The focusing coupled modified Korteweg–de Vries equation with nonzero boundary conditions is investigated by the Riemann–Hilbert approach. Three symmetries are formulated to derive compact exact solutions. The solutions include six different types of soliton solutions and breathers, such as dark–dark, bright–bright, kink–dark–dark, kink–bright–bright solitons, and a breather–breather solution.
We study a diffusion-type equation for the density of a collisionless relativistic gas (Jüttner gas). The rate of diffusion propagation turns out to be finite. We consider problems of the existence and uniqueness of solutions of this equation, as well as some of its generalized solutions.
We investigate the analytic properties of solutions of a system of two first-order nonlinear differential equations with an arbitrary parameter (l) associated with an overdamped Josephson model. We reduce this system to a system of differential equations that is equivalent to the fifth Painlevé equation with the sets of parameters
$$biggl(frac{(1-l)^2}{8}, -frac{(1-l)^2}{8},0,-2biggr), ; biggl(frac{l^2}{8}, -frac{l^2}{8},0,-2biggr).$$We show that the solution of the third Painlevé equation with the parameters ((-2l, 2l-2,1,-1)) can be represented as the ratio of two linear fractional transformations of the solutions of the fifth Painlevé equation (with the parameters in the above sequence) connected by a Bäcklund transformation.
We consider solutions of the cubically nonlinear Schrödinger equation. For a certain class of solutions of the form (Psi(t,z)=(f(t,z)+id(z))e^{iphi(z)}) with (f,phi,dinmathbb{R}), we prove that they are nonexistent in the general case (f_zneq 0), (f_tneq 0), (d_zneq 0). In the three nongeneric cases ((f_zneq 0)), ((f_tneq 0), (f_t=0), (d_z=0)), and ((f_z=0), (f_tneq 0)), we present a two-parameter set of solutions, for which we find the constraints specifying real bounded and unbounded solutions.
We propose a special variant of the inverse scattering transform method to construct and analyze soliton excitations in a semi-infinite sample of an easy-axis ferromagnet in the case of a partial pinning of spins at its surface. We consider the limit cases of free edge spins and spins that are fully pinned at the sample boundary. We find frequency and modulation characteristics of solitons localized near the sample surface. In the case of different degrees of edge spin pinning, we study changes in the cores of moving solitons as a result of their elastic reflection from the sample boundary. We obtain integrals of motion that control the dynamics of magnetic solitons in a semi-infinite sample.
We consider the Cauchy problem for the cubic nonlinear Schrödinger equation with a large gradient of the initial function and a small dispersion parameter. The renormalization method is used to construct an asymptotic solution in the explicit form of integral convolution. An asymptotic analogue of the renormalization group property is established under scaling transformations determined by the dispersion parameter. In the case of a negative focusing coefficient, a clarifying expression is obtained for the asymptotic solution in terms of known elliptic special functions.
Scale transformations of cosmological models based on a statistical system of degenerate fermions with a scalar Higgs interaction are studied. The similarity properties of cosmological models under scale transformations of their fundamental parameters are revealed. The transformation laws for the coordinates of singular points and eigenvalues of the characteristic matrix of the dynamical system of the cosmological model under its scale transformations are established. With the help of the transformation to new variables, the previously studied dynamical system of scalar-charged fermions is modified to a dynamical system with a nondegenerate characteristic matrix; for its nondegenerate branch, the singular points and eigenvalues of the characteristic matrix are found, which coincide with the corresponding values for the vacuum field model. Examples of numerical simulation of such cosmological models are given.
We use a superfield Darboux matrix to study Darboux transformations of an (mathcal N=1) supersymmetric coupled dispersionless integrable system. The notion of quasideterminants is used to obtain superfield (N)-soliton solutions of that system. A superfield Lax representation is used to obtain a superfield Bäcklund transformation via a set of superfield Riccati equations. The Bäcklund and Darboux transformations are further used to compute explicit expressions for superfield soliton solutions of the supersymmetric coupled dispersionless integrable system.
The Benjamin–Ono equation involving the Hilbert transformation has been studied extensively from different standpoints. Its variant forms and multi-component extensions have been proposed. In this paper, we study the classification of two-component Benjamin–Ono-type systems of the general form. Our classification is carried out by developing the perturbative symmetry approach due to Mikhailov and Novikov. As a result, new two-component integrable Benjamin–Ono type systems are obtained.
We develop a method for deriving Hamilton’s equations describing the dynamics of solitons when they move along an inhomogeneous and time-varying large-scale background for nonlinear wave equations that are completely integrable in the Ablowitz–Kaup–Newell–Segur (AKNS) scheme. The method is based on the development of old Stokes’ ideas that allow analytically continuing the relations for linear waves into the soliton region, and is practically implemented in the example of the defocusing nonlinear Schrödinger equation. A condition is formulated under which the external potential is only to be taken into account when describing the evolution of the background, and that this case, the Newton equation is obtained for the soliton dynamics in an external potential.
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