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In this paper, we prove the orbital stability of periodic peaked traveling waves (peakons) for a nonlinear quartic μ -Camassa–Holm equation. The equation is a μ -version of the nonlinear quartic Camassa–Holm equation which was proposed by Anco and Recio (J Phys A Math Theor 52:125–203, 2019). The equation admits the periodic peakons. It is shown that the periodic peakons are orbitally stable under small perturbations in the energy space by finding inequalities related to the three conservation laws with global maximum and minimum of the solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with the initial boundary value problem for viscoelastic Kirchhoff-like plate equations with rotational inertia, memory, p-Laplacian restoring force, weak damping, strong damping, and nonlinear source terms. We establish the local existence and uniqueness of the solution by linearization and the contraction mapping principle. Then, we obtain the global existence of solutions with subcritical and critical initial energy by applying potential well theory. Then, we prove the asymptotic behavior of the global solution with positive initial energy strictly below the depth of the potential well. Finally, we conduct a comprehensive study on the finite time blow-up of solutions with negative initial energy, null initial energy, and positive initial energy strictly below the depth of the potential well and arbitrary positive initial energy, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of periodic peaked solitary waves to a generalized μ-Camassa–Holm–Novikov equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ-version of a linear combination of the Camassa–Holm, modified Camassa–Holm, and Novikov equations. It is shown that the proposed equation admits a single peakons. It is natural extension of the previous results obtained in [Khesin B, Lenells J, Misiolek G. Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms. Math Ann. 2008;342:617–656; Moon B. The existence of the single-peaked traveling waves to the μ-Novikov equation. Appl Anal. 2018;97:1540–1548; Qu CZ, Fu T, Liu Y. Well-posedness, wave breaking and peakons for a modified μ-Camassa–Holm equation. J Funct Anal. 2014;266(2):433–477.] for the μ-Camassa–Holm, modified μ-Camassa–Holm, and μ-Novikov equations, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of peaked traveling wave solution of the generalized μ-Novikov equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ-version of a linear combination of the Novikov equation and Camassa-Hom equation. It is found that the equation admits single peaked traveling wave solutions. [ABSTRACT FROM AUTHOR]

We present a general and effective method, known as the Fokas method, to solve an arbitrary discrete linear evolution equation posed in a finite lattice. The method is based on the simultaneous analysis of both parts of Lax pair, as well as the global relation that involves initial and boundary values. We show that, as in the continuum problems, the method can be applied effectively to solve general linear differential-difference equations in a finite lattice. In particular, we demonstrate the method by addressing a number of significant examples and we discuss the continuum limits of the solution and the boundary values. [ABSTRACT FROM AUTHOR]

Considered herein is a generalized periodic Novikov equation with cubic nonlinearity which is called the -Novikov equation. It is shown that the equation admits a single peaked soliton with a similar character of the -Camassa-Holm equation. [ABSTRACT FROM AUTHOR]

We study the Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic boundary data for large t > 0. We derive an expression for the Dirichlet to Neumann map to all orders in the perturbative expansion of a small ϵ > 0 in the case of the asymptotically periodic boundary data. More precisely, we show that if the unknown Neumann boundary data are asymptotically periodic for large t in the sense that u x ( 0 , t ) and u x x ( 0 , t ) tend to periodic functions g ˜ 1 ( t ) and g ˜ 2 ( t ) for large t, respectively, then the periodic functions g ˜ 1 ( t ) and g ˜ 2 ( t ) can be characterized in terms of the given asymptotically periodic Dirichlet boundary datum u ( 0 , t ). Moreover, we determine effectively the Fourier coefficients of the functions g ˜ 1 ( t ) and g ˜ 2 ( t ) by solving a certain recursive algebraic equations. [ABSTRACT FROM AUTHOR]

Considered herein is the periodic rotation-two-component Camassa–Holm system, which can be derived from the f -plane governing equations for the geophysical water waves with a constant underlying current. The nonlocal nonlinearities on blow-up criteria and wave-breaking phenomena are established. Finally, a sufficient condition for global solutions is obtained by using a method of the Lyapunov function. [ABSTRACT FROM AUTHOR]

The rotation μ -Camassa–Holm equation (R- μ -CH) equation is a nonlinear partial differential equation closely related to the rotation Camassa–Holm (R-CH) equation (Chen et al., 2018; Gui et al., 2019a; Gui et al., 2019b; Tu et al., 2019). We prove that the nonexistence of the periodic peaked traveling wave solutions of the R- μ -CH equation unlike μ -CH equation. [ABSTRACT FROM AUTHOR]

We establish global well-posedness to the generalized weakly dissipative periodic two-component Camassa–Holm system in some special case. [ABSTRACT FROM AUTHOR]

Abstract: The existence of solitary wave solutions of the generalized two-component Hunter–Saxton system is determined. It is also shown that there are peaked and cusped solitary waves with singularities among those smooth solitary wave solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we study the wave-breaking phenomena and global existence for the generalized two-component Hunter–Saxton system in the periodic setting. We first establish local well-posedness for the generalized two-component Hunter–Saxton system. We obtain a wave-breaking criterion for solutions and results of wave-breaking solutions with certain initial profiles. We also determine the exact blow-up rate of strong solutions. Finally, we give a sufficient condition for global solutions. [Copyright &y& Elsevier]

Recently, Zhu et al. (2020) proposed a kind of rotation-Camassa–Holm equation. In this paper, we study the question of nonexistence of periodic peaked traveling wave solution for rotation-Camassa–Holm equation. Indeed, rotation-Camassa–Holm equation has no nontrivial periodic Camassa–Holm peaked solution unlike Camassa–Holm equation, modified Camassa–Holm equation, Novikov equation. [ABSTRACT FROM AUTHOR]

In the present study an asymptotic model for wave propagation in shallow water with the effect of the Coriolis force is derived from the governing equation in two dimensional flows. The transport equation theory is then applied to investigate the local well-posedness and wave breaking phenomena for this model. The nonexistence of the Camassa-Holm-type peaked solution and classification of various traveling-wave solutions to the new system are also established. Moreover it is shown that all the symmetric waves to this model are traveling waves. [ABSTRACT FROM AUTHOR]