##### References:
Sketch of an ideal fluid system for internal wave propagation. The fluid layers are homogeneous with densities $\rho_{1}<\rho_{2}$
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 $c_s = 0.57$
Discrepancy $|1-M_n|$ of the stabilizing factor vs. number of iterations for the BO system using the Petviashvili-type methods (18) and (20)
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
Graph of an approximate solitary-wave solution of the ILW system. The approximate profiles $\zeta,u$ were generated using the Petviashvili method (18) with $\gamma = 0.8$
Propagation of an approximate solitary-wave solution for the BO system. The profile was generated using the Petviashvili method (18) with $c_s = 0.57$
Solitary waves of the BO system (left) and the rBO equation (right) for various values of $\gamma$ and $c_s$
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 $\gamma$
Comparison of solitary-wave solutions of the BO system and the rBO equation with $\gamma = 0.8$
Comparison of some solitary waves of the BO system and the rBO equation with $\gamma = 0.1$
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 $d_2$; $\alpha = 1.2$, $\gamma = 0.8$, $\varepsilon = \sqrt{\mu} = 0.1$
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
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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