Exact traveling wave solutions of the Boussinesq–Burgers equation

The extended homogeneous balance method is used to construct exact traveling wave solutions of the Boussinesq–Burgers equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation. Many exact traveling wave solutions of the Boussinesq–Burgers equation are successfully obtained.     

A Note on the analytic solutions of the Camassa–Holm equation

In this Note we are concerned with the well-posedness of the Camassa–Holm equation in analytic function spaces. Using the Abstract Cauchy–Kowalewski Theorem we prove that the Camassa–Holm equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic, belongs to $H^s(\mathbb{R})$ with $s>3/2$, $6u_0{6}_{L^1}<\infty$ and $u_0?u_{0xx}$ does not change sign, we prove that the solution stays analytic globally in time. To cite this article: M.C. Lombardo et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On the asymptotic boundedness of the energy of solutions of the wave equation u tt ? a(t)Δu?=?0

Applying a class of quadratic forms introduced by Manfrin (Port. Math. 65(4):447?C484, 2008), we study the asymptotic behavior, as t ?? +??, of the energy of the solutions to the Cauchy problem for wave equations with time dependent propagation speed.     

Propagation of traveling wave solutions to the Vakhnenko-Parkes dynamical equation via modified mathematical methods

In this paper, we investigate some new traveling wave solutions to Vakhnenko-Parkes equation via three modified mathematical methods. The derived solutions have been obtained including periodic and solitons solutions in the form of trigonometric, hyperbolic, and rational function solutions. The graphical representations of some solutions by assigning particular values to the parameters under prescribed conditions in each solutions and comparing of solutions with those gained by other authors indicate that these employed techniques are more effective, efficient and applicable mathematical tools for solving nonlinear problems in applied science.     

Bifurcations of travelling wave solutions for the Gilson–Pickering equation

In this paper, the qualitative behavior and exact travelling wave solutions of the Gilson–Pickering equation are studied by using the qualitative theory of polynomial differential system. The phase portraits of the system are given under different parametric conditions. Some exact travelling wave solutions of the Gilson–Pickering equation are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of smooth and non-smooth travelling wave solutions are given.     

Classification of radial solutions to the Emden–Fowler equation on the hyperbolic space

We study the Emden–Fowler equation ?Δu = |u| ^p?1 u on the hyperbolic space ${{\mathbb H}^n}$ . We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is p = (n + 2)/(n ? 2) as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers (Bhakta and Sandeep, Poincaré Sobolev equations in the hyperbolic space, 2011; Mancini and Sandeep, Ann Sci Norm Sup Pisa Cl Sci 7(5):635–671, 2008) consider finite energy solutions, we shall deal here with infinite energy solutions and we determine the exact asymptotic behavior of wide classes of finite and infinite energy solutions.     

Almost automorphic solutions to integral equations on the line

Given a∈L ¹(ℝ) and A the generator of an L ¹-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=∫ _−∞ ^t a(t−s)[Au(s)+f(s,u(s))]ds for each f:ℝ×X→X almost automorphic in t, uniformly in x∈X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a∈L ¹(ℝ) positive, nonincreasing and log-convex is already sufficient.     

Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on ℝ2

In this article, we investigate the initial value problem(IVP) associated with the defocusing nonlinear wave equation on ?² as follows:

$\{\begin{array}{l}{?}_{tt}u-\Delta u=-u^3,\\ u\left(0,x\right)=u_0\left(x\right),{?}_{t}u\left(0,x\right)=u_1\left(x\right),\end{array}$

where the initial data (u₀, u₁) ? H^s(?²) × H^s?1(?²). It is shown that the IVP is global well-posedness in H^s(?²) × H^s?1(?²) for any 1 > s > 2/5. The proof relies upon the almost conserved quantity in using multilinear correction term. The main difficulty is to control the growth of the variation of the almost conserved quantity. Finally, we utilize linear-nonlinear decomposition benefited from the ideas of Roy [1].     

Four types of bounded wave solutions of CH-γ equation

Recently, many authors have studied the following CH-γ equationut c0ux 3uux - α2(uxxt uuxxx 2uxuxx) γuxxx =0,where α2, c0 and γ are paramters. Its bounded wave solutions have been investigated mainly for the case α2 ＞ 0. For the case α2 ＜ 0, the existence of three bounded waves (regular solitary waves,compactons, periodic peakons) was pointed out by Dullin et al. But the proof has not been given.In this paper, not only the existence of four types of bounded waves periodic waves, compacton-like waves, kink-like waves, regular solitary waves, is shown, but also their explicit expressions or implicit expressions are given for the case α2 ＜ 0. Some planar graphs of the bounded wave solutions and their numerical simulations are given to show the correctness of our results.     

Four types of bounded wave solutions of CH-■ equation

Recently, many authors have studied the following CH-γequation: ut c0ux 3uux -α2(wxxt uuxxx 2uxuxx) 4-γuxxx = 0,whereα2, c0 andγare paramters. Its bounded wave solutions have been investigated mainly for the caseα2 > 0. For the caseα2 < 0, the existence of three bounded waves (regular solitary waves, compactons, periodic peakons) was pointed out by Dullin et al. But the proof has not been given. In this paper, not only the existence of four types of bounded waves: periodic waves, compacton-like waves, kink-like waves, regular solitary waves, is shown, but also their explicit expressions or implicit expressions are given for the caseα2 < 0. Some planar graphs of the bounded wave solutions and their numerical simulations are given to show the correctness of our results.     

The heat kernel of the Laplacian defined on a uniform grid

We adapt a method originally developed by E.B. Davies for second order elliptic operators to obtain an upper heat kernel bound for the Laplacian defined on a uniform grid on the plane.     

Numerical approximation of the LQR problem in a strongly damped wave equation

The aim of this work is to obtain optimal-order error estimates for the LQR (Linear-quadratic regulator) problem in a strongly damped 1-D wave equation. We consider a finite element discretization of the system dynamics and a control law constant in the spatial dimension, which is studied in both point and distributed case. To solve the LQR problem, we seek a feedback control which depends on the solution of an algebraic Riccati equation. Optimal error estimates are proved in the framework of the approximation theory for control of infinite-dimensional systems. Finally, numerical results are presented to illustrate that the optimal rates of convergence are achieved.     

Existence of solutions to the nonlinear Schrödinger equation on locally finite graphs

Archiv der Mathematik - Let $$G=(V, E)$$ be a locally finite connected graph and $$\Delta $$ be the usual graph Laplacian operator. According to Lin and Yang (Rev. Mat. Complut., 2022), using...     

Maximizing the L∞ norm of the gradient of solutions to the Poisson equation

We find the maximum of ¦Du _f ¦ _L∞ when u_f is the solution, which vanishes at infinity, of the Poisson equation Δu =f on ? ⁿ in terms of the decreasing rearrangement off. Hence, we derive sharp estimates for ¦Du _f ¦ _L∞ in terms of suitable Lorentz orL ^p norms off. We also solve the problem of maximizing ¦Du _f ^B (0)¦ whenu _f ^B is the solution, vanishing on?B, to the Poisson equation in a ballB centered at 0 and the decreasing rearrangement off is assigned.     

Optimal boundary control of the wave equation with pointwise control constraints

In optimal control problems frequently pointwise control constraints appear. We consider a finite string that is fixed at one end and controlled via Dirichlet conditions at the other end with a given upper bound M for the L ^∞-norm of the control. The problem is to control the string to the zero state in a given finite time. If M is too small, no feasible control exists. If M is large enough, the optimal control problem to find an admissible control with minimal L ²-norm has a solution that we present in this paper.     

Exact loop solutions,cusp solutions,solitary wave solutions and periodic wave solutions for the special CH–DP equation

By using the method of dynamical systems, the travelling wave solutions of a special CH–DP equation are studied. Exact explicit parametric representations of smooth solitary waves, solitary cusp waves, breaking waves and uncountably infinitely many smooth periodic wave solutions are given. In different regions of the parametric plane, different phase portraits are determined. The so called loop soliton solution is discussed.