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Register a new userDynamics of solutions in the generalized Benjamin-Ono equation: a numerical study
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Added by
Kai Yang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We consider the generalized Benjamin-Ono (gBO) equation on the real line, $ u_t + partial_x (-mathcal H u_{x} + tfrac1{m} u^m) = 0, x in mathbb R, m = 2,3,4,5$, and perform numerical study of its solutions. We first compute the ground state solution to $-Q -mathcal H Q^prime +frac1{m} Q^m = 0$ via Petviashvilis iteration method. We then investigate the behavior of solutions in the Benjamin-Ono ($m=2$) equation for initial data with different decay rates and show decoupling of the solution into a soliton and radiation, thus, providing confirmation to the soliton resolution conjecture in that equation. In the mBO equation ($m=3$), which is $L^2$-critical, we investigate solutions close to the ground state mass, and, in particular, we observe the formation of stable blow-up above it. Finally, we focus on the $L^2$-supercritical gBO equation with $m=4,5$. In that case we investigate the global vs finite time existence of solutions, and give numerical confirmation for the dichotomy conjecture, in particular, exhibiting blow-up phenomena in the supercritical setting.
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We consider a higher-dimensional version of the Benjamin-Ono (HBO) equation in the 2D setting: $u_t- mathcal{R}_1 Delta u + frac{1}{2}(u^2)_x=0, (x,y) in mathbb{R}^2$, which is $L^2$-critical, and investigate properties of solutions both analytically and numerically. For a generalized equation (fractional 2D gKdV) after deriving the Pohozaev identities, we obtain non-existence conditions for solitary wave solutions, then prove uniform bounds in the energy space or conditional global existence, and investigate the radiation region, a specific wedge in the negative $x$-direction. We then introduce our numerical approach in a general context, and apply it to obtain the ground state solution in the 2D critical HBO equation, then show that its mass is a threshold for global vs. finite time existing solutions, which is typical in the focusing (mass-)critical dispersive equations. We also observe that globally existing solutions tend to disperse completely into the radiation in this nonlocal equation. The blow-up solutions travel in the positive $x$-direction with the rescaled ground state profile while also radiating dispersive oscillations into the radiative wedge. We conclude with examples of different interactions of two solitary wave solutions, including weak and strong interactions.
In the article a convergent numerical method for conservative solutions of the Hunter--Saxton equation is derived. The method is based on piecewise linear projections, followed by evolution along characteristics where the time step is chosen in order to prevent wave breaking. Convergence is obtained when the time step is proportional to the square root of the spatial step size, which is a milder restriction than the common CFL condition for conservation laws.
We prove that if $u_1,,u_2$ are solutions of the Benjamin-Ono equation defined in $ (x,t)inR times [0,T]$ which agree in an open set $Omegasubset R times [0,T]$, then $u_1equiv u_2$. We extend this uniqueness result to a general class of equations of Benjamin-Ono type in both the initial value problem and the initial periodic boundary value problem. This class of 1-dimensional non-local models includes the intermediate long wave equation. Finally, we present a slightly stronger version of our uniqueness results for the Benjamin-Ono equation.
The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on $L^2({mathbb T})$. The paper shows that the Gibbs measures on bounded balls of $L^2$ satisfy some logarithmic Sobolev inequalities. The space of $n$-soliton solutions of the periodic Benjamin-Ono equation, as discovered by Case, is a Hamiltonian system with an invariant Gibbs measure. As $nrightarrowinfty$, these Gibbs measures exhibit a concentration of measure phenomenon. Case introduced soliton solutions that are parameterised by atomic measures in the complex plane. The limiting distributions of these measures gives the density of a compressible gas that satisfies the isentropic Euler equations.
This paper proposes a new class of mass or energy conservative numerical schemes for the generalized Benjamin-Ono (BO) equation on the whole real line with arbitrarily high-order accuracy in time. The spatial discretization is achieved by the pseudo-spectral method with the rational basis functions, which can be implemented by the Fast Fourier transform (FFT) with the computational cost $mathcal{O}( Nlog(N))$. By reformulating the spatial discretized system into the different equivalent forms, either the spatial semi-discretized mass or energy can be preserved exactly under the continuous time flow. Combined with the symplectic Runge-Kutta, with or without the scalar auxiliary variable reformulation, the fully discrete energy or mass conservative scheme can be constructed with arbitrarily high-order temporal accuracy, respectively. Our numerical results show the conservation of the proposed schemes, and also the superior accuracy and stability to the non-conservative (Leap-frog) scheme.
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