Analysis and Mathematical Physics
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Global existence and blow-up for wave equation of p-Laplacian type
Analysis and Mathematical Physics - - 2023
This article explores a quasilinear wave equation of p-Laplacian type:
$$\begin{aligned} u_{tt}-\Delta _{p}u-\Delta u_t=|u|^{r-1}u \end{aligned}$$ in a
bounded domain $$\Omega \subset \mathbb {R}^{3}$$ and subject to Dirichlet
boundary conditions. The operator $$\Delta _{p}$$ denotes the classical
p-Laplacian with $$2...
Lump and lump-soliton solutions to the $$(2+1)$$ -dimensional Ito equation
Analysis and Mathematical Physics - Tập 8 - Trang 427-436 - 2017
Based on the Hirota bilinear form of the $$(2+1)$$ -dimensional Ito equation,
one class of lump solutions and two classes of interaction solutions between
lumps and line solitons are generated through analysis and symbolic computations
with Maple. Analyticity is naturally guaranteed for the presented lump and
interaction solutions, and the interaction solutions reduce to lumps (or line
solitons) w...
On spectral asymptotic of quasi-exactly solvable quartic potential
Analysis and Mathematical Physics - - 2021
Motivated by the earlier results of Masoero and De Benedetti (Nonlinearity
23:2501, 2010) and Shapiro et al. (Commun Math Phys 311(2):277–300, 2012), we
discuss below the asymptotic of the solvable part of the spectrum for the
quasi-exactly solvable quartic oscillator. In particular, we formulate a
conjecture on the coincidence of the asymptotic shape of the level crossings of
the latter oscillato...
Solving the inverse Sturm–Liouville problem with singular potential and with polynomials in the boundary conditions
Analysis and Mathematical Physics - Tập 13 - Trang 1-35 - 2023
In this paper, we for the first time get constructive solution for the inverse
Sturm–Liouville problem with complex-valued singular potential and with
polynomials of the spectral parameter in the boundary conditions. The uniqueness
of recovering the potential and the polynomials from the Weyl function is
proved. An algorithm of solving the inverse problem is obtained and justified.
More concretely...
Laguerre polynomials and transitional asymptotics of the modified Korteweg–de Vries equation for step-like initial data
Analysis and Mathematical Physics - Tập 9 - Trang 1761-1818 - 2018
We consider the compressive wave for the modified Korteweg–de Vries equation
with background constants $$c>0$$ for $$x\rightarrow -\infty $$ and 0 for
$$x\rightarrow +\infty $$. We study the asymptotics of solutions in the
transition zone $$4c^2t-\varepsilon t0,$$$$\sigma \in (0,1),$$$$\beta >0.$$ In
this region we have a bulk of nonvanishing oscillations, the number of which
grows as $$\frac{\var...
Necessary and sufficient conditions for the bounds of the commutator of a Littlewood-Paley operator with fractional differentiation
Analysis and Mathematical Physics - Tập 9 - Trang 2109-2132 - 2019
For $$b\in L_{\mathrm{loc}}({\mathbb {R}}^n)$$ and $$0<\alpha <1$$, we use
fractional differentiation to define a new type of commutator of the
Littlewood-Paley g-function operator, namely $$\begin{aligned} g_{\Omega ,\alpha
;b}(f )(x) =\bigg (\int _0^\infty \bigg |\frac{1}{t} \int _{|x-y|\le
t}\frac{\Omega (x-y)}{|x-y|^{n+\alpha -1}}(b(x)-b(y))f(y)\,dy\bigg
|^2\frac{dt}{t}\bigg )^{1/2}. \end{alig...
A series expansion for generalized harmonic functions
Analysis and Mathematical Physics - Tập 11 - Trang 1-28 - 2021
We consider a class of generalized harmonic functions in the open unit disc in
the complex plane. Our main results concern a canonical series expansion for
such functions. Of particular interest is a certain individual generalized
harmonic function which suitably normalized plays the role of an associated
Poisson kernel.
Nonlinear splittings on fibre bundles
Analysis and Mathematical Physics - Tập 12 - Trang 1-28 - 2021
We introduce the notion of a nonlinear splitting on a fibre bundle as a
generalization of an Ehresmann connection. We present its basic properties and
we pay attention to the special cases of affine, homogeneous and principal
nonlinear splittings. We explain where nonlinear splittings appear in the
context of Lagrangian systems and Finsler geometry and we show their relation to
Routh symmetry redu...
N-fold Darboux transformations and exact solutions of the combined Toda lattice and relativistic Toda lattice equation
Analysis and Mathematical Physics - Tập 10 Số 3 - Trang 1-21 - 2020
In this paper, the N-fold Darboux transformation (DT) of the combined Toda
lattice and relativistic Toda lattice equation is constructed in terms of
determinants. Comparing with the usual 1-fold DT of equations, this kind of
N-fold DT enables us to generate the multi-soliton solutions without complicated
recursive process. As applications of the N-fold DT, we derive two kinds of
N-fold explicit ex...
Epsilon coherent states with polyanalytic coefficients for the harmonic oscillator
Analysis and Mathematical Physics - Tập 9 - Trang 367-383 - 2017
We construct a new class of coherent states indexed by points z of the complex
plane and depending on two positive parameters m and $$ \varepsilon >0$$ by
replacing the coefficients $$z^{n}/\sqrt{n!}$$ of the canonical coherent states
by polyanalytic functions. These states solve the identity of the states Hilbert
space of the harmonic oscillator at the limit $$\varepsilon \rightarrow 0^{+}$$
and ...
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