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Following a prescription of \cite{4} for a solitonic specialization of the general solutions to the (abelian) periodic Toda field theories, we discuss a construction of the soliton solutions for a wide class of two-dimensional completely…
In this paper, a general bright-dark soliton solution in the form of Pfaffian is constructed for an integrable semi-discrete vector NLS equation via Hirota's bilinear method. One- and two-bright-dark soliton solutions are explicitly…
We review recent results on global wellposedness and long-time behavior of smooth solutions to the derivative nonlinear Schr\"{o}dinger (DNLS) equation. Using the integrable character of DNLS, we show how the inverse scattering tools and…
We propose an ultradiscrete analogue of the vertex operator in the case of the ultradiscrete KdV equation, which maps N-soliton solutions to N+1-soliton ones.
In this paper we announce the result of asymptotic dynamics of solitons of nonlinear Schrodinger equations with external potentials. To each local minima of the potential there is a soliton centered around it. Under some conditions on the…
We announce a detailed numerical investigation for some class of difference schemes, which arises from Euler implicit scheme. Such schemes demonstrate unusual behavior and leads to origin of solitons. Applications to some nonlinear problems…
A class of special solutions are constructed in an intuitive way for the ultradiscrete analog of $q$-Painlev\'e II ($q$-PII) equation. The solutions are classified into four groups depending on the function-type and the system parameter.
The discrete Toda lattice preserves the eigenvalues of tridiagonal matrices, and convergence of dependent variables to the eigenvalues can be proved under appropriate conditions. We show that the ultradiscrete Toda lattice preserves…
In this paper we show that the set of closure relations on a finite poset P forms a supersolvable lattice, as suggested by Rota. Furthermore this lattice is dually isomorphic to the lattice of closed sets in a convex geometry (in the sense…
In this paper, we present a systematic procedure to derive discrete analogues of integrable PDEs via Hirota's bilinear method. This approach is mainly based on the compatibility between an integrable system and its B\"acklund…
The focusing nonlinear Schrodinger equation possesses special non-dispersive solitary type solutions, solitons. Under certain spectral assumptions we show existence and asymptotic stability of solutions with the asymptoic profile (as time…
We are interested in solutions of the nonlinear Klein-Gordon equation (NLKG) in $\mathbb{R}^{1+d}$, $d\ge1$, which behave as a soliton or a sum of solitons in large time. In the spirit of other articles focusing on the supercritical…
We present a simple recipe to construct exactly and quasi-exactly solvable Hamiltonians in one-dimensional `discrete' quantum mechanics, in which the Schr\"{o}dinger equation is a difference equation. It reproduces all the known ones whose…
In the paper we present a functional-discrete method for solving Sturm-Liouville problems with potential including function from L_{1}(0,1) and \delta-function. For both, linear and nonlinear cases the sufficient conditions providing…
New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, it gives an explicit solution to the Ditkin-Prudnikov problem (1966). The 3-term recurrence relations, explicit representations,…
In this paper, we prove the soliton resolution conjecture for general type II solutions to the focusing energy critical wave equation, in space dimension 3,4 or 5, along a sequence of times. This is an important step towards the full…
We introduce a simultaneous ultradiscrete Painleve II equation with parity variables, which is shown to be more suitable for studying two-parameter solutions than the single second-order ultradiscrete Painleve II equation with parity…
Exact stationary soliton solutions of the fifth order KdV type equation $$ u_t +\alpha u^p u_x +\beta u_{3x}+\gamma u_{5x} = 0$$ are obtained for any p ($>0$) in case $\alpha\beta>0$, $D\beta>0$, $\beta\gamma<0$ (where D is the soliton…
The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3\kappa u_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as a model delineating the propagation of shallow water waves, stands as a completely integrable system…
New exact solutions, nonstationary and stationary, of Veselov-Novikov (VN) equation in the forms of linear superpositions of arbitrary number of exact special solutions $u^{(n)}$, $n=1,...,N$ are constructed via $\bar\partial$-dressing…