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Register a new userUniversal power law for the energy spectrum of breaking Riemann waves
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The universal power law for the spectrum of one-dimensional breaking Riemann waves is justified for the simple wave equation. The spectrum of spatial amplitudes at the breaking time $t = t_b$ has an asymptotic decay of $k^{-4/3}$, with corresponding energy spectrum decaying as $k^{-8/3}$. This spectrum is formed by the singularity of the form $(x-x_b)^{1/3}$ in the wave shape at the breaking time. This result remains valid for arbitrary nonlinear wave speed. In addition, we demonstrate numerically that the universal power law is observed for long time in the range of small wave numbers if small dissipation or dispersion is accounted in the viscous Burgers or Korteweg-de Vries equations.
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The interaction of the nonlinear internal waves with a nonuniform current with a specific form, characteristic for the equatorial undercurrent, is studied. The current has no vorticity in the layer, where the internal wave motion takes place. We show that the nonzero vorticity that might be occuring in other layers of the current does not affect the wave motion. The equations of motion are formulated as a Hamiltonian system.
We revisit the problem on the inner structure of shock waves in simple gases modelized by the Boltzmann kinetic equation. In cite{pomeau1987shock}, a self-similarity approach was proposed for infinite total cross section resulting from a power law interaction, but this self-similar form does not have finite energy. Motivated by the work of Pomeau, Bobylev and Cercignani started the rigorous study of the solutions of the spatial homogeneous Boltzmann equation, focusing on those which do not have finite energy cite{bobylev2002self,bobylev2003eternal}. In the present work, we provide a correction to the self-similar form, so that the solutions are more physically sound in the sense that the energy is no longer infinite and that the perturbation brought by the shock does not grow at large distances of it on the cold side in the soft potential case.
The celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off-diagonal pairs of elements, having applications to a wide range of physical and biological systems. Here, we investigate the generalization of this law to random matrices exhibiting higher-order cyclic correlations between $k$-tuples of matrix entries. We show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with $k$-fold rotational symmetry. This hypotrochoid law applies to full matrices as well as sparse ones, and thereby holds with remarkable universality. We further extend our analysis to matrices and graphs with competing cycle motifs, which are described more generally by polytrochoid spectral boundaries.
We give two conditionally exactly solvable inverse power law potentials whose linearly independent solutions include a sum of two confluent hypergeometric functions. We notice that they are partner potentials and multiplicative shape invariant. The method used to find the solutions works with the two Schrodinger equations of the partner potentials. Furthermore we study some of the properties of these potentials.
This is a short review of the two papers on the $x$-space asymptotics of the critical two-point function $G_{p_c}(x)$ for the long-range models of self-avoiding walk, percolation and the Ising model on $mathbb{Z}^d$, defined by the translation-invariant power-law step-distribution/coupling $D(x)propto|x|^{-d-alpha}$ for some $alpha>0$. Let $S_1(x)$ be the random-walk Green function generated by $D$. We have shown that $bullet~~S_1(x)$ changes its asymptotic behavior from Newton ($alpha>2$) to Riesz ($alpha<2$), with log correction at $alpha=2$; $bullet~~G_{p_c}(x)simfrac{A}{p_c}S_1(x)$ as $|x|toinfty$ in dimensions higher than (or equal to, if $alpha=2$) the upper critical dimension $d_c$ (with sufficiently large spread-out parameter $L$). The model-dependent $A$ and $d_c$ exhibit crossover at $alpha=2$. The keys to the proof are (i) detailed analysis on the underlying random walk to derive sharp asymptotics of $S_1$, (ii) bounds on convolutions of power functions (with log corrections, if $alpha=2$) to optimally control the lace-expansion coefficients $pi_p^{(n)}$, and (iii) probabilistic interpretation (valid only when $alphale2$) of the convolution of $D$ and a function $varPi_p$ of the alternating series $sum_{n=0}^infty(-1)^npi_p^{(n)}$. We outline the proof, emphasizing the above key elements for percolation in particular.
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