This paper extends the dual-Petrov-Galerkin method proposed by Shen [21], further developed by Yuan, Shen and Wu [27] to general fifth-order KdV type equations with various nonlinear terms. These fifth-order equations arise in modeling different wave phenomena. The method is implemented to compute the multi-soliton solutions of two representative fifth-order KdV equations: the Kaup-Kupershmidt ...

The bifurcation method of planar systems and simulation method of differential equations are employed to investigate loop soliton solutions of the reduced Ostrovsky equation (ROE). The parameter representation of loop soliton solutions of the ROE are obtained. The planar graphs of the loop soliton solutions is shown under the some parameter. These results are supplement to investigate the ROE.

The existence of solutions with localized solitary wave structures is one the significant characteristics nonlinear integrable systems. Darboux transformation (DT) a well-known method for constructing multi-soliton solutions, using type wave, systems, but there are still no reports on extending DT techniques to construct such fractional models. This article takes coupled Schrödinger (CNLS) equa...

We study soliton solutions of the Kadomtsev-Petviashvili II equation (-4u(t)+6uu(x)+3u(xxx))(x)+u(yy)=0 in terms of the amplitudes and directions of the interacting solitons. In particular, we classify elastic N-soliton solutions, namely, solutions for which the number, directions, and amplitudes of the N asymptotic line solitons as y-->infinity coincide with those of the N asymptotic line soli...

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the exact travelling wave solutions of a generalized Degasperis-Procesi equation ut−uxxt+4uux+γ(u−uxx)x = 3uxuxx+uuxxx. The implicit expression of smooth soliton solutions is given. The explicit expressions of peaked soliton solutions and periodic cuspon solutions are also obtained. Further, we show the r...

In this paper, Whitham–Broer–Kaup (WBK) equations with time-dependent coefficients are exactly solved through Hirota’s bilinear method. To be specific, the WBK equations are first reduced into a system of variable-coefficient Ablowitz–Kaup– Newell–Segur (AKNS) equations. With the help of the AKNS equations, bilinear forms of the WBK equations are then given. Based on a special case of the bilin...

The interaction and propagation of optical pulses in a nonlinear waveguide array is described by the discrete nonlinear Schrödinger equation i∂zψn = −D(ψn+1 + ψn−1 − 2ψn) − γ|ψn|ψn, (1) whereD is a dispersion (or diffraction) coefficient, and γ is a measure of the nonlinearity. By means of the variational approximation, we study the discrete soliton solutions of this equation. We use a trial fu...

We study two members of the multi-component AKNS hierarchy. These are multi-NLS and multi-MKdV systems. derive Hirota bilinear forms these equations obtain soliton solutions. find all possible local nonlocal reductions systems give a prescription to their also (2+1)-dimensional extensions

In this letter, we present a simple and new idea to generate two types of novel integrable multi-Lévy-index mix-Lévy-index (mixed) fractional nonlinear soliton hierarchies, containing multi-index mixed higher-order Schrödinger (NLS) hierarchy, complex modified Korteweg–de Vries (cmKdV) mKdV hierarchy. Their explicit forms can be given using the completeness squared eigenfunctions. Moreover, the...

Analysis of the generalized Weierstrass-Enneper system includes the estimation of the degree of indeterminancy of the general analytic solution and the discussion of the boundary value problem. Several different procedures for constructing certain classes of solutions to this system, including potential, harmonic and separable types of solutions, are proposed. A technique for reduction of the W...