scholarly journals Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds

2020 ◽  
Vol 13 (2) ◽  
pp. 403-433 ◽  
Author(s):  
David Beltran ◽  
Jonathan Hickman ◽  
Christopher D. Sogge
Keyword(s):  
Variable Coefficient ◽  
Wave Equations ◽  
Local Smoothing

Square function inequality for a class of Fourier integral operators satisfying cinematic curvature conditions

2020 ◽  
Vol 32 (6) ◽  
pp. 1375-1394
Author(s):  
Chuanwei Gao ◽  
Changxing Miao ◽  
Jianwei-Urbain Yang
Keyword(s):  
Variable Coefficient ◽  
Wave Equations ◽  
Integral Operators ◽  
Oscillatory Integral ◽  
Fourier Integral ◽  
Acta Math ◽  
Fourier Integral Operators ◽  
Square Function ◽  
Local Smoothing ◽  
Smoothing Estimate

AbstractIn this paper, we establish an improved variable coefficient version of the square function inequality, by which the local smoothing estimate {L^{p}_{\alpha}\to L^{p}} for the Fourier integral operators satisfying cinematic curvature condition is further improved. In particular, we establish almost sharp results for {2<p\leqslant 3} and push forward the estimate for the critical point {p=4}. As a consequence, the local smoothing estimate for the wave equation on the manifold is refined. We generalize the results in [S. Lee and A. Vargas, On the cone multiplier in \mathbb{R}^{3}, J. Funct. Anal. 263 2012, 4, 925–940; J. Lee, A trilinear approach to square function and local smoothing estimates for the wave operator, preprint 2018, https://arxiv.org/abs/1607.08426v5] to its variable coefficient counterpart. The main ingredients in the argument includes multilinear oscillatory integral estimate [J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 2006, 2, 261–302] and decoupling inequality [D. Beltran, J. Hickman and C. D. Sogge, Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds, Anal. PDE 13 2020, 2, 403–433].

Characteristics of solitary waves, quasiperiodic solutions, homoclinic breather solutions and rogue waves in the generalized variable-coefficient forced Kadomtsev–Petviashvili equation

2017 ◽  
Vol 31 (36) ◽  
pp. 1750350 ◽  
Author(s):  
Xue-Wei Yan ◽  
Shou-Fu Tian ◽  
Min-Jie Dong ◽  
Li Zou
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Variable Coefficient ◽  
Wave Equations ◽  
Direct Method ◽  
Dynamical Behavior ◽  
Rogue Waves ◽  
Nonlinear Wave Equations ◽  
Quasiperiodic Solutions ◽  
Petviashvili Equation

In this paper, the generalized variable-coefficient forced Kadomtsev–Petviashvili (gvcfKP) equation is investigated, which can be used to characterize the water waves of long wavelength relating to nonlinear restoring forces. Using a dependent variable transformation and combining the Bell’s polynomials, we accurately derive the bilinear expression for the gvcfKP equation. By virtue of bilinear expression, its solitary waves are computed in a very direct method. By using the Riemann theta function, we derive the quasiperiodic solutions for the equation under some limitation factors. Besides, an effective way can be used to calculate its homoclinic breather waves and rogue waves, respectively, by using an extended homoclinic test function. We hope that our results can help enrich the dynamical behavior of the nonlinear wave equations with variable-coefficient.

scholarly journals Sinc-Chebyshev Collocation Method for a Class of Fractional Diffusion-Wave Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Zhi Mao ◽  
Aiguo Xiao ◽  
Zuguo Yu ◽  
Long Shi
Keyword(s):  
Variable Coefficient ◽  
Fractional Derivatives ◽  
Wave Equations ◽  
Fractional Diffusion ◽  
Algebraic Equations ◽  
Chebyshev Collocation Method ◽  
Diffusion Wave ◽  
Linear Algebraic Equations ◽  
Collocation Technique ◽  
Sinc Functions

This paper is devoted to investigating the numerical solution for a class of fractional diffusion-wave equations with a variable coefficient where the fractional derivatives are described in the Caputo sense. The approach is based on the collocation technique where the shifted Chebyshev polynomials in time and the sinc functions in space are utilized, respectively. The problem is reduced to the solution of a system of linear algebraic equations. Through the numerical example, the procedure is tested and the efficiency of the proposed method is confirmed.

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