Order reduction for problems with traveling wave solutions to reaction–diffusion systems

Some traveling wave solutions of soliton family

Solitons are ubiquitous and exist in almost every area from sky to bottom. For solitons to appear, the relevant equation of motion must be nonlinear. In the present study, we deal with the Korteweg-deVries (KdV), Modied Korteweg-de Vries (mKdV) and Regularised LongWave (RLW) equations using Homotopy Perturbation method (HPM). The algorithm makes use of the HPM to determine the initial expansion...

Homoclinic Solutions for Fourth Order Traveling Wave Equations

We consider homoclinic solutions of fourth order equations u ′′′′ + βu ′′ + Vu(u) = 0 in R , where V (u) is either the suspension bridge type V (u) = eu − 1 − u or SwiftHohenberg type V (u) = 1 4 (u2 − 1)2. For the suspension bridge type equation, we prove existence of a homoclinic solution for all β ∈ (0, β∗) where β∗ = 0.7427 · · · . For the Swift-Hohenberg type equation, we prove existence o...

Traveling Wave Solutions of Fourth Order PDEs for Image Processing

Existence of Traveling-wave Solutions to Boussinesq Systems

In this manuscript, the existence of traveling-wave solutions to Boussinesq systems  ηt + ux + (ηu)x + auxxx − bηxxt = 0, ut + ηx + uux + cηxxx − duxxt = 0, is established. We prove that all the systems with a < 0, c < 0 and b = d exhibit traveling-wave solutions with small propagation speeds. The result complements our earlier work [6] on a restricted family of the systems where both existenc...

Positive solutions for nonlinear systems of third-order generalized sturm-liouville boundary value problems with $(p_1,p_2,ldots,p_n)$-laplacian

In this work, byemploying the Leggett-Williams fixed point theorem, we study theexistence of at least three positive solutions of boundary valueproblems for system of third-order ordinary differential equationswith $(p_1,p_2,ldots,p_n)$-Laplacianbegin{eqnarray*}left { begin{array}{ll} (phi_{p_i}(u_i''(t)))'  +  a_i(t) f_i(t,u_1(t), u_2(t), ldots, u_n(t)) =0 hspace{1cm} 0  leq t leq 1, alpha_i u...