Studies in Applied Mathematics最新文献

We study integrable boundary conditions for the whole hierarchy of nonlinear Schrödinger (NLS) equations defined on the half-line. We find that the even-order and odd-order NLS equations admit rather different integrable boundary conditions. In particular, the odd-order NLS equations admit a new class of integrable boundary conditions that involves time reversal. We establish the integrability of the NLS hierarchy with our new boundary conditions by demonstrating the existence of infinitely many integrals of motion in involution. Moreover, we develop the boundary dressing technique to construct soliton solutions satisfying these new boundary conditions.

Using the original advanced version of the direct method, we efficiently compute the equivalence groupoids and equivalence groups of two peculiar classes of Kolmogorov backward equations with power diffusivity and solve the problems of their complete group classifications. The results on the equivalence groups are double-checked with the algebraic method. Within these classes, the remarkable Fokker–Planck and the fine Kolmogorov backward equations are distinguished by their exceptional symmetry properties. We extend the known results on these two equations to their counterparts with respect to a nontrivial discrete equivalence transformation. Additionally, we carry out Lie reductions of the equations under consideration up to the point equivalence, exhaustively study their hidden Lie symmetries, and generate wider families of their new exact solutions via acting by their recursion operators on constructed Lie-invariant solutions. This analysis reveals eight powers of the space variable with exponents $��-�1�$, 0, 1, 2, 3, 4, 5, and 6 as values of the diffusion coefficient that are prominent due to symmetry properties of the corresponding equations.

We consider the forced vibration of Euler–Bernoulli beam equation with hinged and elastically fixed ends, and study the existence of multiple periodic solutions for such an equation with the nonlinear term having superliner growth but not requiring its homogeneity and monotonicity. At the first step, we need to analyze the asymptotic formula of the eigenvalues and eigenfunctions for the corresponding Sturm–Liouville problem. Then we investigate the fundamental properties of the linear beam operator and the corresponding functional. Finally, in view of the effect of inhomogeneous term, we shall construct a modified functional and make use of the minimax method and Morse index to obtain the existence of infinitely many periodic solutions.

We describe some properties of zeros of the Umemura polynomials associated with a class of rational solutions to the fifth Painlevé equation. A combinatorial formula for the discriminant of the Umemura polynomials is constructed. Based on the result, we investigate the multiplicity of zeros of the Umemura polynomials. We also present relations of Stieltjes type for the rational solutions. Further, we give a modest result for the maximal modulus of zeros of the Umemura polynomials.

In this paper, the Riemann–Hilbert (RH) method is developed to solve the Hirota–Satsuma coupled KdV (HScKdV) system associated with $��4�\times �4�$ matrix spectral problem. Because the spectral matrix is asymmetric and high order, it brings great difficulties to analysis and solution. First, a direct scattering problem is carried out, from which the initial data are mapped to the scattering data. On the basis of introducing the generalized cross product of vectors, the basic meromorphic matrix eigenfunctions of corresponding Lax pairs are expressed by the Jost solutions and adjoint Jost solutions. Then, the inverse scattering problem is characterized as the $��4�\times �4�$ matrix RH problem, which gives the formula for constructing solutions of the HScKdV system. As an example, the RH problem corresponding to the reflectionless case is solved and the soliton solution of the HScKdV system is obtained.

In this paper, the classical and nonlocal semi-discrete nonlinear Schrödinger (sdNLS) equations with nonzero backgrounds are solved by means of the bilinearization-reduction approach. In the first step of this approach, the unreduced sdNLS system with a nonzero background is bilinearized and its solutions are presented in terms of quasi-double Casoratians. Then, reduction techniques are implemented to deal with complex and nonlocal reductions, which yields solutions for the four classical and nonlocal sdNLS equations with a plane wave background or a hyperbolic function background. These solutions are expressed with explicit formulae and allow classifications according to canonical forms of certain spectral matrix. In particular, we present explicit formulae for general rogue waves for the classical focusing sdNLS equation. Some obtained solutions are analyzed and illustrated.