Complex wave excitations and their interaction behaviors for higher-dimensional nonlinear model

Abstract

Taking an approach by means of a special variable separation, some novel and interesting localized (3+1)-dimensional excitations, including embedded-soliton, taper-like soliton, plateau-type soliton and rectangle-soliton, were revealed thanks to the intrusion of appropriate boundary conditions and/or initial qualifications. The properties of the complex wave excitations in the periodic wave background and their interactions were investigated, showing some novel features or interesting behaviors.

Introduction

In the fields of nonlinear science, soliton theory plays an essential role and has been applied in almost all natural sciences especially in physics, such as fluid physics, condensed matter, biophysics, plasma physics, nonlinear optics, quantum field theory and particle physics. It is worthy to note that previous studies demonstrated that there are more localized excitations in (2 + 1) dimensions than those in (1 + 1) dimensions [1], [2], [3]. Unfortunately, extensive studies of the soliton theory in the literature have been related to (1 + 1) and (2 + 1) dimensions because of the difficulties to find exact and explicit general solutions of any higher-dimensional integrable systems. Furthermore, as far as we know, localized structures derived from some solitary wave solutions or rational solutions in (1 + 1) and (2 + 1) dimensions system are usually propagating on a constant ideal background [1], [4], [5], which does not exist actually in the reality since some background waves are always encountered. Given that the real physical space time is in (3+1) dimensions, its localized excitations, especially these relate to propagate on a background wave, have attracted attention of many mathematicians and physicists for years, but little progress has been made in this direction.

Motivated by these reasons, we take the (3+1)-dimensional Nizhnik–Novikov–Veselov (NNV) equation

$$u_{\mathit{yzt}}+{\mathit{au}}_{\mathit{yyy}}{u}_{\mathit{yz}}+{\mathit{bu}}_{\mathit{xz}}{u}_{\mathit{xyz}}+{\mathit{cu}}_{\mathit{yy}}{u}_{\mathit{yyz}}+(9c-2b)u_{\mathit{xx}}{u}_{\mathit{yzz}}+(12c-2b-3a)u_{\mathit{xxz}}{u}_{\mathit{yz}}+{\mathit{eu}}_{\mathit{yyyyz}}+{\mathit{hu}}_{\mathit{xxyzz}}=0\text{,}$$

where a, b, c, e and h are arbitrary constants, as a concrete example to studying the complex wave excitations in the periodic wave background and theirs interactions properties. The Eq. (1) was originally proposed by Lin through means of the realizations of the generalized centerless Virasoro type symmetry algebra [6].

In order to get the special solution of model (1), we are able to rewrite (1) as the following potential form:

$$w_{\mathit{zt}}+{\mathit{aw}}_{\mathit{yy}}{w}_z+{\mathit{bv}}_z{v}_{\mathit{yz}}+{\mathit{cw}}_y{w}_{\mathit{yz}}+(9c-2b)v_x{w}_{\mathit{zz}}+(12c-2b-3a)v_{\mathit{xz}}{w}_z+{\mathit{ew}}_{\mathit{yyyz}}+\frac{3e(7c-b-a)}{a+c}{v}_{\mathit{xyzz}}=0\text{,}{v}_y=w_x\text{,}$$

by using v = u_x, w = u_y.

With the help of standard leading order analysis, the truncated Painleve expansion of the (3+1)-dimensional NNV equation can be written as

$$v=\frac{\mu }{\varphi }+\xi \text{,}w=\frac{\mathrm{\Psi }}{\varphi }+\omega \text{,}$$

where ξ, ω are an arbitrary known soliton solution of (2). This means that Eq. (3) is a Backlund transformation of the (3+1)-dimensional NNV equation. To get some significant solutions, we fix the original seed solutions as

$$\xi =0\text{,}\omega =\omega (y\text{,}z\text{,}t)\text{being arbitrary function of}y\text{,}z\text{,}t\text{.}$$

Substituting Eqs. (3), (4) into (2) and vanishing the coefficients of the different order of ϕ leads to the overdetermined system of equations, and using the Wu elimination method, which is a sufficient method to solve systems of algebra polynomial equations, after finishing some tedious but direct calculations, we finally obtain

$$v=\frac{12\mathit{ek}}{(a+c)(1+\exp (-\mathit{kx}-f))}\text{,}w=\frac{12{\mathit{ef}}_y}{(a+c)(1+\exp (-\mathit{kx}-f))}+\omega \text{,}$$

where a, c, e and k are arbitrary constants, f ≡ f(y, z, t) is an arbitrary x-independent function, and f, ω satisfy

$$8{\mathit{cef}}_z{f}_y^2{f}_{\mathit{yy}}+(72c-18a-12b){\mathit{ek}}^2{f}_z{f}_{\mathit{yz}}+(3a-3c){\mathit{ef}}_z{f}_{\mathit{yy}}^2+6{\mathit{aef}}_y^3{f}_{\mathit{yz}}+(4\mathit{ec}-2\mathit{ae})f_z{f}_y{f}_{\mathit{yyy}}-6{\mathit{eaf}}_y{f}_{\mathit{yy}}{f}_{\mathit{yz}}+(a+c)f_z{f}_y{f}_t+e(a+c)f_z{f}_y^4+(21c-3b-3a){\mathit{ek}}^2{f}_y{f}_z^2+(\mathit{ac}+a^2){\omega }_z{f}_y^3+(c^2+\mathit{ac}){\omega }_y{f}_z{f}_y^2+(9\mathit{ac}-2\mathit{bc}-3a^2)k^2{f}_z{\omega }_z+6{\mathit{aef}}_y^2{f}_{\mathit{yyz}}+(12c^2-2\mathit{ab})k^2{f}_z{\omega }_z+(9b-33c-3a){\mathit{ek}}^2{f}_{\mathit{yzz}}=0\text{.}$$

Remark 1

With the aid of Maple, we have verified all solutions obtained by putting them back into the original equation (1).

The paper is organized as follows. In Section 2, some localized (3+1)-dimensional excitations, such as embedded-soliton, taper-like soliton, plateau-type soliton and rectangle-soliton, were revealed initially by introducing the appropriate boundary conditions and/or initial qualifications. In Section 3, the interaction properties of the complex wave excitations in the periodic wave background were analyzed graphically. A brief discussion and summary is given in the last section.