4 Citations
Finite gap conditions and small dispersion asymptotics for the classical periodic Benjamin–Ono equation
- MathematicsQuarterly of Applied Mathematics
- 2020
In this paper we characterize the Nazarov–Sklyanin hierarchy for the classical periodic Benjamin–Ono equation in two complementary degenerations: for the multiphase initial data (the periodic…
Lax eigenvalues in the zero-dispersion limit for the Benjamin-Ono equation on the torus
- Mathematics, Physics
- 2023
We consider the zero-dispersion limit for the Benjamin-Ono equation on the torus for bell shaped initial data. Using the approximation by truncated Fourier series, we transform the eigenvalue…
References
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- MathematicsPhilosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
- 1972
Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind.…
Integrable viscous conservation laws
- Mathematics
- 2013
We propose an extension of the Dubrovin–Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few…
Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach
- Mathematics
- 2008
AbstractWe obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation
$$u_t+6uu_x+\epsilon^{2}u_{xxx}=0,\quad u(x,t=0,\epsilon)=u_0(x),$$for…
Internal waves of permanent form in fluids of great depth
- Environmental ScienceJournal of Fluid Mechanics
- 1967
This paper presents a general theoretical treatment of a new class of long stationary waves with finite amplitude. As the property in common amongst physical systems capable of manifesting these…
An extension of the steepest descent method for Riemann-Hilbert problems: the small dispersion limit of the Korteweg-de Vries (KdV) equation.
- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1998
This paper extends the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou in a critical new way to small dispersion KdV (Korteweg-de Vries) equation and derives the hyperelliptic asymptotic solution of S. Venakides that describes the oscillations.
A Deformation of the Method of Characteristics and the Cauchy Problem for Hamiltonian PDEs in the Small Dispersion Limit
- Mathematics
- 2015
We introduce a deformation of the method of characteristics valid for Hamiltonian perturbations of a scalar conservation law in the small dispersion limit. Our method of analysis is based on the…
On a functional equation related to the intermediate long wave equation
- Mathematics
- 2004
We resolve an open problem stated by Ablowitz et al (1982 J. Phys. A: Math. Gen. 15 781) concerning the integral operator appearing in the intermediate long wave equation. We explain how this is…
The Small Dispersion Limit of the Korteweg-deVries Equation. I
- Mathematics
- 1982
In Part I the scattering transform method is used to study the weak limit of solutions to the initial value problem for the Korteweg-deVries (KdV) equation as the dispersion tends to zero. In that…
The KdV hierarchy: universality and a Painlevé transcendent
- Mathematics
- 2011
We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where $\e\to 0$. For negative analytic initial data with a single negative hump, we prove that for…
Monodromy- and spectrum-preserving deformations I
- Mathematics
- 1980
A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential…