We consider the perturbed sine-Gordon equation $θ_{tt}-θ_{xx}+\sin θ=
\varepsilon^2 f(\varepsilon x)$, where the external perturbation $\varepsilon^2 f(\varepsilon x)$ corresponds to a small, slowly varying electric field. We show that the initial value problem with an appropriate initial state close enough to the solitary manifold has a unique solution, which follows up to time $1/{\varepsilon}$ and errors of order $\varepsilon^{ 3/4}$ a trajectory on the solitary manifold. The trajectory on the solitary manifold is described by ODEs, which agree approximately up to errors of order $\varepsilon^3$ with Hamilton equations for the restricted to the solitary manifold sine-Gordon Hamiltonian.
We consider the Benjamin-Ono equation with a slowly varying potential $u_t + (Hu_x-Vu + \tfrac12 u^2)_x=0$ with $V(x)=W(hx)$, $0< h \ll 1$, and $W\in C_c^\infty(\mathbb{R})$, and $H$ denotes the Hilbert transform. The soliton profile is $Q_{a,c}(x) = cQ(c(x-a))$, where $Q(x) = \frac{4}{1+x^2}$ and $a\in \mathbb{R}$, $c>0$ are parameters. For initial condition $u_0(x)$ to (pBO) close in $H_x^{1/2}$ to $Q_{0,1}(x)$, we show that the solution $u(x,t)$ to (pBO) remains close in $H_x^{1/2}$ to $Q_{a(t),c(t)}(x)$ and specify the $(a,c)$ parameter dynamics on an $O(h^{-1})$ time scale.
We consider the high-order nonlinear Schrödinger equation derived earlier by Sedletsky [Ukr. J. Phys. 48(1), 82 (2003)] for the first-harmonic envelope of slowly modulated gravity waves on the surface of finite-depth irrotational, inviscid, and incompressible fluid with flat bottom. This equation takes into account the third-order dispersion and cubic nonlinear dispersive terms. We rewrite this equation in dimensionless form featuring only one dimensionless parameter $kh$, where $k$ is the carrier wavenumber and $h$ is the undisturbed fluid depth. We show that one-soliton solutions of the clas…
The Benjamin Ono equation with a slowly varying potential is $$ \text{(pBO)} \qquad u_t + (Hu_x-Vu + \tfrac12 u^2)_x=0 $$ with $V(x)=W(hx)$, $0< h \ll 1$, and $W\in C_c^\infty(\mathbb{R})$, and $H$ denotes the Hilbert transform. The soliton profile is $$Q_{a,c}(x) = cQ(c(x-a)) \,, \text{ where } Q(x) = \frac{4}{1+x^2}$$ and $a\in \mathbb{R}$, $c>0$ are parameters. For initial condition $u_0(x)$ to (pBO) close to $Q_{0,1}(x)$, it was shown in a previous work by Z. Zhang that the solution $u(x,t)$ to (pBO) remains close to $Q_{a(t),c(t)}(x)$ and approximate parameter dynamics for $(a,c)$ were pr…
The soliton resolution conjecture for evolution PDEs of dispersive type states (vaguely) that generic initial data of finite energy give rise asymptotically to a set of receding solitons and a decaying background radiation. In this letter, we investigate a possible extension of this conjecture to discrete lattices of the Fermi-Pasta-Ulam-Tsingou type (rather than PDEs) in two cases: the case of finite energy initial data and a more general case where the initial data are a short range perturbation of a periodic function. In the second case, inspired by rigorous results on the Toda lattice, we …
We use the high-order nonlinear Schrödinger equation (NLSE) derived to model the evolution of slowly modulated wave trains with narrow spectrum on the surface of ideal finite-depth fluid. This equation is the finite-depth counterpart of celebrated Dysthe's equation, which is usually used for the same purpose in the case of infinite depth. We demonstrate that this generalized equation admits bright soliton solutions for depths below the modulation instability threshold $kh\approx 1.363$ ($k$ being the carrier wave number and $h$ the undisturbed fluid depth), which is not possible in the case of…
The three-wave resonant interaction equations are a non-dispersive system of partial differential equations with quadratic coupling describing the time evolution of the complex amplitudes of three resonant wave modes. Collisions of wave packets induce energy transfer between different modes via pumping and decay. We analyze the collision of two or three packets in the semiclassical limit by applying the inverse-scattering transform. Using WKB analysis, we construct an associated semiclassical soliton ensemble, a family of reflectionless solutions defined through their scattering data, intended…
Coulomb Sectors and Scattering for Maxwell-Higgs Fields on Schwarzschild and Slowly Rotating Kerr Backgrounds
Theoretical Physics Laboratory, Theoretical High Energy Physics Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung（万隆理工学院数学与自然科学学院理论物理实验室、理论高能物理研究部）
The graphene superlattice equation, a modified sine-Gordon equation, governs the propagation of solitary electromagnetic waves in a graphene superlattice. This equation has kink solutions without explicit analytical expression, requiring the use of quadrature methods. The inelastic collision of kinks and antikinks with the same but opposite speed is studied numerically for the first time; after their interaction they escape to infinity when its speed is either larger than a critical value or it is inside a series of resonance windows; otherwise, they form a breather-like state that slowly deca…
