Darboux变换求一个新的孤子方程的精确解

以平凡解u=0,v=1作为种子解,代入矩阵谱问题Φx=UΦ,U=(-λ+u v～(1/2) v λ-u),Φt=VΦ,V=(V1 V2 V3 -V1),其中V1=-λ2+u2+1/6ux+1/6(lnv)xx+1/8(lnv)x2,V2=vλ+uv-1/2vx,V3=(vλ)～(1/2)+uv～(1/2)+vx/(4v～(1/2)).求出基本解.选取两个基本解φ(λj)=(coshξjβjsinhξj+λj coshξj),ф(λj)=(sinhξjβjcoshξj+λj sinhξj),其中ξj=βj(x+λj t),βj=(λj2+1)～(1/2),(1≤j≤N-1).再利用克莱姆法则和达布变换求出方程的非平凡解,最后又具体给出N=1和N=2两种情形.

The Soliton Solution of a New Equation from Darboux Transformations

To obtian the new soliton solutions of the new equation,chose the seed solution u = O, v = 1, Substing them into spectral problem ;以平凡解φ,=Uφ,U=（-λ=u/√v v/λu）,φ1=Vφ,V=（V1/V3 -V2/V1）,here V1=-λ^2＋u^2＋6/1ux＋6/1（1nv）xx＋8/1（lnv）^x2,V2=vλ＋vu-2/1vx,V3=√vλ＋u√v＋4√v/vx.φ（λ1）=（coshε1/β1sinhε1＋λjcoshεj）,φ（λj）=（sinhδj / βjcoshεj＋λjsinhεj）,其中εj= βj（x＋λjt）, β1=√λ1^2＋1,（1≤j≤N-1）,Then to obtian the new soliton solutions of the new equation from the Darboux Transformations. Last to discuss the two special case N=1 and N=2.