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Register a new userRiemann-Hilbert problem, integrability and reductions
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The present paper is dedicated to integrable models with Mikhailov reduction groups $G_R simeq mathbb{D}_h.$ Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on the realization of the $G_R$-action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with $mathbb{D}_h$ symmetries are presented.
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The main goal of this paper is to put together: a) the Whitham theory applicable to slowly modulated $N$-phase nonlinear wave solutions to the focusing nonlinear Schrodinger (fNLS) equation, and b) the Riemann-Hilbert Problem approach to particular solutions of the fNLS in the semiclassical (small dispersion) limit that develop slowly modulated $N$-phase nonlinear wave in the process of evolution. Both approaches have their own merits and limitations. Understanding of the interrelations between them could prove beneficial for a broad range of problems involving the semiclassical fNLS.
Reductions of the KP-Whitham system, namely the (2+1)-dimensional hydrodynamic system of five equations that describes the slow modulations of periodic solutions of the Kadomtsev-Petviashvili (KP) equation, are studied. Specifically, the soliton and harmonic wave limits of the KP-Whitham system are considered, which give rise in each case to a four-component (2+1)-dimensional hydrodynamic system. It is shown that a suitable change of dependent variables splits the resulting four-component systems into two parts: (i) a decoupled, independent two-component system comprised of the dispersionless KP equation, (ii) an auxiliary, two-component system coupled to the mean flow equations, which describes either the evolution of a linear wave or a soliton propagating on top of the mean flow. The integrability of both four-component systems is then demonstrated by applying the Haantjes tensor test as well as the method of hydrodynamic reductions. Various exact reductions of these systems are then presented that correspond to concrete physical scenarios.
We consider a matrix Riemann-Hilbert problem for the sextic nonlinear Schr{o}dinger equation with a non-zero boundary conditions at infinity. Before analyzing the spectrum problem, we introduce a Riemann surface and uniformization coordinate variable in order to avoid multi-value problems. Based on a new complex plane, the direct scattering problem perform a detailed analysis of the analytical, asymptotic and symmetry properties of the Jost functions and the scattering matrix. Then, a generalized Riemann-Hilbert problem (RHP) is successfully established from the results of the direct scattering transform. In the inverse scattering problem, we discuss the discrete spectrum, residue condition, trace formula and theta condition under simple poles and double poles respectively, and further solve the solution of a generalized RHP. Finally, we derive the solution of the equation for the cases of different poles without reflection potential. In addition, we analyze the localized structures and dynamic behaviors of the resulting soliton solutions by taking some appropriate values of the parameters appeared in the solutions.
This paper discusses some general aspects and techniques associated with the long-time asymptotics of steplike solutions of the Korteweg-de Vries (KdV) equation via vector Riemann--Hilbert problems. We also elaborate on an ill-posedness of the matrix Riemann-Hilbert problems for the KdV case. To the best of our knowledge this is the first time such ill-posedness is discussed in applications of Riemann--Hilbert theory. Furthermore, we rigorously justify the asymptotics for the shock wave in the elliptic zone derived previously.
The Riemann-Hilbert (RH) problem is first developed to study the focusing nonlinear Schr{o}dinger (NLS) equation with multiple high-order poles under nonzero boundary conditions. Laurent expansion and Taylor series are employed to replace the residues at the simple- and the second-poles. Further, the solution of RH problem is transformed into a closed system of algebraic equations, and the soliton solutions corresponding to the transmission coefficient $1/s_{11}(z)$ with an $N$-order pole are obtained by solving the algebraic system. Then, in a more general case, the transmission coefficient with multiple high-order poles is studied, and the corresponding solutions are obtained. In addition, for high-order pole, the propagation behavior of the soliton solution corresponding to a third-order pole is given as example.
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