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In this paper, we are concerned with a class of Schrodinger-Poisson systems with the asymptotically linear or asymptotically 3-linear nonlinearity. Under some suitable assumptions on V, K, a, and f, we prove the existence, nonexistence, and asymptotic behavior of solutions via variational methods. In particular, the potential V is allowed to be sign-changing for the asymptotically linear case.

This paper studies the dynamical behavior of a weakly damped forced Korteweg–de Vries (KdV) equation in a wavelet basis and introduces the wavelet approximate inertial manifold and wavelet Galerkin solution of weakly damped forced KdV equation. The results includes theorems that for the KdV equation the wavelet approximate inertial manifold (WAIM) exists and sets up the wavelet Galerkin method of the equation. Error estimates are given by using the WAIM.

This paper develops a new approach in the analysis of a coupled Camassa-Holm equations. A continuous semigroup of global conservative solutions and a continuous semigroup of global dissipative solutions are obtained, respectively. The solutions are conservative, in the sense that the total energy equals to a constant, for almost every time. While the solutions are dissipative, in the sense that energy loss occurs through wave breaking. Compared to the approaches used by Bressan, A. and Constantin, A., “Global conservative solutions of the Camassa-Holm equation,” Arch. Ration. Mech. Anal. 183, 215–239 (2007)10.1007/s00205-006-0010-z and Bressan, A. and Constantin, A., “Global dissipative solutions of the Camassa–Holm equation,” Anal. Appl. (Singapore) 5, 1–27 (2007)10.1142/S0219530507000857 for the Camassa-Holm equation, two characteristics are introduced and a new set of independent and dependent variables are applied to the coupled Camassa-Holm equations.

In this paper, we prove that the modified Camassa–Holm equation is Painleve integrable. We also study the orbital stability problem of negative solitary waves for this integrable equation. It is shown that the negative solitary waves are stable for arbitrary wave speed of propagation.

The Dullin–Gottwald–Holm equation can be described as the unidirectional propagation of surface waves in a shallow water regime. In this paper, we study the orbital stability problem of the peaked solitons (peakons) for the periodic Dullin–Gottwald–Holm equation on the line. By constructing a function only depending on three important conservative laws, we prove that the shapes of peakons are stable under small perturbations. We also demonstrate that the solutions of the Cauchy problem for this equation converge to those of the corresponding periodic Camassa–Holm equation as the linear dispersive parameter converges to zero.

This paper studies low-regularity solutions of the periodic Fornberg–Whitham equation with initial value. The existence and the uniqueness of solutions are proved. The results are illustrated by considering the periodic peakons of the periodic Fornberg–Whitham equation.

This paper is concerned with the global well posedness and limit behavior of the solutions to the viscous Degasperis–Procesi equation. We prove the global existence and uniqueness of solution to the viscous Degasperis–Procesi equation for u0∊L2(R). The local well posedness for u0∊L2(R) has been proven based on the energy estimate. The strong and weak limit behaviors of solution to the viscous Degasperis–Procesi equation are obtained.

This paper studies the problem for optimal control of the viscous Degasperis-Procesi equation. The existence and uniqueness of weak solution to the equation are proved in a short interval. The optimal control of the viscous Degasperis-Procesi equation under boundary condition is given and the existence of optimal solution to the equation is proved.