Bilinear Auto-Bäcklund Transformations and Similarity Reductions for a (3+1)-dimensional Generalized Yu-Toda-Sasa-Fukuyama System in Fluid Mechanics and Lattice Dynamics

Abstract

Recent investigations on the liquids and lattices are both active. In this paper, with symbolic computation, we consider a (3+1)-dimensional generalized Yu-Toda-Sasa-Fukuyama system for the interfacial waves in a two-layer liquid or elastic waves in a lattice, with two sets of the bilinear auto-Bäcklund transformations hereby built up. Moreover, we construct one set of the similarity reductions, from that system to a known ordinary differential equation. As for the amplitude or elevation of the relevant wave, our results rely on the coefficients in that system.

1 Introduction

Fluid mechanics and lattice dynamics deal with fluid behaviors/interactions under various forces and with the vibrations of atoms inside crystals, basic to such fields as the oceanic, atmospheric, mineral and solid state sciences [1,2,3,4,5,6,7,8]. In connection with fluid mechanics and lattice dynamics, in this paper, we consider a (3+1)-dimensional generalized Yu-Toda-Sasa-Fukuyama system for the interfacial waves in a two-layer liquid or elastic waves in a lattice [7] (and references therein), i.e.,

$$\begin{aligned}&\left( h_1 v_{t}+h_2 v_{xxz}+h_4 v_{x} u_{z}+h_5 v v_{z}\right) _{x} +h_3 v_{yy}+h_6 v_{xy}+h_7 v_{xz}+h_8 v_{yz}=0,\qquad \end{aligned}$$

(1a)

$$\begin{aligned}&u_{x}=v, \end{aligned}$$

(1b)

where t is the scaled temporal coordinate, x, y and z denote the scaled spatial coordinates, v(x, y, z, t) indicates the amplitude or elevation of the relevant wave, while v(x, y, z, t) and u(x, y, z, t) mean two real differentiable functions as for x, y, z and t [7]. In addition, $h_{\Upsilon }$’s are the real constants, with $\Upsilon = 1, ..., 8$.

Some special cases of System (1) were studied previously [8,9,10,11,12,13,14,15,16,17]: for instance,

• 1. when $h_6=h_7=h_8=0$, describing the interfacial waves in a two-layer liquid, a (3+1)-dimensional Yu-Toda-Sasa-Fukuyama system [8], i.e.,

  $$\begin{aligned}&\left( h_1 v_{t}+h_2 v_{xxz}+h_4 v_{x} u_{z}+h_5 v v_{z}\right) _{x} +h_3 v_{yy}=0 , \end{aligned}$$

  (2a)
  $$\begin{aligned}&u_{x}=v; \end{aligned}$$

  (2b)

• 2. when $h_1=h_2=1$, $h_5=6$, $h_4=h_6=h_7=h_8=0$ and $z=x$, describing the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in a fluid¹, a (2+1)-dimensional generalized Kadomtsev-Petviashvili equation [9, 10],

  $$\begin{aligned} \left( \, v_{t}+v_{xxx}+6 v v_{x} \, \right) _{x}+h_3 v_{yy}=0; \end{aligned}$$

  (3)

• 3. when $h_1=h_2=h_3=h_5=1$, $h_4=h_6=h_7=h_8=0$ and $z=x$, describing the weakly transverse water waves in the long wave regime with small surface tension, a (2+1)-dimensional Kadomtsev-Petviashvili equation [11, 12],

  $$\begin{aligned} \left( \, v_{t}+v_{xxx}+v v_{x} \, \right) _{x}+v_{yy}=0; \end{aligned}$$

  (4)

• 4. when $h_1=h_2=1$, $h_5=-6$, $h_3=h_4=h_6=h_7=h_8=0$ and $z=x$, describing the long waves in shallow water under the gravity, waves in a nonlinear lattice, ion-acoustic and magneto-acoustic waves in a plasma, and also applying to nonlinear optics² and quantum mechanics, a (1+1)-dimensional Korteweg-de Vries equation [13, 14],

  $$\begin{aligned} v_{t}+v_{xxx}-6 v v_{x}=0; \end{aligned}$$

  (5)

• 5. when $h_1=h_5=1$, $h_2=\frac{1}{4}$, $h_4=\frac{1}{2}$ and $h_3=h_6=h_7=h_8=0$, describing the (2+1)-dimensional interaction between a Riemann wave propagating along the z axis and a long wave propagating along the x axis, a (2+1)-dimensional integrable Calogero-Bogoyavlenskii-Schiff system [15, 16],

  $$\begin{aligned}&v_{t}+\frac{1}{4} v_{xxz}+\frac{1}{2} v_{x} u_{z}+ v v_{z} =0, \end{aligned}$$

  (6a)
  $$\begin{aligned}&u_{x}=v; \end{aligned}$$

  (6b)

• 6. when $h_1=-4$, $h_2=1$, $h_3=3$, $h_4=2$, $h_5=4$ and $h_6=h_7=h_8=0$, describing the interfacial waves in a two-layer liquid or elastic quasiplane waves in a lattice, a (3+1)-dimensional Yu-Toda-Sasa-Fukuyama system [17], i.e.,

  $$\begin{aligned}&\left( -4 v_{t}+ v_{xxz}+2 v_{x} u_{z}+4 v v_{z}\right) _{x} +3 v_{yy}=0, \end{aligned}$$

  (7a)
  $$\begin{aligned}&u_{x}=v\;\;. \end{aligned}$$

  (7b)

For System (1), currently interesting, under the coefficient constraints

$$\begin{aligned} h_1=1, \qquad h_4=h_5, \end{aligned}$$

(8)

on account of the transformations

$$\begin{aligned} v=\frac{6 h_2}{h_4} (\ln f)_{xx}, \qquad u=\frac{6 h_2}{h_4} (\ln f)_{x}, \end{aligned}$$

(9)

Shen et al. [7] have presented a bilinear form, i.e.,

$$\begin{aligned} \left( D_x D_t+h_2 D_x^3 D_z+h_3 D_y^2+h_6 D_x D_y+h_7 D_x D_z+h_8 D_y D_z\right) f\cdot f=0, \end{aligned}$$

(10)

where f stands for a real differentiable function in respect of x, y, z and t, while the bilinear notations $D_x$, $D_y$, $D_z$ and $D_t$ are explained in the Appendix. Besides, bilinear auto-Bäcklund transformation³, breather and periodic solutions for System (1) have been worked out [7].

Hereby with symbolic computation [54,55,56,57,58], on the one hand, for System (1), we will build up two sets of the bilinear auto-Bäcklund transformations, which are different from the one presented in Ref. [7], through the Hirota method [10, 59,60,61,62]. On the other hand, we will construct a set of the similarity reductions for System (1).

2 Bilinear Auto-Bäcklund Transformations for System (1)

The bilinear notations $D_x$, $D_y$, $D_z$ and $D_t$ can be found in the Appendix.

Because of the existing bilinear form under coefficient constraints, i.e., (10) under (8), employing the Hirota method, assuming that g stands for another solution of Form (10) and taking into account the expression⁴

$$\begin{aligned}&f^2 \big [ \left( D_x D_t+h_2 D_x^3 D_z+h_3 D_y^2+h_6 D_x D_y+h_7 D_x D_z+h_8 D_y D_z\right) g\cdot g \big ] \nonumber \\&\quad - g^2 \big [ \left( D_x D_t+h_2 D_x^3 D_z+h_3 D_y^2+h_6 D_x D_y+h_7 D_x D_z\right. \nonumber \\&\quad \left. +h_8 D_y D_z\right) f\cdot f \big ]=0, \end{aligned}$$

(11)

we get

$$\begin{aligned}&f^2 \big [ \left( D_x D_t+h_2 D_x^3 D_z+h_3 D_y^2+h_6 D_x D_y+h_7 D_x D_z+h_8 D_y D_z\right) g\cdot g \big ]\\&\qquad - g^2 \big [ \left( D_x D_t+h_2 D_x^3 D_z+h_3 D_y^2+h_6 D_x D_y+h_7 D_x D_z+h_8 D_y D_z\right) f\cdot f \big ]\\&\quad =\big [\, f^2 \big (D_x D_t \, g\cdot g\big ) -g^2 \big (D_x D_t \, f\cdot f\big ) \,\big ] \\&\qquad +h_2 \big [\, f^2 \big (D_x^3 D_z \, g\cdot g\big ) -g^2 \big (D_x^3 D_z \, f\cdot f\big ) \,\big ]\\&\qquad +h_3 \big [\, f^2 \big (D_y^2 \, g\cdot g\big ) -g^2 \big (D_y^2 \, f\cdot f\big ) \,\big ] \\&\qquad +h_6 \big [\, f^2 \big (D_x D_y \, g\cdot g\big ) -g^2 \big (D_x D_y \, f\cdot f\big ) \,\big ]\\&\qquad +h_7 \big [\, f^2 \big (D_x D_z \, g\cdot g\big ) -g^2 \big (D_x D_z \, f\cdot f\big ) \,\big ] \\&\qquad +h_8 \big [\, f^2 \big (D_y D_z \, g\cdot g\big ) -g^2 \big (D_y D_z \, f\cdot f\big ) \,\big ]. \end{aligned}$$

Then, making use of the exchange formulae⁵ [10]

$$\begin{aligned}&F^2 \left( D_x D_t \, G \cdot G\right) -G^2 \left( D_x D_t \, F \cdot F\right) =2 \, D_x \left( D_t \, G \cdot F\right) \cdot \left( F G\right) , \end{aligned}$$

(12a)

$$\begin{aligned}&F^2 \left( D^3_x D_z \, G \cdot G\right) -G^2 \left( D^3_x D_z \, F \cdot F\right) \nonumber \\&\quad =2 \, D_z \left( D^3_x \, G \cdot F\right) \cdot \left( F G\right) -6 \, D_x \left( D_x D_z \, G \cdot F\right) \cdot \left( D_x \, G \cdot F\right) , \end{aligned}$$

(12b)

$$\begin{aligned}&F^2 \left( D^3_x D_z \, G \cdot G\right) -G^2 \left( D^3_x D_z \, F \cdot F\right) \nonumber \\&\quad =\frac{1}{2} \, D_z \left( D_x^3 \, G \cdot F\right) \cdot \left( F G\right) +\frac{3}{2} \, D_x \left( D_x^2 D_z \, G \cdot F\right) \cdot \left( F G\right) \nonumber \\&\qquad -3 \, D_x \left( D_x D_z \, G \cdot F\right) \cdot \left( D_x \, G \cdot F\right) -\frac{3}{2} \, D_x \left( D_x^2 \, G \cdot F\right) \cdot \left( D_z \, G \cdot F\right) \nonumber \\&\qquad -\frac{3}{2} \, D_z \left( D_x^2 \, G \cdot F\right) \cdot \left( D_x \, G \cdot F\right) , \end{aligned}$$

(12c)

$$\begin{aligned}&F^2 \left( D_y^2 \, G \cdot G\right) -G^2 \left( D_y^2 \, F \cdot F\right) =2 \, D_y \left( D_y \, G \cdot F\right) \cdot \left( F G\right) , \end{aligned}$$

(12d)

$$\begin{aligned}&F^2 \left( D_x D_y \, G \cdot G\right) -G^2 \left( D_x D_y \, F \cdot F\right) =2 \, D_x \left( D_y \, G \cdot F\right) \cdot \left( F G\right) , \end{aligned}$$

(12e)

$$\begin{aligned}&F^2 \left( D_x D_y \, G \cdot G\right) -G^2 \left( D_x D_y \, F \cdot F\right) =2 \, D_y \left( D_x \, G \cdot F\right) \cdot \left( F G\right) , \end{aligned}$$

(12f)

$$\begin{aligned}&F^2 \left( D_x D_z \, G \cdot G\right) -G^2 \left( D_x D_z \, F \cdot F\right) =2 \, D_x \left( D_z \, G \cdot F\right) \cdot \left( F G\right) , \end{aligned}$$

(12g)

$$\begin{aligned}&F^2 \left( D_x D_z \, G \cdot G\right) -G^2 \left( D_x D_z \, F \cdot F\right) =2 \, D_z \left( D_x \, G \cdot F\right) \cdot \left( F G\right) , \end{aligned}$$

(12h)

$$\begin{aligned}&F^2 \left( D_y D_z \, G \cdot G\right) -G^2 \left( D_y D_z \, F \cdot F\right) =2 \, D_y \left( D_z \, G \cdot F\right) \cdot \left( F G\right) , \end{aligned}$$

(12i)

$$\begin{aligned}&F^2 \left( D_y D_z \, G \cdot G\right) -G^2 \left( D_y D_z \, F \cdot F\right) =2 \, D_z \left( D_y \, G \cdot F\right) \cdot \left( F G\right) , \end{aligned}$$

(12j)

we could build up two sets of the bilinear auto-Bäcklund transformations for System (1):

Set 1:

The exchange formulae, i.e., (12a), (12b), (12d), (12f), (12g) and (12j), bring about

$$\begin{aligned}&f^2 \big [ \left( D_x D_t+h_2 D_x^3 D_z+h_3 D_y^2+h_6 D_x D_y+h_7 D_x D_z+h_8 D_y D_z\right) g\cdot g \big ]\\&\qquad -g^2 \big [ \left( D_x D_t+h_2 D_x^3 D_z+h_3 D_y^2+h_6 D_x D_y+h_7 D_x D_z+h_8 D_y D_z\right) f\cdot f \big ]\\&\quad =\big [\, f^2 \big (D_x D_t \, g\cdot g\big ) -g^2 \big (D_x D_t \, f\cdot f\big ) \,\big ] \\&\qquad +h_2 \big [\, f^2 \big (D_x^3 D_z \, g\cdot g\big ) -g^2 \big (D_x^3 D_z \, f\cdot f\big ) \,\big ]\\&\qquad +h_3 \big [\, f^2 \big (D_y^2 \, g\cdot g\big ) -g^2 \big (D_y^2 \, f\cdot f\big ) \,\big ] \\&\qquad +h_6 \big [\, f^2 \big (D_x D_y \, g\cdot g\big ) -g^2 \big (D_x D_y \, f\cdot f\big ) \,\big ]\\&\qquad +h_7 \big [\, f^2 \big (D_x D_z \, g\cdot g\big ) -g^2 \big (D_x D_z \, f\cdot f\big ) \,\big ] \\&\qquad +h_8 \big [\, f^2 \big (D_y D_z \, g\cdot g\big ) -g^2 \big (D_y D_z \, f\cdot f\big ) \,\big ]\\&\quad =2 \, D_x \big (D_t \, g \cdot f\big ) \cdot \big (f g\big ) +h_2 \, \big [ 2 \, D_z \big (D^3_x \, g \cdot f\big ) \cdot \big (f g\big )\\&\qquad -6 \, D_x \big (D_x D_z \, g \cdot f\big ) \cdot \big (D_x \, g \cdot f\big ) \big ]+2 \, h_3 \, D_y \big (D_y \, g \cdot f\big ) \cdot \big (f g\big )\\&\qquad +2 \, h_6 \, D_y \big (D_x \, g \cdot f\big ) \cdot \big (f g\big ) +2 \, h_7 \, D_x \big (D_z \, g \cdot f\big ) \cdot \big (f g\big )\\&\qquad +2 \, h_8 \, D_z \big (D_y \, g \cdot f\big ) \cdot \big (f g\big )\\&\quad =2 \, D_x \big [\big (D_t+h_7 D_z\big ) \, g \cdot f\big ] \cdot \big (f g\big ) +2 \, D_z \big [\big (h_2 D_x^3+h_8 D_y\big ) \, g \cdot f\big ] \cdot \big (f g\big )\\&\qquad -6 \, h_2 \, D_x \big (D_x D_z \, g \cdot f\big ) \cdot \big (D_x \, g \cdot f\big ) +2 \, D_y \big [\big (h_3 D_y+h_6 D_x\big ) \, g \cdot f\big ] \cdot \big (f g\big ). \end{aligned}$$

Under the coefficient constraints, i.e., (8), assumptions that

$$\begin{aligned}&D_x \big [\big (D_t+h_7 D_z\big ) \, g \cdot f\big ] \cdot \big (f g\big )=0, \end{aligned}$$

(13a)

$$\begin{aligned}&D_z \big [\big (h_2 D_x^3+h_8 D_y\big ) \, g \cdot f\big ] \cdot \big (f g\big )=0, \end{aligned}$$

(13b)

$$\begin{aligned}&D_x \big (D_x D_z \, g \cdot f\big ) \cdot \big (D_x \, g \cdot f\big )=0, \end{aligned}$$

(13c)

$$\begin{aligned}&D_y \big [\big (h_3 D_y+h_6 D_x\big ) \, g \cdot f\big ] \cdot \big (f g\big )=0, \end{aligned}$$

(13d)

develop into

$$\begin{aligned}&v=\frac{6 h_2}{h_4} (\ln f)_{xx}, \end{aligned}$$

(14a)

$$\begin{aligned}&u=\frac{6 h_2}{h_4} (\ln f)_{x}, \end{aligned}$$

(14b)

$$\begin{aligned}&v_0=\frac{6 h_2}{h_4} (\ln g)_{xx}, \end{aligned}$$

(14c)

$$\begin{aligned}&u_0=\frac{6 h_2}{h_4} (\ln g)_{x}, \end{aligned}$$

(14d)

$$\begin{aligned}&\big (D_t+h_7 D_z\big ) \, g \cdot f=0, \end{aligned}$$

(14e)

$$\begin{aligned}&\big (h_2 D_x^3+h_8 D_y\big ) \, g \cdot f=0, \end{aligned}$$

(14f)

$$\begin{aligned}&D_x D_z \, g \cdot f =\mu _1 D_x \, g \cdot f, \end{aligned}$$

(14g)

$$\begin{aligned}&\big (h_3 D_y+h_6 D_x\big ) \, g \cdot f=0, \end{aligned}$$

(14h)

with $u_0$ and $v_0$ as another set of the solutions of System (1), while $\mu _1$ as a real constant.

Theorem 2.1

Equations (14) comprise a set of the bilinear auto-Bäcklund transformations⁶ for System (1), because of their mutual consistency.

Corollary 2.1

Describing certain interfacial waves in a two-layer liquid or elastic quasiplane waves in a lattice, modelling the amplitude or elevation of the relevant wave, Bilinear Auto-Bäcklund Transformations (14) depend on $h_2$, $h_3$, $h_4$, $h_6$, $h_7$ and $h_8$, the coefficients in System (1), under (8), the coefficient constraints.

In order to confirm the mutual consistence of Bäcklund Transformations (14), using symbolic computation, under the variable-coefficient constraints

$$\begin{aligned} h_7 \ne 0, \qquad h_2 h_3 h_6 h_8 > 0, \end{aligned}$$

(15)

and with the choice of

$$\begin{aligned} \mu _1=\sigma _1, \end{aligned}$$

(16)

we could find certain analytic solutions of Bäcklund Transformations (14), i.e.,

$$\begin{aligned}&v=0, \qquad \end{aligned}$$

(17a)

$$\begin{aligned}&u=0, \qquad \end{aligned}$$

(17b)

$$\begin{aligned}&v_0=\frac{6 h_2}{h_4} \bigg \{ \ln \bigg [1+\delta _1 \mathrm {exp}\bigg (\pm \sqrt{\frac{h_6 h_8}{h_2 h_3}} x \mp \frac{h_6}{h_3} \sqrt{\frac{h_6 h_8}{h_2 h_3}} y+\sigma _1 z-h_7 \sigma _1 t\bigg )\bigg ]\bigg \}_{xx}, \end{aligned}$$

(17c)

$$\begin{aligned}&u_0=\frac{6 h_2}{h_4} \bigg \{ \ln \bigg [1+\delta _1 \mathrm {exp}\bigg (\pm \sqrt{\frac{h_6 h_8}{h_2 h_3}} x \mp \frac{h_6}{h_3} \sqrt{\frac{h_6 h_8}{h_2 h_3}} y+\sigma _1 z-h_7 \sigma _1 t\bigg )\bigg ]\bigg \}_{x}, \end{aligned}$$

(17d)

where $\sigma _1$ denotes a real non-zero constant, while $\delta _1$ represents a positive constant.

Theorem 2.2

There stand Analytic Solutions (17c) and (17d)⁷ for System (1).

Corollary 2.2

Describing certain interfacial waves in a two-layer liquid or elastic quasiplane waves in a lattice, modelling the amplitude or elevation of the relevant wave, Analytic Solutions (17c) and (17d) depend on $h_2$, $h_3$, $h_4$, $h_6$, $h_7$ and $h_8$, the coefficients in System (1), under (8) and (15), the coefficient constraints.

Set 2:

The exchange formulae, i.e., (12a), (12c), (12d), (12e), (12h) and (12i), result in

$$\begin{aligned}&f^2 \big [ \left( D_x D_t+h_2 D_x^3 D_z+h_3 D_y^2+h_6 D_x D_y+h_7 D_x D_z+h_8 D_y D_z\right) g\cdot g \big ]\\&\qquad -g^2 \big [ \left( D_x D_t+h_2 D_x^3 D_z+h_3 D_y^2+h_6 D_x D_y+h_7 D_x D_z+h_8 D_y D_z\right) f\cdot f \big ]\\&\quad =\big [\, f^2 \big (D_x D_t \, g\cdot g\big ) -g^2 \big (D_x D_t \, f\cdot f\big ) \,\big ] \\&\qquad +h_2 \big [\, f^2 \big (D_x^3 D_z \, g\cdot g\big ) -g^2 \big (D_x^3 D_z \, f\cdot f\big ) \,\big ]\\&\qquad +h_3 \big [\, f^2 \big (D_y^2 \, g\cdot g\big ) -g^2 \big (D_y^2 \, f\cdot f\big ) \,\big ] \\&\qquad +h_6 \big [\, f^2 \big (D_x D_y \, g\cdot g\big ) -g^2 \big (D_x D_y \, f\cdot f\big ) \,\big ]\\&\qquad +h_7 \big [\, f^2 \big (D_x D_z \, g\cdot g\big ) -g^2 \big (D_x D_z \, f\cdot f\big ) \,\big ] \\&\qquad +h_8 \big [\, f^2 \big (D_y D_z \, g\cdot g\big ) -g^2 \big (D_y D_z \, f\cdot f\big ) \,\big ]\\&\quad =2 \, D_x \big (D_t \, g \cdot f\big ) \cdot \big (f g\big )+\frac{1}{2} \, h_2 \big [ D_z \big (D_x^3 \, g \cdot f\big ) \cdot \big (f g\big ) \\&\qquad +3 \, D_x \big (D_x^2 D_z \, g \cdot f\big ) \cdot \big (f g\big )-3 \, D_x \big (D_x^2 \, g \cdot f\big ) \cdot \big (D_z \, g \cdot f\big ) \\&\qquad -6 \, D_x \big (D_x D_z \, g \cdot f\big ) \cdot \big (D_x \, g \cdot f\big ) -3 \, D_z \big (D_x^2 \, g \cdot f\big ) \cdot \big (D_x \, g \cdot f\big )\big ] \\&\qquad +2 \, h_3 \, D_y \big (D_y \, g \cdot f\big ) \cdot \big (f g\big ) +2 \, h_6 \, D_x \big (D_y \, g \cdot f\big ) \cdot \big (f g\big )\\&\qquad +2 \, h_7 \, D_z \big (D_x \, g \cdot f\big ) \cdot \big (f g\big ) +2 \, h_8 \, D_y \big (D_z \, g \cdot f\big ) \cdot \big (f g\big )\\&\quad =\frac{1}{2} \, D_x \big [\big (4 D_t+3 h_2 D_x^2 D_z+4 h_6 D_y\big ) \, g \cdot f\big ] \cdot \big (f g\big )\\&\qquad +\frac{1}{2} \, D_z \big [\big (h_2 D_x^3+4 h_7 D_x\big ) \, g \cdot f\big ] \cdot \big (f g\big )\\&\qquad +2 \, D_y \big [\big (h_3 D_y+h_8 D_z\big ) \, g \cdot f\big ] \cdot \big (f g\big ) -\frac{3}{2} \, h_2 \, D_z \big (D_x^2 \, g \cdot f\big ) \cdot \big (D_x \, g \cdot f\big )\\&\qquad -\frac{3}{2} \, h_2 \, D_x \big (D_x^2 \, g \cdot f\big ) \cdot \big (D_z \, g \cdot f\big ) -3 \, h_2 \, D_x \big (D_x D_z \, g \cdot f\big ) \cdot \big (D_x \, g \cdot f\big ). \end{aligned}$$

Assumptions that

$$\begin{aligned}&D_x \big [\big (4 D_t+3 h_2 D_x^2 D_z+4 h_6 D_y\big ) \, g \cdot f\big ] \cdot \big (f g\big )=0, \end{aligned}$$

(18a)

$$\begin{aligned}&D_z \big [\big (h_2 D_x^3+4 h_7 D_x\big ) \, g \cdot f\big ] \cdot \big (f g\big )=0, \end{aligned}$$

(18b)

$$\begin{aligned}&D_y \big [\big (h_3 D_y+h_8 D_z\big ) \, g \cdot f\big ] \cdot \big (f g\big )=0, \end{aligned}$$

(18c)

$$\begin{aligned}&D_z \big (D_x^2 \, g \cdot f\big ) \cdot \big (D_x \, g \cdot f\big )=0, \end{aligned}$$

(18d)

$$\begin{aligned}&D_x \big (D_x^2 \, g \cdot f\big ) \cdot \big (D_z \, g \cdot f\big )=0, \end{aligned}$$

(18e)

$$\begin{aligned}&D_x \big (D_x D_z \, g \cdot f\big ) \cdot \big (D_x \, g \cdot f\big )=0, \end{aligned}$$

(18f)

give rise to

$$\begin{aligned}&v=\frac{6 h_2}{h_4} (\ln f)_{xx}, \end{aligned}$$

(19a)

$$\begin{aligned}&u=\frac{6 h_2}{h_4} (\ln f)_{x}, \end{aligned}$$

(19b)

$$\begin{aligned}&v_0=\frac{6 h_2}{h_4} (\ln g)_{xx}, \end{aligned}$$

(19c)

$$\begin{aligned}&u_0=\frac{6 h_2}{h_4} (\ln g)_{x}, \end{aligned}$$

(19d)

$$\begin{aligned}&\big (4 D_t+3 h_2 D_x^2 D_z+4 h_6 D_y\big ) \, g \cdot f=0, \end{aligned}$$

(19e)

$$\begin{aligned}&\big (h_2 D_x^3+4 h_7 D_x\big ) \, g \cdot f=0, \end{aligned}$$

(19f)

$$\begin{aligned}&\big (h_3 D_y+h_8 D_z\big ) \, g \cdot f=0, \end{aligned}$$

(19g)

$$\begin{aligned}&D_x^2 \, g \cdot f =\mu _2 D_x \, g \cdot f, \end{aligned}$$

(19h)

$$\begin{aligned}&D_x^2 \, g \cdot f =\mu _3 D_z \, g \cdot f, \end{aligned}$$

(19i)

$$\begin{aligned}&D_x D_z \, g \cdot f =\mu _4 D_x \, g \cdot f, \end{aligned}$$

(19j)

with $\mu _2$, $\mu _3$ and $\mu _4$ as three real constants.

Theorem 2.3

Equations (19) comprise a set of the bilinear auto-Bäcklund transformations for System (1), because of their mutual consistency.

Corollary 2.3

Describing certain interfacial waves in a two-layer liquid or elastic quasiplane waves in a lattice, modelling the amplitude or elevation of the relevant wave, Bilinear Auto-Bäcklund Transformations (19) depend on $h_2$, $h_3$, $h_4$, $h_6$, $h_7$ and $h_8$, the coefficients in System (1), under (8), the coefficient constraints.

In order to confirm the mutual consistence of Bäcklund Transformations (19), making use of symbolic computation, under the variable-coefficient constraints

$$\begin{aligned} h_2 h_7 < 0, \qquad h_3 h_8 \ne 0, \qquad 3 h_3 h_7+h_6 h_8 \ne 0, \end{aligned}$$

(20)

and with the choices of

$$\begin{aligned} \mu _2=\pm 2 \sqrt{-\frac{h_7}{h_2}}, \qquad \mu _3=-\frac{4 h_7}{h_2 \sigma _2}, \qquad \mu _4=\sigma _2, \end{aligned}$$

(21)

we are able to obtain some analytic solutions of Bäcklund Transformations (19), i.e.,

$$\begin{aligned}&v=0, \qquad \end{aligned}$$

(22a)

$$\begin{aligned}&u=0, \qquad \end{aligned}$$

(22b)

$$\begin{aligned}&v_0=\frac{6 h_2}{h_4} \bigg \{\ln \bigg [1+\delta _2 \mathrm {exp}\bigg (\pm 2 \sqrt{-\frac{h_7}{h_2}} x-\frac{h_8 \sigma _2}{h_3} y+\sigma _2 z+\frac{3 h_3 h_7+h_6 h_8}{h_3} \sigma _2 t\bigg )\bigg ]\bigg \}_{xx} , \end{aligned}$$

(22c)

$$\begin{aligned}&u_0=\frac{6 h_2}{h_4} \bigg \{\ln \bigg [1+\delta _2 \mathrm {exp}\bigg (\pm 2 \sqrt{-\frac{h_7}{h_2}} x-\frac{h_8 \sigma _2}{h_3} y+\sigma _2 z+\frac{3 h_3 h_7+h_6 h_8}{h_3} \sigma _2 t\bigg )\bigg ]\bigg \}_{x} , \end{aligned}$$

(22d)

where $\sigma _2$ means a real non-zero constant, while $\delta _2$ denotes a positive constant.

Theorem 2.4

There stand Analytic Solutions (22c) and (22d) for System (1).

Corollary 2.4

Describing certain interfacial waves in a two-layer liquid or elastic quasiplane waves in a lattice, modelling the amplitude or elevation of the relevant wave, Analytic Solutions (22c) and (22d) depend on $h_2$, $h_3$, $h_4$, $h_6$, $h_7$ and $h_8$, the coefficients in System (1), under (8) and (20), the coefficient constraints.

3 Similarity Reductions for System (1)

Our purpose is to build up some similarity reductions for System (1) following the similar approaches to the ones in Refs. [65,66,67,68,69,70,71,72,73] in the form

$$\begin{aligned}&u(x,y,z,t)=\theta (x,y,z,t)+\alpha (x,y,z,t) p[r(x,y,z,t)] , \end{aligned}$$

(23a)

$$\begin{aligned}&v(x,y,z,t)=\delta (x,y,z,t)+\kappa (x,y,z,t) q[r(x,y,z,t)], \end{aligned}$$

(23b)

with p(r) and q(r) implying the real differentiable functions, while $\theta (x,y,z,t)$, $\alpha (x,y,z,t)$, $\delta (x,y,z,t)$, $\kappa (x,y,z,t)$ and r(x, y, z, t) standing for the real differentiable functions to be determined.

Thinking of the case of $r_{x}=r_{y}=r_{z}=0$, $\alpha (x,y,z,t) \ne 0$, $\kappa (x,y,z,t) \ne 0$, $p[r(x,y,z,t)] \ne 0$ and $q[r(x,y,z,t)] \ne 0$, employing symbolic computation and substituting Assumptions (23) into System (1), we obtain

$$\begin{aligned}&h_1 \kappa _{x} s(t) q'+h_4 \left( \alpha _{xz} \kappa _{x} +\alpha _{z} \kappa _{xx}\right) p q+h_5 \left( \kappa _{x} \kappa _{z} +\kappa \kappa _{xz}\right) q^2\nonumber \\&\quad +h_4 \left( \alpha _{xz} \delta _{x} +\alpha _{z} \delta _{xx}\right) p +\left[ h_1 \kappa _{xt} \right. +h_2 \kappa _{xxxz}+h_3 \kappa _{yy} +h_4 \left( \theta _{z} \kappa _{xx}+\theta _{xz} \kappa _{x}\right) \nonumber \\&\quad +h_5 \left( \kappa _{x} \delta _{z}+\kappa _{z} \delta _{x} +\kappa \delta _{xz}+\kappa _{xz} \delta \right) +h_6 \kappa _{xy}+h_7 \kappa _{xz} \left. +h_8 \kappa _{yz}\right] q\nonumber \\&\quad +h_1 \delta _{xt}+h_2 \delta _{xxxz} +h_3 \delta _{yy}+h_4 \left( \theta _{z} \delta _{xx} +\theta _{xz} \delta _{x}\right) +h_5 \left( \delta _{x} \delta _{z}+\delta \delta _{xz}\right) \nonumber \\&\quad +h_6 \delta _{xy}+h_7 \delta _{xz}+h_8 \delta _{yz}=0 , \end{aligned}$$

(24a)

$$\begin{aligned}&\!\! -\kappa q+\alpha _{x} p+\left( \theta _{x}-\delta \right) =0 , \end{aligned}$$

(24b)

with the apostrophe indicating the differentiation with respect to r, while $s(t)=dr(t)/dt$.

Once it is required that Eqs. (24) stand for a couple of the real ordinary differential equations (ODEs) with respect to p(r) and q(r), the ratios of the coefficients of different derivatives and powers of p(r) and q(r) must represent certain real functions as for r only.

We then make use of the coefficients of $q'$ in Eq. (24a) and q in Eq. (24b), respectively, as the normalizing coefficients in Eqs. (24), to get

$$\begin{aligned} \Omega _{1}(r) h_1 \kappa _{x} s(t)&=h_4 \left( \alpha _{xz} \kappa _{x}+\alpha _{z} \kappa _{xx}\right) , \end{aligned}$$

(25a)

$$\begin{aligned} \Omega _{2}(r) h_1 \kappa _{x} s(t)&=h_5 \left( \kappa _{x} \kappa _{z} +\kappa \kappa _{xz}\right) , \end{aligned}$$

(25b)

$$\begin{aligned} \Omega _{3}(r) h_1 \kappa _{x} s(t)&=h_4 \left( \alpha _{xz} \delta _{x} +\alpha _{z} \delta _{xx}\right) , \end{aligned}$$

(25c)

$$\begin{aligned} \Omega _{4}(r) h_1 \kappa _{x} s(t)&=h_1 \kappa _{xt} +h_2 \kappa _{xxxz}+h_3 \kappa _{yy} +h_4 \left( \theta _{z} \kappa _{xx}+\theta _{xz} \kappa _{x}\right) +h_6 \kappa _{xy} \nonumber \\&\quad +h_5 \left( \kappa _{x} \delta _{z}+\kappa _{z} \delta _{x} +\kappa \delta _{xz} +\kappa _{xz} \delta \right) +h_7 \kappa _{xz} +h_8 \kappa _{yz} , \end{aligned}$$

(25d)

$$\begin{aligned} \Omega _{5}(r) h_1 \kappa _{x} s(t)&=h_1 \delta _{xt}+h_2 \delta _{xxxz} +h_3 \delta _{yy} +h_4 \left( \theta _{z} \delta _{xx} +\theta _{xz} \delta _{x}\right) \nonumber \\&\quad +h_5 \left( \delta _{x} \delta _{z}+\delta \delta _{xz}\right) +h_6 \delta _{xy}+h_7 \delta _{xz}+h_8 \delta _{yz} , \end{aligned}$$

(25e)

and

$$\begin{aligned}&-\Gamma _{1}(r) \kappa =\alpha _{x} , \end{aligned}$$

(26a)

$$\begin{aligned}&-\Gamma _{2}(r) \kappa =\theta _{x}-\delta , \end{aligned}$$

(26b)

with $\Omega _{i}(r)$’s and $\Gamma _{j}(r)$’s meaning certain real to-be-determined functions as for r, while $i=1,...,5$ and $j=1,2$.

On the basis of the remarks in Ref. [65], a set of the conditions for $\theta (x,y,z,t)$, $\alpha (x,y,z,t)$, $\delta (x,y,z,t)$, $\kappa (x,y,z,t)$ and r(t) are figured, any solution of which could come to, at least, a similarity reduction.

On account of the second freedom in Remark 3 in Ref. [65], Eqs. (25b) and (26a) bring about

$$\begin{aligned}&\alpha (x,y,z,t)=- \! \int \! \kappa (x,y,z,t) dx, \qquad h_5 \ne 0, \end{aligned}$$

(27a)

$$\begin{aligned}&\kappa (x,y,z,t)=\frac{h_1}{h_5} s(t) z+\eta _0(x,y,t), \qquad \Omega _{2}(r)=\Gamma _{1}(r)=1, \end{aligned}$$

(27b)

and then Eq. (25a) results in

$$\begin{aligned} \eta _0(x,y,t)=\eta _1(y,t) x+\eta _2(y,t), \qquad \Omega _{1}(r)=-\frac{h_4}{h_5}, \end{aligned}$$

(28)

with $\eta _0(x,y,t)$ as a real differentiable function of x, y and t, while $\eta _1(y,t)$ and $\eta _2(y,t)$ as two real differentiable functions of y and t.

By reason of the first freedom in Remark 3 in Ref. [65], Eqs. (25c) and (26b) make for

$$\begin{aligned} \theta (x,y,z,t)=\xi _0 x, \qquad \delta (x,y,z,t)=\xi _0, \qquad \Omega _{3}(r)=\Gamma _{2}(r)=0, \end{aligned}$$

(29)

so that Eqs. (25d) and (25e) develop into

$$\begin{aligned}&\eta _1(y,t)=\xi _1, \qquad \eta _2(y,t)=\xi _2 y+\xi _3(t),\nonumber \\&r(t)=\xi _4 t+\xi _5, \qquad \Omega _{4}(r)=\Omega _{5}(r)=0, \end{aligned}$$

(30)

with $\xi _0$, $\xi _1 \ne 0$, $\xi _2 \ne 0$, $\xi _4 \ne 0$ and $\xi _5$ as the real constants, while $\xi _3(t)$ as a real function of t.

Until now, System (1) can be simplified to the following ODEs:

$$\begin{aligned}&q'-\frac{h_4}{h_5} p q+q^2=0 , \end{aligned}$$

(31a)

$$\begin{aligned}&p+q=0 . \end{aligned}$$

(31b)

For the purpose of transforming ODEs (31) into a single ODE, we get

$$\begin{aligned} p=-q, \end{aligned}$$

(32)

and then find

$$\begin{aligned} q'+\left( 1+\frac{h_4}{h_5}\right) q^2=0. \end{aligned}$$

(33)

In short, under the variable-coefficient constraints

$$\begin{aligned} h_1 \ne 0,\qquad h_5 \ne 0,\qquad h_4+h_5 \ne 0, \end{aligned}$$

(34)

we conclude with a set of the similarity reductions for System (1), written as

$$\begin{aligned}&u(x,y,z,t) =\xi _0 x+\left[ \frac{1}{2} \xi _1 x^2+\xi _2 x y +\xi _3(t) x+\frac{h_1}{h_5} \xi _4 x z\right] q[r(t)] , \end{aligned}$$

(35a)

$$\begin{aligned}&v(x,y,z,t)=\xi _0+\left[ \xi _1 x+\xi _2 y+\xi _3(t) +\frac{h_1}{h_5} \xi _4 z\right] q[r(t)] , \end{aligned}$$

(35b)

$$\begin{aligned}&r(t)=\xi _4 t+\xi _5 , \end{aligned}$$

(35c)

with q(r) satisfying

$$\begin{aligned} q'+\left( 1+\frac{h_4}{h_5}\right) q^2=0. \end{aligned}$$

(36)

ODE (36) indicates a known ODE, the information of which has been reported in Ref. [74].

Theorem 3.1

There lie Similarity Reductions (35) with ODE (36) for System (1).

Corollary 3.1

Describing certain interfacial waves in a two-layer liquid or elastic quasiplane waves in a lattice, modelling the amplitude or elevation of the relevant wave, Similarity Reductions (35) with ODE (36) depend on $h_1$, $h_4$ and $h_5$, the coefficients in System (1), under (34), the coefficient constraints.

4 Conclusions

To date, studies on the liquids and lattices have appeared interesting. In this paper, with symbolic computation, we have considered System (1), a (3+1)-dimensional generalized Yu-Toda-Sasa-Fukuyama system for the interfacial waves in a two-layer liquid or elastic waves in a lattice.

For System (1), we have built up Bilinear Auto-Bäcklund Transformations (14) and Bilinear Auto-Bäcklund Transformations (19), both of which are different from that in Ref. [7]. On the other hand, we have constructed Similarity Reductions (35) with ODE (36), i.e., from System (1) to a known ODE. As for the amplitude or elevation of the relevant wave, our results have been presented to rely on the coefficients in System (1).

Appendix

The Hirota bilinear operators $D_x$, $D_y$, $D_z$ and $D_t$ have been defined as [10]

$$\begin{aligned}&D_{x}^{m_1}D_{y}^{m_2}D_{z}^{m_3}D_{t}^{m_4} G(x,y,z,t)\cdot F(x,y,z,t)= \left( \frac{\partial }{\partial x}-\frac{\partial }{\partial \tilde{x}}\right) ^{m_1}\,\left( \frac{\partial }{\partial y}-\frac{\partial }{\partial \tilde{y}}\right) ^{m_2}\, \nonumber \\&\quad \left( \frac{\partial }{\partial z}-\frac{\partial }{\partial \tilde{z}}\right) ^{m_3}\, \left( \frac{\partial }{\partial t}-\frac{\partial }{\partial \tilde{t}}\right) ^{m_4}\,G(x,y,z,t)\,F(\tilde{x},\tilde{y},\tilde{z},\tilde{t}) \bigg |_{\tilde{x}=x,\,\tilde{y}=y,\,\tilde{z}=z,\,\tilde{t}=t}, \nonumber \\ \end{aligned}$$

(A.1)

with $\tilde{x}$, $\tilde{y}$, $\tilde{z}$ and $\tilde{t}$ indicating four formal variables, G(x, y, z, t) denoting a $C^{\infty }$ function of x, y, z and t, $F(\tilde{x},\tilde{y},\tilde{z},\tilde{t})$ representing a $C^{\infty }$ function of $\tilde{x}$, $\tilde{y}$, $\tilde{z}$ and $\tilde{t}$, while $m_1$, $m_2$, $m_3$ and $m_4$ implying four non-negative integers [10].

Recent applications of the Hirota bilinear operators include the ones to certain Bose-Einstein condensates with the dipole-dipole attractions and repulsions [75], liquids with the gas bubbles [76], time-dependent radiative transfer problems [77], nonlinear reduced fluid models for some plasmas [78] and ion-acoustic wave structures with the effects of magnetic fields in plasma physics [79]. Other recent vigorous references include, e.g., Refs. [22, 26, 27, 43, 80].