Related papers: Wave phenomena of the Toda lattice with steplike i…
The asymptotic stability of rarefaction wave for 1-d relaxed compressible isentropic Navier-Stokes equations is established. For initial data with different far-field values, we show that there exists a unique global in time solution.…
In 1978, A. C. Newell [SIAM J. Appl. Math. 35(4) (1978) 650-664] proposed an exactly solvable model called Newell equation, which simulates the investigation of significant interaction mechanism between long and short waves. Nearly fifty…
We extend a recent result of [13] for the KdV hierarchy to the Toda lattice hierarchy. Namely, for an arbitrary solution to the Toda lattice hierarchy, we define a pair of wave functions, and use them to give explicit formulae for the…
We consider a nonlinear field equation which can be derived from a binomial lattice as a continuous limit. This equation, containing a perturbative friction-like term and a free parameter $\gamma$, reproduces the Toda case (in absence of…
In integrable models, stationary equations for higher symmetries serve as one of the main sources of reductions consistent with dynamics. We apply this method to the non-Abelian two-dimensional Toda lattice. It is shown that already the…
We use the combinatorics of toric networks and the double affine geometric $R$-matrix to define a three-parameter family of generalizations of the discrete Toda lattice. We construct the integrals of motion and a spectral map for this…
A system of linear differential equations with oscillatory decreasing coefficients is considered. The coefficients has the form $t^{-\alpha}a(t)$,~$\alpha>0$, where $a(t)$ is trigonometric polynomial with an arbitrary set of frequencies.…
Large-time asymptotic properties of solutions to a class of semilinear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are…
This paper presents an asymptotic analysis of the Boltzmann equations (Riccati differential equations) that describe the physics of thermal dark-matter-relic abundances. Two different asymptotic techniques are used, boundary-layer theory,…
We present a new and relatively elementary method for studying the solution of the initial-value problem for dispersive linear and integrable equations in the large-$t$ limit, based on a generalization of steepest descent techniques for…
A new class of integrable two-dimensional dilaton gravity theories, in which scalar matter fields satisfy the Toda equations, is proposed. The simplest case of the Toda system is considered in some detail, and on this example we outline how…
We have derived a non-abelian analog for the two-dimensional discrete Toda lattice which possesses solutions in terms of quasideterminants and admits Lax pairs of different forms. Its connection with non-abelian analogs for several…
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction…
Solitary waves in a general nonlinear lattice are discussed, employing as a model the nonlinear Schr\"odinger equation with a spatially periodic nonlinear coefficient. An asymptotic theory is developed for long solitary waves, that span a…
A method for solving the half-space Sommerfeld problem is proposed, which allows us to obtain exact solutions in the form of Sommerfeld integrals, as well as their short-wave asymptotics. The first carried out by reducing the Sommerfeld…
We discuss the algebro-geometric initial value problem for the Toda hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we…
We establish spectral, linear, and nonlinear stability of the vanishing and slow-moving travelling waves that arise as time asymptotic solutions to the Fisher-Stefan equation. Nonlinear stability is in terms of the limiting equations that…
We investigate the time-asymptotic stability of planar rarefaction wave for the three-dimensional Boltzmann equation, based on the micro-macro decomposition introduced in [24, 22] and our new observations on the underlying wave structures…
A fairly complete list of Toda-like integrable lattice systems, both in the continuous and discrete time, is given. For each system the Newtonian, Lagrangian and Hamiltonian formulations are presented, as well as the 2x2 Lax representation…
Results on the finite nonperiodic Toda lattice are extended to some generalizations of the system: The relativistic Toda lattice, the generalized Toda lattice associated with simple Lie groups and the full Kostant-Toda lattice. The areas…